International Journal of Acoustics and Vibration, Vol

Home , Jens Blauert, Mechanical resonance

EDITORIAL OFFICE International Journal of EDITOR-IN-CHIEF Acoustics and Vibration Malcolm J. Crocker A quarterly publication of the International Institute of Acoustics and Vibration MANAGING EDITOR Marek Pawelczyk ASSOCIATE EDITORS Dariusz Bismor Volume 22, Number 2, June 2017 Nickolay Ivanov Zhuang Li ASSISTANT EDITORS Teresa Glowka EDITORIAL Sebastian Kurczyk ICSV24: London Calling Karol Jablonski Jian Kang and Barry Gibbs ...... 142 Jan Wegehaupt EDITORIAL ASSISTANT ARTICLES Neha Patel Modeling and Vibration Suppression of Flexible Spacecraft Using Higher-Order Margarita Maksotskaya Sandwich Panel Theory Milad Azimi, Morteza Shahravi and Keramat Malekzadeh Fard ...... 143 Free Vibration Analysis of Functionally Heterogeneous Hollow Cylinder Based on Three Dimensional Elasticity Theory Masoud Asgari ...... 151 EDITORIAL BOARD Electrically Forced Vibrations of Partially Electroded Rectangular Quartz Plate Piezoelectric Resonators Jorge P. Arenas Hui Chen, Ji Wang, Jianke Du and Jiashi Yang ...... 161 Valdivia, Chile Damping Performance of Dynamic Vibration Absorber in Nonlinear Simple Beam with 1:3 Internal Resonance Jonathan D. Blotter Yi-Ren Wang and Hsueh-Ghi Lu ...... 167 Provo, USA Energy Harvesting Estimation from the Vibration of a Simply Supported Beam Leonid Gelman A. Rajora, A. Dwivedi, A. Vyas, S. Gupta and A. Tyagi ...... 186 Cranﬁeld, UK Preparation and Experimental Study of Magnetorheological Fluids for Vibration Control Samir Gerges Ying-Qing Guo, Zhao-Dong Xu, Bing-Bing Chen, Cheng-Song Ran Florianopolis,´ Brazil and Wei-Yang Guo ...... 194 Colin H. Hansen A Subjective Related Measure of Airborne Sound Insulation Adelaide, Australia Reinhard O. Neubauer and Jian Kang ...... 201 Mechanical Fault Diagnosis Method Based on LMD Shannon Entropy and Hugh Hunt Improved Fuzzy C-means Clustering Cambridge, England Shaojiang Dong, Xiangyang Xu and Jiayuan Luo ...... 211 Analysis of Transverse Vibration Acceleration for a High-speed Elevator with Dan Marghitu Random Parameter Based on Perturbation Theory Auburn, USA Chen Wang, Ruijun Zhang and Qing Zhang ...... 218 Manohar Lal Munjal Model-Based Algebraic Approach to Robust Parameter Estimation in Uncertain Dynamics Rotating Machinery Bangalore, India Pablo Izquierdo, Higinio Rubio, Manuel P. Donsion and J.C. Garcia-Prada ... 224 Kazuhide Ohta Optimal Vibration Control for Structural Quasi-Hamiltonian Systems with Noised Observations Fukuoka, Japan Zu-guang Ying, Rong-chun Hu and Rong-hua Huan ...... 233 Goran Pavic Detection of Damage in Spot Welded Joints Using a Statistical Energy Analysis-like Villeurbanne, France Approach Achuthan C. Pankaj, Marandahalli V. Shivaprasad and S. M. Murigendrappa .. 242 Subhash Sinha Probes Design and Experimental Measurement of Acoustic Radiation Resistance Auburn, USA Xiaoqing Wang and Yang Xiang ...... 252 Thermoviscoelastic Vibrations of a Micro-Scale Beam Subjected to Sinusoidal Pulse Heating D. S. Mashat, A. M. Zenkour and A. E. Abouelregal ...... 260 Noise and Vibration Analysis of a Heat Exchanger: a Case Study Thiago A. Fiorentin, Alexandre Mikowski, Olavo M. Silva and Arcanjo Lenzi .. 270 About the Authors ...... 276 Editor’s Space

ICSV24: London Calling

The International Institute of Acoustics and Vibration Institute of Technology, China): Advances in flutter analy- (IIAV) and the UK’s Institute of Acoustics (IOA) are pleased sis and control of aircraft structures; Jeremie Voix (Univer- to invite scientists and engineers from all over the world to at- sité du Québec, Canada): The ear beyond hearing: from tend the 24th International Congress on Sound and Vibration smart earplug to in-ear brain computer interfaces; Jens Blauert (ICSV24) to be held in London from July 23th to 27th this (Ruhr-Universität Bochum, Germany): Reading the world year. This is the first major international congress to be held with two ears; Andrei Metrikine (Delft University of Technol- in London covering the whole spectrum of acoustics and vi- ogy, Netherlands): Nonlinear dynamics offshore structures in bration since the 1974 International Congress on Acoustics. It marine currents and ice; and Timothy Leighton (University of will be an important opportunity for scientists and engineers to Southampton, UK): Climate change, dolphins, spaceships on share their latest research results and exchange ideas on theo- antimicrobial resistance – the impact of bubble acoustics. ries, technologies, and applications in acoustics and vibration. The Technical Exhibition will be centrally located in the In total, 1271 abstracts, from 60 countries, have been accepted, Congress Hotel and will provide delegates with a wide range including abstracts of 236 papers that were submitted for peer of products and expert services on instrumentation, software, review. and consultancy. Cosmopolitan London is an exciting, dynamic, and wel- A range of social and cultural activities are planned for coming destination. It is a city where delegates can enjoy a ICSV24 delegates as well as accompanying persons. The ac- rich selection of culture including theatres, concerts, muse- tivities include a visit to the university city of Cambridge on ums, art galleries, night-life, gastronomy, markets, shopping, Sunday, 23 th July 2017; the opening ceremony and the wel- and sports. In short, London offers something that will appeal come reception on Monday, 24th July, at the Westminster Park to every delegate. The ICSV24 venue this year is the centrally Plaza Hotel; the Westminster Soundwalk on Tuesday, 25th July located Park Plaza Westminster Bridge Hotel that overlooks will start and finish at the Park Plaza Hotel; the Millennium the iconic Big Ben. With its views across the River Thames, Bridge visit (an IIAV Members only event) on Tuesday, 25th this hotel offers an ideal location that is within walking dis- July; the IOA Young Members Group social event on Tuesday, tance to the city’s most well-known attractions, including the 25th July; the Gala Dinner on the evening of Wednesday, 26th Houses of Parliament, the National Theater, the London Eye, at the Sheraton Park Lane Hotel; and lastly the closing cere- and the London Aquarium. mony will take place at the Park Plaza Hotel on of Thursday The Scientific Programme is structured into sixteen Themes, 27th. each containing Structured Sessions and Regular Sessions. Students specialising in sound and vibration are especially The Themes include: acoustical measurement and instrumen- welcome at ICSV24 and can enjoy reduced fees as well as an tation, active noise and vibration control, aero acoustics and opportunity to win the Best Paper Award. aviation noise, community and environmental noise, physical acoustics, industrial and occupational noise and vibration, ma- Along with the IIAV Officers and the Local Organising chinery noise and vibration, materials for noise and vibration Committee, we look forward to welcoming you in London this control, human response psycho-physio-bio acoustics, road July. and rail traffic noise and vibration, room acoustics, signal processing, simulation and numerical methods, underwater and maritime noise, building acoustics, policy, education and Eu- ropean projects, and musical acoustics. Jian Kang Barry Gibbs Distinguished plenary speakers will present lectures on re- ICSV24 General Chair Technical Programme cent developments in important topics of sound and vibra- Coordinator tion. The lecturers are: Eric Heller (Harvard University, USA): The miracle of resonance and formants; Haiyan Hu (Beijing

142 https://doi.org/10.20855/ijav.2017.22.2E84 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Modeling and Vibration Suppression of Flexible Spacecraft Using Higher-Order Sandwich Panel Theory

Milad Azimi Department of Mechanical Engineering, Islamic Azad University East Tehran Branch, Tehran, Iran, 18661–13118

Morteza Shahravi and Keramat Malekzadeh Fard Department of Aerospace and Mechanical Engineering, Malek Ashtar University of Technology, Tehran, Iran, 15875–1774

(Received 16 May 2014; accepted: 1 March 2017) This paper presents a novel approach for the modeling and vibration suppression of a flexible spacecraft during a large angle attitude maneuver. A Higher-Order theory is used to model the elastic behavior of solar panel appendages with surface bounded piezoelectric (PZT) patches that capture the transverse shear deformation effects through the thickness of the smart sandwich panel. With the implementation of an appropriate time scales transformation technique and using Singular Perturbation Theory (SPT), the spacecraft dynamic behavior has been divided into double slow and fast subsystems. Modified Sliding Mode (MSM) and Strain Rate Feedback (SRF) control theory have been used for attitude and vibration control simultaneously by global stability proof of the overall system, while the controllers accomplished their missions in coupled rigid/flexible dynamic domain without parasitic parameter interactions. Numerical simulation assesses the benefits of the proposed approach.

NOMENCLETURE u(x, y) Displacement in the longitudinal x- direction of the appendages V Potential strain energy w(x, y) Lateral deflection of the appendage a Hub dimension x Starting x-coordinate of the k-th PZT C Structural damping matrix k patches d Equivalent PZT coefficient 31 kyj The distance from strating point of PZT D PZT material electric charge density dis- 3 with respect to neutral axis of core placement δ Variation e Attitude error θ ε Singular perturbation parameter E PZT material electric field strength 3 εT PZT material permitivity E Young’s Modulus 3 η , η Sensor and actuator voltages g , g PZT sensors and actuator amplifier gains, s a s a $ PZT width respectively P θ Hub angle G Shear modulus θ Desired hub angle h , h , h , h Bottom, Core, PZT, Top layers thickness, d b c p t ρ, ρ Appendage and PZT density, respectively respectively P σ PZT material stress I Moment of inertia 1 ψ(x) Finite element shape functions J Hub moment of inertia h τ Shear stress k (i = 1 : 4) Positive design parameters to stabilize the yx i (˙) Time differentiation system K Stiffness matrix Lb,LP Length of the flexible pannel and PZT 1. INTRODUCTION patches, respectively M Mass matrix Recently, control problems for flexible space structures have n Number of elements attracted significant attention because of the important de- P Boundary layer width of the sliding surface mands for decreasing the consumption of energy and limitation q General coordinates of the mass. Today’s modern spacecrafts consist of a rigid hub R Reduced in magnitude and flexible solar sandwich panel appendages. These sand- S1 PZT material strain wich panel constructions consist of metal or composite face H Sθ Sliding surface sheets and metallic honeycombs that play an important role to j k j increase the ratio of stiffness to weight in structure configura- T , TP Kinetic energy hub with attached main j-th pannel and k-th PZT pairs on j-th pannel, tion. A flexible space structure is required to not only execute respectively precise motion for the purpose of pointing, tracking, etc., but International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 143–150) https://doi.org/10.20855/ijav.2017.22.2459 143 M. Azimi, et al.: MODELING AND VIBRATION SUPPRESSION OF FLEXIBLE SPACECRAFT USING HIGHER-ORDER SANDWICH PANEL. . . also to be stabilized and dampen the vibrations that are natu- rally excited along the motion as fast as possible at the terminal point.1 In order to accurately predict the vibrational behavior of such a structure, a more sophisticated model with less restrictive assumptions is needed. Smart sandwich structures with distributed sensors and actuators have the capability to actively respond to a changing environment while offering significant weight savings and additional passive controllability through ply tailoring. Piezoelectric sensing and actuation of composite laminates is the most promising concept due to the Figure 1. Flexible spacecraft model and parameters. static and dynamic control capabilities. Among various types of sandwich structure modeling approaches such as classical In this paper, a new hybrid control strategy is extended theory with no shear deformation (CPT), first order shear de- for flexible spacecraft rotational maneuver control as well formation theory (FOSDT), Ordinary theories (OSPT), various as vibration suppression of the flexible sandwich appendages Higher-Order approaches, the one that is suitable for sandwich equipped by PZT patches by using singular perturbation the- structures with metallic honeycomb core is the Higher-Order ory. By introducing a novel approach by means of applying Sandwich Panel Theory (HSAPT).2 It has long been recog- higher-order theories, the spacecraft structural domain is en- nized that higher order theories provide an effective solution tered in its real configuration. To perform a desired maneuver tool for accurately and efficiently predicting the deformation while suppressing the vibration of the elastic modes, a time- behavior of sandwich structures subjected to bending loads. A scale decomposition approach, where the coupled dynamics review and assessment of various theories for modeling sand- of the system are separated into fast (structural vibration) and wich composites with application to sandwich beams can be slow (attitude dynamics) subsystems, is used. The derivation found in some references.3, 4 Many authors have investigated of the sliding mode control systems with switching functions the higher-order theories to model the free vibration response imposes drastic chattering effects and excites low frequency of sandwich structures5, 6 but no researches have reported on modes of the systems. Hence, the modified SMC is adopted the analysis of the dynamical behavior of sandwich panels in for the spacecraft attitude control to overcome this drawback. attitude maneuver yet. This modification of the attitude controller yields a shorter set- Many critical studies for the spacecraft attitude control tling time and also decreases the residual vibration. In order problem by sliding mode control (SMC) approach have been to validate the performance and accuracy of the proposed ap- done.7, 8 A critical key point of controller design based on the proach, a comparative study is performed. Furthermore, nu- SMC is that it uses nonlinear feedback control with discon- merical simulations are obtained to demonstrate the effective- tinues behavior of sliding surface in the state space. However, ness and satisfaction of the proposed approach. because of the robust nature of this kind of controller due to the switching functions of the structure, it is an effective method 2. MATHEMATICAL MODELING to attenuate disturbances and uncertainties. It has also been applied in a wide range of maneuvering control of flexible space- In the present study, we consider a system of spacecraft craft. The discontinuous nature of SMC designs is well known model which contain both rigid main body and the flex- for exciting high frequency un-modeled dynamics, especially ible sandwich panels, containing surface, bounded piezo- when applied to physical systems that are modeled imperfectly. electric sensors, and actuators, as shown in Fig. 1. To To avoid these hazards, several techniques such as defining identify the spacecraft attitude relative to an inertial frame continuous switching functions,9 improve the system’s perfor- FN {nx, ny, nz}, a main body fixed frame FB{bx, by, bz} is mance. However, the elastic motion of the flexible spacecraft defined. will be affected by the attitude maneuvering operation, which The velocity of a typical deformed point on each appendage causes undesirable vibrations. For the flexible spacecraft ma- with respect to the body of the reference frame is given by: neuvering, the attitude regardless of the system flexibility or i i ˙ ~ control on the flexible members, large amplitude transient, and v (x, y, t) = u˙ (x, y, t) − w(x, y, t)θ(t) bx + steady state oscillations may occur. i ˙ ~ Attitude maneuver of rigid spacecraft can be done without w˙ (x, y, t) + x+a+u (x, y, t) θ(t) by; (1) a lot of vibration problems after reaching its desired attitude. The task of vibration suppression of a flexible spacecraft by where x and y are the auxiliary variables measured from the means of smart structures such as PZT sensor/actuator pairs root of the hub along and perpendicular to the un-deformed has been studied by control system designers.10 Active con- appendage axis. Henceforth, a similar system of superscript trol methods can be used to damp out undesirable vibrations notation will be used in this paper. The deflections of the of a flexible structure.11, 12 Some of the various methods used face sheets u(x, y, t) and w(x, y, t) are longitudinal and ver- for vibration control in systems are the Strain Rate Feedback tical displacement components measured along the body fixed (SRF),10, 11, 13 Positive Position Feedback (PPF) , periodic out- frame, respectively. Moreover, the following assumptions are put feedback control, sliding mode control, fast output sam- made: (1) Regarding to the ordinary form of higher-order sand- pling feedback control, wave suppression method, indepen- wich panel2 and related refined theories,14, 15 a Quasi-layer dent modal space control method, modified independent modal wise hypothesis is considered in which the displacement fields space control method, mode localization, and optimum control of the upper and lower layers ut,b(x, y, t) and wt,b(x, y, t) sat- based on the minimization of power supplied to the beam. isfy the Euler-Bernoulli theory, and the core only undergoes

144 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 M. Azimi, et al.: MODELING AND VIBRATION SUPPRESSION OF FLEXIBLE SPACECRAFT USING HIGHER-ORDER SANDWICH PANEL. . .

n W w(x, y, t) = w(x, t) = wP +(x, t) = wt(x, t) = wc(x, t) = wb(x, t) = wP −(x, t);

 Actuator Layer up+(x, y, t) = ut(x, y, t) −−−−−−−−−−−−−−−→ h ≤ y ≤ h + h ;  t t p  t Upper Face Sheet  t t hc−ht dw (x,t) hc hc u (x, y, t) = u (x, t) − y − 2 dx −−−−−−−−−−−−−−−−→ 2 ≤ y ≤ 2 + ht;  2 2 c ht dw(x,t) y (3hc−2y) d τ t y c Core hc hc U u (x, y, t) = −y − − 2 + u (x, t) + τ −−−−−−−−−→ ≤ y ≤ − ; (2) 2 dx 12Ec dx Gc 2 2  b  b b hc+hb dw (x,t) Lower Face Sheet hc hc u (x, y, t) = u (x, t) − y + −−−−−−−−−−−−−−−−→ −hb − ≤ y ≤ − ;  2 dx 2 2  p− b Sensor Layer u (x, y, t) = u (x, y, t) −−−−−−−−−−−−−−→ −hp − hb ≤ y ≤ −hb.

c shear stresses τyx; (2) the displacement field in the core was with assumed to be linear in the vertical direction wc(x, y, t) and c c 1 b t c dw(x, t) nonlinear in the longitudinal direction u (x, y, t) in conformity γ = u (x, t)−u (x, t) −h + 0.5(ht +hb) ; with higher-order theory assumptions; (3) each layer has the hc dx 16 k j 2 ! same transverse deflection; (4) the mass and stiffness of the 2 h kΓj = kyj + kyjkhj + p . (6) adhesive between the layers is neglected and; (5) loading only p 3 in the bx − by plane is considered, which solely caused deflections designed by u and w. The work done by the external torques τ˜ and PZT patches17 is The displacement of the smart panel including face sheets calculated by: and the core, which consider Quasi Layer wise Theories (QLT) 2 2 n assumptions, are given by the Eq. (2). X X Xk W = W + kW j Using Eq. (1), neglecting the radial deformations and non- nc τ˜ p j=1 j=1 linear radial velocity, the total kinetic energy of the spacecraft k=1 1 model including PZT patches can be calculated by: = {η}T [=]{η} − {q}T [R]{q} +τθ ˜ ; (7) 2

2 2 nk X j X X k j where T = T + TP j=1 j=1 k=1 k j 1 j 2 j nk j [=] = diag ξp ;[R] = {

hc 2 a+Lb t,b t2 1 Z Z 1 X Z G γ2 dxdy + kEj k$j h · δ(T −V ) + δW dt = 0. (10) 2 c c 2 P p p nc i hc a t1 − 2

xk+LP Z ∂2w(x, t)2 ∂ui(x, t)2 Integration over the spatial domains leads to the global kΓj + dx; (5) mass, the stiffness, and the force matrices. In order to con- ∂x2 ∂x xk sider the structural damping effects in spacecraft dynamics, the International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 145 M. Azimi, et al.: MODELING AND VIBRATION SUPPRESSION OF FLEXIBLE SPACECRAFT USING HIGHER-ORDER SANDWICH PANEL. . .

Rayleigh’s dissipation function may be given as: can be restated as follows h i−1 1 θ¨ = M − M M−1 M · T = q˙ T Cq˙ . (11) RR RF FF FR d 2 f f −1 2 R −1 ˙ τ˜ − MRF MFF − ε CFF KFF Z − Z − ε[R][ga]{ηa} ; Using Eqs. (3), (5), (7), (11), and the extended Hamilton’s (15) principle, the attitude dynamic model of the ﬂexible spacecraft −1 ¨ −2 R h −1 −1 i can be obtained in the following form: Z = ε KFF MFF − MFRMRRMRF − 2 R −1 ˙ −1 ε[R][ga]{ηa} − ε CFF KFF Z − Z − MFRMRRτ˜ . MRR MRF θ¨ 0 0 θ˙ + + (16) MFR MFF q¨f 0 CFF q˙ f 0 0 θ τ˜ On setting in Eqs. (15) and (16), the slow subsystem be- = ; 0 KFF qf −[R][ga]{ηa} comes −1 T {ηS} = [gs][=] [R] {q}. (12) −1 ZSlow = MFRMRRτ˜ − ε[R][ga]{ηa} . (17)

As it can be seen from Eq. (12), R and = matrices can be de- Substituting Eq. (17) into Eq. (15), results in composed into sensor and actuator parts corresponding to the −1 sensor/actuator voltages ηS and ηa. Controlling rigid-flexible ¨ −1 θSlow = MRR − MRF MFF MFR · systems is difficult because they are under-actuated systems in Slow which all modes of flexure in each appendage have to be con- −1 −1 trolled by adjusting a single actuating torque. This essentially 1 − MRF MFF MFRMRR τ˜ . (18) Slow uses a perturbation parameter to divide the complex dynamic systems into simpler subsystems at different time scales. It has To obtain the fast subsystem, a fast time scale defined by also been successfully applied to controlling systems with ei- Φ = t/ε and setting ZFast = Z−ZSlow. Note that the slow vari- ther flexible links or joints in combination with rigid bodies ables remain constant in the fast time scale18 or due to the time dynamic. scale introduced, the slow variables act as parameters. Also, by 2 The crucial assumption to be made is that the spectrum of considering ε = 0 in Eqs. (15) and (16), the fast subsystem the flexible modes is well separated from the spectrum of the can be calculated as rigid modes. In this way, two reduced order systems are iden- d2 h i−1 tified: a slow subsystem involves the hub variables as the slow Z = KR M − M M −1 M × dΦ2 Fast FF FF FR RR RF state variables and the fast subsystem that contains the gen- d eralized flexible coordinates as the fast state variables. Then, M−1 − εC Z −Z −ε[R][g ]{η } + FF FF Fast Fast a a Fast a composite controller is designed to control these separated dΦ time-scale subsystems, such that when it is applied to the sys- −1 M τ˜ . (19) tem with coupled rigid-flexible dynamics, the desired trajec- FR Fast tory will be tracked with simultaneous control of appendages vibration. For this case, the dynamic model of the system is The equation of the motion, Eq. (12) is rewritten as transformed into two-time-scale SPT model by defining a com- " # µRR 0 ¨ 0 0 ˙ mon scale factor kMin, which is the minimum of all the con- θS θS 0 µ ¨ + 0 D ˙ + stants of the stiffness matrix KFF . With this common scale FF ZF FF ZF factor, the stiffness matrix KFF elements with large magni- ( −1 −1 ) 0 0 θ 1 − MRF M MFRM τ˜ tude in comparison to the other coefficients, in equation of the S = FF RR . 0 κFF ZF −1 motion, can be scaled such that:18 −ε[R][ga]{ηa} − MFRMRRτ˜ (20) R [KFF ]{qf } = kMin[KFF ]{qf } = {Z} =⇒ where µRR, µFF , DFF , and κFF are directly obtained from R 1 Eqs. (18) and (19). What can be deduced from the Eq. (20) [KFF ]{qf } = {Z}. (13) kMin is that the slow subsystem is not affected by PZT actuators. This feature of the system has enabled the torque devices, such Introducing a new variable as a singular perturbation param- as reaction wheels, to only stabilize the slow subsystem and 0.5 prevent counteracting effects of two types of sub controllers. eter ε = (1/kMin) , Eq. (13) can be expressed as:

R 2 3. CONTROLLER DESIGN [KFF ]{qf } = ε {Z}. (14) The ensuing tracking error is interpreted in SMC as a devi- √ ation of the system state from the nonlinear sliding surface S By comparing k

˙ H H The reduced order feedback control that would turn the slid- The sliding condition should satisfy Sθ = 0 and Sθ = H H ing surface Sθ into an invariant manifold for the system is 0. In the case of Sθ = 0, where e = −k2e˙θ − ˙ H 2 2 called the equivalent control ueq to ensure that Sθ = 0. Ad- k1e˙θ tanh{e˙θ/p }/k3, we have e¨ = −k3e/˙ ℵ, then Eq. (28) 19 ditionally, the variable structure part uVS, is selected in order yields H n o e˙ to guarantee that Sθ = 0, thus the designing surface is attrac- ˙ ˙ T θ V (t) = −2 e˙θ ZF µˆ ; (29) tive and the desired condition is accessible. The control law is Z˙ F written as where µRRk3 k4f(e ˙θ ) u = ueq + uVS. (22) + 0 µˆ = ℵ e˙θ . (30) The dynamics in sliding mode can be written as 0 DFF

d d Lemma: Let S be the Schur complement of µˆ(1, 1) in µˆ, that S˙ H = 0 → (e ) + f(e ˙ ) = 0. (23) θ dt θ dt θ is −1 By solving the above equation for the control input, an expres- µRRk3 k4f(e ˙θ) sion for u is obtained as follows S = DFF − µˆ(1, 2) + µˆ(2, 1). (31) eq ℵ e˙θ

k3e˙θA Then µˆ is positive definite if µ (1, 1) > 0 and S > 0. ueq = − ; (24) RR ℵ Utilizing the SPT technique discretize the equation of the mo- where tion, thereafter the prescribed control law has implemented on θ coordinate which only needs µRR(1, 1) > 0. Considering d −1 −1 −1 all possible values for ki, it can be seen from Schur comple- k2 ≤ ℵ = f(e ˙θ); A = MRR −MRF MFF MFR . de˙θ Slow ment method that the Eq. (29) is a negative definite function (25) of e˙θ. It can be found that the only equilibrium point satisfies Defining the variable structure component uVS yields V˙ (t) =∼ 0. So the designed sliding surface converges to origin asymptotically. u = k SH ; VS 4 θ P–Amp (26)

k SH 4. SIMULATION RESULTS AND where 4 is a positive number and the term θ P–Amp shows that the sliding surface’s proportionate gain has been amplified DISCUSSION by positive numbers k3. This technique is employed to reduce In this section, a comparative numerical simulation is used the settling time in actuator which placed on the rigid main to demonstrate the validity of the flexible spacecraft perfor- body. mance and the effectiveness of the proposed hybrid controller with the new substructure model approach (SPT technique). A Theorem: Let the control objective to force the rigid body uniform symmetrical sandwich panel with PZT layers which modes to follow some pre-specified trajectories. Then, the con- are bonded at the bottom and top surfaces is considered as H trol law Eq. (22) can achieve this objective and ensures that Sθ flexible parts of a spacecraft. The spacecraft simulation in- tends to zero as time tends to infinity. volves a single-axis 60◦ rest-to-rest maneuver. The parameters for the simulated flexible spacecraft are tabulated in Ta- Proof: Consider the following Lyapunov function candidate: ble 1. All computations presented in the paper are calculated by using MATLAB/Simulink software package. The results n ˙ T o e˙θ V (t) = e˙θ Z µ + are summarized comparing the case with conventional flexible F Z˙ F appendages, modeled by Euler-Bernoulli beams in the present T h i eθ study, and those available in open literature. In all the simu- eθ Z diag(k , κ ) . (27) F 4 FF Z T F lations, the initial conditions are considered as x0 = [0 0] , θ = 60◦, θ˙ = θ¨ = 0. The positive weighting coefficients k are included to allow d d d i Figure 2 shows the results of implementing the proposed relative emphasis on the four contributors to the error energy hybrid controller in which the SMC using hyperbolic tangent of the system. This function (Eq. (27)) is one of many possible function is utilized for the slow subsystem. Moreover, the im- ways to weight the mechanical system error energy and merely posed desired angular position is accurately achieved as well provides one illustration of an approach. It is obvious by in- (Fig. 3). The controller gains k , k , k , and k are selected to spection that imposing k > 0 in Eq. (27) guarantees V (t) ≥ 0 1 2 3 4 i be 100, 10, 0.7 and 1.2, respectively. and that the global minimum of V (t) = 0 occurs only at the As seen in Fig. 2, the convergence time for desired angular desired state. displacement of 60◦ and velocity of 0◦/s is about 40 seconds. The desired state implies that when the rigid hub begins at The proposed controller, containing the proportionate ampli- the rest and rotates to a new angular position θ = θ = const, d fication term is considered and the results of the simulations the time derivative to the Eq. (27) yields: for the case that exclude the current term, are shown in these n o figures. As it can be seen, the settling time period in the case ˙ ˙ T e¨θ V (t) = 2 e˙θ ZF µ + in which the P-Amp term is excluded, is much longer than the Z¨ F proposed controller. h i T e˙θ Figure 3 shows the required equivalent control torque for 2 eθ ZF diag(k4, κFF ) ˙ . (28) ZF performing the maneuver. For comparison, the conventional

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 147 M. Azimi, et al.: MODELING AND VIBRATION SUPPRESSION OF FLEXIBLE SPACECRAFT USING HIGHER-ORDER SANDWICH PANEL. . .

Table 1. Parameters of the ﬂexible spacecraft. Flexible appendage Parameters PZT Layer Face Sheets Honeycomb Core Young’s Modulus / Shear Modulus (GPa) Et,b = 70 Gc = 0.26 Ep = 68 3 Density (kg/m ) ρt,b = 2700 ρc = 83.3 ρp = 7700 Thickness (m) ht,b = 0.0005 hc = 0.02 hp = 0.00035 Width (m) $t,b = 0.50 $c = 0.5 $P = 0.08 Length (m) Lt,b = 2 Lc = 2 Lp = 0.08 −12 PZT Strain Constant (m/V) — — d31 = 125 × 10 −3 PZT Stress Constant (Vm/N) — — e31 = 10.5 × 10 Hub dimension (m) a = 0.3 — — 2 Hub inertia (kg m ) Jh = 27.2 — —

Figure 3. Time history of control actions applied on the slow subsystem.

titude controller. This is an important characteristic for actual implementation of the controller. Convergence of the flexible body coordinates and PZT actuation voltage are shown in Figs. 4 to 6. In the present study, actuators and sensors are placed as col- located sets of PZT at each appendage, and play the role of a MIMO control system. The potential capabilities of the PZT pairs are considered in two different instances: a condition in which the attitude controller and the PZT voltages are applied to the dynamic model and in order to compare the effects of PZT voltages on residual vibration suppression of the elastic appendages so an active control system is inactivated. These two cases are depicted in Figs. 5 and 6. By comparing Fig. 4 with Fig. 5, it can be deduced that the effect of PZT actions is noteworthy. Notice that the main quali- Figure 2. The attitude maneuver control integrated with SRF using the pro- tative feature of the vibrational behavior analysis of the present posed and conventional controller; (a) The time history of attitude dynamics (slow subsystem) and (b) The time history of attitude velocity of the hub. study is based on the higher-order quasi-layer wise sandwich panel theory that is more easily perceived comparing with sliding mode control (CSMC),19 has been implemented. The those formal with predefined configuration. Figures 4a and 4b CSMC that is designed in the time domain is hardly applica- reveal that considering the shear stress and higher-order lon- ble as the inherent elastic modes of the flexible systems will gitudinal deflection pattern for the core results to an inherent be unduly excited by the switching control input. In order to vibration profile. Furthermore, significant deviations are ob- overcome this drawback, the sgn(.) term present in the con- served between the higher order theory and the classical panel ventional sliding surface design is replaced by a hyperbolic theory in the vibrational behavior of the elastic appendages tangent function in order to reduce the effects of chattering. which indicates the importance of the higher order terms in Active control technique (SRF) and using PZT sen- the displacement field. sor/actuator patches in conjunction with an attitude controller The primary conclusion drawn from Figs. 4 and 5 is that (SMC) but in two separate sub controllers were used to re- the HOSPM with a prescribed formulation yields accurate reduce the residual vibration. Additionally, active suppression sponses for three-layer sandwich substructures. Moreover, the of structural vibration causes smooth and fine actuation of at- assumption adopted by different researchers in literature is

148 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 M. Azimi, et al.: MODELING AND VIBRATION SUPPRESSION OF FLEXIBLE SPACECRAFT USING HIGHER-ORDER SANDWICH PANEL. . .

Figure 4. Time history of the appendage deflection without an active vibration Figure 5. Time history of the appendage deflection with active vibration con- control; (a) Modified SMC/Higher-Order SPM and SMC/CPT and (b) Modi- trol; (a) Modified SMC/Higher-Order SPM and SMC/CPT and (b) Modified fied SMC/Higher-Order SPM and Modified SMC/CPT. SMC/Higher-Order SPM and Modified SMC/CPT. based on that the dynamics of the space structures are exactly theory. Moreover, modifying the sliding surface reduces the known and is examined and reported in Figs. 4a, 5a, and 6b. issues of the excitation of high frequency elastic modes that In the attitude control law, the modification to sliding mani- frequently happen in the conventional sliding mode, whereas fold for slow subsystem ensures that the spacecraft follows the expedites the error convergence. Global stability of the cou- shortest possible path to the sliding manifold and highly reduce pled rigid/flexible system has been guaranteed via Lyapunov the switching action. As seen in Figs. 6a and 6b, this reduction approach by implementation of singular perturbation theory causes the PZT actuation to be suppressed. during attitude and vibration control design procedure.

5. CONCLUSIONS REFERENCES

Higher-order sandwich panel theory has been used for dy- 1 Koch, R. M. Structural dynamics of large space struc- namic modeling of smart flexible appendages of flexible space- tures having random parametric uncertainties, International craft during attitude maneuver. The obtained equations decom- Journal of Acoustics and Vibration, 8 (2), 95–103, (2003). posed into two different time scales and used singular pertur- https://dx.doi.org/10.20855/ijav.2003.8.2135 bation theory. Based on the governing equations of motion, a hybrid control strategy has been developed for simultane- 2 Frostig, Y. and Baruch, M. Free vibrations of sandwich ous attitude and vibration control during a large targeting ma- beams with a transversely flexible core: a high order ap- neuver. Using higher-order sandwich panel theory for model- proach, Journal of Sound and Vibration, 176 (2), 195–208, ing of elastic deflection of the flexible appendages improves (1994). https://dx.doi.org/10.1006/jsvi.1994.1368 the flexible/rigid body interaction effects of the spacecraft during attitude maneuver. It is found that the proposed model is 3 Hu, H., Belouettar, S., Potier-Ferry, M, and useful to monitor extra vibrational behavior of the smart flex- Daya, E. M. Review and assessment of vari- ible appendages accurately from the view point of layer wise ous theories for modeling sandwich composites,

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 149 M. Azimi, et al.: MODELING AND VIBRATION SUPPRESSION OF FLEXIBLE SPACECRAFT USING HIGHER-ORDER SANDWICH PANEL. . .

8 Hu, Q. Robust adaptive sliding mode attitude control and vibration damping of flexible spacecraft subject to unknown disturbance and uncertainty, Transactions of the Institute of Measurement and Control, 34 (4), 436–447, (2012). https://dx.doi.org/10.1177/0142331210394033 9 Hu, Q. Variable structure maneuvering control with time-varying sliding surface and active vibration damping of flexible spacecraft with input saturation, Acta Astronautica, 64 (11), 1085–1108, (2009). https://dx.doi.org/10.1016/j.actaastro.2009.01.009 10 Song, G. and Kotejoshyer, B. Vibration reduction of flexible structures during slew operations, International Jour- nal of Acoustics and Vibration, 7 (2), 105–109, (2002). https://dx.doi.org/10.20855/ijav.2002.7.2107 11 Song, G. and Sethi V. Comparative study of active control of a large composite I-beam, International Jour- nal of Acoustics and Vibration, 8 (4), 231–238, (2003). https://dx.doi.org/10.20855/ijav.2003.8.4149 12 Horodinca, M., Seghedin, N.-E., Carata, E., Boca, M., Filipoaia, C., and Chitariu, D. Dynamic characterization of a Piezoelectric actuated cantilever beam using energetic parameters, Mechanics of Advanced Materials and Structures, 21 (2), 154–164, (2014). https://dx.doi.org/10.1080/15376494.2012.680668 13 Hu, Q. Sliding mode attitude control with L2-gain performance and vibration reduction of flexible spacecraft with actuator dynamics, Acta Astronautica, 67 (5), 572–583, (2010). https://dx.doi.org/10.1016/j.actaastro.2010.04.018 14 Carrera, E. and Petrolo, M. Refined one-dimensional formulations for laminated structure analysis, AIAA Journal, 50 (1), 176–189, (2012). https://dx.doi.org/10.2514/1.j051219 Figure 6. Time history of PZT actuation voltage; (a) Modified SMC/Higher- Order SPM and (b) SMC/CPT. 15 Giunta, G., Biscani, F., Belouettar, S., Ferreira, A. J. M., and Carrera, E. Free vibration analy- Composite Structures, 84 (3), 282–292, (2008). sis of composite beams via refined theories, Compos- https://dx.doi.org/10.1016/j.compstruct.2007.08.007 ites Part B: Engineering, 44 (1), 540–552, (2013). https://dx.doi.org/10.1016/j.compositesb.2012.03.005 4 Li, Z. and Crocker M. J. A review on vibration damping in sandwich composite structures, International Jour- 16 Sainsbury, M. G. (Experimental and theoretical techniques nal of Acoustics and Vibration, 10 (4), 159–169, (2005). for the) Vibration analysis of damped complex structures, https://dx.doi.org/10.20855/ijav.2005.10.4184 PhD dissertation, Imperial College London, University of London, (1976). 5 Damanpack, A. and Khalili, S. High-order free vi- 17 bration analysis of sandwich beams with a flex- Meeker, T. Publication and Proposed Revision of ible core using dynamic stiffness method, Com- ANSI/IEEE Standard 176-1987 ANSI/IEEE Standard on posite Structures, 94 (5), 1503–1514, (2012). Piezoelectricity, IEEE Transactions on Ultrasonics, Ferro- https://dx.doi.org/10.1016/j.compstruct.2011.08.023 electrics, and Frequency Control, 43 (5), 717–772, (1996). https://dx.doi.org/10.1109/tuffc.1996.535477 6 Khalili, S., Kheirikhah, M., and Fard, K. M. Buck- 18 Mirzaee, E., Eghtesad, M., and Fazelzadeh, S. Maneuver ling analysis of composite sandwich plates with flexible control and active vibration suppression of a two-link flex- core using improved high-order theory, Mechanics of Ad- ible arm using a hybrid variable structure/Lyapunov con- vanced Materials and Structures, 22 (4), 233–247, (2015). trol design, Acta Astronautica, 67 (9), 1218–1232, (2010). https://dx.doi.org/10.1080/15376494.2012.736051 https://dx.doi.org/10.1016/j.actaastro.2010.06.054 7 Shahravi, M., Kabganian, M., and Alasty, A. Adaptive ro- 19 Hu, Q. and Ma, G. Variable structure control and active bust attitude control of a flexible spacecraft, International vibration suppression of flexible spacecraft during attitude Journal of Robust and Nonlinear Control, 16 (6), 287–302, maneuver, Aerospace Science and Technology, 9 (4), 307– (2006). https://dx.doi.org/10.1002/rnc.1051 317, (2005). https://dx.doi.org/10.1016/j.ast.2005.02.001

150 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Free Vibration Analysis of Functionally Heterogeneous Hollow Cylinder Based on Three Dimensional Elasticity Theory

Masoud Asgari Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Pardis Street, Molla-Sadra Avenue, Vanak Square, Tehran, Iran

(Received 16 November 2014; accepted 13 January 2016) A two-dimensional functionally heterogeneous thick hollow cylinder with a ﬁnite length is considered and its natural modes and frequencies are determined. Since mode shapes of a thick cylinder are three-dimensional, even with axisymmetric conditions, three-dimensional theory of elasticity implemented for problem formulation. The axisymmetric conditions are assumed for the cylinder. The material properties of the two-dimensional functionally graded material (2D-FGM) cylinder are varied in the radial and axial directions with power law functions. Effects of volume fraction distribution on the different types of anti-symmetric mode shape conﬁguration and vibrational behaviour of a simply supported cylinder are analyzed. Three-dimensional equations of motion are used and the eigen value problem is developed based on direct variational method. The study shows that the 2D-FGM cylinder exhibit interesting vibrational characteristics and mode shapes when the constituent volume fractions are varied.

1. INTRODUCTION Few studies have been conducted for thick hollow circular cylinders. They require a three-dimensional analysis, which Recently, the composition of several different materials is is based on the theory of elasticity. As a result, in the lit- often used in structural components in order to optimize the eratures, the study of free vibrations of thick circular cylin- responses of structures subjected to severe loadings. For re- ders using three-dimensional theory of elasticity is relatively ducing the local stress concentrations induced by abrupt tran- scarce in comparison to the study of thin shells using other sitions in material properties, the transition between different shell theories. Studies on shells based on three-dimensional materials is made gradually. This idea leads to the concept of theory of elasticity have been presented by some researchers functionally graded materials (FGMs).1 The mechanical prop- for infinitely long cylindrical shells.9–12 For finite-length thick erties of FGMs vary continuously between several different cylindrical shells, different methods such as the finite element materials. Most research in this area is concerned with thermo- method, series solution, and the Ritz energy method have been elastic and residual stress analysis. In many applications of used by some researchers for both solid and hollow homoge- these materials the vibrational characteristics are of great im- neous cases.13–17 A three-dimensional energy formulation was portance in addition to stress considerations. The vibration of used by Liew et. al.18 to compute frequencies and develop FGM cylindrical structures has been studied by a number of graphical representation for three-dimensional mode shapes researchers.2–7 Different studies on the vibration of cylindrical of a homogeneous hollow cylinder. Loy and Lam8 also pre- shells made of a FGM based on Love’s and some other clas- sented an approximate analysis by using a layerwise approach sical shell theories have been done and usually Rayleigh-Ritz to study the vibration of thick circular cylindrical shells on the and finite element methods used for solving governing equa- basis of three-dimensional theory of elasticity. Buchanan and tions.2–7 Yii19 investigated the effect of symmetrical boundary condi- While the vibrational behaviour of a thick walled cylinder is tions on the vibration of thick hollow cylinders using finite ele- of considerable engineering importance, all of the previously ment method. Other researchers20–23 have also presented stud- discussed papers are mainly focused on cylindrical shells us- ies using three-dimensional theory of elasticity based on the ing the classical equations of thin shell theories, except Chen Rayleigh-Ritz method for homogeneous and laminated cylin- et al.7 who investigated a thick hollow cylinder using equa- ders. tions of piezoelasticity based on laminate model as opposed Most of the referred studies considered ways of determining to classical shell theories. Classical or thin-shell theories are the frequencies of cylinders. However, very few of the au- based on the simplifying assumptions of Kirchhof-Love’s hy- thors give a description of the mode shapes of the thick cylin- pothesis. This omission makes the thin-shell theories highly ders. While mode shapes are also very important sources of inadequate for the analysis of even slightly thick shells.8 The information for understanding and controlling the vibration of higher order shell theories are better than the thin-shell theo- a structure.26 Singal and Williams24 combined experimental ries for the analysis of slightly thick shells but are still inade- results with a Ritz energy method of analysis to compare fre- quate for the analysis of moderately thick shells. To analyze quencies for free–free cylinders. They gave a description for moderately thick shells, the transverse normal stress and strain the mode shapes of thick- walled hollow cylinders and rings. components, which are ignored in the higher ordered shell the- Wang and Williams25 have studied extensively frequencies and ories, have to be accounted for and only an analysis based on mode shapes of finite length hollow cylinder using a commer- the three-dimensional theory of elasticity would account for all cial finite element code and three-dimensional block elements the transverse stress and strain components.8 for their analysis. Singhal et al.26 presented theoretical and

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 151–160) https://doi.org/10.20855/ijav.2017.22.2460 151 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . . experimental modal analysis by using a thick-walled circular cylinder model to obtain its natural frequencies and mode shapes. On the other hand, in the previous discussed literature, vibration analysis of moderately thick- walled hollow cylinders are limited to isotropic and laminated cylinders and functionally graded thick hollow cylinders with finite length were not seen in the literatures. Also the functionally graded cylindrical shells considered using thin shell theories. So, investigation of a functionally graded thick hollow cylinder can Figure 1. Axisymmetric cylinder with two dimensional material distribu- be of great importance. Additionally, in all of the discussed tions.37 cases, the variation of volume fraction and properties of the FGMs are one-dimensional and the properties vary continuously through the elements, accuracy can be improved without ously from the inner surface to the outer one with a prescribed refining the mesh size.39, 40 Based on these facts the graded fi- function. But a conventional functionally graded material may nite element developed by the author41 is used for modeling of also not be so effective in some design problems since all the present problem. outer or inner surfaces will have the same composition distribution while in advanced machine elements, load distribu- 2. PROBLEM FORMULATION tion may change in two or three directions.27 Therefore, if the FGM has two-dimensional dependent material properties In this section, the volume fraction distributions in the two more effective material resistance can be obtained. Based on radial and axial directions are introduced. The 3D govern- this fact, a two-dimensional FGM whose material properties ing equations of motion in cylindrical coordinates are obtained are bi-directionally dependent is introduced. Recently a few and the graded finite element is used for modeling the non- authors have investigated 2D-FGM especially its stress analy- homogeneity of the material. sis.27–36 The author also has investigated the natural frequencies of 2.1. Volume Fraction and Material a thick hollow cylinder with finite length made of 2D-FGM.37 Distribution in 2D-FGM Cylinder The influence of constituent volume fractions on natural fre- In the conventional one dimensional functionally graded quencies was studied by varying the volume fractions of the cylinder, the cylinder’s material is graded through the radial constituent metals and ceramics. Furthermore, the effects of direction. The cylinder is made of a combined metal-ceramic length and thickness of the cylinder on fundamental natu- material for which the mixing ratio is varied continuously in ral frequency were considered in different types of 2D-FGM the r-direction from pure ceramic in the inner surface to pure cylinder. metal in the outer surface, or vice versa. In such cases, the So far, investigation of mode shapes configuration of FGM volume fraction variation of the metal is proposed as a power thick finite cylinder has not been considered in previous stud- law relation.37 Using the rule of mixtures, the distribution of ies. Hence, in order to extend the previous studies and to material properties is obtained. investigate the mode shape configuration of an FGM thick Significant advances in fabrication and processing tech- hollow cylinder, the free vibration analysis and all types of niques have made it possible to produce FGMs using processes mode shapes of a thick hollow cylinder with finite length that allow FGMs with complex properties and shapes, in- made of 2D-FGM on the basis of three- dimensional theory cluding two-and three- dimensional gradients using computer- of elasticity has been considered in this study. The mate- aided manufacturing techniques. 2D-FGMs are usually made rial properties of the cylinder are varied in the radial and ax- by continuous gradation of three or four distinct material ial directions with power law functions. Effects of volume phases that one or two of them are ceramics and the others are fraction distribution and FGM configuration on the natural metal alloy phases. The volume fractions of the constituents frequencies and three-dimensional mode shapes of a simply vary in a predetermined composition profile. Now consider supported functionally graded thick hollow cylinder are ana- the volume fractions of 2D-FGM at any arbitrary point in the lyzed. The influence of constituent volume fractions is stud- axisymmetric cylinder of internal radius ri, external radius ro, ied by varying the volume fractions of the constituent met- and finite length L shown in Fig. 1 In the present cylinder the als and ceramics. A functionally graded cylinder with two- inner surface is made of two distinct ceramics and the outer dimensional gradation of distribution profile has been investi- surface from two metals. The variables c1, c2, m1, and m2 gated as well as the one-dimensional gradation of material dis- denote first ceramic, second ceramic, first metal and second tribution. Three-dimensional equations of motion are used and metal, respectively. the eigen value problem is developed based on Rayleigh-Ritz The volume fraction of the first ceramic material is changed variational method and all non- axisymmetric mode shapes are from 100% at the lower surface to zero at the upper surface considered. The finite element method with graded material by a power law function. Additionally, this volume fraction is characteristics within each element of the structure is used for changed continuously from inner surface to the outer surface. the solution. Using conventional finite element formulations The volume fractions of the other materials change similar to such that the property field is constant within an individual ele- the mentioned one in two directions. The function of volume ment for dynamic problems leads to significant discontinuities fraction distribution of each material can be explained as37 and inaccuracies.38 These inaccuracies will be more significant nr n in 2D-FGM cases. On the other hand, by using graded finite r − ri h z z i Vc1(r, z) = 1 − 1 − ; (1a) element in which the material property field is graded continu- r0 − ri L 152 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . . Table 1. Basic constituents of the 2D-FGM cylinder. E(r, z) [D(r, z)] = Constituents Material E (Gpa) ρ (kg/m3) (1 + ν)(1 − 2ν) m1 Ti6Al4V 115 2715   m2 Al 1100 69 4515 1 − ν ν ν 0 0 0 c1 SiC 440 3210  ν 1 − ν ν 0 0 0  c2 Al2O3 150 3470  ν ν 1 − ν 0 0 0     1−2ν  ; (7) 0 0 0 2 0 0  1−2ν   0 0 0 0 2 0  nr n 1−2ν r − ri h z z i 0 0 0 0 0 Vc2(r, z) = 1 − ; (1b) 2 r0 − ri L where ν denotes the Poison’s ratio, which is uniform through nr n r − ri h z z i the cylinder and E(r, z) is Young’s modulus that depends on r Vm1(r, z) = 1 − ; (1c) and z coordinates. The strain-displacement equations are:42 r0 − ri L

nr n ∂u r − ri z z εr = ; (8a) Vm2(r, z) = ; (1d) ∂r r0 − ri L where n and n are non-zero parameters that represent the ∂v u r z ε = + ; (8b) basic constituent distributions in r and z directions. Material θ r∂θ r properties at each point can be obtained by using the linear rule of mixtures, in which a material property P at any arbitrary ∂w εz = ; (8c) point (r, z) in the 2D-FGM cylinder is determined by linear ∂z combination of volume fractions and material properties of the ∂u ∂w basic materials as37 γ = + ; (8d) rz ∂z ∂r P = Pc1Vc1 + Pc2Vc2 + Pm1Vm1 + Pm2Vm2. (2) ∂v ∂w γ = + ; (8e) zθ ∂z r∂θ The basic constituents of the 2D-FGM cylinder are presented in Table 1. It should be noted that Poisson’s ratio is ∂u ∂v v γ = + − . (8f) assumed to be constant through the body. This assumption is rθ r∂θ ∂r r reasonable because of the small differences between the Pois- The cylinder is simply supported on its two end edges. So son’s ratios of basic materials. mechanical boundary conditions on upper and lower edges are assumed as 2.2. Governing Equations v(r, 0, θ, t) = v(r, L, θ, t) = w(r, 0, θ, t) = w(r, L, θ, t) = 0. Consider a 2D-FGM thick hollow cylinder of internal ra- (9) dius, r external radius r , and finite length L. Because of the i o A solution that satisfies the circumferential displacement and axisymmetry of geometry and material distribution profile, co- defines a circular frequency is19 ordinates r, z, and θ are used in the analysis. Neglecting body forces, the equations of motion in cylindrical coordinates are iωt u(r, θ, z, t) = ψ1(r, z) cos(mθ)e ; (10a) obtained as

2 ∂σrr ∂τrz ∂τrθ σrr − σθθ ∂ u iωt + + + = ρ(r, z) ; (3a) v(r, θ, z, t) = ψ2(r, z) sin(mθ)e ; (10b) ∂r ∂z r∂θ r ∂t2

w(r, θ, z, t) = ψ (r, z) cos(mθ)eiωt; ∂τ ∂σ ∂τ 2τ ∂2v 3 (10c) rθ + θθ + zθ + rθ = ρ(r, z) ; (3b) ∂r r∂θ ∂z r ∂t2 where m is the circumferential wave number and ω is the circular frequency. Also considering the circumferential symmetry ∂τ ∂τ ∂σ τ ∂2w of the cylinder about the coordinate θ, the displacement ampli- rz + zθ + zz + rz = ρ(r, z) ; (3c) ∂r r∂θ ∂z r ∂t2 tude functions can be written as ψ1(r, z), ψ2(r, z) and ψ3(r, z). It is obvious that m = 0, which donates the axisymmetric vi- where u, v, and w are radial, circumferential, and axial com- bration. Certain specified uniform boundary conditions along ponents of displacement respectively those are functions of the two ends can be satisfied by choosing the displacement am- (r, z, θ, t) and ρ(r, z) is the mass density that depends on r and plitude functions properly. z coordinates. The constitutive equations for FGM are written as In order to solve the governing equations the finite element method with graded element properties is used. For this pur- {σ } = [D(r, z)]{ε }; (4) ij ij pose, the variational formulation is considered. In conven- where the stress and strain components and the coefficients of tional finite element formulations a predetermined set of mate- elasticity are rial properties are used for each element such that the property field is constant within an individual element. For modeling {σij} = σrr σθθ σzz τzθ τrz τrθ ; (5) a continuously non-homogeneous material, the material property function must be discretized according to the size of elements mesh. This approximation can provide significant dis- {εij} = εrr εθθ εzz γzθ γrz γrθ ; (6) continuities. In addition, variation of material properties in

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 153 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . . two directions such as the present problem makes this effect three-dimensional and axisymmetric. Vector of nodal displace- more considerable. Based on these facts the graded finite elements (degrees of freedom) is ment is strongly preferable for modeling of the present prob- e lem. Hamilton’s principle for the present problem is {Q } = u v w u v w u v w u v w T t2 i i i j j j k k k l l l ; Z (20) δ(Π − T )dt = 0; (11) t1 where subscripts i, j, k, l are related to four nodes of each element. where Π and T are potential energy and kinetic energy respec- The cylinder will be divided into some brick-like subdivi- tively. These functions and their variations are sions in radial, axial, and circumferential directions as well ! as making a tetrahedral mesh through the use of brick sub- 1 ZZZ ∂u2 ∂v 2 ∂w 2 T = ρ(r, z) + + dV ; divisions. In this case 10*10*12 brick subdivisions produced 2 ∂t ∂t ∂t that leads to 2400 elements including 4356 degrees of free- vol dom. Applying Hamilton’s principle for each element, it can (12) be achieved as ZZZ ∂2u ∂2v   ZZZ δT = ρ(r, z) 2 δu + 2 δv+ e T T e ∂t ∂t δ{Q }  [B(r, z, θ)] [D(r, z)][B(r, z, θ)]dV  {Q }+ vol e ∂2w V + ∂w dV ;   2 (13) ZZZ ∂t e T T e +δ{Q }  ρ(r, z)[N] [N]dV  {Q¨ } = 0; ZZZ V e 1 T Π = {σij} {εij}dV ; (14) (21) 2 vol where V is the volume of element and [B] is the operation 1 ZZZ matrix of strain-nodal displacement. In graded finite element, δΠ = {σ }T {δε }dV ; (15) 2 ij ij the interpolation function for the displacements within the el- vol ements and strain-displacement relations are the same as stan- where V denotes the area and volume of the domain under con- dard conventional finite. In this way the constitutive relation is sideration. Substituting Eqs. (12) to (15) in Hamilton’s principle and applying side conditions, δu(t1) = δu(t2) = δv(t1) = {σij} = [D(r, z)]{εij}; (22) δv(t2) = δw(t1) = δw(t2) = 0, by part integration have where the components of [D(r, z)] could be explicit functions describing the actual material property gradient in which ZZZ ZZZ ∂2u T E(r, z) is determined at each point through the element using {σij} {δεij}dV + ρ(r, z) 2 δu+ ∂t distribution function of this property based on rule of mixtures vol vol as ∂2v ∂2w + 2 δv + 2 ∂w dV = 0; (16) ∂t ∂t E(r, z) = Ec1Vc1(r, z) + Ec2Vc2(r, z)+

+Em1Vm1(r, z) + Em2Vm2(r, z). (23) The strain-displacement relations can be written as43 Also, the mass density ρ(r, z) is, in general, a function of {ε} = [L]{u}; (17) position as well as the mechanical properties. Therefore, in the graded finite element the mass density distribution should be where [L] relates displacements into strain components. And assigned into the element formulation as   u ρ(r, z) = ρc1Vc1(r, z) + ρc2Vc2(r, z)+ {u} = v . (18)   +ρm1Vm1(r, z) + ρm2Vm2(r, z). (24) w Since δ{Qe} is the variational displacement of the nodal points Four node tetrahedral element is used to discrete the domain. and is arbitrary, it can be omitted from Eq. (23), and then this By taking the nodal values of u, v, and w as the degrees of equation can be written as freedom a linear displacement model can be assumed as [M]e{Që} + [K]e{Qe} = 0; (25) e u e iωt where the characteristic matrices are given as v  = [N]{Q0}e ; (19) w ZZZ [K]e = [B(r, z, θ)]T [D(r, z)][B(r, z, θ)]dV ; (26) where [N] is the matrix of assumed shape functions and satisfy V e e certain specified boundary conditions and {Q0} is the nodal displacement vector of element. The matrix of interpolation ZZZ functions corresponding to elements which derived in terms [M]e = [N(r, z, θ)]T [N(r, z, θ)]ρ(r, z)dV. (27) of global coordinates of nodes of elements its components are V e 154 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 2. Dimensionless frequencies φ = (ωH/π) pρ/G of axisymmetric vibration for freely supported isotropic cylinders (m = 0, H/R = 0.4, ν = 0.3).

H/L Method φ1 φ2 φ3 φ4 φ5 φ6 (Armenakas et al., 1969) 0.20495 0.34765 1.07205 1.82336 2.09588 3.00073 0.2 (Cheung and Wu, 1972) 0.20495 0.34765 1.07312 1.82688 2.10257 3.02850 Present method 0.20492 0.34763 1.07309 1.82686 2.10254 3.02848 (Armenakas et al., 1969) 0.27540 0.67185 1.23591 1.76178 2.25224 2.99980 0.4 (Cheung and Wu, 1972) 0.27544 0.67188 1.23712 1.76587 2.25874 3.02742 Present method 0.27539 0.67182 1.23704 1.76580 2.25868 3.02735 (Armenakas et al., 1969) 0.42022 0.98133 1.44860 1.75005 2.43159 3.00691 0.6 (Cheung and Wu, 1972) 0.42038 0.98145 1.45008 1.75433 2.43851 3.03437 Present method 0.42032 0.98138 1.45001 1.75421 2.43842 3.03428

For ﬁnding the components of characteristic matrices, the materials are two distinct ceramics and two distinct metals de- integral must be taken over the elements’ volume considering scribed in Table 1 and Poison’s ratio ν = 0.3. Volume fractions Eqs. (23) and (24). As [D(r, z)] and ρ(r, z) are not constant, of materials are distributed according to Eqs. (1a) to (1d). Vi- these matrices are evaluated by numerical integration for each bration characteristics of cylinder for some different powers element. of material composition proﬁle n r and n z are presented and Now by assembling the element matrices, the global matrix compared. Dimensionless frequency parameter is assumed as equation for the structure can be obtained as r ρ Ω = (r ω) . (29) [M]{Q¨} + [K]{Q} = {0}. (28) o G

Once the finite element equations are established, Q = The longitudinal and radial modes are uncoupled from pure iωt Q0e was substituted into Eq. (28) which resulted in an eigen torsional modes when the circumferential wave number is value equation that can be solved using standard eigen value taken as m = 0.19 For symmetric boundary conditions, the extraction procedures. To get a better illustration of the mode mode shapes are either symmetric or antisymmetric. shapes, the numerical results of the displacements relate to In order to investigate the effect of material distribution pro- each eigen value on each node were transferred into the proper file in the case of non-symmetrical modes, some selected mode coordinate system and the radial, tangential, and longitudinal shapes were computed for comparison, are shown in following directions of the nodes determined. Tables. Three-dimensional antisymmetric vibrational modes related 3. NUMERICAL RESULTS AND DISCUSSION to non-zero values for circumferential wave numbers (m 6= 0) will be described in the following illustrations. Mode shapes To verify the present solution, as similar works to the present are according to the classification of the modes of thick cylin- 25 work are few, a finite length of a homogeneous thick cylinder ders used by Wang and Williams. Axial bending modes, that can be found with the existing literature is used. A fi- where the circumferential cross section segments bend oppo- nite element for axisymmetric elasticity is formulated directly sitely in the axial direction, and the radial motion with shear- in the cylindrical coordinates to study the vibration of hol- ing modes; for this kind the cylinder no longer retains a con- low, isotropic, and homogeneous finite length cylinders and stant cross sectional along its length. Circumferential in these frequencies are computed for free-free end boundary condi- modes means adjacent segmental elements expand or contract tions in the reference19 and compared with the reference.16 For one by one in the circumferential direction. The median cir- solving the aforementioned problem using the graded finite el- cumferential length of an expanding segment becomes longer ement method developed here, we considered a thick hollow and the length of the contracting segment becomes shorter. cylinder with freely supported end conditions in which the ma- Global modes for these modes is the thick cylinder can be con- terial distribution is uniform. Therefore, the volume fraction sidered to behave as a simple beam vibrating in a transverse exponent and property coefficients in the 2D-FGM are taken direction, a bar vibrating in torsion or as a rod vibrating in a as: nz = 0, nr = 0, Pc1 = Pc2 = Pm1 = Pm2 = P , where longitudinal direction. P is a uniform material properties of the cylinder. Comparison The variations of the mode shapes the circumferential wave of the results for this case in Table 2 shows good agreement numbers m for different values of longitudinal mode numbers between the two methods again. for nr = 0.2 and nz = 0 is shown in Table 3. Although the displacements of each nodes of the cylinder are calculated, just A thick hollow cylinder of inner radius ri = 0.5 m, outer some cross sections of the cylinder on inner and outer radii are radius ro = 1 m and length L = 2 m is considered. Effects of volume fraction distribution on the natural frequencies and shown due to the nature of the illustration. mode shapes configuration of a simply supported functionally The variations of the natural dimensionless frequencies pa- graded thick hollow cylinder are analyzed. The influence of rameter with the circumferential wave numbers, m for differ- constituent volume fractions is studied by varying the volume ent values of radial power exponent while the axial power ex- fractions of the constituent metals and ceramics. A function- ponent nz = 0 is shown in Fig. 2. It is clear that the effect ally graded cylinder with two-dimensional gradation of distri- of variation of radial power exponent is more considerable for bution profile has been investigated as well as the case where higher natural frequencies. The same results for nr = 5 and the axial power law exponent is assumed to be zero, i.e., nz = nz = 0 is shown in Table 4. 0, and the results of one-dimensional gradation of material dis- The effect of variation of axial power exponent is considered tribution can be obtained in the hollow cylinder. The basic in Tables 5 and 6. Mode shapes for two different axial power materials are as explained in the previous section. Constituent exponents while the radial power exponent is zero (nr = 0)

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 155 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 3. Antisymmetric modes in 1D-FGM cylinder, nz = 0, nr = 0.2. Mode 1 Mode 2 Mode 3 Mode 4

m=1

m=2

m=3

Table 4. Antisymmetric modes in 1D-FGM cylinder, nz = 0, nr = 5. Mode 1 Mode 2 Mode 3 Mode 4

m=1

m=2

m=3 are shown in these tables. The same investigation for natural the influence of the value of nz can be seen here. It is clear that frequency is also indicated in Fig. 3. It is clear that effect of as nz increased, the natural frequencies increased. The natural axial power exponent on the natural frequencies is insignificant frequencies and the mode shapes of different kinds of modes when the radial power exponent is zero. vary with the changing material’s distribution profile. In order to further study the mode configuration due to the It is clear from the results that the natural frequencies and power exponent, the 2D-FGM cylinder with both non-zero ex- mode shapes, are strongly influenced by the material compo- ponents is considered in Tables 7 and 8. sition profile. The constituent volume fractions of the con- Variation of natural frequencies with a circumferential wave stituent materials affect antisymmetric mode shapes particu- number for different values of axial power exponent, while the larly in higher mode numbers. It should be noted that although radial power exponent is not zero (nr = 2) is shown in Fig. 4; the manufacturing of multidimensional FGM may seem to

156 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 5. Antisymmetric modes in 1D-FGM cylinder, nr = 0, nz = 0.2. Mode 1 Mode 2 Mode 3 Mode 4

m=1

m=2

m=3

Table 6. Antisymmetric modes in 1D-FGM cylinder, nr = 0, nz = 5. Mode 1 Mode 2 Mode 3 Mode 4

m=1

m=2

m=3 be costly or difficult, these technologies are relatively new, 4. CONCLUSIONS processes such as three-dimensional printing (3DPTM) and Laser Engineering Net Shaping (LENS R ) can currently pro- Based on the three-dimensional mode shapes, a study on duce FGMs with relatively arbitrary tree-dimensional grad- the free vibration of simply supported thick hollow cylinder ing.33 With further refinement, FGM manufacturing meth- with finite length made of 2D-FGM is presented. Material ods may provide designers with more control of the composi- properties are graded in the thickness and longitudinal direction profile of functionally graded components with reasonable tions of the cylinder according to a volume fraction power cost. law distribution. The equations of motion are based on three- dimensional elasticity theory and the graded finite element method, which has some advantages to the conventional finite

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 157 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Table 7. Antisymmetric modes in 2D-FGM cylinder, nr = 0.5, nz = 5. n=1 n=2 n=3 n=4

m=1

m=2

m=3

Table 8. Antisymmetric modes in 2D-FGM cylinder, nr = 5, nz = 0.5. n=1 n=2 n=3 n=4

m=1

m=2

m=3 element method and is employed for the solution. The effects varied. Based on the achieved results, 2D-FGMs have a power- of two-dimensional material distribution on the mode shapes ful potential for designing and optimization of structures under characteristics are considered and compared with conventional multi-functional requirements. one-dimensional FGM. Variation of natural frequencies and anti-symmetric three-dimensional mode shapes with circum- REFERENCES ferential wave number associated with the numerous values of radial and axial power exponents are calculated for 1D- FGM 1 Koizumi, M. The concept of FGM Ceramic Transaction, cylinder as well as 2D-FGM cylinder. The study shows that the Functionally Graded Materials, 34, 3–10, (1993). 2D-FGM cylinder exhibits an interesting frequency and mode shape characteristics when the constituent volume fractions are 2 Loy, C. T., Lam, K. Y., and Reddy, J. N. Vibra-

158 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

Figure 2. Natural frequencies for various radial power law exponents in 1D- Figure 4. Natural frequencies for various axial power law exponents in 2D- 37 FGM cylinder, nz = 0. FGM cylinder, nr = 2. 7 Chen, W. Q., Bian, Z. G., Lv, C. F., and Ding, H. J. 3D free vibration analysis of a functionally graded piezoelectric hollow cylinder filled with compressible fluid, Inter- national Journal of Solids and Structures, 41, 947–964, (2004). https://dx.doi.org/10.1016/j.ijsolstr.2003.09.036 8 Loy, C. T. and Lam, K. Y. Vibration of thick cylindrical shells on the basis of three-dimensional theory of elasticity, Journal of Sound and vibration, 226 (4), 719–737, (1999). https://dx.doi.org/10.1006/jsvi.1999.2310 9 Greenspon, J. E. Flexural vibrations of a thick-walled cylinder according to the exact theory of elasticity, Journal of Aerospace Sciences, 27, 1365–1373, (1957). https://dx.doi.org/10.2514/8.8370 10 Gazis, D. C. Three-dimensional investigation of the propagation of waves in hollow circular cylinders, Journal of Figure 3. Natural frequencies for various axial power law exponents in 1D- the Acoustical Society of America, 31, 568–578, (1959). FGM cylinder, nr = 0. https://dx.doi.org/10.1121/1.1907754 tion of functionally graded cylindrical shells, Int Jour- 11 Nelson, R. B., Dong, S. B., and Kalra R. D. Vibra- nal of Mechanical Sciences, 41, 309–324, (1999). tions and waves in laminated orthotropic circular cylin- https://dx.doi.org/10.1016/s0020-7403(98)00054-x ders, Journal of Sound and Vibration, 18, 429–444, (1971). https://dx.doi.org/10.1016/0022-460x(71)90714-0 3 Pradhan, S. C., Loy, C. T., Lam, K. Y., and Reddy, J. N. Vibration characteristics of function- 12 Armenakas, A. E., Gazis, D. S., and Herrmann G. ally graded cylindrical shells under various boundary Free vibrations of Circular Cylindrical Shells, Pergamon conditions, Applied Acoustics, 61, 111–129, (2000). Press, Oxford, (1969). https://dx.doi.org/10.1016/b978-0- https://dx.doi.org/10.1016/s0003-682x(99)00063-8 08-011732-4.50012-x 13 4 Zhao, X., Ng, T. Y., and Liew K. M. Free vi- Gladwell, G. and Vijay, D. K. Natural frequen- bration of two-side simply-supported laminated cylin- cies of free finite-length circular cylinders, Journal drical panels via the mesh-free kp-Ritz method, Int of Sound and Vibration, 42 (3), 387–397, (1975). Journal of Mechanical Sciences, 46, 123–142, (2004). https://dx.doi.org/10.1016/0022-460x(75)90252-7 https://dx.doi.org/10.1016/j.ijmecsci.2004.02.010 14 Gladwell, G. and Vijay, D. K. Vibration analysis of ax- 5 Patel, B. P., Gupta, S. S., Loknath, M. S., and isymmetric resonators, Journal of Sound and Vibration, Kadu, C. P. Free vibration analysis of functionally 42 (2), 137–145, (1975). https://dx.doi.org/10.1016/0022- graded elliptical cylindrical shells using higher-order 460x(75)90211-4 theory, Composite Structures 69, 259–270, (2005). 15 Hutchinson, J. R. and El-Azhari, S. A. Vibrations of free https://dx.doi.org/10.1016/j.compstruct.2004.07.002 hollow circular cylinders, Journal of Applied Mechanics, 53, 641–646, (1986). https://dx.doi.org/10.1115/1.3171824 6 Kadoli, R. and Ganesan, N. Buckling and free vibration analysis of functionally graded cylindrical shells sub- 16 So, J. and Leissa, A. W. Free vibrations of thick hollow jected to a temperature specified boundary condition, circular cylinders from three-dimensional analysis, Trans Journal of Sound and Vibration, 289, 450–480, (2006). ASME, Journal of Vibrations and Acoustics, 119, 89–95, https://dx.doi.org/10.1016/j.jsv.2005.02.034 (1997). https://dx.doi.org/10.1115/1.2889692

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 159 M. Asgari: FREE VIBRATION ANALYSIS OF FUNCTIONALLY HETEROGENEOUS HOLLOW CYLINDER BASED ON THREE-DIMENSIONAL. . .

17 So, J. Three dimensional vibration analysis of elastic bodies 31 Cho, J. R. and Ha, D.Y. Optimal tailoring of 2D volume- of revolution, PhD dissertation, The Ohio State University, fraction distributionsfor heat-resisting functionally graded Columbus, (1993). materials using FDM, Comput Method Appl Mech Eng, 191, 3195–3211, (2002). https://dx.doi.org/10.1016/s0045- 18 Liew, K. M., Hung, K. C., and Lim, M. K. Vibration of 7825(02)00256-6 stress-free hollow cylinders of arbitrary cross section, Trans ASME, Journal of Applied Mechanics, 62, 718–724, (1995). 32 Hedia, H. S., Midany, T. T., Shabara, M. N., and https://dx.doi.org/10.1115/1.2897005 Fouda N. Development of cementless metal-backed ac- etabular cup prosthesis using functionally graded ma- 19 Buchanan, G. R. and Yii, C. B. Y., Effect of symmet- terial, Int J Mech Mater Des, 2, 259–267, (2005). rical boundary conditions on the vibration of thick hol- https://dx.doi.org/10.1007/s10999-006-9006-y low cylinders, Applied Acoustics, 63, 547–566, (2002). https://dx.doi.org/10.1016/s0003-682x(01)00048-2 33 Goupee, A. J. and Vel, S. Optimization of natu- 20 Cheung, Y. K. and Wu, C. I., Free vibrations of thick, lay- ral frequencies of bi-directional functionally graded ered cylinders having finite length with various boundary beams, Struct Multidisc Optim, 32 (6), (2006). conditions, Journal of Sound and Vibration, 24, 189–200, https://dx.doi.org/10.1007/s00158-006-0022-1 (1972). https://dx.doi.org/10.1016/0022-460x(72)90948-0 34 Asgari, M., Akhlaghi M., and Hosseini S. M. Dy- 21 Soldatos, K. P. and Hadjigeorgiou, V. P. Three-dimensional namic Analysis of Two-Dimensional Functionally Graded solution of the free vibration problem of homoge- Thick Hollow Cylinder with Finite Length under Im- neous isotropic cylindrical shells and panels, Jour- pact Loading, Acta Mechanica, 208, 163–180, (2009). nal of Sound and Vibration, 137, 369–384, (1990). https://dx.doi.org/10.1007/s00707-008-0133-4 https://dx.doi.org/10.1016/0022-460x(90)90805-a 35 Asgari, M. and Akhlaghi M. Thermo-Mechanical Analysis 22 Jiang, X. Y. 3-D vibration analysis of fiber rein- of 2D-FGM Thick Hollow Cylinder using Graded Finite El- forced composite laminated cylindrical shells, Jour- ements, Advances in Structural Engineering, 14 (6), (2011). nal of Vibration and Acoustics, 119, 46–51, (1997). https://dx.doi.org/10.1260/1369-4332.14.6.1059 https://dx.doi.org/10.1115/1.2889686 36 Asgari, M. and Akhlaghi, M. Transient thermal stresses in 23 Ye, J. Q. and Soldatos, K. P. Three-dimensional vibrations two-dimensional functionally graded thick hollow cylinder of cross-ply laminated hollow cylinders with clamped edge with finite length, Arch Appl Mech, 80, 353–376, (2010). boundaries, Journal of Vibration and Acoustics, 119, 317– 37 Asgari M. and Akhlaghi M. Free Vibration Anal- 323, (1997). https://dx.doi.org/10.1115/1.2889726 ysis of 2D-FGM Thick Hollow Cylinder Based on 24 Singal, R. K. and Williams, K. A theoretical and experimen- Three Dimensional Elasticity Equations, European tal study of vibrations of thick circular cylindrical shells Journal of Mechanics A/Solids, 30, 72–81, (2011). and rings, Trans ASME, Journal of Vibrations, Acoustics, https://dx.doi.org/10.1016/j.euromechsol.2010.10.002 Stress, and Reliability in Design, 110, 533-537, (1988). 38 Santare, M. H., Thamburaj, P., and Gazonas, A. The https://dx.doi.org/10.1115/1.3269562 use of graded finite elements in the study of elastic 25 Wang, H. and Williams, K. Vibrational modes wave propagation in continuously non-homogeneous ma- of thick cylinders of finite length, Journal of terials, Int J Solids Structures, 40, 5621–5634, (2003). Sound and Vibration, 191 (5), 955–971, (1996). https://dx.doi.org/10.1016/s0020-7683(03)00315-9 https://dx.doi.org/10.1006/jsvi.1996.0165 39 Santare, M. H. and Lambros, J. Use of a graded finite 26 Singhal, R. K., Guan, W., and Williams, K. Modal anal- element to model the behavior of non-homogeneous ysis of a thick-walled circular cylinder, Mechanical Sys- materials, J Appl Mech, 67, 819–822, (2000). tems and Signal Processing, 16 (1), 141–153, (2002). https://dx.doi.org/10.1115/1.1328089 https://dx.doi.org/10.1006/mssp.2001.1420 40 Kim, J. H. and Paulino, G. H. Isoparametric graded 27 Nemat-Alla, M. Reduction of thermal stresses by definite elements for nonhomogeneous isotropic and or- veloping two dimensional functionally graded materi- thotropic materials, J Appl Mech, 69, 502–514, (2002). als, Int. J. Solids Structures, 40, 7339–7356, (2003). https://dx.doi.org/10.1115/1.1467094 https://dx.doi.org/10.1016/j.ijsolstr.2003.08.017 41 Asgari, M. and Akhlaghi M. Thermo-mechanical 28 Dhaliwal, R. S. and Singh, B. M. On the theory of elasticity Stresses in Two-Dimensional Functionally Graded of a non-homogeneous medium, J Elasticity, 8, 211–219, Thick Hollow Cylinder with Finite Length, Ad- (1978). https://dx.doi.org/10.1007/bf00052484 vances in Structural Engineering, 14 (6), (2012). 29 Clements, D. L. and Budhi, W. S. A boundary element https://dx.doi.org/10.1177/0021998315622051 method for the solution of a class of steady-state prob- 42 Boresi, B., Peter, A., and Ken, P. Elasticity in Engineering lems for anisotropic media, Heat Transfer, 121, 462–465, Mechanics, 2nd ed. Wiley, New York, (1999). (1999). https://dx.doi.org/10.1115/1.2826000 43 30 Zienkiewicz, O. C. and Taylor, R. L. The Finite element Abudi, J. and Pindera, M. J. Thermoelastic theory for method, VII: Soloid mechanics, 5th edition, Butterworth- the response of materials functionally graded in two di- Heinemann, Oxford, (2000). rections, Int. J. Solids Structures, 33, 931–966, (1996). https://dx.doi.org/10.1016/0020-7683(95)00084-4

160 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Electrically Forced Vibrations of Partially Electroded Rectangular Quartz Plate Piezoelectric Resonators Hui Chen, Ji Wang and Jianke Du Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang 315211, China

Jiashi Yang Piezoelectric Device Laboratory, School of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang 315211, China Department of Mechanical and Materials Engineering, the University of Nebraska-Lincoln, Lincoln, NE 68588- 0526, USA

(Received 3 October 2014; accepted 16 February 2015) We performed a theoretical analysis on electrically forced, coupled thickness-shear, and flexural vibrations of a rectangular AT-cut quartz resonator, which is electroded in its central part only. Mindlin’s first-order theory for piezoelectric plates is used and simplified by the flexure-twist approximation due to Mindlin and Spencer. An analytical solution is then obtained. The admittance of the resonator under a time-harmonic driving voltage, which is an important property in resonator design, is calculated and examined. The mode shapes at resonances are also presented and discussed. It is found that for a perfectly symmetric resonator only cylindrical modes depending on one of the two in-plane spatial coordinates can be excited by the applied voltage. This supports the conventional approximate analyses based on the assumption of cylindrical deformations.

1. INTRODUCTION be impractical or impossible, except in the case of cylindrical motions where the fields vary along one of the two in-plane Piezoelectric crystals are widely used to make resonators for directions of a plate only. When the vibration varies in both time-keeping, frequency generation and operation, telecom- of the in-plane directions of a plate, there are very few theo- munication, and sensing. Quartz is the most widely used retical results on partially electroded resonators from the 2-D crystal for resonator applications. Due to material anisotropy equations, especially for electrically forced vibrations. and electromechanical coupling, a theoretical analyses of crys- For coupled TSh and flexural motions of a plate, a pro- tal resonators presents considerable mathematical challenges. cedure called flexure-twist approximation was developed11 Since many piezoelectric resonators operate with thickness- for anisotropic elastic plates, which describes a plate shear- 1–4 1–4 shear (TSh) vibration modes of plates, Mindlin et al. de- deformable in one of the two in-plane directions of the plate veloped and refined the two-dimensional (2-D) equations for and is shear-rigid in the perpendicular direction. The flexure- motions of piezoelectric plates. The 2-D equations lead to twist approximation was later extended to the case of piezo- many theoretical results useful for the understanding and de- electric plates.12 This approximation simplifies the 2-D plate sign of piezoelectric resonators. For example, the behaviors of equations significantly, with which the free vibration solution cylindrical TSh motions or the so-called straight-crested waves of a rectangular quartz plate was obtained by Mindlin and depending on one of the two in-plane spatial variables of the Spencer11 which provides basic information on the resonant 5–7 crystal plates only are now reasonably well understood. Fi- frequencies and mode shapes of the plate. However, the under- nite element procedures based on the 2-D plate equations were standing of the behaviors of a partially electroded, rectangular 8, 9 also developed. More references can be found in a review resonator in electrically forced vibrations is still very much in 10 article by Wang and Yang. need. This is because resonators are used as elements of elec- Even with the 2-D plate equations, there are practically im- trical circuits of alternating currents. Two basic properties of portant situations for which the available theoretical results are a resonator, its resonant frequency and admittance, are both very limited, which means there is a lack of basic understand- of primary interest for circuit analyses. A free vibration anal- ing. This is because real crystal resonators actually used in ysis can only produce resonant frequencies and modes. The various electronic equipment are with partial electrodes. When admittance of a resonator can only be obtained from an electri- using the 2-D plate equation to analyze these resonators, the cally forced vibration analysis. Recently, there has been grow- electroded and unelectroded regions need to be treated sepa- ing interest in forced vibration analyses of piezoelectric plates rately using slightly different equations with continuity condi- and the computation of resonator admittance,13–15 which are tions between them. This is complicated and sometimes it can for cylindrical motions. In this paper, we use Mindlins first-

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 161–166) https://doi.org/10.20855/ijav.2017.22.2461 161 H. Chen, et al.: AN ANALYSIS OF ELECTRICALLY FORCED VIBRATIONS OF PARTIALLY ELECTRODED, RECTANGULAR QUARTZ PLATE. . .

2 γ55 = c55 − c56/c66 = 1/s55; (3b)

ψ11 = d11γ11 + d13γ13; (4a)

ψ35 = e35 − e36c56/c66 = d35γ55. (4b) 11, 12 κ1 is a correction factor given by

π2 8 κ2 = 1 + R − k2 ; (5a) 1 12 π2 26

2 2 e26 k26 = ; (5b) ε22cˆ66 2 e26 cˆ66 = c66 + . (5c) Figure 1. A partially electroded piezoelectric resonator of monoclinic crystals. ε22 When there is an alternating voltage V exp(iωt) applied order plate theory3 simplified by the flexure-twist approxima- across the electrodes, the ϕ(1) in Eq. (1) is related to the volt- tion11, 12 to analyze electrically forced, coupled TSh and flexu- age by3 ral vibrations of a partially electroded rectangular piezoelectric 1 ϕ(1) = V exp(iωt). (6) resonator of monoclinic crystals. This includes the commonly 2b used rotated Y-cut quartz and langasite crystal plates as special The substitution of Eqs. (2) and (6) into Eq. (1) gives cases. b2 κ2c u(0) + u(1) + γ u(1) − u(0) = 2. GOVERNING EQUATIONS 1 66 2,11 1,1 3 55 1,313 2,3113 (0) Consider the partially electroded plate of monoclinic crys- = ρ(1 + R)ü2 ; (7a) tals shown in Fig. 1. Its thickness and mass density are 2b and ρ, respectively. The thickness and mass density of the elec- γ u(1) + γ u(1) − u(0) − trodes are 2b0 and ρ0, respectively. The mass ratio between the 11 1,11 55 1,33 2,313 electrodes and the crystal plate is R = 2b0ρ0/(bρ). Mindlin’s V −3b−2 κ2c u(0) + u(1) + κ e exp(iωt) = 2-D plate equations are slightly different for electroded and 1 66 2,1 1 1 26 2b unelectroded plates. We begin with an electroded plate. For (1) (1) = ρ(1 + 3R)ü1 . (7b) TSh resonators the dominating displacement is u1 (x1, x3, t), which is usually coupled to the unwanted flexural displace- We consider steady-state, time-harmonic motions. All fields (0) ment u2 (x1, x3, t). Under the flexure-twist approximation, have the same time-harmonic factor. The free charge Qe on the 11 the TSh and flexural motions are governed by upper electrode and the current I flowing into this electrode 12 (0) (0) (0) can be calculated from T12,1 + T32,3 = 2bρ(1 + R)ü2 ; (1a) a c Z Z 3 (1) (1) (0) 2b (1) Qe = − D2dx1dx3; (8a) T11,1 + T31,3 − T21 = ρ(1 + 3R)ü1 ; (1b) 3 −a −c where the relevant plate resultants describing shear forces and I = Q˙ = iωQ ; (8b) bending/twisting moments take the following form:12 e c where (0) h 2 (0) (1) (1)i T = 2b κ c66 u + u + κ1e26ϕ ; (2a) 12 1 2,1 1 (0) ∼ D2 (0) (1) (1) D2 = = κ1e26 u + u − ε22φ . (9) 3 2,1 1 (0) (1) 2b h (1) (0) (1)i 2b T32 = T13,1 = γ55 u1,31 − u2,311 + ψ35ϕ,31 ; 3 Then the frequency-dependent admittance of the resonator is (2b) 3 given by (1) 2b (1) (1) T = γ11u + ψ11ϕ ; (2c) A = I/V. 11 3 1,1 ,1 (10) 2b3 h i Quartz is a material with very weak piezoelectric coupling. T (1) = γ u(1) − u(0) + ψ ϕ(1) . (2d) 31 3 55 1,3 2,31 35 ,3 In an unelectroded plate the electric field is small because it is produced by mechanical fields through the small piezoelectric In Eq. (2), the plate material constants are related to the coupling instead of a driving voltage. Therefore, we neglect usual three-dimensional elastic constants cpq (or spq), piezo- the small electric field and treat an unelectroded plate as an electric constants eip, and dielectric constants εij through anisotropic elastic plate. This approximation is frequently used in the analysis of quartz resonators.11 Therefore, for an unelec- 2 γ11 = s33/ s11s33 − s13 ; (3a) troded plate, formally Eqs. (1), (2), and (7) still apply, with the 162 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 H. Chen, et al.: AN ANALYSIS OF ELECTRICALLY FORCED VIBRATIONS OF PARTIALLY ELECTRODED, RECTANGULAR QUARTZ PLATE. . .

2 2 3ρ(1 + R)ω2 − 3κ2c ξ − γ b2ξ ζ2 γ b2ξζ2 − 3κ2c ξ 1 66 55 55 1 66 = 0; (15) 2 2 2 2 2 2 2 2 2 2 γ55b ξζ − 3κ1c66ξ ρω (1 + 3R)b − γ11ξ b − γ55ζ b − 3κ1c66 2 2 2 2 2 2 2 2 2 3ρω − 3κ1c66ξ − γ55b ξ ζ γ55b ξζ − 3κ1c66ξ 2 2 2 2 2 2 2 2 2 2 = 0; (18) γ55b ξζ − 3κ1c66ξ ρω b − γ11ξ b − γ55ξ b − 3κ1c66

2 2 2 2 2 2 2 2 2 3ρω − 3κ1c66ξ − γ55b ξ ζ −γ55b ξζ + 3κ1c66ξ 2 2 2 2 2 2 2 2 2 2 = 0; (20) −γ55b ξζ + 3κ1c66ξ ρω b − γ11ζ b − γ55ζ b − 3κ1c66

(1) 2 mass ration R=0, the applied voltage V = 0 or φ = 0, and In Eq. (13), Aj are undetermined constants. ξj are the two 2 2 the correction factor κ1 = π /12. roots of the Eq. (15) (see on top of the page), and The boundary conditions for the free edges of the plate in 2 2 2 2 2 2 Fig. 1 are11 3ρ(1 + R)ω − 3κ1c66ξj − γ55b ξj ζ αj = − . (16) (0) (1) 2 2 2 γ55b ξ ζ − 3κ c66ξ T12 + T31,3 = 0; (11a) j 1 j

(1) In Eq. (13), the vibration varies along both the x1 and x3 di- T11 = 0; (11b) rections. The variation along x3 is described by cos ζx3 with 11 x1 = ±(a + d); (11c) ζ = pπ/(2c) where p = 0, 2, 4, ··· . Thus, the boundary conditions at x3 = ±c in (11b) are satisﬁed. cos ζx3 with (1) T31 = 0; (11d) p = 0, 2, 4, ··· form a complete set of base functions for symmetric function of x over −c < x < c. Strictly speaking, a x = ±c. (11e) 3 3 3 summation with respect to p resenting a series should be used At the junctions between the electroded and unelectroded parts in Eq. (13). Since every term of the series satisﬁes the bound- in Fig. 1, we have the following continuity conditions:11 ary conditions at x3 = ±c in Eq. (11), we are treating the series term by term, which is simpler in writing. In general the par- h i (0) ticular solution in Eq. (14) should also be expressed as a series u2 = 0; (12a) of cos ζx3, but in our case, this series has only one constant h i term corresponding to p = 0, which is the particular solution u(1) = 0; (12b) 1 itself.

h (1)i For the unelectroded part of the plate with a < x1 < a + d, T11 = 0; (12c) the governing equations corresponding to Eq. (7) are homoge- h i neous. The general solution can be written as T (0) + T (1) = 0; (12d) 12 31,3 2 (0) X u = (Bj sin ξjx1 + Cj cos ξjx1) cos ζx3; (17a) x1 = ±a; (12e) 2 j=1 where the square brackets represent the jumps of the ﬁelds 2 across the junctions. (1) X u1 = (βjBj cos ξjx1 + γjCj sin ξjx1) cos ζx3. (17b) j=1 3. FORCED VIBRATION SOLUTION 2 In Eq. (17), Bj are undetermined constants and the related ξj From the symmetries present in the structure in Fig. 1, the and βj are determined by Eq. (18) (see on top of the page) and electrically excited TSh vibration is symmetric about both the 2 2 2 2 2 2 3ρω − 3κ1c66ξj − γ55b ξj ζ x1 and x3 axes. We only need to consider the right half of the βj = − 2 2 2 . (19) γ55b ξjζ − 3κ1c66ξj plate with x1 > 0. In the electroded region with 0 < x1 < a, since Eq. (7) is inhomogeneous, its general solution can be Similarly, Cj are also undetermined constants and the corre- 2 written as the sum of the general solution of the corresponding sponding ξj and γj are determined by Eq. (20) (see on top of homogeneous equation and a particular solution, i.e., the page) and

2 2 2 2 2 2 2 3ρω − 3κ1c66ξj − γ55b ξj ζ (0) X γ = . (21) u = A sin ξ x cos ζx ; j 2 2 2 2 j j 1 3 (13a) γ55b ξjζ − 3κ1c66ξj j=1 The substitution of Eqs. (13) and (17) into the two bound- 2 ary conditions at x1 = a + d in (11a) and the four continuity (1) X (1) u1 = αjAj cos ξjx1 cos ζx3 +u ˜1 ; (13b) conditions at x1 = a in Eq. (12) leads to j=1 2 2 X where the particular solution is given by 2bκ1c66 [(ξj + βj) Bj cos ξj(a + d)+ j=1 3κ e V u˜(1) = 1 26 . (14) 1 2 2 2 +(γj − ξj)Cj sin ξj(a + d)] = 0; (22a) ρ(1 + 3R)ω b − 3κ1c66 2b International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 163 H. Chen, et al.: AN ANALYSIS OF ELECTRICALLY FORCED VIBRATIONS OF PARTIALLY ELECTRODED, RECTANGULAR QUARTZ PLATE. . .

2 X ξj [−βjBj sin ξj(a + d) + γjCj cos ξj(a + d)] = 0; j=1 (22b) 2 2 X X Aj sin ξja = (Bj sin ξja + Cj cos ξja); (22c) j=1 j=1

2 X (1) αjAj cos ξja +u ˜1 = j=1 2 X (βjBjξj sin ξja + γjCj sin ξja); (22d) j=1

2 X − αjξjAj sin ξja = Figure 2. Admittance versus driving frequency near TSh resonances. j=1 2 fundamental TSh frequency given by Eq. (23) in Fig. 2. There X = (−βjBjξj sin ξja + γjCjξj cos ξja); (22e) are two resonances (0.9920 and 0.9955 as normalized by (23)) j=1 in the frequency range of interest, i.e., when the normalized frequency is close to one from below because the electrode inertia lowers the resonant frequencies. Ideally, an isolated reso- 2 X nance peak is desired. However, due to coupling with flexure, 2bκ2c ξ + α A cos ξ a +u ˜ + κ e V = 1 66 j j j j 1 1 26 there are usually secondary peaks near the main peak and what j=1 2 is shown in Fig. 2 is typical. 2 X At the two resonances, the distributions of the TSh and = 2bκ1c66 [(ξj + βj) Bj cos ξja + (γj − ξj) Cj sin ξja]; j=1 flexural displacement components at the plate top surface are (22f) shown in Figs. 3 and 4, respectively. The TSh displacement is from the operating mode. It is much larger than the unwanted which are six linear algebraic equations for the six undeter- flexural displacement. This indicates a reasonably good res- mined constants Aj, Bj and Cj. These equations are inhomo- onator with a TSh dominated vibration and weak coupling to geneous and driven by V . They are solved on a computer using flexure. This coupling cannot be avoided totally because it is MATLAB. related to the wave reflections at the plate edges. The main TSh displacement is large in the electroded central region, and de- 4. NUMERICAL RESULTS AND DISCUSSION cays to almost zero near the plate edges. This is the so-called energy trapping of TSh modes by electrodes. Energy trapping As a numerical example, consider an AT-cut quartz plate resis fundamental to resonator mounting which can be designed onator with a typical thickness described by b = 0.3173 mm. near the plate edges without affecting the vibration of the plate. When the small piezoelectric coupling in quartz is neglected, We note that the small flexural displacement is sinusoidal and an infinite and unelectroded plate with such a thickness has the therefore is not trapped. This causes a small leakage of vibra- following fundamental TSh frequency: tion energy into mounting which is part of the overall damping r in resonators. We also note that the TSh displacement is a long π c66 ω0 = = 16.384 MHz; (23) wave and the flexural displacement is a relatively short wave. 2b ρ Figure 5 shows the effect of the electrode/plate mass ratio which will be used as a frequency unit. In our numerical exam- R on the admittance, which seems to be relatively simple. For ple, the electrode/plate mass ratio R = 0.005 is fixed except in thicker or heavier electrodes represented by larger values of R, Fig. 5. Both a = 15b = 4.7591 mm and d = 10b = 3.1727 mm the resonant frequencies become lower as expected because of are fixed except in Fig. 6. Material damping is introduced more electrode inertia. At the same time, the magnitude of the by allowing the relevant elastic constants to assume complex admittance and the shapes of the two peaks do not seem to have values, which can represent viscous damping. In our nu- changed. merical calculation, the elastic constants cpq is replaced by If we vary the electrode length 2a only and all other parame- −1 cpq(1 + iQ ), where Q is a real, positive, and large num- ters, including the total length of the plate are kept the same as ber. We fixed Q = 1000 except in Fig. 7. This is a relatively those used in Fig. 2, the result is shown in Fig. 6 in which larger small Q for the material damping of quartz. A relatively small electrodes result in large admittance as expected. Interestingly, value of Q is used as a representation of the total damping in while the location of the first peak does not have any notice- the system. able change as the electrode length increases, the location of We ploted the absolute value of the complex admittance cal- the second peak moves to the left in the process. This shows culated from Eq. (10) versus the driving frequency near the that the electrode length has a clear effect on mode coupling.

164 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 H. Chen, et al.: AN ANALYSIS OF ELECTRICALLY FORCED VIBRATIONS OF PARTIALLY ELECTRODED, RECTANGULAR QUARTZ PLATE. . .

a) a)

b) b)

Figure 3. Plate surface displacement distributions at the ﬁrst resonance in Figure 4. Plate surface displacement distributions at the second resonance in Fig. 2. Fig. 2.

This is not surprising because the coupling between TSh and x3 dependent modes cannot be excited by a thickness electric flexure is known to be sensitive to the plate length/thickness field in a structurally symmetric resonator. In this case, the 17 ratio. In a partially electroded plate as shown in Fig. 1, since usual analyses of cylindrical motions independent of x3 is a the vibration is mainly in the electroded central region, what very good approximation. In practice, structural imperfections matters to mode coupling is the length of the electroded cen- may change the situation and introduce x3 dependence. If the tral region instead of the plate total length. electrodes do not cover the entire width 2c of the resonator and In Fig. 7, the damping described by Q is varied while all cover the central part along x3 only, an x3 dependent analysis other parameters are kept the same as those used for plotting becomes necessary, which remains mathematically challeng- Fig. 2. A larger value of Q represents less damping in the sys- ing. tem. This should imply a stronger mechanical resonance with a larger TSh deformation and more charges on the electrodes ACKNOWLEDGEMENTS through piezoelectric coupling, and a larger admittance which is confirmed in Fig. 7. This work was supported by the National Natural Science Foundation of China (Nos. 11272161 and 11372145), the Y. 5. CONCLUSIONS K. Pao Visiting Professorship at Ningbo University, and the K. C. Wong Magana Fund through Ningbo University. The admittance is large at TSh resonances and is sensitive to system parameters. For electrically forced vibrations of par- REFERENCES tially electroded, rectangular, AT-cut quartz plates with free edges, the operating TSh mode is weakly coupled to the flex- 1 Mindlin, R. D. High frequency vibrations of crystal plates, ural mode. The TSh wave is long and is mainly trapped un- Quart. Appl. Math., 19 (1), 51–61, (1961). der the electrodes. The flexural wave is relatively short and not trapped. The in-plane TSh mode variation at resonances 2 Tiersten, H. F. and Mindlin, R. D. Forced vibrations of is mainly along x1. In the x3 direction, within the first- or- piezoelectric crystal plates, Quart. Appl. Math., 20 (2), der Mindlin plate theory and its flexure-twist approximation, 107–119, (1962).

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 165 H. Chen, et al.: AN ANALYSIS OF ELECTRICALLY FORCED VIBRATIONS OF PARTIALLY ELECTRODED, RECTANGULAR QUARTZ PLATE. . .

Figure 5. Effect of mass ratio on admittance. Figure 7. Effect of damping on admittance.

9 Yong, Y. K. and Stewart, J. T. Mass-frequency inﬂuence surface, mode shapes, and frequency spectrum of a rectangular AT-cut quartz plate, IEEE Trans. Ultrason. Ferro- electr. Freq. Control, 38 (1), 67–73, (1991).

10 Wang, J. and Yang, J. S. Higher-order theories of piezoelectric plates and applications, Appl. Mech. Rev., 53 (4), 87–99, (2000).

11 Mindlin, R. D. and Spencer, W. J. Anharmonic, thickness- twist overtones of thickness-shear and ﬂexural vibrations of rectangular, AT-cut quartz plates, J. Acoust. Soc. Am., 42 (6), 1268–1277, (1967).

12 Zhang, C. L., Chen, W. Q., and Yang, J. S. Electrically forced vibration of a rectangular piezoelectric plate of monoclinic crystals, International Journal of Applied Electro- Figure 6. Effect of electrode length on admittance. magnetics and mechanics, 31 (4), 207–218, (2009). 3 Mindlin, R. D. High frequency vibrations of piezoelectric 13 Zhang W. P. Analytical modeling of resistance for AT-cut crystal plates, Int. J. Solids Struct., 8 (7), 895–906, (1972). quartz strips, Proc. IEEE International Frequency Control 4 Lee, P. C. Y., Syngellakis, S., and Hou J. P. A two- Symposium, (1998). dimensional theory for high-frequency vibrations of piezo- 14 Zhang, W. P. and Doyle, M. Analytical modeling of resis- electric crystal plates with or without electrodes, J. Appl. tance for AT-cut quartz strips, in: Yang, J. S. and Maugin, Phys., 61 (4), 1249–1262, (1987). G. A. (Ed.), Mechanics of Electromagnetic Materials and 5 Mindlin, R. D. and Lee, P. C. Y. Thickness-shear and flexu- Structures, IOS Press, 147–158, (2000). ral vibrations of partially plated, crystal plates, Int. J. Solids 15 Wang, J., Zhao, W. H., and Du, J. K. The determination of Struct., 2 (1), 125–139, (1966). electrical parameters of quartz crystal resonators with the 6 Wang, J., Lee, P. C. Y., and Bailey, D. H. Thickness-shear consideration of dissipation, Ultrasonics, 44 Supplement, and flexural vibrations of linearly contoured crystal strips e869–e873, (2006). Comput. Struct. with multiprecision computation, , 70 (4), 16 Bottom, V. E. Introduction to Quartz Crystal Unit Design. 437–445, (1999). New York: Van Nostrand Reinhold, (1982). 7 Wang, J. and Zhao, W. H. The determination of the optimal 17 Mindlin, R. D. Thickness-shear and flexural vibrations of length of crystal blanks in quartz crystal resonators, IEEE crystal plates, J. Appl. Phys., 22 (3), 316–323, (1951). Trans. Ultrason. Ferroelectr. Freq. Control, 52 (11), 2023– 2030, (2005).

8 Lee, P. C. Y. and Tang, M. S. H. Initial stress ﬁeld and resonance frequencies of incremental vibrations in crystal resonators by ﬁnite element method, Proc. 40th Annual Fre- quency Control Symposium, (1986).

166 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Damping Performance of Dynamic Vibration Absorber in Nonlinear Simple Beam with 1:3 Internal Resonance Yi-Ren Wang Department of Aerospace Engineering, Tamkang University, NewTaipei City, Tamsui Dist., 25137, Taiwan.

Hsueh-Ghi Lu RD Center-Environmental Reliability QA Div., Enterprise Products Testing Department, Pegatron Corporation, Taipei City 11261, Taiwan.

(Received 20 October 2014; accepted 17 November 2016) This study investigated the damping effects of a dynamic vibration absorber (DVA) attached to a hinged-hinged nonlinear Euler-Bernoulli beam. The model constructed in this study was used to simulate suspended nonlinear elastic beam systems or vibrating elastic beam systems on a nonlinear Winkler-type foundation. This makes the modeling in this study applicable to suspension bridges, railway tracks, and even carbon nanotubes. The hinged-hinged beam in this study includes nonlinear stretching effects, which is why we adopted the method of multiple scales (MOMS) to analyze the frequency responses of fixed points in various modes. The use of amplitudes and vibration modes made it possible to examine the internal resonance. Our results indicate that particular elastic foundations or suspension systems can cause 1:3 internal resonance in a beam. The use of 3D maximum amplitude contour plots (3DMACPs) enabled us to obtain a comprehensive understanding of various DVA parameters, including mass, spring coefficient, and the location of DVA on the beam, and thereby combine them for optimal effect. Our results were verified using numerical calculations.

1. INTRODUCTION upon they applied the Galerkin method to the analysis of dynamic responses in a nonlinear beam. Analysis of vibrations in an elastic beam involves many factors. Van Horssen and Engineering from the scale of small mechanical components Boertjens6 considered a suspension bridge under the influence to that of aircraft wings and bridges must take into account of nonlinear aerodynamic effects using the theoretical model the effects of vibration, which can cause fatigue, loosening of of a Euler-Bernoulli beam with linear springs to simulate vi- parts, and failure. Vibration can be linear or nonlinear, with brations in the bridge. They discovered that energy can shift the latter involving internal resonance that is complex and dif- between the first and third modes and that this is typical of non- ficult to analyze. As described by Nayfeh and Mook,1 in the linear internal resonance. Their study prompted a great deal event that the natural vibration frequencies of various degrees of research based on theoretical models. Theories similar to of freedom (DOFs) are multiples of one another, excitation this, in which a beam is placed on an elastic foundation, have at high frequencies can increase amplitudes at low frequen- been widely applied to civil, mechanical, and aerospace engi- cies. This study sought to account for this unique nonlinear neering, by researchers such as Mundrey.7 Fu et al.8 reported phenomenon. Nayfeh and Balachandran2 defined several sta- nonlinear vibrations in embedded carbon nanotubes (CNTs), ble conditions in nonlinear systems and presented a number based on research involving the use of nonlinear 2D Euler- of methods that could be used to determine system stability, Bernoulli beams to simulate vibrations in CNTs on elastic ma- and are therefore highly valuable as a reference in the anal- trices. Shen9 used the model of a nonlinear 2D Euler-Bernoulli ysis of system stability. Nayfeh and Pai3 investigated vibra- beam to analyze vibrations in a post-buckling beam placed on a tions in nonlinear Euler-Bernoulli beams. According to New- double-layered elastic foundation, thereby demonstrating that ton’s laws, Euler’s angle transformation, and the Karman-type the stiffness of elastic foundations has a significant influence strain-displacement relationship in order to derive 3D and 2D on the vibration behavior of nonlinear beams. equations associated with nonlinear beams. These theoretical models have greatly benefited related research. Nayfeh Wang and Chen10 developed a damping approach to vibra- and Mook1 also proposed a number of methods with which tion reduction without the need to change the damper, but to solve nonlinear systems, including the Poincare´ method, the merely enable its relocation by having a mass-spring-damper Lindstedt method, the average method, and MOMS, which is vibration absorber hung from the optimal damping location on highly conducive to the analysis of damped vibration systems. a rotating mechanism (a CD-ROM drive or the coupling sys- Ji and Zu4 studied the rotating shaft system of a Timoshenko tem of rotary wings and swashplates). Wang and Chang11 stud- beam using MOMS to analyze the natural frequency responses ied nonlinear vibrations in a rigid plate using cubic springs of nonlinear systems. Nayfeh and Nayfeh5 employed MOMS to simulate supports at the four corners with two single- to identify nonlinear modes and nonlinear frequencies, where- mass dampers hung from beneath the plate. The locations

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 167–185) https://doi.org/10.20855/ijav.2017.22.2462 167 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . . of the dampers were adjusted to achieve optimal damping ef- termine the frequency response of the temperature and material fects. Dynamic vibration absorber (DVA) is a practical passive properties. A 2-DOF nonlinear system (the main body and the damping device for vibrating systems. For a linear vibrating controller (absorber)) with quadratic and cubic nonlinearities beam, DVA can be pre-tuned to the modal frequency of the under external and parametric bounded excitations was con- vibrating structure or the disturbance frequency to damp out sidered in the work of Sayed and Kamel.21 The method of the beam vibration. Wang and Chang12 discussed a hinged- multiple scales was applied to determine the amplitudes and free linear Euler-Bernoulli beam placed on a nonlinear elas- phases in the existence of internal resonance. The stability tic foundation. They found that placing a DVA of appropriate of the controlled system was studied. Ghayesh et al.22 in- mass could prevent internal resonance and suppress vibrations vestigated the axially moving beam with a three-to-one inter- in the beam. Nambu et al.13 studied the smart self-sensing nal resonance. The coupled longitudinal and transverse dis- active DVA attached on a flexible beams supported string. A placements of the beam were studied numerically. The fre- small cantilevered piezoelectric transducer was used to real- quency response of the nonlinear forced system was solved ize this active DVA. They showed improvement control ro- by the pseudo-arc length continuation method and direct time bustness upon passive DVAs. Tso et al.14 proposed a novel integration. Ghayesh and Amabili23 provided a detailed dis- design method of hybrid dynamic vibration absorber (HVA) to cussion of an axially moving Timoshenko beam with an inter- suppress broadband vibration of the whole structure instead of nal resonance. They used the Galerkin’s scheme to discretize just one point of the structure. Samani and Pellicano15 identi- the nonlinear partial differential equations into a set of non- fied the optimal locations of a DVA applied to a simple beam linear ordinary differential equations. The pseudo-arc length subjected to regularly spaced and concentrated moving loads. continuation method was also employed to find the frequency The nonlinear stiffness of a nonlinear DVA possesses com- response of the nonlinear forced system. The dynamic and res- plex dynamic properties such as quasi-periodic, chaotic, and onant response were examined by the Poincare´ maps. Stabil- sub-harmonic responses. They discovered that the damping ity properties with and without internal resonance were pre- effects of a nonlinear non-symmetric dissipation DVA are su- sented. Ansari et al.24 investigated the forced vibration of perior to those of a linear DVA. In contrast, Wang and Lin16 nonlinear magneto-electro-thermo elastic (METE) nanobeams. used internal resonance contour plots (IRCPs) and flutter speed The Galerkin technique was applied to obtain a time-varying contour plots (FSCPs) to analyze nonlinear dynamic stability. set of ordinary differential equations. The pseudo-arc length Their results demonstrated that changing the location of the continuum scheme was also used to solve this nonlinear beam damper can prevent internal resonance and inhibit vibrations system. Ansari et al.25 applied the similar numerical strategies in the main body. Wang and Kuo17 examined the vibrations in the analysis of the forced vibration behavior of reinforced of a hinged-free nonlinear beam placed on a nonlinear elastic single-walled carbon nanotubes (SWCNTs). Different effects foundation. The 1:3 internal resonance in the first and second of SWCNT parameters were examined by the frequency re- modes of the beam was found. An optimal location of an at- sponses. Their results showed that the amplitude peak reduces tached DVA was proposed to prevent internal resonance and when the nanotube volume fraction or dimensionless damping suppress vibrations. Wang and Tu18 investigated the damp- parameter gets larger. Ansari et al.26 also invested the suring effects of a tuned mass damper (TMD) on a fixed-free 3D face stress effect on the vibrations of nanobeams. The general- nonlinear beam resting on a nonlinear elastic foundation with ized differential quadrature method and the shifted Chebyshev- existing internal resonances. They proposed that the locations Gauss-Lobatto grid points were employed to solve the nonlin- of the maximum amplitudes between node points in the mode ear problem. The frequency response of nanobeams including shapes and near the free end of the beam display the best damp- the effect of surface stress was investigated. Ansari et al.27 ing effect for the fixed-free 3D beam. In the works of Wang performed a numerical study of two-phase BiTiO3-CoFe2O4 and Kuo17 and Wang and Tu,18 the nonlinear geometry and composites nanobeams subjected to various boundary condi- nonlinear inertia were included in these two nonlinear beams. tions. The Galerkin and pseudo arc-length methods were em- The conditions to trig the internal resonance were also inves- ployed for solving nonlinear problems. They also provided a tigated in their researches. The various beam boundary condi- novel technique of periodic time differential operators for the tions were examined for better understanding the DVA damp- time domain discretization. Ansari and Gholami28 further in- ing effects on beam vibrations. Wang and Liang19 investigated vestigated the surface stress and surface inertia effects on the the damping effects of vibration absorbers with a lumped mass rectangular nanoplates. The equations of motion were solved on a hinged-hinged beam. This kind of vibration absorber is numerically for the rectangular Si and Al nanoplates with variable to mitigate vibrations in mechanical or civil engineering ous boundary supports by means of the generalized differential structures on an elastic foundation; however, they are not ap- quadrature method. The hardening nonlinearity and surface ef- plicable to all suspension systems. It is this shortcoming that fects showed considerable influences on nanoplates. we sought to address in this research. This study considered a nonlinear hinged-hinged Euler- Further consideration of resonance analysis of structures us- Bernoulli beam supported or suspended using nonlinear ing the analytical and numerical strategies can be found in the springs, including internal resonance and the coupling of mul- work of Ansari et al.20 They derived a geometrically nonlinear tiple modes. This system is able to simulate a vibrating me- beam model to simulate the nanobeam vibration under ther- chanical system suspended or placed on an elastic medium, mal environment. The material properties of nanobeams vary such as a suspension bridge or the tracks of subways or high- through the thickness direction with functionally graded (FG) speed trains. This system can also be used to simulate CNTs distribution. The method of multiple scales was applied to de- placed on an elastic matrix. The main body in this study was

168 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

(a)

(b)

(c)

Figure 1. A schematic of the hinged-hinged beam system, (a) suspension beam, (b) Winkler type foundation, (c) relationship between beam displacement and rotation angle. subjected to a harmonic force. We used a mass-spring DVA to develop a complete model including the elastic foundation to reduce vibrations, taking into account the effects of stretch- and the DVA. We first considered the initial status of the beam ing. The influence of parameters, such as the mass, elasticity when it is straight and assumed that each cross-section is a coefficient, and location of the DVA on damping performance plane that follows the stress-strain laws. According to Nayfeh were analyzed extensively. We used 3D maximum amplitude and Pai9 and from Fig. 1c, the axial strain e and the rotation an- 0 q 02 contour plots (3DMACPs) to identify the optimal combination gle θ are related to u0 and W as e = (1 + u0)2 + W − 1, of parameters as well as numerical calculations to verify the 0 0 accuracy of the results. cos θ = (1 + u )/(1 + e), and sin θ = W /(1 + e). As- suming u and W are small but finite, then e and θ can be 02 02 further expanded as e = u0 + W /2 − (u0W )/2 ... and 2. DEVELOPMENT OF THE THEORETICAL 0 0 0 0 0 θ = tan−1(W /(1 + u0)) = W − u0W + u02W + u02W − MODEL 03 W /3 ... . Using Newton’s Laws, Euler’s angle transforma- The main body in this study comprised of a nonlinear elastic tion, and Taylor series expansion, we obtained the basic equa- beam supported or suspended by nonlinear springs to simulate tions of motion for the nonlinear beam. We excluded any ro- an elastic foundation or suspension cables. Figure 1 illustrates tations in the beam, limiting it to planar motions. Thus, the the coordinate system, where m denotes the mass of the elastic equations of motion for the 2D beam are as follows: beam per unit length; A, E, and IA are the cross-section area, Young’s modulus, and moment of inertia of the beam, respec- 0 1 02 02 tively; k is the liner spring constant of the elastic foundation; β mu¨ − EAu00 = EA W − u0W + signifies the nonlinear spring constant; µ is the linear damping 2 0 coefficient of the beam; and f and g denote the linear spring h 0 000 000 0 00 00 0 000i s s EIA W W − u W − 2u W − 3u W ; (1) constant and the damping coefficient of the DVA, respectively. 0 ¨ iv 0 0 02 0 1 03 mW − EIAW = EA u W − u W + W + 2.1. Deduction and Non-Dimensionalization 2 00 of Nonlinear Equations of Motion 0 000 0 0 02 02 000 EIA u W + u W − u − W W −

Figures 1a and 1b exhibit the hinged-hinged nonlinear beam 00 0 000 0 1 03 employed in this study. We referred to the nonlinear beam the- u0 u0W − u02W − W ; (2) ory proposed by Nayfeh and Pai9 and used Newtonian’s Laws 3

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 169 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . . where ()0 and (˙) denote d/dx and d/dt, respectively. The The non-dimensionalized equation of motion of the DVA is u and W represent the beam displacement in the longitudinal ∗∗ h ∗ i direction (x-axis) and transversal direction (vertical-axis), re- m0W D(τ) − ek(u) + λ(u) = 0; (9) spectively. The system considered in this study is a slender elastic beam; therefore, the longitudinal inertia force mu¨ in where Eq. (1) can be disregarded. The beam is hinged at both ends, u(τ) = W (lD, τ) − WD(τ). (10) and no longitudinal forces were imposed. This made it neces- Equation (7) is the non-dimensionalized equation of motion of sary to consider the effects of stretching. The boundary condi- the main body, in which ()0 and (∗) represent d/dx and d/dτ; tions are as follows: ω2 is the ratio between the linear elasticity coefficient of the support or suspension springs and the elasticity coefficient of u(0, t) = 0; u(l, t) = 0; W (0, t) = 0; the elastic beam; µ is the non-dimensionalized beam damping 00 00 W (l, t) = 0; W (0, t) = 0; W (l, t) = 0. (3) coefficient; K is the ratio between the nonlinear vibration frequency of the supports or suspension springs and the vibration 00 From Eq. (1), we can express the u as frequency of the elastic beam, lD is the non-dimensionalized DVA position, Aˆ is the non-dimensional beam rigidity; and ek 0 1 02 λ u00 = − W + .... (4) and are the non-dimensionalized spring constant and damp- 2 ing coefficient of the DVA, respectively. Appendix 1 contains the definitions of the other non-dimensionalized coefficients. The integral of Eq. (4) is 2.2. Application of MOMS 1 02 u0 = − W + c (t); 2 1 This study adopted MOMS to analyze the frequency re- l sponse and fixed points of the nonlinear equation, which 1 Z 02 u = − W dx + c1(t)x + c2(t). (5) involves dividing the time scale into fast and slow time 2 0 scales. Suppose T0 is the fast-time term, T1,T2,... are Substituting Eq. (5) into the boundary conditions in Eq. (3) the slow-time terms, and W (x, τ, ε) = W0(x, t0, t1,...) + produces εW1(x, t0, t1,...), where ε is the time scale of small disturbances and is an infinitesimal value. We also considered the l 1 Z 02 damping and nonlinear terms as well as the external force as c2(t) = 0; c1(t) = W dx. (6) small disturbance terms and set the order at ε1 to facilitate anal- 2l 0 ysis. We substituted these principles into Eq. (7) and disre- After substituting Eqs. (5) and (6) into Eq. (2), we can simplify garded the influence of high-order terms such as ε2, ε3,... on the equations of motion to an equation in which vibrations are the system. The terms of ε0 are as follows: presented in the W direction. We included the structural damp- 2 4 ˙ ∂ W0 ∂ W0 2 ing term µW of the elastic beam, the linear and nonlinear elas- 2 + 4 + ω W0 = 0. (11) 3 ∂T ∂x tic terms k[W + βW ] of the support or suspension springs, 0 and the uniform distributed load F = F eiΩt , where µ is the The terms of ε1 are as follows: structural damping coefficient; k and β represent the linear and " # ∂2W ∂4W ∂2W 1 Z 1∂W 2 nonlinear elastic constant of the support or suspension springs, 1 1 2 0 ˆ 0 2 + 4 + ω W1 = 2 A dx + respectively. The hanging DVA was regarded an external force ∂T0 ∂x ∂x 2 0 ∂x ˙ ˙ 2 {f s[W (x, t)−W D]+gs[W (x, t)−W D]} δ[x−lD], where f s h ∗ i ∂ W0 F − ek(u) + λ(u) δ[x − lD] − 2 − and gs denote the spring constant and the damping coefficient ∂T0∂T1 of the DVA, respectively. It is also noted that l represents the ∂W0 3 µ − KW0 . (12) beam length and the W D and lD represent the displacement ∂T0 and the location of the DVA, respectively. We used Newton’s 0 1 laws to deduce the equation of motion before reorganizing and The boundary conditions corresponding to ε and ε equations 00 nondimensionalizing it, thereby obtaining the following: are as follows: W (0, τ) = 0, W (0, τ) = 0, W (l, τ) = 0, W 00(l, τ) = 0. Considering that the DVA is linear, MOMS ∗∗ ∗ 0 iv 2 3 n need not be used to divide the time scale, such that only ε is W + W + ω W + µW + KW + ek W (x, τ) − WD + considered in the DVA equation of motion. ∗ ∗ o λ W (x, τ) − W D δ[x − lD] = 3. ANALYSIS OF INTERNAL RESONANCE " # 1 Z l Aˆ W 02dx W 00 + F eiΩτ . (7) CONDITIONS 2 0 3.1. Equation of Motion for the Nonlinear The boundary conditions in Eq. (3) are thus Beam Without DVA The process of modeling the system without the DVA is the W (0, τ) = 0; W 00(0, τ) = 0; same as the development of the theoretical model in Section 2. 00 W (l, τ) = 0; W (l, τ) = 0. (8) Thus, we only provide a brief description below.

170 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

After dropping the DVA from the equation of motion in the Using the orthogonal method, we multiply Eq. (15) by φn(x) previous section, we obtain the following: and integrate it from 0 to l (l = 1 (normalized)) and obtain ∗∗ 4 2 ∗∗ ξ0m(τ) + γm + ω ξ0m(τ) = 0; (20) iv 2 W + W + ω W = ∗∗ 4 2 ! ξ1m(τ) + γm + ω ξ1m(τ) = 1 Z l ∗ ˆ 02 00 3 iΩτ " ∞ ! # A W dx W − µW − KW + F e . (13) 1 Z 1 X 2 0 − γ2 ξ Aˆ γ2ξ2 cos2γ x dx + 2 m 0m p 0p p 0 p=1 R 1 2 In this model, the two ends are hinged; therefore, the boundary 0 φmdx ∂ ξ0m(τ) ∂ξ0m(τ) F 1 − 2 − µ − conditions are R 2 ∂T0∂T1 ∂T0 0 φmdx ∞ ! ∞ ! ∞ ! K Z 1 X X X W (0, τ) = 0; W 00(0, τ) = 0; ξ φ ξ φ ξ φ φ dx; R 1 0i i 0j j 0k k m φ2 dx 0 W (l, τ) = 0; W 00(l, τ) = 0. (14) 0 m i j k (21)

Note that in this non-dimensionalized circumstance, l = 1. and set the natural frequencies of the various modes in the p 4 2 system as ωn = γm + ω . To determine the existence of internal resonance in the beam system, we must derive the ω2 conditions capable of causing internal resonance. Among 3.2. Conditions of Internal Resonance in the these conditions, Eq. (21) contains ξ0iξ0jξ0k, an infinite se- Beam System without DVA ries of nonlinear terms. This hinders the determination of internal resonance with ω2; therefore, we used the method We applied MOMS and the terms comprising ε0 are as fol- proposed by Van Horssen6 and followed the procedure de- lows: scribed by Wang and Liang19 to derive the internal resonance conditions created by ω2. We first observe the inte- 2 4 ∂ W0 ∂ W0 2 R 1 + + ω W = 0. (15) gration term φi(x)φj(x)φk(x)φm(x)dx on the right side ∂T 2 ∂x4 0 0 0 of Eq. (21), which is the orthogonal product of φm(x) and φi(x)φj(x)φk(x). Based on the orthogonal method, the inte- Terms comprising ε1 are as follows: gration term in Eq. (21) may contribute to the secular terms if it does not equal 0 and m = ±i ± j ± k. Further- more, as the dominant term in φ (x) is sin γ x, we can 2 4 2 "Z 1 2 # m m ∂ W1 ∂ W1 1 ∂ W0 ∂W0 2 ˆ only observe whether ±γi ± γj ± γk equals γm in the prod- 2 + 4 + ω W1 = A 2 dx + ∂T0 ∂x 2 ∂x 0 ∂x R 1 uct terms in 0 φi(x)φj(x)φk(x)φm(x)dx in order to confirm 2 ∂ W0 ∂W0 3 the existence of orthogonality before addressing the question F − 2 − µ − KW0 . (16) ∂T0∂T1 ∂T0 of whether the harmonic frequencies extending from the harmonic combination are secular terms. As γm = nπ > 0, we −γ = γ + γ − γ −γ = γ − γ + γ Suppose that the general solution to W (x, τ) is can eliminate m i j k and m i j k from the possibilities in ±γm = ±γi ± γj ± γk. Since γm = γ + γ − γ , −γ = γ − γ − γ , and γ = γ − γ + γ are i j k m i j k m i j k −iζ iωT0 iζ −iωT0 W (x, τ) = A(T1)e e +A(T1)e e φ(x); (17) similar; therefore, we need only to discuss three combinations: γm = γi +γj +γk, γm = γi −γj −γk, and γm = γi +γj −γk from the possibilities in ±γm = ±γi ± γj ± γk. Further- where ζ is the phase angle. Also, let more, in the product ξ0iξ0jξ0k on the right side of Eq. (21), there exists secular terms that are equal to the harmonics on φ(x) = E1 sin γx + E2 cos γx + E3 sinh γx + E4 cosh γx. the left side, which prevents the system from reaching con- (18) vergence. Of these terms, only those that exist in the form By substituting the equation above into the boundary condi- of ei(±ωi±ωj ±ωk)T0 are possible. Hence, we only considered tions, we obtain the eigenvalues γn = nπ/l, n = 1, 2, 3,... , the combinations in ωm = ±ωi ± ωj ± ωk, which is why we and l = 1, whereas the mode shape φn(x) can be expressed as discussed the following combinations according to conditions φn(x) = sin γnx. After using orthogonal method, W0n(x, τ) described above: can be written as ( γm = γi + γj + γk (A) 1 1 1 1 ; γ4 +ω22 = ±γ4 +ω22 ±γ4 +ω22 ±γ4 +ω22 W (x, τ) = m i j k 0n ( ∞ γm = −γi − γj + γk 1 1 1 1 X −iζn iωnT0 iζn −iωnT0 (B) ; Bn(T1)e e + Bn(T1)e e φn(x) = 4 22 4 22 4 22 4 22 γm +ω = ± γi +ω ± γj +ω ± γk +ω n=1 ( ∞ γm = γi + γj − γk X (C) 1 1 1 1 . ξ0n(τ)φn(x). (19) 4 22 4 22 4 22 4 22 γm +ω = ± γi +ω ± γj +ω ± γk +ω n=1 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 171 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

We determined the following ranges: when 0 ≤ ω2 ≤ 10π4 right side of Eq. (21), we select terms proportional to eiω1T0 i(ω3−2ω1)T0 and γi, γj, γk ≥ π, solutions contributing to secular terms can and e (these are called secular terms): be found in these combinations; thus, the discussion on the 1 Z 1 ˆ 4 2 −iζ1 combinations and probabilities is as follows. − A γ1 cos γ1xdx 3B1B1B1e + 2 0 Case (A) γ = γ + γ + γ . Z 1 m i j k 2 2 2 −iζ1 1 1 1 γ1 γ3 cos γ3xdx 2B1B3B3e + 4 22 4 22 4 22 Combination (1): γm + ω = γi + ω + γj + ω + 0 1 1 0 −iζ1 0 −iζ1 4 22 2 4 22 2 A21F 1 − 2iω1B1(T1)e − 2ω1ζ1B1(T1)e − γk + ω . Using inequality h < h + ω ≤ h − 1 1 2 4 22 2 2 2 2 Z a + a + ω and let γm = γ + γ + γ + λ, where −iζ1 2 2 −iζ1 i j k µiω1B1(T1)e − A31K 6B1B3B3 φ1φ3dx e + λ = 2(γiγj + γjγk + γiγk). This enables us to derive the 0 1 1 1 2 4 22 2 2 2 Z Z γ < γ + ω ≤ γ + γ + γ − 2 4 −iζ 2 3 i(2ζ −ζ ) following inequality: m m i j k 3B B φ dx e 1 + 3B B φ φ dx e 1 3 . 1 1 1 1 1 3 1 3 2 4 22 2 0 0 3π +3 π + ω , such that 2(γiγj +γjγk+γiγk) < −3π + 1 1 (22) 4 22 2 2 4 22 3 π + ω . We also know that if γm − π + π + ω ≥ 1 1 rd 4 22 2 2 2 2 4 22 In terms of the 3 mode (m = 3), we must select the secular γm + ω > γi + γj + γk, we can get π − π + ω < 1 terms with harmonics ω3 and 3ω1. Thus, from the right side of 2 4 22 λ < −3π + 3 π + ω and γi, γj, γk ≥ π, which means Eq. (21) and let m = 3, we select terms proportional to eiω3T0 2 2 4 that λ ≥ 6π . If ω ≤ 8π , then the inequality cannot and ei3ω1T0 (these are called secular terms) as shown below: be satisfied. If γi = γj = γk = π, then γm = 3π and 1 1 1 Z 1 4 22 4 22 ˆ 4 2 −iζ3 (3π) + ω = 3 π + ω , whereby we have the solu- − A γ3 cos γ3xdx 3B3B3B3e + tion ω2 = 9π4. 2 0 1 1 1 1 4 22 4 22 4 22 Z Combination (2): γm + ω = γi + ω + γj + ω − 2 2 2 −iζ3 1 γ1 γ3 cos γ1xdx 2B3B1B1e + 4 22 0 γk + ω . This combination has no solution. We can use the 0 −iζ3 0 −iζ3 analysis method from Combination (1); however, the symmet- A23F 3 − 2iω3B3(T1)e − 2ω3ζ3B3(T1)e − ric relationship means that equations with a negative sign have Z 1 −iζ3 3 3 −i3ζ1 no solution. µiω3B3(T1)e − A33K B1 φ1φ3dx e + 1 1 1 0 4 22 4 22 4 22 Combination (3): γm + ω = γi + ω − γj + ω − Z 1 Z 1 1 2 2 −iζ3 2 4 −iζ3 γ4 + ω22 . This combination has no solution. We can use the 6B1B1B3 φ1φ3dx e + 3B3 B3 φ3dx e . k 0 0 analysis method from Combination (1); however, the symmet- (23) ric relationship means that equations with two negative signs have no solution. The selected secular terms are designated as being equal to 0 in order to derive a solvability condition. Below, we dis- st Case (B) γm = −γi − γj + γk is the same as Case (A), cuss the conditions where we excite the 1 mode. We multi- st iζ1 ˆ irτ because m and k are symmetric. ply the secular terms of the 1 mode by e and let f1e = ˆ iσT1 iω1T0 f1e e , ΓA = σT1 + ζ1, and ΓB = 3ζ1 − ζ3. The 2 Case (C) γm = γi + γj − γk has solutions, but when ω = periodic solutions of the beam correspond to the constant so- 9π4, internal resonance cannot occur within any of the modes lutions (also called fixed points in nonlinear dynamics (see with m = 1 ∼ 4. Chapter 5 in Nayfeh and Pai book3), which correspond to Γ0 = σ + ζ0 = 0, Γ0 = 3ζ0 − ζ0 = 0, and ∂B1 = ∂B3 = 0 Based on the above analysis, we can see that within the lim- A 1 B 1 3 ∂T1 ∂T1 ited range of ω2 < 10π4, 1:3 internal resonance occurs only in Eq. (23). Thus, the real part can be written as 2 4 in modes m = 1 and m√ = 3 when ω √= 9π . When m = 1, 1 Z 1 p 4 2 4 4 2 ˆ 4 2 ω1 = γ1 + ω = π + 9π = 10π ; when m = 3, − A γ1 cos γ1xdx 3B1B1B1 + p √ 2 0 ω = γ4 + ω2 = p(3π)4 + 9π4 = 3 10π2, which means 3 3 Z 1 that 3ω1 = ω3. We need only consider the internal resonance 2 2 2 γ1 γ3 cos γ3xdx 2B1B3B3 + 2ω1σB1(T1) − in the 1st and 3rd modes. 0 Z 1 Z 1 2 2 2 4 A31K 6B1B3B3 φ1φ3dx + 3B1 B1 φ1dx + 3.3. Frequency Responses in Beam System 0 0 without DVA Z 1 2 3 ˆ 3B1B3 cos ΓB φ1φ3dx = −A21f1 cos ΓA; (24) To analyze the frequency responses of the system and derive 0 the fixed point plots, we assume that the elastic beam is subject and the imaginary part as to a uniform distributed force: Z 1 2 3 ˆ irτ ˆ i(ωm+εσ)T0 ˆ iεσT0 iωmT0 − µω B (T ) − A K 3B B sin Γ φ φ dx = fme = fme = fm e e = 1 1 1 31 1 3 B 1 3 0 ˆ iσT1 iωmT0 ˆ fme e . − A21f1 sin ΓA. (25) With regard to the 1st mode (m = 1), we must select the sec- After calculating the sum of the squares of Eqs. (24) and (25) ular terms with harmonics ω1 and ω3 − 2ω1. Thus, from the and eliminating the terms associated with time, we obtain the

172 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . . following: was the 3rd mode that was being excited, the amplitudes in the 1st mode were greater than those of the 3rd mode, due to the Z 1 1 4 2 unique internal resonance associated with nonlinear systems. − Aˆ γ cos γ1xdx 3B1B1B1 + 1 st rd 2 0 This proves that 1:3 internal resonance occurs in the 1 and 3 Z 1 modes of the system. 2 2 2 γ1 γ3 cos γ3xdx 2B1B3B3 + 2ω1σB1(T1) − 0 Z 1 Z 1 3.5. Numerical Verification 2 2 2 4 A31K 6B1B3B3 φ1φ3dx + 3B1 B1 φ1dx + To ensure the internal resonance in the system, we employed 0 0 Z 1 2 numerical calculations to verify the results in the fixed point 2 3 3B1B3 cos ΓB φ1φ3dx + − µω1B1(T1) − plots without the DVA (i.e., the solutions of Eqs. (26) to (28)). 0 Since Eqs. (26) to (28) are the solvability conditions from Z 1 2 2 3 2 ˆ2 Eq. (21), we substituted the vibration mode obtained using A31K 3B1B3 sin ΓB φ1φ3dx = A21f1 . (26) 0 Eq. (21) into Eq. (13) and employed the orthogonal properties for its integration, the result of which was then used to excite We then multiple the secular terms of the 3rd mode (Eq. (23)) the 1st mode, thus producing by eiζ3 . The real part can be written as ∗∗ ∗ ˆ 4 2 k Z 1 ξ + γ ξ1 + µξ + ω ξ1 + Γ1 + 1 ˆ 4 2 1 1 1 R 1 2 − A γ3 cos γ3xdx 3B3B3B3 + 0 φ1dx 2 0 Z 1 R 1 Z 1 1 2 2 2 2 2 2 2 0 φ1dx 2 2 2 γ ξ1 γ ξ cos γ1x + γ ξ cos γ3x dx = F ; γ γ cos γ xdx 2B B B + 6ω σB (T ) − 1 1 1 3 3 R 1 2 1 3 1 3 1 1 3 3 1 2 0 φ dx 0 0 1 Z 1 (29) 3 3 A33K B1 φ1φ3dx cos(−ΓB) + 0 where Z 1 Z 1 2 2 2 4 Z 1 6B1B1B3 φ1φ3dx + 3B3 B3 φ3dx = 0; (27) 3 3 2 2 2 2 3 3 0 0 Γ1 = ξ1 φ1 + 3ξ1 φ1ξ3φ3 + 3ξ1φ1ξ3 φ3 + ξ3 φ3 φ1dx. 0 and the imaginary part as (30) When it was used to excite the 3rd mode, the outcome was Z 1 3 3 −µω B (T ) − A K B φ φ dx sin(−Γ ) = 0. ∗∗ ∗ ˆ 3 3 1 33 1 1 3 B 4 2 k 0 ξ + γ ξ3 + µξ + ω ξ3 + Γ3 + 3 3 3 R 1 2 (28) 0 φ3dx Z 1 R 1 1 2 2 2 2 2 2 2 0 φ3dx We solve Eqs. (26) to (28) using numerical methods, which γ ξ3 γ ξ cos γ1x + γ ξ cos γ3x dx = F ; 2 3 1 1 3 3 R 1 2 enables us to plot the fixed points plots for amplitudes B1 and 0 0 φ3dx B3 (since the solutions are corresponding to the fixed points of (31) ∂B1 = ∂B3 = 0) and tuned frequency σ in the system in order ∂T1 ∂T1 to observe the internal resonance. Due to limitations regarding where the length of the manuscript, we will not go into details of the Z 1 rd 3 3 2 2 2 2 3 3 analysis where we excite the 3 mode, which is similar to that Γ3 = ξ1 φ1 + 3ξ1 φ1ξ3φ3 + 3ξ1φ1ξ3 φ3 + ξ3 φ3 φ3dx. st 0 of the 1 mode. (32) For the integration coefficients, please refer to Appendix 2. 3.4. Verification of Internal Resonance We used the Runge-Kutta (RK4) method to solve the dynamic Using the relationship between amplitude and frequency in equations (Eqs. (29) and (31)) and then drew time response the nonlinear system without a DVA, we plotted the fixed point graphs and Poincare´ maps. The right side of Fig. 2 presents plots to search for 1:3 internal resonance. The left part of Fig. 2 the fixed point plot and the numerical verification graph of the displays the fixed points plots when the 1st mode (lower mode) 1st mode; the upper graph displays the time responses, and the is excited by a harmonic force. The horizontal axis measures lower graph is the Poincare´ map. The horizontal axis of a fixed the tuned frequency near the natural frequency in this mode, points plot represents the tuned frequency near the beam’s nat- and the vertical axis measures the amplitude of vibration in the ural frequency. Since the external forcing function was de- ˆ irτ ˆ i(ωm+εσ)T0 ˆ iεσT0 iωmT0 beam. As can be seen in Fig. 2, multiple amplitudes may corre- fined as fme = fme = fm e e = ˆ iσT1 iωmT0 spond to a single frequency. A system that resides within this fme e , we can see that when σ = 0, the dimension- unstable region for long periods of time can undergo fatigue less forcing frequency r equals the beam’s mth natural fre- rd and damage. Figure 3 presents the fixed points plots of the 3 quency ωm. Physically, the fixed point plot displays the nonlin- mode. When the 1st mode is being excited, the amplitudes of ear steady-state (fixed point) frequency response near the sys- the 1st mode (Fig. 2) are greater than those of the 3rd mode tem’s natural frequencies. The upper graph of Fig. 2 displays (Fig. 3), which is normal. Fixed point plots of the 3rd mode the time response graph obtained when σ = 4. The conver- under excitation are presented in the left part of Fig. 4, and gence value in this figure is the same as that of the fixed point the concurrent fixed point plots of the 1st mode are presented plot in the left side in Fig. 2 when σ = 4. The Poincare´ map in Fig. 5. A comparison of the two revealed that although it also displays unstable in this case (chaos). The right side of

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 173 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

Figure 2. The 1st mode’s ﬁxed point plots when the 1st mode is excited (No DVA).

the effects of the DVA ﬁxed at various locations according to the frequency responses of the overall system. The results are presented in Section 5.

4.1. Analysis of DVA Equation We assumed that the DVA motions constitute a single-point force (xD denotes the location of the DVA). To enhance the in- ﬂuence of the DVA and simplify the setup, we did not consider the damping term of the DVA. Thus, the motion of equation for ε1 can be written as

2 4 1 ∂ W1 ∂ W1 2 ε : 2 + 4 + ω W 1 = ∂T0 ∂X 2 Z 1 2 ! 2 ∂ W0 1 ∂W0 ∂ W0 2 dx + F − 2 − ∂X 2 0 ∂X ∂T0∂T1

∂W0 ˆ 3 µ − kW − ek W0(x, t)−WD δ(x−xD). (33) ∂T 0 Figure 3. The 3rd mode’s ﬁxed point plots when the 1st mode is excited (No 0 DVA). The DVA equation is as follows:

Fig. 4 presents the fixed point plot and the numerical verifica- " ∞ # tion graph of the 3rd mode; again, the upper graph displays the ∗∗ X m0W D(t) − ek Wm(x, t) − WD = 0. (34) time responses, and the lower graph is the Poincare´ map for m=1 σ = 4. The convergence value of time response is the same as in the left side in Fig. 4 when σ = 4. The Poincare´ map In Section 3, we established that 1:3 internal resonance occurs also displays unstable in this case (chaos). Both Figs. 2 and in the main body between the modes of m = 1 and m = 3 4 show that the internal resonance predicted in the fixed point when ω = 3π2. Thus, we only discussed the 1st and the 3rd plots with no DVA is reasonable. modes (m = 1 and m = 3) of the system. It is noted that the spring force of the DVA is decided by the relative motion 4. VIBRATION REDUCTION ANALYSIS OF of the DVA (WD) and the beam (Wm). It can be expressed nd DVA as ek (Wm − WD). By using the Newton’s 2 law, this force can be added on the beam shown in Eq. (33). As the DVA is This section expands on the results in Sections 2 and 3 by considered an external force, we first solved the displacement analyzing the damping effects of a DVA on a suspended beam of the DVA and then substituted it into the equation of motion or a beam supported on an elastic foundation. We examined for the main body in Eq. (33). With Eq. (34), let m = 1 and

174 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

Figure 4. The 3rd mode’s ﬁxed point plots when the 3rd mode is excited (No DVA).

which, when substituted into Eq. (35), produce the following:

∗∗ k k e e −iζ1 iω1T0 W D + WD = φ1(xD)B1e e + m0 m0 k k e −iζ1 −iω1T0 e −iζ3 iω3T0 φ1(xD)B1e e + φ3(xD)B3e e + m0 m0 k e −iζ3 −iω3T0 φ3(xD)B3e e . (37) m0

We can assume that the solution to WD is

−iζ1 iω1T0 iζ1 −iω1T0 WD = C1e e + C1e e +

−iζ3 iω3T0 iζ3 −iω3T0 C3e e + C3e e . (38)

After substituting Eq. (38) into Eq. (37) and comparing the coefﬁcients, we can derive the following:

Figure 5. The 1st mode’s ﬁxed point plots when the 3rd mode is excited (No ek . 2 DVA). Cn = Bn φn(xD) ek − m0ωn ; m0

ek . 2 Cn = Bn φn(xD) ek − m0ωn ; n = 1, 3; (39) m = 3. Then, the equation can be expanded into m0 −iζ1 iω1T0 iζ1 −iω1T0 WD = C1e e + C1e e +

−iζ3 iω3T0 iζ3 −iω3T0 C3e e + C3e e . (40) ∗∗ m0W D(t)−ek φ1(xD)ξ1(t)+φ3(xD)ξ3(t)−WD = 0. (35) Therefore, we can substitute Eqs. (39) and (40) back into Eq. (33) to determine the secular terms.

Using Eq. (11), we can assume that the solutions for ξ10 and ξ30 are 4.2. Frequency Response of The Beam System

As before, we assumed that the external force is evenly dis- ξ = B e−iζ1 eiω1T0 + B eiζ1 e−iω1T0 ; 10 1 1 ˆ irτ ˆ i(ωm+εσ)T0 ˆ iεσT0 iωmT0 tributed: fme = fme = fm e e = −iζ3 iω3T0 iζ3 −iω3T0 ξ30 = B3e e + B3e e ; (36) ˆ iσT1 iωmT0 st fme e . In terms of the 1 mode (m = 1), we selected

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 175 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

rd iζ3 ω1 and ω3 − 2ω1 for the secular terms: We multiplied all of the secular terms of the 3 mode by e . The real part can be written as 1 Z 1 − Aˆ γ4 cos2γ xdx 3B B B e−iζ1 + 1 1 1 1 1 1 2 0 1 Z − Aˆ γ4 cos2γ xdx 3B B B + Z 1 2 3 3 3 3 3 γ2γ2 cos2γ xdx 2B B B e−iζ1 + 0 1 3 3 1 3 3 1 0 Z γ2γ2 cos2γ xdx 2B B B − 2ω ζ0 B (T ) − 0 −iζ1 0 −iζ1 1 3 1 3 1 1 3 3 3 1 A21F 1 − 2iω1B1(T1)e − 2ω1ζ1B1(T1)e − 0 Z 1 Z 1 −iζ1 2 2 −iζ1 3 3 µiω1B1(T1)e − A31K 6B1B3B3 φ1φ3dx e + A33K B1 φ1φ3dx cos(−ΓB) + 0 0 1 1 Z 1 Z 1 Z 2 Z 2 4 −iζ1 3 i(2ζ1−ζ3) 2 2 2 4 3B1 B1 φ1dx e + 3B1B3 φ1φ3dx e − 6B1B1B3 φ1φ3dx + 3B3 B3 φ3dx − 0 0 0 0 ( " #) kφ (x ) ek3ekφ3(xD) −iζ1 −iζ1 e 1 D ek1 B1φ1(xD)e − B1e . ek3B3φ3(xD) + B3 = 0; (45) 2 m k − m ω2 m0 ek − m0ω1 0 e 0 3 (41) and the imaginary part written as rd For the 3 mode (m = 3), we selected ω3 and 3ω1 as the 0 secular terms: − 2ω3B3(T1) − µω3B3(T1) − Z 1 1 Z 1 A K B3 φ3φ dx sin(−Γ ) = 0. ˆ 4 2 −iζ3 33 1 1 3 B (46) − A γ3 cos γ3xdx 3B3B3B3e + 0 2 0 Z 1 2 2 2 −iζ3 To obtain the frequency responses of the system at the ﬁxed γ1 γ3 cos γ1xdx 2B3B1B1e + 0 0 0 0 points, we let ΓA = σ + ζ1 = 0 =⇒ ζ1 = −σ, 0 0 0 0 ∂B1 ∂B3 0 −iζ3 0 −iζ3 Γ = 3ζ − ζ = 0 =⇒ ζ = −3σ, and = = 0 A23F 3 − 2iω3B3(T1)e − 2ω3ζ3B3(T1)e − B 1 3 3 ∂T1 ∂T1 Z 1 before substituting them into the solvability condition. After −iζ3 3 3 −i3ζ1 calculating the sum of the squares of the real and imaginary µiω3B3(T1)e − A33K B1 φ1φ3dx e + 0 portions of the solvability conditions for the 1st mode and elim- Z 1 Z 1 2 2 −iζ3 2 4 −iζ3 inating the terms associated with time, we obtain 6B1B1B3 φ1φ3dx e + 3B3 B3 φ3dx e − 0 0 ( " #) 1 Z 1 kφ (x ) ˆ 4 2 −iζ3 −iζ3 e 3 D − A γ1 cos γ1xdx 3B1B1B1 + ek3 B3φ3(xD)e − B3e . 2 m k − m ω2 0 0 e 0 3 Z 1 2 2 2 (42) γ1 γ3 cos γ3xdx 2B1B3B3 + 2ω1σB1(T1) − 0 0 Z 1 Z 1 We then designated that the secular terms are equal to in 2 2 2 4 order to derive a solvability condition. Below, we discuss A31K 6B1B3B3 φ1φ3dx + 3B1 B1 φ1dx + st 0 0 the circumstances when an external force excites the 1 mode 1 2 Z ˆ irτ ˆ iσT1 iω1T0 3 (f1e = f1e e ). We multiplied all of the secular 3B1B3 cos ΓB φ1φ3dx − ek1B1φ1(xD) + st iζ1 0 terms of the 1 mode by e and let ΓA = σT1 + ζ1 and 2 ΓB = 3ζ1 − ζ3. The real part can be written as ek1ekφ1(xD) B1 + − µω1B1(T1) − 2 m0 ek − m0ω 1 Z 1 1 ˆ 4 2 1 2 − A γ1 cos γ1xdx 3B1B1B1 + 2 Z 2 0 3 2 ˆ2 A31K 3B1B3 sin ΓB φ1φ3dx = A21f1 . (47) Z 1 0 2 2 2 0 γ1 γ3 cos γ3xdx 2B1B3B3 − 2ω1ζ1B1(T1) − 0 The real part of the solvability conditions for the 3rd mode is Z 1 Z 1 A K 6B B B φ2φ2dx + 3B2B φ4dx + 31 1 3 3 1 3 1 1 1 Z 1 0 0 1 ˆ 4 2 1 − A γ3 cos γ3xdx 3B3B3B3 + 2 Z 2 3B B cos Γ φ3φ dx − k B φ (x ) + 0 1 3 B 1 3 e1 1 1 D Z 1 0 2 2 2 γ1 γ3 cos γ1xdx 2B3B1B1 + 6ω3σB3(T1) − ek1ekφ1(xD) ˆ 0 B1 = −A21f1 cos ΓA; (43) 2 Z 1 m0 ek − m0ω1 3 3 ek3B3φ3(xD) − A33K B1 φ1φ3dx cos(−ΓB) + 0 and the imaginary part can be written as Z 1 Z 1 2 2 2 4 6B1B1B3 φ φ dx + 3B B3 φ dx + 0 1 3 3 3 − 2ω1B1(T1) − µω1B1(T1) − 0 0 1 2 Z ek3ekφ3(xD) 3 ˆ B3 = 0; (48) A31K 3B1B3 sin ΓB φ1φ3dx = −A21f1 sin ΓA. (44) 2 0 m0 ek − m0ω3

176 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

(a) (b)

Figure 6. Fixed point plots (with DVA), (a) The 1st mode’s plot, when the 1st mode is excited, (b) The 3rd mode’s plot, when the 1st mode is excited, (c) The 1st mode’s plot, when the 3rd mode is excited, and (d) The 3rd mode’s plot, when the 3rd mode is excited. and the imaginary part is modes are strongly coupled in cases where internal resonance exists, energy is transferred continuously during motion. The Z 1 rd 3 3 case of no DVA we considered in Figs. 4 and 5, when the 3 −µω3B3(T1) − A33K B1 φ1φ3dx sin(−ΓB) = 0. st 0 mode is excited, the 1 mode’s amplitude is larger than the (49) 3rd mode. This is a typical internal resonance phenomenon in a nonlinear system. In the works of Wang and Kuo17 and Finally, we combine Eqs. (47) to (49) to get the numerical so- Wang and Tu18 for the hinged-free and ﬁxed-free beams, the lutions. We can thus draw ﬁxed-point plots for amplitudes B1 1:3 internal resonance was found in the 1st and the 2nd modes. and B3 and σ in order to observe the frequency responses. Due In the present study, a different boundary condition was ex- to limitations regarding the length of the manuscript, we will amined for a hinged-hinged nonlinear beam and a 1:3 internal rd not go into details of the analysis of the 3 mode, which is resonance was found in the 1st and the 3rd modes. The elas- st similar to that of the 1 mode. tic foundation dimensionless spring constant was found analytically to be 9π4 to trig the internal resonance. The effects 5. RESULTS AND DISCUSSION of various beam boundary conditions in the internal resonance were shown evidently and should be studied individually. At- 5.1. Internal Resonance Analysis of Beam taching a TMD introduces an additional frequency between the System two internal resonant modes to the beam-dampened system, which breaks the multiple integer frequency relationship and Internal resonance occurs in vibration systems with modal thereby mitigates the exchange of energy between the two cou- frequencies that are multiple integers of each other. When two

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 177 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

Table 1. Normalized maximum amplitude of the 1st mode. Table 2. Normalized maximum amplitude of the 3rd mode.

m0 ek XD = 0.1 XD = 0.125 XD = 0.25 XD = 0.5 m0 ek XD = 0.1 XD = 0.125 XD = 0.25 XD = 0.5 0.02 0.1 0.693029 0.652387 0.516591 0.424662 0.02 0.1 1.464852 1.728147 1.478419 1.778185 0.02 0.2 0.698114 0.652374 0.51657 0.426835 0.02 0.2 1.07284 1.229764 1.638418 2.274189 0.02 0.3 0.698111 0.65237 0.516564 0.424631 0.02 0.3 1.467375 1.726061 1.78791 2.119784 0.02 0.4 0.695346 0.652365 0.516559 0.424626 0.02 0.4 1.467676 1.725644 2.104751 2.481469 0.02 0.6 0.698107 0.652366 0.516557 0.424624 0.02 0.6 1.027626 1.228464 2.177923 2.555767 0.02 0.7 0.695345 0.652364 0.516557 0.424623 0.02 0.7 1.417364 1.725503 1.597176 2.685515 0.05 0.1 0.693113 0.652348 0.519716 0.424711 0.05 0.1 1.40992 1.770635 1.378139 1.668663 0.05 0.2 0.692806 0.651563 0.516551 0.42456 0.05 0.2 1.026374 1.26479 1.392248 1.772926 0.05 0.3 0.695348 0.65175 0.51651 0.424826 0.05 0.3 1.439894 1.747833 1.431815 1.72715 0.05 0.4 0.69299 0.651725 0.515925 0.424794 0.05 0.4 1.442745 1.74515 1.531946 1.839284 0.05 0.6 0.692987 0.651722 0.515917 0.424779 0.05 0.6 1.018009 1.255225 1.556726 1.864086 0.05 0.7 0.692984 0.651718 0.516463 0.424772 0.05 0.7 1.444224 1.744312 1.597176 2.048476 0.07 0.1 0.828863 0.779646 0.620958 0.509231 0.07 0.1 1.055878 1.606127 2.179459 2.278483 0.07 0.2 0.693079 0.653774 0.515912 0.424777 0.07 0.2 1.404675 1.296659 1.34855 1.663102 0.07 0.3 0.69303 0.65152 0.519795 0.424681 0.07 0.3 1.416868 1.765247 1.357211 1.646024 0.07 0.4 0.692987 0.652433 0.515934 0.426812 0.07 0.4 1.423382 1.759675 1.39794 1.691371 0.07 0.6 0.692982 0.652428 0.515919 0.424587 0.07 0.6 0.966909 1.274386 1.409035 1.702337 0.07 0.7 0.695312 0.652421 0.515907 0.42677 0.07 0.7 1.426958 1.757671 1.427694 1.722562 0.1 0.1 0.830796 0.780316 0.616474 0.506321 0.1 0.1 0.004963 0.001511 0.00313 0.001322 0.1 0.2 0.693703 0.65157 0.516605 0.424568 0.1 0.2 0.836822 1.365858 1.33042 1.579634 0.1 0.3 0.693088 0.652413 0.516048 0.424679 0.1 0.3 1.373034 1.797928 1.304739 1.588934 0.1 0.4 0.692999 0.65231 0.515911 0.424543 0.1 0.4 1.389869 1.783871 1.290711 1.573621 0.1 0.6 0.692989 0.651602 0.515879 0.424484 0.1 0.6 0.927665 1.305452 1.28927 1.572192 0.1 0.7 0.695312 0.651801 0.51662 0.424457 0.1 0.7 1.398384 1.779369 1.287593 1.570295 0.12 0.1 0.828966 0.779547 0.618058 0.506593 0.12 0.1 1.458676 1.184319 1.436823 1.59492 0.12 0.2 0.693809 0.652407 0.516627 0.426164 0.12 0.2 0.799082 1.437811 1.339805 1.548611 0.12 0.3 0.69366 0.65175 0.515989 0.424614 0.12 0.3 1.334119 1.824971 1.288543 1.571323 0.12 0.4 0.693019 0.651601 0.516551 0.42473 0.12 0.4 1.363762 1.801805 1.295016 1.527046 0.12 0.6 0.693004 0.651583 0.516505 0.424644 0.12 0.6 0.891536 1.327901 1.240718 1.520059 0.12 0.7 0.698082 0.655397 0.51609 0.426171 0.12 0.7 1.377175 1.794745 1.229752 1.508142 0.15 0.1 0.829058 0.780938 0.617327 0.506605 0.15 0.1 1.33262 1.341661 1.690585 1.822654 0.15 0.2 0.693002 0.651557 0.51656 0.424586 0.15 0.2 1.003977 1.6405 1.401304 1.521638 0.15 0.3 0.693782 0.651755 0.516527 0.424735 0.15 0.3 1.25403 1.879512 1.28088 1.563133 0.15 0.4 0.693064 0.652478 0.51659 0.424734 0.15 0.4 1.316419 1.832141 1.20584 1.481179 0.15 0.6 0.698139 0.65245 0.516517 0.424599 0.15 0.6 0.887034 1.364301 1.192195 1.467781 0.15 0.7 0.69301 0.652413 0.516082 0.424536 0.15 0.7 1.341841 1.819752 1.17053 1.444635 pled modes, thus resulting in a reduction in internal resonance. established that adding a DVA with another DOF to the model We employed numerical analysis to solve Eqs. (47) through damages the natural vibration frequency ratios of the various (49), thereby obtaining the fixed points plots of the system. modes, due to the fact that the additional DOF couples with the Figure 6 (the fixed point plots including the DVA) shows that equation of motion of the beam. As a result, internal resonance when the 3rd mode is excited, the amplitudes of the 1st mode does not occur. In this section, we examine various parameters (Fig. 6c) do not exceed those of the 3rd mode (Fig. 6d). Clearly, of the DVA, including mass ratio, spring constant ratio, and lo- internal resonance does not occur. By comparing Fig. 6 with cation, to identify the combination with the optimal damping Figs. 2 through 5, we can see that the inclusion of the DVA effects. Using the solvability condition and fixed point plots in in the system prevents 1:3 internal resonance in the 1st and Section 4, we derived the maximum amplitude in the 1st mode 3rd modes. It is possible that the damping effects of the DVA when excited, and then normalized the results using the maxi- are directly associated with the vibrations of the main beam. mum amplitude obtained in cases without a DVA (0.598675). In Eq. (7), the term ek[W (x, τ) − WD]δ[x − lD] couples with To achieve this, we considered the mass ratio (the mass of the W (x, t), which then merges with terms W iv and w2 to influ- DVA / the mass of the beam) and the spring constant ratio (the ence (positively or negatively) the previous integer ratio of the spring constant of the DVA / the spring constant of the elas- frequencies, 1:3. As a result, 1:3 internal resonance does not tic foundation). The normalized maximum amplitudes of the occur. 1st mode are shown in Table 1. Similarly, we normalized the maximum amplitude in the 3rd mode when excited by using 5.2. Analyzing The Effectiveness of DVA the maximum amplitude in cases without a DVA (0.11921). Again, the results are shown in Table 2. To avoid the diffi- Damping culties associated with an excessive number of DVA parame- Using MOMS, eigen analysis, and fixed point plots, we es- ters in the tables, we used 3D maximum amplitude projections tablished that 1:3 internal resonance occurs in the 1st and 3rd (3DMACPs) to reveal the influence of parameters on damping modes of a main body without a DVA. A comprehensive view performance. The results of 3DMACPs will be shown in next of the two modes show that regardless of whether the 1st or 3rd two paragraphs. mode is excited, the maximum amplitudes in the 1st and 3rd This study uses the 3D projections to project the maximum modes are 0.598675 and 0.11921, respectively. We thus used amplitudes of various parameter combinations onto the mass- these amplitudes as the basis for comparisons used to evaluate spring constant plane and combine various locations into a 3D the effectiveness of damping. In previous sections, we already maximum amplitude contour plot (3DMACP). Physically, the

178 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

st st Figure 7. A 3D MACP chart of the 1 mode, XD = 0.1. Figure 9. A 3D MACP chart of the 1 mode, XD = 0.25.

st st Figure 8. A 3D MACP chart of the 1 mode, XD = 0.125. Figure 10. A 3D MACP chart of the 1 mode, XD = 0.5.

3DMACPs provide the information regarding the maximum and 0.7, in accordance with the function proposed by Samani 15 amplitudes of each combination of DVA’s mass ratio (m0) and and Pellicano for the optimal DVA parameter combination of dimensionless spring constant (ek) for a ﬁxed DVA location linear hinged-hinged beams: (XD). Different colors of the contour levels represent different 2 amplitude values. This is a novel technique to display a set of ω1 mL ek = m0 ; me = ; µ = m0/me. complicated data and display a rather simple plot for better un- 1 + µ 2 sin2(πd/L) derstanding. Figures 7 through 10 present the 3DMACPs con- (50) taining the normalized maximum amplitudes of the 1st mode According to Samani and Pellicano,15 we can convert the DVA st when the 1 mode is excited and the DVA is at XD = 0.1, mass ratio range from 0.02 to 0.15 into the elastic constant ra- 15 0.125, 0.25, and 0.5, respectively. To differentiate the damping tio estimated by Samani and Pellicano. The range of m0 in effects of the DVA at various locations, we drew the contour of this study roughly corresponds to ek values between 0.4 and the four above combinations with the amplitude ranging from 0.55. However, the function in Eq. (50) is applicable to only 0.425 to 0.82. Comparison of Figs. 7 through 10 shows that the linear beams, as it does not consider the effects of stretching. effectiveness of the DVA increased as it approached the center This effect can cause additional amplitudes in the transverse of the beam. Furthermore, we selected the mass ratios between direction of the beam. In theory, this situation would require a 0.02 and 0.15 because if the mass ratio were too small, it would DVA with greater damping effects in order to achieve accept- have no effect on the hinged beams. In most civil engineering able results. We therefore widened the range of the elastic con- or mechanical structures, a mass ratio greater than 0.15 would stant ratio from 0.4 ∼ 0.55 to 0.1 ∼ 0.7. As for the amplitudes mean an excessive waste of materials on the DVA. In addition, in the 1st mode when the 1st mode is excited, Figs. 7 through the spring constants considered in this study ranged from 0.1 10 show that the DVA being more effective when it is closer

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 179 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

rd rd Figure 11. A 3D MACP chart of the 3 mode, XD = 0.1. Figure 13. A 3D MACP chart of the 3 mode, XD = 0.25.

rd rd Figure 12. A 3D MACP chart of the 3 mode, XD = 0.125. Figure 14. A 3D MACP chart of the 3 mode, XD = 0.5. to the center of the beam a higher mass ratios do not actually contribute to poorer damping performance. Of course, plac- have significant damping effects. On the contrary, excessively ing the DVA near the center of the beam is most effective, but rd high mass ratios can magnify the amplitudes in transverse vi- for the 3 mode, this is not true. A comprehensive view of brations in the beam (because a DVA is included). We also Figs. 7 through 14 shows that if we take into account the op- st rd found that such DVA designs are more effective in damping timal damping effects in both the 1 and 3 modes, it is best the 1st mode. Similarly, damping effects are better when the to place the DVA in the center of the beam with m0 and ek be- DVA is closer to the center of the beam. Nonetheless, coming approximately 0.1 and 0.2, respectively. Clearly, the linear beam results (Eq. (50)) in Samani and Pellicano15 are not ap- binations with m0 between 0.05 and 0.15 and ek less than 0.2 must still be avoided. plicable to nonlinear beams. Furthermore, we considered the coupling of the 1st and 3rd modes in our model, which compli- For the 3rd mode (Figures 11 through 14), we created cates the design of the DVA. Nevertheless, the optimal damp- 3DMACPs for the various locations X and set the range of D ing combination can still be identified using the 3DMACP. In the maximum amplitude to between 1.0 and 2.4. Note that a the next section, we verify the accuracy of 3DMACPs using color closer to light green indicates that the maximum ampli- numerical methods. tude is greater than 1 and that the damping effect is worse than if there were no DVA. We also discovered that with mass ra- 5.3. Analysis of Numerical Verification tios of approximately 0.1 and elastic constants less than 0.2, minimum amplitudes occur at each location. This means that To conduct the numerical analysis of the beam equation in- said combinations have optimal damping effects in terms of cluding the DVA, we can use the mode shape in Eq. (21), sub- the 3rd mode. In Figs. 7 through 10, we can also see that near stituting it into Eq. (7), and integrating it using the orthogonal the end (XD = 0.1 in Fig. 7), a smaller m0 and a greater ek properties to produce the following equations of motion to ex- 180 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

(a)

(b)

Figure 15. Numerical veriﬁcation, (a) m0 = 0.15, ek = 0.1, XD = 0.1, (b) m0 = 0.02, ek = 0.7, XD = 0.1. cite the 1st mode: and to excite the 3rd mode:

∗∗ ∗ ˆ 4 2 k ξ 3 + γ3 ξ3 + µξ3 + ω ξ3 + 1 Γ3 + ∗∗ ∗ ˆ R 2 4 2 k 0 φ3dx ξ + γ ξ1 + µξ + ω ξ1 + Γ1 + 1 1 1 R 1 2 Z 1 0 φ1dx 1 2 2 2 2 2 2 2 γ ξ3 γ ξ cos γ1x + γ ξ cos γ3x dx Z 1 2 3 1 1 3 3 1 2 2 2 2 2 2 2 0 γ1 ξ1 γ1 ξ1 cos γ1x + γ3 ξ3 cos γ3x dx R 1 2 0 0 φ3dx fs [ξ1φ1(XD) + ξ3φ3(XD) − WDφ(XD)] = F ; R 1 R 1 2 0 φ1dx 0 φ3dx fs [ξ1φ1(XD) + ξ3φ3(XD) − WDφ(XD)] = F ; R 1 2 (53) 0 φ1dx (51) where Z 1 3 3 2 2 2 2 3 3 Γ3 = ξ1 φ1 + 3ξ1 φ1ξ3φ3 + 3ξ1φ1ξ3 φ3 + ξ3 φ3 φ3dx. where 0 (54) Please refer to Appendix 2 for the relevant integration coefﬁ- Z 1 3 3 2 2 2 2 3 3 cients. We employed the RK4 approach to calculate displace- Γ1 = ξ1 φ1 + 3ξ1 φ1ξ3φ3 + 3ξ1φ1ξ3 φ3 + ξ3 φ3 φ1dx; ments in the nonlinear beam with various parameter and loca- 0 (52) tion combinations when excited by a given external frequency.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 181 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

(a)

(b)

Figure 16. Numerical veriﬁcation, (a) m0 = 0.15, ek = 0.1, XD = 0.5, (b) m0 = 0.02, ek = 0.7, XD = 0.5.

To verify the results from 3DMACPs (Figs. 7 to 14), we per- plot of Fig. 14 with m0 = 0.1, ek = 0.1, and XD = 0.5 formed the numerical analysis and cross-referred the numeri- lain over the numerical results of the maximum amplitudes, cal results, fixed points plots and Poincare´ maps (Figs. 15 to and Fig. 18b shows the same with m0 = 0.02, ek = 0.4, and 18). The left part of Fig. 15a displays the fixed point plot of XD = 0.5. Comparing the 3DMACPs with the fixed point Fig. 7 with m0 = 0.15, ek = 0.1, and XD = 0.1 lain over the plots and numerical method (time response), for each speci- numerical results of the maximum amplitudes. The right upper fied value of m0, ek, and XD, the maximum value of the fixed graph of Fig. 15a displays the numerical result of time response point plots agrees with the 3DMACP and the numerical results. and the lower graph is the numerical result of Poincare´ map. We can see that the results are consistent, thereby demonstrat- Figure 15b shows the same with m0 = 0.02, ek = 0.7, and ing the accuracy of the model developed in this study and the XD = 0.1. The left part of Fig. 16a exhibits the fixed point plot concept of 3DMACPs. of Fig. 10 with m0 = 0.15, ek = 0.1, and XD = 0.5 lain over the numerical results of the maximum amplitudes. The right 6. CONCLUSIONS upper graph of Fig. 16a displays the time response, and the lower graph is the Poincare´ map. Figure 16b shows the same In this study, we considered a hinged-hinged nonlinear beam with m0 = 0.02, ek = 0.7, and XD = 0.5. As before, Fig. 17a supported or suspended by nonlinear springs to simulate a sys- displays the fixed point plot of Fig. 11 with m0 = 0.1, ek = 0.1, tem that is suspended or placed on an elastic foundation. We and XD = 0.1 lain over the numerical results of the maximum subjected the main body to a harmonic force. Both ends of amplitudes, and Fig. 17b shows the same with m0 = 0.02, the nonlinear elastic beam are hinged; therefore, we had take ek = 0.7, and XD = 0.1. Figure 18a displays the fixed point into account the effects of stretching. We added a DVA to

182 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

(a)

(b)

Figure 17. Numerical veriﬁcation, (a) m0 = 0.1, ek = 0.1, XD = 0.1, (b) m0 = 0.02, ek = 0.7, XD = 0.1. the beam system and investigated its effectiveness in damping ACKNOWLEDGEMENTS when place in various locations. Using some elastic constant combinations, 1:3 internal resonance occurred in the 1st and This research was supported by the Ministry of Science 3rd modes. Placing the DVA at suitable locations was shown and Technology of Taiwan, Republic of China (grant number: to prevent this, such that the damping effects of the DVA are MOST 103-2221-E-032-047). more apparent in the 1st mode than in the 3rd mode. Damping performance was more signiﬁcant when the DVA was closer REFERENCES to the center of the beam; however, 3DMACPs showed that 1 Nayfeh, A. H. and Mook, D. T. Nonlinear Oscillations, optimal parameter combinations could also be found when the Wiley-Interscience, New York, (1979). DVA was placed in other locations. For the 1st mode, smaller elastic constants and larger mass ratios are not recommended. 2 Nayfeh, A. H. and Balachandran, B. Applied Nonlinear Dy- rd As for the 3 mode, damping performance is better when the namics, Wiley-Interscience, New York, 158–172, (1995). DVA is near the center of the beam; however, the best results are obtained when the mass ratio is approximately 0.1 and the 3 Nayfeh, A. H. and Pai, P. F. Linear and Nonlinear Struc- elastic constant is less than 0.2. For the best damping perfor- tural Mechanics, New York, (2004). st rd mance in both the 1 and 3 modes, the ideal combination 4 includes the DVA at the center of the beam with the mass ratio Ji, Z. and Zu, J. W. Method of multiple scales and elastic constant of approximately 0.1 and 0.2, respectively. for vibration analysis of rotor-shaft systems with non-linear bearing pedestal model, Journal of

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 183 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

(a)

(b)

Figure 18. Numerical veriﬁcation, (a) m0 = 0.1, ek = 0.1, XD = 0.5, (b) m0 = 0.02, ek = 0.4, XD = 0.5.

Sound and Vibration, 218 (2), 293–305, (1998). 9 Shen, H. S. A novel technique for nonlinear anal- https://dx.doi.org/10.1006/jsvi.1998.1835 ysis of beams on two-parameter elastic foundations, International Journal of Structural Sta- 5 Nayfeh, A. H. and Nayfeh, S. A. On nonlinear modes bility and Dynamics, 11 (6), 999–1014, (2011). of continuous systems, Transactions of the ASME, Jour- https://dx.doi.org/10.1142/S0219455411004440 nal of Vibration and Acoustics, 116 (1), 129–136, (1994). https://dx.doi.org/10.1115/1.2930388 10 Wang, Y. R. and Chen, T. H. The vibration reduction 6 Boertjens, G. J. and Van Horssen, W. T. On mode analysis of a nonlinear rotating mechanism deck sys- interactions for a weakly nonlinear beam equa- tem, Journal of Mechanics, 24 (3), 253–266, (2008). tion, Nonlinear Dynamics, 17 (1), 23–40, (1998). https://dx.doi.org/10.1017/S1727719100002318 https://dx.doi.org/10.1023/A:1008232515070 11 Wang, Y. R. and Chang, M. H. On the vibration reduction of 7 Mundrey, J. S. Railway Track Engineering, Tata McGraw- a nonlinear support base with dual-shock-absorbers, Jour- Hill, New Delhi, (2000). nal of Aeronautics, Astronautics and Aviation, Series A, 8 Fu, Y. M., Hong, J. W., and Wang, X. Q. Analysis of 42 (3), 179–190, (2010). https://dx.doi.org/10.6125/JAAA nonlinear vibration for embedded carbon nanotubes, Jour- nal of Sound and Vibration, 296 (4–5), 746–756, (2006). 12 Wang, Y. R. and Chang, C. M. Elastic beam with nonlin- https://dx.doi.org/10.1016/j.jsv.2006.02.024 ear suspension and a dynamic vibration absorber at the free

184 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-R. Wang, et al.: DAMPING PERFORMANCE OF DYNAMIC VIBRATION ABSORBER IN NONLINEAR SIMPLE BEAM WITH 1:3 INTERNAL. . .

end, Transaction of the Canadian Society for Mechanical 24 Ansari, R., Hasrati, E., Gholami, R., and, Sadeghi, Engineering (TCSME), 38 (1), 107–137, (2014). F. Nonlinear analysis of forced vibration of non-

13 local third-order shear deformable beam model Nambu, Y., Yamamoto, S., and Chiba, M. A smart of magneto–electro–thermo elastic nanobeams, dynamic vibration absorber for suppressing the vibra- Composites: Part B, 83, 226–241, (2015). tion of a string supported by flexible beams, Smart https://dx.doi.org/10.1016/j.compositesb.2015.08.038 Materials and Structures, 23 (2), 025032, (2014). https://dx.doi.org/10.1088/0964-1726/23/2/025032 25 Ansari, R., Mohammadi, V., Faghih Shojaei, M., Gholami, R., and Sadeghi, F. Nonlinear forced 14 Tso, M. H., Yuan, J., and Wong, W. O. Suppression of ran- vibration analysis of functionally graded car- dom vibration in flexible structures using a hybrid vibration bon nanotube-reinforced composite Timoshenko absorber, Journal of Sound and Vibration, 331 (5), 974– beams, Composite Structures, 113, 316–327, (2014). 986, (2012). https://dx.doi.org/10.1016/j.jsv.2011.10.017 https://dx.doi.org/10.1016/j.compstruct.2014.03.015 15 Samani, F. S. and Pellicano, F. Vibration reduction 26 Ansari, R., Faghih Shojaei, M., Mohammadi, V., Gholami, of beams under successive traveling loads by means R., and Sahmani, S. On the forced vibration analysis of of linear and nonlinear dynamic absorbers, Journal of Timoshenko nanobeams based on the surface stress elas- Sound and Vibration, 331 (10), 2272–2290, (2012). ticity theory, Composites: Part B, 60, 158–166, (2014). https://dx.doi.org/10.1016/j.jsv.2012.01.002 https://dx.doi.org/j.compositesb.2013.12.066 16 Wang, Y. R. and Lin, H. S. Stability analysis and 27 Ansari, R., Gholami, R., and Rouhi, H. Size- vibration reduction for a two-dimensional non- dependent nonlinear forced vibration analysis linear system, International Journal of Structural of magneto-electro-thermo-elastic Timoshenko Stability and Dynamics, 13 (5), 1350031, (2013). nanobeams based upon the nonlocal elasticity the- https://dx.doi.org/10.1142/S0219455413500314 ory, Composite Structures, 126, 216–226, (2015). 17 Wang, Y. R. and and Kuo, T. H. Effects of a dynamic https://dx.doi.org/10.1016/j.compstruct.2015.02.068 vibration absorber on nonlinear hinged-free beam, 28 Ansari, R. and Gholami, R. Surface effect on the ASCE Journal of Engineering Mechanics, 142 (4), large amplitude periodic forced vibration of first-order (2016). https://dx.doi.org/10.1061/(ASCE)EM.1943- shear deformable rectangular nanoplates with various 7889.0001039 edge supports, Acta Astronautica, 118, 72–89, (2016). 18 Wang, Y. R. and Tu, S. C. Influence of tuned mass damper https://dx.doi.org/10.1016/j.actaastro.2015.09.020 on fixed-free 3D nonlinear beam embedded in nonlinear elastic foundation, Meccanica, 51 (10), 2377–2416, (2016). https://dx.doi.org/10.1007/s11012-016-0372-8 APPENDIX 1 19 Wang, Y. R. and Liang, T. W. Application of lumped-mass vibration absorber on the vibration reduction of a non- 1/2 !1/2 W t EIA EIA linear beam-spring-mass system with internal resonances, W = ; τ = ; ω = 4 ; l l2 ρA ρAl Journal of Sound and Vibration, 350, 140–170, (2015). 2 x l mul https://dx.doi.org/10.1016/j.jsv.2015.04.002 x = ; l = = 1; µ = ; l l 1/2 20 (ρAEIA) Ansari, R., Pourashraf, T., and Gholami, R. An exact so- EAr2 lution for the nonlinear forced vibration of functionally Aˆ = — non-dimensional beam rigidity, and r is the EIA graded nanobeams in thermal environment based on sur- 1/2 2 ρA face elasticity theory, Thin-Walled Structures, 93, 169–176, beam cross-section radius; Ω = Ωl ; (2015). https://dx.doi.org/10.1016/j.tws.2015.03.013 EIA 4 2 4 2 kl βr kl mˆ 21 Sayed, M. and Kamel, M. 1 to 2 and 1 to 3 internal reso- ω = ; K = ; m0 = ; EIA EIA ρAl nance active absorber for non-linear vibrating system, Ap- f g plied Mathematical Modelling, 36 (1), 310–332, (2012). k = s ; λ = s . e 2 https://dx.doi.org/10.1016/j.apm.2011.05.057 ω ρAl ωρAl 22 Ghayesh, M. H., Kazemirad, S., and Amabili, M. Coupled longitudinal-transverse dynamics of an APPENDIX 2 axially moving beam with an internal resonance, R 1 ek sin γmxdx Mechanism and Machine Theory, 52, 18–34, (2012). k = ; A = 0 ; em R 1 2 2m R 1 2 https://dx.doi.org/10.1016/j.mechmachtheory.2012.01.008 0 sin γmxdx 0 sin γmxdx 1 23 Ghayesh, M. H. and Amabili, M. Nonlinear dynamics A3m = . R 1 sin2γ xdx of an axially moving Timoshenko beam with an internal 0 m resonance, Nonlinear Dynamics, 73 (1), 39–52, (2013). https://dx.doi.org/10.1007/s11071-013-0765-3

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 185 Energy Harvesting Estimation from the Vibration of a Simply Supported Beam

Aviral Rajora Delft University of Technology, Netherlands

Ajit Dwivedi Steel Authority of India Limited, Bhilai, India

Ankit Vyas Ministry of Health and Family Welfare, New Delhi, India

Satyam Gupta Infosys Limited, Mysore, India

Amit Tyagi Indian Institute of Technology (Banaras Hindu University), Varanasi, India

(Received 1 November 2014; accepted 19 April 2016) Vibration-based energy harvesting has been investigated in this paper with the goal to utilize the ambient vibration energy to power small electronic components by converting vibration energy into electrical energy. A simply supported beam with a bonded high density piezoelectric patch to the surface is considered for the analysis. Analytical model for free vibration analysis is developed by starting with the linear constitutive relations for the beam and the patch. The equation of motion for transverse vibration of the beam is developed by considering the elastic as well as electrical properties in the generalized Hookes law and accordingly a transverse displacement function satisfying the simply supported boundary conditions is used for achieving the modal frequencies. Additionally, an analytical model is developed in order to estimate the energy generated under the action of a harmonic force applied on the surface of the patch. The results of the analytical model are validated using simulation software ANSYS and COMSOL. The developed analytical model is used to study the behavior of a simply supported harvester with various patch dimensions and locations. This paper throws light on parametric studies of eigen frequencies as well as extracted power corresponding to operating conditions.

NOMENCLATURE M, Mass Matrix; F , Applied force; D, Electric charge density displacement matrix; 0 V , Voltage across the load resistance; e, Coupling coefﬁcient for stress-charge form; 0 C , Capacitance of the energy harvesting circuit; , Strain; p R, Load Resistance for energy harvesting circuit; p, Electric permittivity; P , Power Generated. Ez, Electric Field ; M(x, t), Moment on the beam at location x, time t; F , Applied Force; 1. INTRODUCTION

ρ1, Density of the material 1; The growing demand for energy in various sectors has moti- ρ , Density of the material 2; 2 vated researchers to look into alternative forms of energy gen- E , Youngs modulus of material 1; 1 eration at both large and small scales. Various devices have E2, Youngs modulus of material 2; become miniature with the advancement of nanotechno- logy. l, Length of the beam; However, this decrease in size is limited due to sizeable bat- b − a, Length of patch starting at location a; teries. Thus, it is becoming essential to find a way to re- h1, Height of the beam; place bulky conventional batteries in order to facilitate devel- 1 h2, Height of the patch; oping micro-electronic mechanical devices. Roundy et al. has C, Damping coefficient; shown a comparison of power scavenging from various energy K, Stiffness Matrix; sources like vibrations, solar, and various chemical batteries 186 https://doi.org/10.20855/ijav.2017.22.2463 (pp. 186–193) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. Rajora, et al.: ENERGY HARVESTING ESTIMATION FROM THE VIBRATION OF A SIMPLY SUPPORTED BEAM and found that ambient vibrations are a potential source of energy for the applications where continuous power is desired with long life. Miniature wireless electronic devices require a very low output power; ambient vibration energy can be used to power these devices. In light of this, our attention is focused on vibration energy harvesting. There are different transduction mechanisms for converting vibration energy into useful electrical energy. These are piezo- 1 2 3 electric, electromagnetic, and electrostatic. The comparison Figure 1. Reference figure for theoretical analysis. between maximum energy densities of these three transducers 4 is given in Table 1. In the present study, piezoelectric materi- R als have been used because they have relatively simple configurations and a high conversion efficiency. V0 Various work has been done in the past decade for estimat- V sin(ωt) C ing the vibration energy harvested using piezoelectric material. The effect of piezoelectricity on the elasticity is considered by applying a constitutive relation, as discussed by Tyagi and Ghosh.5 Sodano et al.6 presented the review of the research that has been performed in the area of power harvesting and the Figure 2. Circuit Diagram for Analysis. future goals that must be achieved for power harvesting sys- Since there is no applied voltage, i.e., D = 0, from Eq. (1), we tems to find their way into everyday use. In another paper, the get: 7 same authors developed a model of the PZT power harvest- e13εxx Ez = − (2) ing device, which simplifies the design procedure necessary p33 for determining the appropriate size and vibration levels nec- And we know that: essary for sufficient energy to be produced and supplied to the electronic devices. Chen et al.8 proposed a novel piezoelec- du d2w εxx = = −z . (3) tric cantilever bimorph micro transducer electro-mechanical dx dx2 energy conversion model both analytically and experimentally. They noted that the vibration induced voltage is inversely pro- So finally, d2w portional to the length of the cantilever beam. The review ar- ze13 dx2 9 Ez = . (4) ticle by Priya provides a thorough review of developments p33 in the area of piezoelectric energy harvesting. Erturk and In- According to Euler-Bernoulli beam theory: man10 presented the exact analytical solution of a cantilevered piezoelectric energy harvester with Euler-Bernoulli beam as- ∂2w(x, t) 11 M (x, t) = EI(x) . (5) sumptions. Xu et al. experimentally studied a high perfor- ∂x2 mance bi-stable piezoelectric harvester based on simply supported buckled beam. Erturk and Inman12 evaluated the per- And equilibrium equations are: formance of the bimorph device extensively for the short cir- 2 2 ∂Nx ∂ Mx ∂ w(x, t) cuit and open circuit resonance frequency excitations and the = 0; − ρh = 0. (6) ∂x dx2 dt2 accuracy of the model has been shown in all the cases. It is generally accepted that a simplified analytical model Hooke’s law: for a simply supported beam will help understand the basic physics behind working of piezoelectric energy harvester. In σx,1 = E1εxx (For beam); the present work an analytical model is developed for calcu- σx,2 = E2εxx − e13Ez (For patch) . (7) lating the modal frequencies using the Euler-Bernoulli Beam Equation. An analytical model is also developed for calculation of energy harvested by the piezoelectric patch bonded to a simply supported beam and the results are verified using AN- SYS and COMSOL.

2. FREE VIBRATION ANALYSIS

A simply supported beam with thickness h1 with an elastic patch of thickness h2 that is centrally bonded to the beam surface is considered for the analysis. The linear constitutive relation for the beam with piezoelectric patch is given by: Figure 3. Simply supported beam vibrating in mode I. D = e13εxx + p33Ez. (1) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 187 A. Rajora, et al.: ENERGY HARVESTING ESTIMATION FROM THE VIBRATION OF A SIMPLY SUPPORTED BEAM

It is assumed that the patch is perfectly bonded to the beam and To seek a solution, the following transverse displacement func- contributes to the force and moment. The equilibrium Eq. (6) tion is used for the simply supported beam: for the force and moment are written as: X mπx iωt w = Am sin e . (12) ∂N ∂2M ∂2w(x, t) l x x − (8) = 0; 2 ρh 2 = 0; ∂x dx dt Putting w from Eq. (12) into Eq. (11), we get: where,Nx = ∫ σx, dz + ∫ σx, R dz, Mx = ∫ σx, zdz + 1 2 1 1 4 4 π π XAm mπx ∫ σx,2R1zdz, and ρh = ρ1h1 + R1ρ2h2. −D − D R m4 sin 1 l4 2 l4 1 l Where R1 is a location function which can be expressed in 2 A 2 terms of a Heaviside function: π X m 2 mπx ∂ R1 + D2 m sin ( l2 l ∂x2 3 0; when x < a or x > b π XAm mπx ∂R1 R1 = . − 2D m3 cos = 1; when a ≤ x ≤ b 2 l3 l ∂x XAm mπx ω2ρh sin . (13) Therefore, from equation Eq. (7) and (8), l To obtain the modal solution, both the sides of equation h1 h1+2h2 2 2 e13εxx Eq. (13) are multiplied by and integrated over the length l of Mx = ∫ E1εxxz∂z+ ∫ E2εxx + e13 R1z∂z. the host beam. −h1 h1 p33 2 2 (9) 4 l A 2 π X m mπx pπx ∂ w − D ∫ m4 sin sin dx Now from equation Eq. (2), εxx = −z 2 . 1 4 ∂x l 0 l l Therefore, 4 l A π X m 4 mπx pπx − D2 ∫ R1 m sin sin dx h1 4 2 2 l 0 l l 2 ∂ w Mx = − ∫ E1z ∂z 2 l 2 A 2 π ∂ R1 X m 2 mπx pπx − h1 ∂x ∫ 2 + D2 2 2 m sin sin dx l 0 ∂x l l h1+2h2 2 2 2 3 l A e13 2 ∂ w π ∂R1 X m 3 mπx pπx − ∫ E2 + R1z ∂z; − ∫ 2 2D2 3 m sin sin dx = h1 p33 ∂x l 0 ∂x l l 2 l A 2 3 2 X m mπx pπx ∂ w h1 ω ρh ∫ sin sin dx. (14) Mx = − E 1 ∂x2 12 0 l l ! e 2 ∂2w (h + 2h )3 − h 3 We make use of following mathematical relations: − 13 1 2 1 E2 + R1 2 ; p33 ∂x 24 ( 0; when x < a or x > b 2 2 if R = , ∂ w ∂ w ≤ ≤ Mx = − D − D R ; (10) 1; when a x b 1 ∂x2 2 1 ∂x2

3 2 3 3 E1h1 e13 (h1+2h2) −h1 where, D1 = ; D2 = E2 + . 12 p33 24 b dR Differentiating equation Eq. (10) with respect to x, we get: then ∫ f(x)dx = a dx ∞ ∂M ∂3w ∂R ∂2w ∂3w x = − D − D 1 + R ; ∫ {δ (x − b) − δ (x − a)} f (x) dx = ∂x 1 ∂x3 2 ∂x ∂x2 1 ∂x3 −∞ ∂2M ∂4w f (b) − f(a); x = − D ∂x2 1 ∂x4 b d2R ∫ 2 2 3 4 2 = ∂ R1 ∂ w ∂R1 ∂ w ∂ w a dx − D + 2 + R . 2 ∂x2 ∂x2 ∂x ∂x3 1 ∂x4 ∞ ∫ {δ0 (x − b) − δ0 (x − a)} f (x) dx = 2 −∞ Putting this value of ∂ Mx in equation Eq. (6), we get: ∂x2 f 0 (a) − f 0(b). ∂4w ∂2R ∂2w ∂R ∂3w ∂4w − − 1 1 Upon solving equation Eq. (14), we get Eq. (15). D1 4 D2 2 2 + 2 3 + R1 4 = ∂x ∂x ∂x ∂x ∂x ∂x Equation (15) is solved using MATLAB to get the modal fre- ∂2w quencies. The results are veriﬁed using modeling in ANSYS ρh . (11) ∂t2 and COMSOL software. The results obtained are discussed Boundary conditions for the simply supported beam are: later in this paper. ∂2w 3. ENERGY HARVESTING x = 0; w = 0; = 0; ∂x2 ∂2w The moment equation considering the externally applied x = l; w = 0; = 0. distributed force on the patch area is given as: ∂x2 188 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. Rajora, et al.: ENERGY HARVESTING ESTIMATION FROM THE VIBRATION OF A SIMPLY SUPPORTED BEAM

 (m−p)πb (m−p)πa   (m+p)πb (m+p)πa  4 sin − sin sin − sin π XAm  l l   l l  − 4 − D2 4 m   l  2π(m − p)   2π(m + p) 

 n mπb pπb mπa pπa o  3 A p sin cos − sin cos π X m 2 l l l l − D2 m  n o  l3 pπb mπb pπa mπa +m sin l cos l − sin l cos l 3 π XAm pπa mπa pπb mπb + 2D m3 sin cos − sin cos = 2 l3 l l l l  (m−p)πb (m−p)πa   (m+p)πb (m+p)πa  A sin − sin sin − sin 2 X m  l l   l l  − ρ2h2ω  −  when m =6 p;  2π(m − p)   2π(m + p) 

4 4 " ( 2mπb 2mπa )# π l X 4 π X 4 b − a sin l − sin l − D Amm − D Amm − l 1 l4 2 2 l4 2 4mπ 3 π XAm 2mπb 2mπa − D m3 sin − sin 2 l3 l l 3 π XAm 2mπb 2mπa + D m3 sin − sin 2 l2 l l "( 2mπb 2mπa )# l XAm b − a sin − sin = −ρ h ω2 − ρ h ω2 − l l l when m = p. (15) 1 1 2 2 2 2 4mπ

Under the action of a harmonic force applied on the surface 2 2 ∂ Mx ∂ ω of the patch, the frequency of the voltage developed and trans- = ρh − R2Fo; (16) ∂x2 ∂t2 verse displacement of the beam will also be same. The expres- iωt where R2 is the location function of applied force Fo, which sion of force, voltage, and displacement are:F = Foe ; V = iωt P mπx iωt can be expressed in terms of the Heviside function as: Voe ; ω = Am sin l e . Substituting these expressions in the equation Eq. (16), the equation of motion is ( 0; when x < c or x > d obtained as: R2 = ; 1; when c ≤ x ≤ d 4 4 π π XAm mπx − − 4 D1 4 D2R1 4 m sin where Mx from equation Eq. (10) can be written as: l l l 2 3 A ∂ R1 π X m 3 mπx ∂R1 + D Vo + 2D m cos 3 ∂x2 3 l3 l ∂x 2 ∂ ω 2 A 2 2 M = −D π X m 2 mπx ∂ R1 ∂ R1 x 1 2 + D m sin + D = ∂x 3 l2 l ∂x2 4 ∂x2 2 ∂ ω R1e13Vo h 2 i Am − − 2 (17) 2 X mπx D3R1 2 + (h1 + 2h2) h1 ; ω ρh m sin − R2Fo. (20) ∂x 8h2 l

The above equation is arranged in the matrix form as: ∂M ∂3ω x = −D ∂x 1 ∂x3 3 2 ∂ ω ∂R1 ∂ ω ∂R1 2 − D R + + D ; (18) {K} − ω {M} Am − {θ} Vo = 3 1 ∂x3 ∂x ∂x2 4 ∂x l pπd pπc Fo cos − cos ; (21) mπ l l ∂2M ∂4ω x = −D where, K = Stiffness matrix and M = Mass matrix. Account- ∂x2 1 ∂x4 ing for the damping, if present, Eq. (21) becomes: ∂4ω ∂R ∂3ω ∂2R ∂2ω ∂2R − D R + 2 1 + 1 + D 1 ; 3 1 ∂x4 ∂x ∂x3 ∂x2 ∂x2 4 ∂x2 {K} + iω {C} − ω2 {M} A − {θ} V = (19) m o l pπd pπc Fo cos − cos ; (22) 3 E (h +2h )3−h 3 E1h1 2[ 1 2 1 ] mπ l l where, D1 = 12 ; D3 = 24 ; D4 = 2 2 e13V0[(h1+2h2) −h1 ] . where C is damping coefﬁcient. 8h2 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 189 A. Rajora, et al.: ENERGY HARVESTING ESTIMATION FROM THE VIBRATION OF A SIMPLY SUPPORTED BEAM

Now from equation Eq. (1), putting Ez = Vo/R, we get: The equation for the above circuit can be written as:

Vo ∫ i dt Dz = e13ε13 − p33 . + iR = V sin ωt. h2 C This can also be written as: We get the following equation in Laplace form: ∂2ω V D = −h e − p o . z 2 13 2 33 I(s) V ω ∂x h2 + I(s)R = ; Cs s2 + ω2 The electric charge ﬂow across the electrodes is given by 1 + RCs V ω 2 I(s) = ; b b ∂ ω V 2 2 ∫ ∫ − − o Cs s + ω q = Dzsdx = h2e13 2 p33 sdx. a a ∂x h2 V Cωs I (s) = . Therefore, current I is given as, (1 + RCs)(s2 + ω2) dq b ∂3ω dV The voltage across resistor R is: ∫ − − o (23) I = = h2e13 2 sdx Cp ; dt a ∂x ∂t dt s Vo (s) = I (s) R = V RCω . here capacitance Cp of the patch and is deﬁned as: (1 + RCs)(s2 + ω2)

p33s(b − a) On inverse transformation of the Laplace equation, we get: Cp = . h2 −V RCω h −t i Therefore, from equation Eq. (23), Vo = e RC + cos (ωt) + RCω sin (ωt) . (1 + R2C2ω2) V dV b ∂3ω (27) o o − ∫ (24) + Cp = h2e13 2 sdx. The magnitude of the energy output (P ) from the vibration of R dt a ∂x ∂t a simply supported beam under the action of a harmonic force Where, R is the resistance of the energy harvesting circuit. is given by: Using the value of ω Eq. (24) can also be re-written as V 2 P = o . (28) b R Vo mπ 2 mπx +Cp (iω) Vo = ∫ h2se13 (iω) Am sin dx. R l a l The Voltage across the load resistance is calculated analytically by solving Eq. (26) in MATLAB. The power across the After integration, we get: load can be calculated using Eq. (28). The results are veri- ﬁed using COMSOL software. The software gives the voltage Vo + Cp (iω) Vo = R across the patch, which can be used to determine COMSOL mπb mπb mπ voltage across the load using Eq. (27). The results obtained cos − cos h2se13 (iω) Am. are discussed below. l l l

Substituting the value of Am in Eq. (22), we get: 4. MODELLING WITH ANSYS AND COMSOL − 2 1 [K] + iω [C] ω [M] Vo R + iωCp 4.1. ANSYS − [θ] Vo = smπ h e iω cos mπb − cos mπb l 2 13 l l To model the beam in ANSYS12, we used BEAM 3 ele- l pπd pπc Fo cos − cos . (25) ment and solved the 2-D beam problem. The real constants for mπ l l the beam elements are given in terms of area = 2 × 10−5m2, × −12 4 The above equation can be rearranged to get Vo as: moment of inertia = 1.667 10 m and height = 1 mm. The real constants for the patch of thickness 0.2 mm are iV1 −6 2 Vo = ; (26) given in terms of area = 4 × 10 m , moment of inertia V2 + iV3 = 1.45333 × 10−12m4 and height = 0.2 mm. Here we con- where, sider the width of both patch and beam as 1 cm. The material properties are considered as isotropic and are given in Table 1. mπb mπb The patch is constrained with the beam by coupling the degrees V1 = Foh2e13ω cos − cos l l of freedom for both. The constraint applied at both of the two pπd pπc ends has a displacement of 0 and the moment is 0, which satis- · cos − cos ; l l ﬁes the condition for simply supported beam. The mesh size is 200 elements over the whole length of the beam, i.e., 1 element 2 1 2 V = [K] − ω [M] − ω [C] Cp; 2 R per mm of beam length. 2 ω [C] V = [K] − ω [M] ωCp + 3 R 4.2. COMSOL smπ mπb mπb The beam is modeled in 3-D in COMSOL 4.3. The modal − [θ] h2e13ω cos − cos . l l l analysis is done by choosing the solid mechanics physics and

190 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. Rajora, et al.: ENERGY HARVESTING ESTIMATION FROM THE VIBRATION OF A SIMPLY SUPPORTED BEAM

Table 1. Comparison of Energy Density.4 Type Energy Density (mJ/cm3) Assumption Piezoelectric 35.4 PZT 5 H Electromagnetic 24.8 0.25 T Electrostatic 4 3x107 V m-1

Table 2. The material properties. Parameter Host Beam Patch (PZT) Density(kg/m3) 2800 7800 Modulus of elasticity(N/m2) 68.3e9 12e10 Poisson’s ratio 0.3 0.2 Piezoelectric constant e13 =e23(C/m2) - -5.2 Permittivity(p) - 1.5e-8

Figure 4. Voltage across the PZT patch. eigen frequency as the study. The constraint is applied with Rigid connectors at both the end faces to make the beam sim- The PZT patch with thickness 0.2 mm and length 20 mm ply supported with the rotation constrained in two directions is placed 90 mm from the support in a 200 mm long simply supported beam. From the analytical model, the potential dif- Rx = 0 and Rz = 0 and all the three displacement components as 0. The mesh is generated using free tetrahedral el- ference across the beam is found to be 14.248 V while from ements and the setting size is extra fine. The solution is then COMSOL model, it is found to be 14.911 V (see Fig. 4). Sim- computed for getting natural frequency, as tabulated in Table 3. ilarly, the voltage across the load resistance of 100 kΩ is found To carry out power calculations, transient analysis is done analytically to be 7.1072 V; while using COMSOL, it is found with COMSOL physics as piezoelectric devices and study as to be 7.438 V. The slight difference is due to the reason that time-dependent. A boundary load, which is harmonic in na- there would be some damping due to dissipation of heat across ture (= 2500sin(wt) Pa), is applied on the top of piezoelectric the load, which has not been taken into account in COMSOL patch and power across the top and bottom of the patch is com- model. puted when the load resistance is 1000 Ω. Again, the mesh is created using free tetrahedral elements with an extra fine size. 5.1. Parametric Study The results from COMSOL are compared with results of our This section deals with a parametric study of several vari- analytical model and are discussed in next section. ables in order to determine how the natural frequency and power are affected by the variables. After a certain thickness, 5. RESULTS AND DISCUSSION the natural frequency starts decreasing for all modes because of the dominance of the mass and the stiffness of the patch in Table 2 shows the material properties for the beam and the comparison to stiffness contribution by the host beam because patch. Results for various cases have been taken, which have of the electrical properties (see Fig. 5). As depicted from the been divided in two sections. Equations (15)a and (15)b are moment and force equations, electrical properties contribute solved through MATLAB program to get the natural frequency towards stiffness of the plate which results in increase in the of the beam with the bonded patch. The results obtained were frequency for all the modes (see Fig. 6). The natural frequency verified using ANSYS software package by modeling the beam increases as the length of the patch increases. The location of with the bonded patch. The results obtained were in close the patch also affects the natural frequency of the combination, agreement with the results of ANSYS hence validating the the- as shown in Fig. 7. The placement of piezoelectric patch at the oretical work. This is tabulated in Table 3. centre of the beam is the maximum displacement position for Equations 26 and 28 are solved through MATLAB program the first mode which in turn gives more contribution to stiff- to get the power extracted across the load resistance from the ness. patched beam. The results obtained were verified using COM- For energy harvesting, the PZT length and thickness, along SOL software package by modeling the patched beam. The with the beam thickness, has been studied. The position of the results obtained were in close agreement with the results of external forcing function will be optimized for maximum out- COMSOL, hence validating the theoretical work. put as well. To calculate the energy output one has to do free Table 3. Result validation. vibration analysis first in order to evaluate the fundamental frequencies and the results are used in the calculations of energy Mode Theoretical ANSYS (Hz) COMSOL (Hz) harvesting. Figures 8, 9, and 10 show the maximum energy of Frequency Frequency (Hz) Frequency (Hz) Frequency (Hz) harvested (Pmax) for patch of thickness 0.2 mm and 0.5 mm 1 56.821 56.75 56.95 respectively against the load resistance for various areas. This 2 224.028 223.99 229.96 means that the kinetic energy input due to external excitation 3 518.743 518.00 520.27 remains constant whereas the energy harvested varies with the 4 897.412 896.83 924.75 PZT patch length, patch thickness and with the resistance of 5 1450.408 1448.40 1462.38 energy harvesting circuit.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 191 A. Rajora, et al.: ENERGY HARVESTING ESTIMATION FROM THE VIBRATION OF A SIMPLY SUPPORTED BEAM

Figure 5. Effect of Patch Thickness on Natural Frequency (a=90mm and b- Figure 8. Power with Rl at different patch thicknesses. a=2mm).

Figure 6. Effect of Patch Length on the Natural Frequency (h2=0.2mm and Figure 9. Power with different locations of the excitation force at different a=90mm). patch thicknesses.

Figure 7. Effect of Patch Location on Natural Frequency (h2=0.2mm and b-a=20mm). Figure 10. Power with length of patch at different patch thicknesses.

192 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. Rajora, et al.: ENERGY HARVESTING ESTIMATION FROM THE VIBRATION OF A SIMPLY SUPPORTED BEAM

As the PZT length increases, it starts to affect the over- 2 Williams, C. B., and Yates, R. B., Analy- all characteristics of the beam system, changing the effective sis of a micro-electric generator for Microsys- cross-section, Young’s modulus, and natural frequencies. This tems, Stockholm, Sweden, (1995), 369–372. will have adverse effects, reducing the beam deflections, strain https://dx.doi.org/10.1109/sensor.1995.717207 experienced by the PZT, and overall power produced.10 3 Roundy, S., Wright, P. K., and Pister, K. S. J., The farther away the forcing function is from the beam’s Micro-electrostatic vibration-to-electricity converters, hinged end, the larger the moment applied to the beam through New Orleans, LA, United States, (2002), 487–496. force. So, it follows that the optimal location for the force is at https://dx.doi.org/10.1115/IMECE2002-39309 the centre of the beam, which creates the largest moment. 4 Roundy, S. and Wright, P.K., A piezoelectric vibra- 6. CONCLUSIONS tion based generator for wireless electronics, Smart Materials and Structures, 13, 1131–1142, (2004). The above study is carried out with the help of the gener- https://dx.doi.org/10.1088/0964-1726/13/5/018 alized Hooke’s Law and by considering electrical properties 5 of the PZT. Theoretical analysis went forward from Euler- Tyagi, A. and Ghosh, M. K., Modal analysis of transverse Bernoulli Beam Theory and moment equation. The results ob- vibrations of a thin rectangular plate with a bonded piezo- tained are verified with the help of MATLAB, ANSYS, and electric patch, Int. J. Vehicle Structures and Systems, 3 (4), COMSOL. Similar study can be carried out for the cantilever 234–240, (2011). https://dx.doi.org/10.4273/ijvss.3.4.04 beam, as it will generate more power output but in the cost of 6 Sodano H. A., Inman D. J., and Park G., A review of reduced safety factor. power harvesting from vibration using piezoelectric ma- The effect of electrical properties of the patch on the trans- terials, The Shock and Vibration Digest, 36(3), 197–205, verse vibration of a simply supported beam with PZT patch (2004). https://dx.doi.org/10.1177/0583102404043275 has been investigated for varying thickness and for different positions taken by the patch. The formulation and analytical 7 Sodano H. A., Park G., and Inman D. J., Es- determination of the coupling effects were demonstrated to be timation of electric charge output for piezoelec- related to modal frequencies. Electrical properties significantly tric energy harvesting, Strain, 40, 49–58, (2004). affected the response of transverse vibration frequencies. As https://dx.doi.org/10.1111/j.1475-1305.2004.00120.x the length of piezoelectric patch increased, an increase in the 8 Chen S. N., Wang G. J., and Chien M. C., Ana- modal frequency was achieved experimentally. lytical modeling of piezoelectric vibration-induced micro Similarly, the parametric study with various patch thickness, power generator, Mechatronics, 16, 379–387, (2006). patch length, load resistance and location of excitation force https://dx.doi.org/10.1016/j.mechatronics.2006.03.003 was done. It was found that the energy harvested varies with the PZT patch length, patch thickness and with the resistance 9 Priya, S., Advances in energy harvesting using low profile of energy harvesting circuit. piezoelectric transducers, J. Electroceram, 19, 165–182, Some of the unsolved problems with this project are as- (2007). https://dx.doi.org/10.1007/s10832-007-9043-4 sumption of isotropy in the analysis part and accumulation 10 of continuous power generated by PEG. Piezoelectric mate- Erturk A. and Inman D. J., A distributed param- rial is not isotropic, and there might be some numerical differ- eter electromechanical model for cantilevered piezo- ences if an anisotropic analysis is used. The stiffness differs electric energy harvesters, Journal of Vibration slightly in different directions, and the Poisson’s ratio differs and Acoustics, 130 / 041002, 1–15, (August 2008). by a very small amount. It is unlikely that this anisotropic https://dx.doi.org/10.1115/1.2890402 problem causes a major difference in the power generation 11 Xu, C., Liang, Z., Ren, B., Di, W., Luo, H., Wang, from the above analysis with an isotropic patch assumption, D., Wang, K., and Chen, Z. Bi-stable energy harvest- but there is an argument for carrying out the analysis. ing based on a simply supported piezoelectric buckled beam, Journal of Applied Physics, 114, 114507, (2013). REFERENCES https://dx.doi.org/10.1063/1.4821644 1 Roundy, S., Wright, P. K., and Rabaey, J. , A study of 12 Erturk A. and Inman D. J., An experimentally validated low level vibrations as a power source for wireless sensor bimorph cantilever model for piezoelectric energy har- nodes, Computer Communications, 26, 1131–1144, (2003). vesting from base excitations, Smart Mater. Struct., 18 https://dx.doi.org/10.1016/S0140-3664(02)00248-7 (025009), 1–18, (2009). https://dx.doi.org/10.1088/0964- 1726/18/2/025009

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 193 Preparation and Experimental Study of Magnetorheological Fluids for Vibration Control

Ying-Qing Guo Mechanical and Electronic Engineering College, Nanjing Forestry University, Nanjing 210037, China

Zhao-Dong Xu and Bing-Bing Chen Key Laboratory of C&PC Structures of the Ministry of Education, Southeast University, Nanjing, 210096, China

Cheng-Song Ran Sichuan Provincial Architectural Design and Research Institute, Chengdu 610017, China

Wei-Yang Guo Nanjing Dongrui Damping Control Technology Co, Ltd, Nanjing 210033, China

(Received 1 May 2015; accepted 19 April 2016) Magnetorheological (MR) fluids are a new kind of smart vibration mitigation material for vibration control, whose shear yield stress can change in magnetic field, and the change can occur only in a few milliseconds. The rheological properties, anti-settlement stability, and redispersibility are the very important properties of MR fluids. In this paper, a kind of preparation process of MR fluids is introduced and MR fluids with different grain diameter ratios of carbonyl iron particles are produced. Then, the properties of self-prepared MR fluids are tested, including the sedimentation stability test, viscosity test, and shear yield stress test. The results of the MR fluid property tests show that the adding of 10 µm-carbonyl iron particles will improve the magnetic effect of MR fluids, increase the zero magnetic field viscosity of MR fluids, and increase shear yield stress of MR fluids in same magnetic field, but the anti-settlement properties will be degraded.

1. INTRODUCTION the properties of MR fluids. In 2004, Ulicny and Mance6 studied the anti-oxidation property of MR fluids and proposed MR fluids are a kind of controllable fluid that were identified the method with plating ferromagnetic particles coated in a by the US National Bureau of Standards in 1948.1 MR fluids layer of nickel to improve the long-term oxidation resistance usually consist of micrometer-sized magnetic particles, a di- of MR fluids. In 2010, Jiang et al.7 coated the poly (methyl electric carrier fluid, and some additives. When they are placed methacrylate) (PMMA) to the surface of carbonyl iron (CI) in an adjustable magnetic field, their yield stress changes with particles getting the composite particle CI-PMMA. The pre- the magnetic field intensity changes in a few milliseconds. Be- pared MR fluids based on CI-PMMA particles had better sed- cause of this kind of smart feature of MR fluids is that they imentation stability. In 2010, Du et al.8 chose the surfactant have been concerned by more and more researchers. Further- by hydrophilic-lipophilic balance (HLB) parameters and dis- more, MR fluids can be designed as MR dampers to reduce covered that the surfactant could hardly change the magnetic the different vibration. And MR fluids have been successfully properties of magnetic particles and rheological properties of used in intelligent vibration control, such as the building struc- MR fluids, while increasing the sedimentation stability of MR tures, the bridge structures, the automobile suspension system, fluids significantly. In 2012, Iglesias et al.9 tested the influence the prosthetic limb, some military equipment or magnetorheo- of the volume fraction of nanoscale magnetic particles on the logical fishing. A large-scale and 20 ton MR fluid damper has sedimentation stability. Powell et al.10 synthesized MR fluids been designed and built to reduce vibration for civil engineer- by replacing a part of the magnetic particles with nonmagnetic ing applications.2 Carlson3 used the MR damper to develop microscale glass bead, and the sedimentation rate was reduced an artificial limb. Lord Corporation4 used MR dampers for the by about 4%. In 2011, Lee et al.11 applied tribological charac- vibration control of armored vehicles. Xu et al.5 used the MR teristic to improve the stability and performance of MR fluids damper for mitigating earthquake responses of building struc- by mixing and adding the additives. In 2013, Liu et al.12 pro- ture under the neural networks control strategy. posed a preparation method of silicone oil-based MR fluids and The properties of MR fluids will directly influence the vi- tested the properties of the prepared MR fluids. However, due bration absorption performance of MR dampers, so many re- to the density mismatch between the ferromagnetic particles searchers have studied how to prepare MR fluids to improve and the carrier liquids, the particle sedimentation caused by

194 https://doi.org/10.20855/ijav.2017.22.2465 (pp. 194–200) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-Q. Guo, et al.: PREPARATION AND EXPERIMENTAL STUDY OF MAGNETORHEOLOGICAL FLUIDS FOR VIBRATION CONTROL gravitational forces came to be a serious drawback in the use of MR fluids. Sutrisno et al.13 prepared a high viscosity MR fluids based on grafted poly-iron particles and the fluids have an excellent thermo-oxidative stability with a nearly constant viscosity. Pramudya et al.14 developed novel MR fluids using modified silicone oil, composite polyurethane microsphere additives, and surface-coated iron particles; the addition of composite polyurethane microsphere increased the strength of the fluids, providing well dispersion stability. In 2014, Susan- Resiga et al.15 studied the influence of composition on the yield stress and flow behavior of concentrated ferrofluid-based MR fluids and micrometer-sized iron particles to a concentrated ferrofluid without any supplementary stabilizing agent. The results showed that a direct and simple way to control the magnetorheological and magnetoviscous behavior, as well as the saturation magnetization of the resulting nano-micro composite fluid to fulfill the requirements of their use in various MR control and rotating seal devices. Shah et al.16 prepared a new kind of low sedimentation MR fluid based on plate-like iron particles and its salient properties were evaluated using a small-sized damper. The results showed that plate-like micron size iron particles played an important role in improving stability against rapid sedimentation as well as in enhancing the Figure 1. Preparation process of MR fluids. value of the yield stress. 2.1.2. Instruments In this paper, a kind of preparation process of MR fluids is introduced and MR fluids with different grain diameter ratio The adjustable mixer, the vacuum drying oven, the mortar, of carbonyl iron particles are produced. Then, the properties the standard sieve (100 mesh), the planetary ball mill, the ul- of self-prepared MR fluids are tested, including sedimentation trasonic dispersion instrument and a high precision electronic stability test, viscosity test, and shear yield stress test. The balance are used to prepare MR fluids. results of property tests of MR fluids show that the adding of 10 µm-carbonyl iron particles will improve the magnetic effect 2.2. The Preparation Process of the MR of MR fluids, increase the zero magnetic field viscosity of MR Fluid fluids, and increase the shear yield stress of MR fluids in same The preparation process is an important factor that affects magnetic field. However, the anti-settlement properties will be the performance of MR fluids. As shown in Fig. 1, the prepa- degraded. ration process can be described in detail as follows:

1. The surface treatment of CI particles: Firstly, put CI par- 2. PREPARATION OF THE MR FLUID ticles in a beaker of anhydrous ethanol, then add surfactant. Secondly, put the beaker in the ultrasonic dispersion 2.1. Materials and instruments instrument to make the mixture evenly. Then, place the mixture in vacuum drying oven to keep low temperature The materials and instruments for preparation of MR ﬂuids drying. Lastly, crush the obtained dry CI particles in a are as follows: mortar and then sift the particles through a standard sieve (100 mesh).

2.1.1. Materials 2. Preparation of mineral oil-based fluid: Take mineral oil into a beaker, add the dispersant, thixotropic agent, and Materials for preparation of MR fluids include 3 µm- solid lubricant in order while stirring at room temperature carbonyl iron (CI) particles (produced by Tianyi Co., Ltd. for 2 hours. Jiangsu, China; average grain diameter: 3 µm; density: 7.85 g/cm3), 10 µm-CI particles (produced by Tianyi Co., 3. Synthesis of MR fluids: Put the 3 µm-CI particles and the Ltd. Jiangsu, China; average grain diameter: 10 µm; den- 10 µm-CI particles after surface treatment into the ob- sity: 7.85 g/cm3), mineral oil, organobentonite, polyethylene tained mineral oil-based fluid and stir well. According to glycol, graphite, polyvinylpyrrolidone, sodium dodecyl ben- the above mentioned process of MR fluids, four samples zene sulfonate and oleic acid. Note that all the materials were numbered A, B, C and D were prepared. The mass frac- adopted directly. tion of the total CI particles in each sample was 80% and

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 195 Y.-Q. Guo, et al.: PREPARATION AND EXPERIMENTAL STUDY OF MAGNETORHEOLOGICAL FLUIDS FOR VIBRATION CONTROL

Table 1. The samples formula. The mass percentage Materials Sample A Sample B Sample C Sample D CI 3 µm 72.0 74.0 76.0 78.0 particles 10 µm 8.0 6.0 4.0 2.0 The ratio of 3-µm CI particles to 9:1 37:3 19:1 39:1 10-µm CI particles Mineral oil 16.5 16.5 16.5 16.5 Thixotropic agent 0.6 0.6 0.6 0.6 Antioxidant 0.4 0.4 0.4 0.4 Solid lubricant 0.5 0.5 0.5 0.5 Dispersing agent 1.0 1.0 1.0 1.0 Surface active agent 1.0 1.0 1.0 1.0

the ratio of the 3 µm-CI particles to the 10 µm-CI particles are 9:1, 37:3, 19:1 and 39:1, respectively. That is, (a) The sedimentation observation. the 10 µm-CI particles in the proportion of the total CI particles is 10%, 7.5%, 5.0% and 2.5%, respectively. As shown in Table 1, the type and amount of other additives were exactly the same.

3. PROPERTY TEST ON MR FLUIDS

The sedimentation stability, rheological properties, and high shear yield stress are three main indexes to evaluate the performance of MR ﬂuids. Thus, the prepared four samples of MR ﬂuids in this study were tested to evaluate these properties.

3.1. Sedimentation stability test The sedimentation stability of the MR fluids can be represented by the sedimentation rate, which can be defined as the (b) The sedimentation rate. ratio of the supernatant fluid volume to the mixture volume within a fixed period. The sedimentation ratio is defined as: Figure 2. Sedimentation stability test of MR fluids.

Sedimentation ratio (%) = the grain diameter of the CI particles. That is, the larger the volume of the supernatant liquid ratio of 3 µm-CI particles, the slower the settlement. × 100%. (1) volume of the entire mixture liquid 3.2. Viscosity Test In this paper, the four samples of MR fluids were placed in the four 10-ml graduated glass tubes marked in 0.2-ml, respec- In the absence of an external magnetic field, the viscosity of tively, as shown in Fig 2a. The volume of the supernatant liq- MR fluids mainly demonstrates the flow characteristic of MR uid was obtained by observing the phase boundary between fluids under different shear rates, and MR fluids can be thought the supernatant liquid and the concentrated suspension until as Newtonian fluids. An NDJ-1 rotational viscometer pro- this volume reached an asymptotic value. Figure 2b shows the duced by Shanghai Jingchun Instrument Equipment Co., Ltd. sedimentation stability of the MR fluids samples with different was adopted to perform the viscosity tests, as shown Fig. 3a. amount of 10 µm-CI particles. It can be seen from the figure All the samples were carried out the viscosity test under differ- that the sedimentation rates of all the samples were relatively ent circular shear rates, 6.28 rad/s, 12.57 rad/s, 31.42 rad/s, and large during the first three days and then the rate slowed and 62.83 rad/s. The circular shear rate is produced when the ro- tended to be constant after ten days. At the same time, it can be tational viscometer rotates to measure the zero magnetic field seen that the initial sedimentation rate of the sample D is much viscosity of MR fluids. less than that of the sample C, B, and A. The average final Figure 3b shows the viscosity of the samples under the dif- sedimentation rate of the sample A, B, C, and D were 18.2%, ferent shear rates with the zero magnetic field. The zero mag- 14.9%, 14.1%, and 9.8%, respectively. And the average final netic field viscosities of samples A, B, C, and D are 5.20 Pa·s, sedimentation rate of the sample D was 8.4%, 5.1%, 4.3% less 4.74 Pa·s, 4.52 Pa·s, and 4.2 Pa·s, respectively with the shear than that of the sample A, B, C, respectively. The results show rate of 6.28 rad/s. It shows that with the increase of the propor- that the sedimentation stability of the MR fluids is relevant to tion of 10 µm-CI particles, the zero magnetic field viscosity

196 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-Q. Guo, et al.: PREPARATION AND EXPERIMENTAL STUDY OF MAGNETORHEOLOGICAL FLUIDS FOR VIBRATION CONTROL

Figure 4. MR ﬂuids.

(a) Viscosity test.

Figure 5. Self-developed shear yield stress test device.

shown in Fig. 5) was used to measure shear yield stress of the sample A, B, C, and D under the different magnetic induction strengths and shear velocities. The shear velocity was produced when the self-developed shear stress device moved rec- tilinearly to measure the shear yield stress of MR fluids. The (b) The zero magnetic field viscosity of MR fluids under shear velocities are: 3.65 m/s, 5.48 m/s, 7.31 m/s, 9.14 m/s, different shear rates. and 10.97 m/s. At the same time, the magnetic induction intensity are: 10 mT, 20 mT, 25 mT, 65 mT, and 115 mT. Fig- Figure 3. Viscosity test of MR fluids under different shear rates. ure 6 shows the shear yield stress of the MR fluids samples of MR fluids increases. At the same time, with an increase in as a function of external magnetic induction intensity under the shear rate, the zero magnetic field viscosity of MR fluids five different shear velocities, 3.65 m/s, 5.48 m/s, 7.31 m/s, decreased gradually. This phenomenon conforms to the theory 9.14 m/s, and 10.97 m/s, respectively. It can be seen from of shear-thinning. Fig. 6 that the shear yield stress of the MR fluid samples A, B, C, and D increase with an improvement in magnetic induction 3.3. Shear Yield Stress Test intensity. Additionally, the shear yield stress value of sample A increases to 80 kPa when the magnetic induction intensity is In an external magnetic field, the CI particles will line up 115 mT, which shows that the MR fluids has a high shear yield to form a chain-like microstructure, restrict the motion of the stress. Meanwhile, the MR fluids exhibit a typical monotonic fluid, and increase in their ability to support shear stress, as improvement in shear yield stress with an increase of magnetic shown in Fig. 4. The shear yield stress of MR fluids will induction intensity as well as Bingham plastic behavior char- change with the change of the different magnetic fields and acterized by a field dependent yield stress. At the same time, it shear rates. The shear stress, τcan be described as Bingham also can be seen that the shear yield stress of sample A and D 17 model, is respectively the biggest and minimum value when magnetic τ = τy(H) sgn(γ ˙ ) + ηγ˙ ; (2) induction intensity and shear rate are given. It demonstrates where γ˙ is the shear rate, η is the field-independent plastic vis- that the shear yield stress of the MR fluids improve with the cosity defined as the slope of the measured shear stress versus increase of the amount of 10 µm-CI particles. shear rate, τy is the shear yield stress, and H is the magnetic Figure 7 shows the change in shear yield stress of the MR induction intensity. The value of the shear yield stress will em- fluids samples as a function of shear rate under five different body the damping of the damper with MR fluids. magnetic induction intensity, 10 mT, 20 mT, 25 mT, 65 mT, In this paper, a self-developed shear stress test device (as and 115 mT, respectively. The shear yield stress under the

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 197 Y.-Q. Guo, et al.: PREPARATION AND EXPERIMENTAL STUDY OF MAGNETORHEOLOGICAL FLUIDS FOR VIBRATION CONTROL

(a) 3.65 m/s (b) 5.48 m/s

(e) 10.97 m/s

Figure 6. Shear yield stress under different magnetic fields and shear rates. magnetic induction intensity ranging from 10 mT to 115 mT 4. CONCLUSIONS fluctuates at a limited amplitude with an increase of shear ve- In this study, MR fluids were prepared with the appropriate locity, which indicates that the shear velocity has little impact components and process. The 10 µm-CI particles were used to on the shear yield stress. The result implies that the chain study the effect of the size on the MR fluids performance. To structure of the MR fluids constructed within fluids under the investigate the performance of the MR fluids and determine the applied magnetics induction intensity are continuously broken effect of 10 µm-CI particles, some tests, including sedimenta- and then rapidly reformed, so that MR fluids are relatively in- tion stability, rheological properties, and high shear yield stress sensitive to shear velocity. were carried out. The following conclusions can be drawn.

198 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Y.-Q. Guo, et al.: PREPARATION AND EXPERIMENTAL STUDY OF MAGNETORHEOLOGICAL FLUIDS FOR VIBRATION CONTROL

(a) Sample A (b) Sample B

Figure 7. Shear yield stress versus shear rates under different magnetic induction intensity.

1. The settlement results show that the sedimentation stabil- Jiangsu Province Outstanding Youth Natural Science Founda- ity of MR fluids are relevant to the grain diameter of the tion (BK20140025), Jiangsu Province Natural Science Foun- CI particles. The smaller the particle size, the slower the dation (BK20141086), Scientific Research Foundation for Re- settlement. turned Scholars of the Ministry of Education of China, and Jiangsu Province Qing Lan Project. The support is gratefully 2. The rheological properties tests indicate that the addition acknowledged. of 10 µm-CI particles can lead to an increase in zero magnetic field viscosity of MR fluids, which shows the grain REFERENCES diameter of CI particles has much effect on rheological 1 properties of MR fluids. Rabinow, J. The magnetic fluid clutch, Elec- trical Engineering, 67 (12), 1167–1167, (1948). 3. The shear yield stress of MR fluids improve with the in- https://dx.doi.org/10.1109/ee.1948.6444497 crease of the amount of 10 µm-CI particles, which proves 2 Yang, G., Spencer, B. F., and Jung H. Dy- that the 10 µm-CI particles can improve the magnetic ef- namic modeling of large-scale magnetorheological damper fect of MR fluids. At the same time, the shear velocity has systems for civil engineering applications, Jour- little impact on the shear yield stress under the different nal Engineering Mechanics ASCE, 130 (9), 1107– magnetic induction strengths. 1114, (2004). https://dx.doi.org/10.1061/(asce)0733- 9399(2004)130:9(1107) ACKNOWLEDGEMENTS 3 Carlson, J. D., Matthis, W., and Toscano, J. R. This study was financially supported by National Sci- Smart prosthetics based on magnetorheological fluence Fund for Distinguished Young Scholars (51625803), ids, SPIE 8th Annual Symposium on Smart Struc-

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 199 Y.-Q. Guo, et al.: PREPARATION AND EXPERIMENTAL STUDY OF MAGNETORHEOLOGICAL FLUIDS FOR VIBRATION CONTROL

tures and Materials, Newport Beach, CA, (2001). 11 Lee, D. W., Choi, J. Y., Cho, M. W., Lee, C. H., https://dx.doi.org/10.1117/12.429670 Cho, W. O., and Yun, H. C. Tribological characteristics in modified magneto-rheological fluid, Ap- 4 Lord Corporation. MR suspension technol- plied Mechanics and Materials, 110, 225–231, (2011). ogy, http://mutualhosting.com/ lordfulfillment/ up- https://dx.doi.org/10.4028/www.scientific.net/amm.110- load/PB7138.pdf, (2009). 116.225 5 Xu, Z. D., Shen, Y.P., and Guo, Y.Q. Semi-active control of 12 Liu, X. H., Lu, H., Chen, Q. Q., Wang, D. D., and structures incorporated with magnetorheological dampers Zhen, X. J. Study on the preparation and properties of using neural networks, Smart Materials and Structures, silicone oil-based magnetorheological fluids, Materials 12 (1), 80–87, (2003). https://dx.doi.org/10.1088/0964- and Manufacturing Processes, 28 (6), 631–636, (2013). 1726/12/1/309 https://dx.doi.org/10.1080/10426914.2013.773017 6 Ulicny, J. C. and Mance, A. M. Evaluation of electroless 13 Sutrisno, J., Fuchs, A., Sahin, H., and Gordanine- nickel surface treatment for iron powder used in MR fluids, jad, F. Surface coated iron particles via atom trans- Materials Science and Engineering: A, 369 (1-2), 309–313, fer radical polymerization for thermal-oxidatively sta- (2004). https://dx.doi.org/10.1016/j.msea.2003.11.039 ble high viscosity magnetorheological fluid, Journal 7 Jiang, W. Q., Zhu, H., Guo, C. Y., Li, J. F., Xue, Q., Feng, of Applied Polymer Science, 128 (1), 470–480, (2013). J. H., and Gong X. L. Poly (methyl methacrylate)coated https://dx.doi.org/10.1002/app.38199 carbonyl iron particles and their magnetorheological char- 14 Pramudya, I., Sutrisno, J., Fuchs, A., Kavlicoglu, B., Sahin, acteristics, Polymer International, 59 (7), 879–883, (2010). H., and Gordaninejad, F. Compressible magnetorheologi- https://dx.doi.org/10.1002/pi.2794 cal fluids based on composite polyurethane microspheres, 8 Du, C. B., Chen, W. Q., and Wan, F. X. Influ- Macromolecular Materials and Engineering, 298 (8), 888– ence of HLB parameters of surfactants on prop- 895, (2013). https://dx.doi.org/10.1002/mame.201200156 erties of magneto-rheological fluid, Advanced 15 Susan-Resiga, D. and Vks, L. Yield stress and flow be- Materials Research, 97–101, 843–847, (2010). havior of concentrated ferrofluid-based magnetorheological https://dx.doi.org/10.4028/www.scientific.net/amr.97- fluids: The influence of composition, Rheologica Acta, 53 101.843 (8), 645–653, (2014). https://dx.doi.org/10.1007/s00397- 9 Iglesias, G. R., Lopez-Lopez, M. T., Duran, J. 014-0785-z D. G., Gonzalez-Caballero, F., and Delgado, A. 16 Shah, K., Seong, M. S., Upadhyay, R. V., and Choi, S. B. A V. Dynamic characterization of extremely bidis- low sedimentation magnetorheological fluid based on plate- perse magnetorheological fluids, Journal of Col- like iron particles, and verification using a damper test, loid and Interface Science, 377 (1), 153–159, (2012). Smart Materials and Structures, 23 (2), 027001, (2014). https://dx.doi.org/10.1016/j.jcis.2012.03.077 https://dx.doi.org/10.1088/0964-1726/23/2/027001 10 Powell, L. A., Wereley, N. M., and Ulicny, J. Mag- 17 Philips, R. W. Engineering applications of fluids with a netorheological fluids employing substitution of nonmag- variable yield stress. California: University of California, netic for magnetic particles to increase yield stress, IEEE Berkeley, (1969). Transactions on Magnetics, 48 (11), 3764–3767, (2012). https://dx.doi.org/10.1109/tmag.2012.2202885

200 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A Subjective Related Measure of Airborne Sound Insulation Reinhard O. Neubauer and Jian Kang School of Architecture, University of Shefﬁeld, Western Bank, Shefﬁeld S10 2TN, UK

(Received 1 May 2015; accepted 19 April 2016) In many practical cases, the objective measures of airborne sound insulation using standard procedures do not agree with subjective assessments. This paper describes a calculation scheme based on the loudness level linked to the specific fluctuation strength and yields a weighted normalized loudness level difference. Evidence has been presented through a subjective evaluation that the model can be considered to be a link between an objective and subjective evaluation. The stimuli offered in the experiment were electronically filtered sound samples representing the sound insulation of interest. Steady-state and non-steady state signals are used as stimuli. To differentiate the signal in terms of psychoacoustic measures, investigations of music type signals were focused on specific fluctuation strength. An assessment of identical airborne sound insulation experimental results has shown that steady-state signals were assessed to be significantly quieter than non-steady-state signals, which also yield greater specific fluctuation strength. As expected, sound insulation was judged differently for different sound samples. A simple level difference is shown not to exhibit the effects of a given signal to the frequency-dependent airborne sound insulation curve. This study supports findings in the literature that airborne sound insulation performance is significantly dependent on what type of sound signal is used.

1. INTRODUCTION crepancies among sound insulation and the weighted normalized loudness level difference. The quality of sound insulation in buildings is generally described using a single number rating of the sound insulation and has an important bearing on the comfort, health, and 2. EXPERIMENT ON THE SUBJECTIVE general amenity of the residents.1–3 In many cases, however, single-number ratings do not correlate well with subjective ex- EVALUATION ON SOUND SIGNALS pectations.4 Comparing single number quantities of airborne To determine the subjective assessment of different test sig- sound insulation with subjectively estimated airborne sound nals for different sound insulation values, hearing tests were insulation frequently yields significant differences.4, 5 In the conducted. The main goal for this investigation is to find evi- literature, complaints have even been made regarding parti- dence that the perceived sound is judged differently if the sig- tions fulfilling specific requirements with respect to airborne nal is changed and if the spectrum of the airborne sound insu- sound insulation.6–9 Therefore, it is necessary to establish a lation is different. It is, therefore, vital to know how the model better understanding of airborne sound insulation through the depicts differences in sound signals and spectra and how the use of psychoacoustics. Neubauer and Kang introduced a con- differences are related to subjective assessments. cept describing a frequency-dependent weighted normalized loudness level difference.10–12 This concept is intended to be a connection between the objective measure of airborne sound 2.1. Excitation Signals insulation and the psychoacoustic measures of the loudness level and fluctuation strength. From the literature,13–15 it is known that music is reported as This study investigates the previously introduced calculation one of the most frequently detected noises even in dwellings scheme describing the airborne sound insulation in terms of a that fulfil sound insulation requirements. Therefore, the in- probability of the insulation’s “best fit.” This means that if fluence of using different signals is investigated by using a certain airborne sound insulation is compared with its “un- two categories of signals: steady-state and non-steady-state biased” airborne sound insulation, as e.g. calculated, the air- signals. The steady-state signals are broadband noise sig- borne sound insulation of the real construction is biased by nals: “pink noise” (PN) and “white noise” (WN). These sig- means of resonances, leakages or other effects, which can in- nals are selected because they are recommended in standards fluence the airborne sound insulation. for measuring airborne sound insulation. The non-steady- This paper first describes the inapplicability of standards state signals, i.e., the transient signals, were music samples, to rate airborne sound insulation in terms of a subjective as- namely rap (Eminem: “Lose Yourself”) (E) and classic mu- sessment, and second discusses the new model of a weighted sic (Beethoven’s Symphony No. 9: Poco Allegro, Stringendo normalized loudness level difference. Finally, it validates the Il Tempo, Sempre Piu Allegro-Prestissimo) (B). This type of model by comparing subjective test results and clarifies the dis- music was also investigated earlier.16–18

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 201–210) https://doi.org/10.20855/ijav.2017.22.2465 201 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION

Figure 2. Results and boxplot of the response distribution for the data samples Figure 1. Hypothetical sound insulation over frequency used as the respective of white noise, pink noise, Beethoven, and Eminem. filter. 2.2. Perception of Damped Sound Signals A pilot test was conducted in order to find an indication whether a sound signal will be differently judged if the damping and the sound signals used as a source signal is different. Nine untrained participants, five females and four males were asked to listen to some sound samples via headphone (Sennheiser HD 280 pro) and judge the sound by answering pre-coded questions. The headphone was closed-back ensur- ing a 32 dB attenuation of external noise. The ear coupling was circumaural and its frequency response is 8 Hz – 25,000 Hz. The acoustic stimuli were played in different sequences for the participants to decrease the order effects. The background noise level was less than 25 dB(A) . The participants had normal hearing abilities and the median of age was 34. The stim- Figure 3. Mean of the grouped and overall grouped response distribution. uli used were the electronically filtered sound samples of the sound signal described in section 2.1 using a filter function was judged “louder” (from data in Fig. 2: 5.4 ± 0.7) than representing the sound insulation of interest. Beethoven (from data in Fig. 2: 5.3 ± 0.7). Overall, the mu- The insulation curves did not differ in their shape. In Fig. 1, sic group (B, E) was judged as: can hear / can clearly hear, the applied filter functions are depicted. The participants were while the noise group (WN, PN) was judged as: can hear when asked to select one of the following answers: 0 - I do not hear a concentrate on it / can hear. Both groups differ in judgment of sound; 1 - I can hear a weak sound; 2 - I hardly hear a sound; the subject by one category, which means noise sound samples 3 - Yes, I can hear a sound but not easily; 4 - Yes, I can hear are judged not as loud as music sound samples. From Fig. 2, a sound when I concentrate on it; 5 - Yes, I can hear a sound; the pink noise was observed to be judged louder than the white and 6 - Yes I can clearly hear a sound. noise, and “Eminem” was judged to be heard slightly clearer The source signals, white and pink noise and Eminem and than “Beethoven”. Beethoven as described above, were used. Due to the small As shown in Fig. 3, the overall grouped response distribution sample (n = 9), the non-parametric Wilcoxon test, i.e., the of the two different sound samples was assessed differently. Wilcoxon signed-rank test, was applied instead of the most The results of this experiment demonstrate that different commonly applied t-test. In contrast to the t-test, the Wilcoxon sound samples are judged differently with respect to loudness. test does not require the data set to be normally distributed. A To summarize the results, music was found to be judged as be- summary of the results is shown in Fig. 2 where the boxplot ing heard clearer than a noise type sound signal. This finding of response distribution for the data samples is shown. The was observed for the increasing Rw-values. At a low airborne sound samples were observed to be generally judged similar. sound insulation of approximately 20 and 30 dB, little differ- However, as shown in Fig. 2, where the calculated mean for ence was observed. For this experiment, it was concluded that different Rw-values is shown, the music signal type was gener- broadband noise was not as “audible” as music if the damp- ally judged to be heard clearer than the broadband noise signal ing increases beyond 40 dB. A comparison of the music type type. sound sources revealed that Eminem was judged more audi- From Fig. 2, it is seen that white noise has lowest me- ble at high sound insulations compared with the Beethoven. dian whereas pink noise and the music type signals have the Although the data sample is small in this experiment, certain same. From comparison it is seen that in summery Eminem conclusions can still be drawn. This is in line with results pub-

202 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION

Figure 5. Results of the second experiment displayed as a boxplot of the response distribution data comparing all the results. Figure 4. Power spectral density (PSD) as a function of frequency in the range 21, 22 of 20 Hz to 20 kHz. Depict is Beethoven (B), Eminem (E), pink noise (PN), the literature (Soeta et al.; Jeon et al.). white noise (WN), and party sound (PS), all having sound pressure level of 60 The obtained data was compared using the t-test for related dB, and duration 5 s. samples and the Wilcoxon test to determine whether the mean lished by Zwicker and Feldkeller,19 Takeshima et al.20 and differences between the sounds (if the sounds were quieter or Soeta et al.,21 where subjective experiments were conducted louder) were significant. The analyses demonstrated that with with small numbers of participants. It is thought that music is regard to the significant (p < 0.05) differences between the heard stronger compared with broadband noise; this result is sounds, the white noise (WN) was judged highly significantly consistent with everyday experiences. (p < 0.001) “quieter” compared with the other sound samples. A summary of the results is shown in Fig. 5 where the boxplot 2.3. Perception of Different Sound Signals of the response distribution for the data samples is shown. The data from Fig. 5 indicates that, as the value decreases, Following the pilot study performed in the first test, the aim the particular sound will be perceived as quieter by all partici- of this second listening test was to find evidence that the type pants. White noise was judged overall to be quietest. Eminem of sound signal heard is judged different in loudness depending was the loudest. This confirmed the results of the first pilot on the type of signal. For this test, 100 untrained participants, survey shown in Fig. 2. It is noted that a differentiation be- 8 females, and 92 males were asked to listen to sound sam- tween gender has not been investigated which is in line with ples via loudspeakers and to judge the loudness. The partici- literature.21–24 pants had normal hearing abilities and the median of age was 46. This different presentation method of the acoustical stimuli 2.4. Detecting Different Airborne Sound was needed because the test was conducted for all participants simultaneously. The acoustic stimuli were played back in dif- Insulations ferent sequences for the participants to reduce the order effects. The goal of the third listening test was to find evidence that The experiment involved 5 different sounds: WN, PN, E, B, the perceived sound level after transmission differs with dif- and PS with three different sound levels (i.e. 40, 50, and 60 dB ferent frequencies depending on the airborne sound insulation SPL) and was designed such that every sound was compared that has the same single Rw-value. The equipment used and against all others. In Fig. 4 an example of the power spec- the procedure of this test was the same as that for the first test, tral density over frequency is shown for sound signals having although the number of participants was 11, there were 3 fe- sound pressure level of 60 dB. males and 8 males. The participants had normal hearing and The participants were presented with two sound signals one the median age was 42. Two types of airborne sound insula- after the other with both signals having the same sound pres- tion, i.e., the filter types had to be judged and were labelled “I sure level. The duration of one sound sample was 5 s. Each and II.” The single values of the filters are I: Rw(C; Ctr) = sound pair was played in a row, and the participant was asked 50(−1; −1)dB and II: Rw(C; Ctr) = 50(−3; −7), respec- to decide whether the latter sound was louder or quieter than tively (the frequency depending airborne sound insulation is the former and was asked to rate the sound from -5 to +5, where depicted in the lower panel of Fig. 13). These filter functions -5 indicates “much quieter,” and +5 “much louder.” Zero indi- were calculated according to ISO 717-1.25 cates on the other hand “equally loud”. In the experiment, all 5 All source signals had sound pressure levels of 85 dB and sounds were joined in 12 pair comparisons (such as WN: WN had a duration of 15 seconds. The participants were asked to vs. PN, WN vs. E, WN vs. B and WN vs. PS; at 40 dB, 50 dB listen to the sound and to rank the sounds from the quietest to 60 dB). The evaluation in a pair comparison is in line with to the loudest. The subjects could listen to the sound samples

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 203 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION

not yet related to a subjective assessment. Therefore, the new model is discussed to demonstrate the discrepancy between the standard measure of a weighted sound reduction index (Rw) and a subjective observation of affected persons, such as an in- habitant in a dwelling for example. The new model proposed by Neubauer and Kang11 aims to integrate both discrepancies. This section ﬁrst discusses the new measure of a weighted normalized loudness level difference.

3.1. The Normalized Level Difference When measuring a transmission loss, it is common to measure a sound pressure level difference. The frequency depending sound pressure level (L) is then transformed into a frequency depending loudness level by following the procedure outlined in ISO 226.23

Figure 6. Results of the third experiment displayed as a boxplot of the re- L(f) → LN (f). (1) sponse distribution for the data samples, Case I: Rw = 50(−1; −1) dB and Case II: Rw = 50(−3; −7) dB. The level difference characterised by the weighted sound reduction index (Rw) without a dip (L0) and with a dip (Lm) as often as they wanted. The participants were asked to se- provides a set of loudness level differences. The level differ- lect one of the following answers: 0 - quietest; 1 - quiet; 2 ence of the idealized (i.e. hypothetical or computed) airborne - equal; 3 - loud; and 4 - loudest. All sounds per filter were sound insulation for third-octave bands is given by Eq. (2): combined and averaged for the data evaluation. This was done because the analysis concerns the comparison of the two filters ∆L0(f) = LN1(f) − LN2(f),0 . (2) and not the comparison of the sound signal. From the data collected from the listening tests, an evaluation was performed. The level LN1 is the loudness level in the source room and The data collected was summarized and described quantita- LN2 is the loudness level in the receiving room. The idealised tively in Fig. 6. Considering the sample size (n = 11), the airborne sound insulation to obtain (LN2,0) may be found us- 27 non-parametric Wilcoxon test was applied as opposed to the ing a prediction model, as provided by e.g., EN 12354, or t-test. It is seen in Fig. 6 that case “II” was judged to be quieter by assuming a reference curve e.g., ISO 717. The level differ- than case “I.” Case “II” is statistically significantly (p < 0.05) ence of an actual (i.e., measured or simulated) airborne sound different from filter “I.” This result is a strong indication that insulation for third-octave bands is given by Eq. (3): the airborne sound insulation is judged differently (i.e., subjectively) depending on the frequency dependent Rw-value. Com- ∆Lm(f) = LN1(f) − LN2(f),m; (3) paring the spectra of the respective sound insulation and the where LN2,m is the loudness level in the receiving room ob- resulting C-values indicates that case “II” has little sound in- tained by the measured or simulated sound pressure level. sulation at low frequencies and a dip at 800 Hz, whereas case The normalized level difference with respect to the idealized ‘I” has high sound insulation at low frequencies and a dip at level difference for the third-octave band values is then the fol- 2.5 kHz. This could be an indication that low frequencies do lowing: not contribute significantly to the subjective assessment of a 26 sound insulation. This finding is in line with that of Hongisto ∆Lm(f) Lnor(f) = . (4) and is worthy of further investigations in the field of subjective ∆L0(f) assessment tests. A method for determining a single value of a sound in terms of a loudness level is given in ISO 532 B28 and in DIN 45631.29 3. MODEL BASED ON WEIGHTED NORMAL- The loudness level can be obtained for any sound.30 The single IZED LOUDNESS LEVEL DIFFERENCE number quantity for the normalized loudness level difference (Lnor) is written as the quotient of the differences of the total From the section above, it is observed that a transmitted loudness levels (LN ), which yields the following: sound is judged differently given different types of signals and LN1 − LN2,m different sound insulation values. The result of the third lis- Lnor = . (5) tening test has shown that the participants judged the sounds LN1 − LN2,0 differently, although the damping was the same. There seems where LN1 is the single number value of the loudness level to be a discrepancy between the standard calculation proce- in the source room, LN2,0 is the single number value of the dure to derive an airborne sound insulation value, such as the loudness level of the idealised or calculated airborne sound in- weighted sound reduction index Rw, and subjective observa- sulation in the receiving room, and LN2,m is the single number tion. It is, however, noted that the spectrum adaptation terms value of the loudness level obtained by the measured or simu- indicated a dependency on a subjective assessment, which is lated sound pressure level.

204 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION 3.2. The Weighting The weighting function in the model, as presented in the I) Lnor > 1 ∧ w > 1 ⇒ Lnor,w > 1 literature,11 is based on some considerations that psycholog- II) L < 1 ∧ w < 1 ⇒ L < 1 ical effects, like annoyance, cannot be fully evaluated by the nor nor,w measurement of the sound pressure level.31 The weighting III) + IV ) Lnor > 1 ∧ w < 1 ⇒ Lnor,w < 1 ∨ Lnor,w > 1 is judged as an awareness of noise, i.e., annoyance. For this V ) + VI) Lnor < 1 ∧ w > 1 ⇒ Lnor,w < 1 ∨ Lnor,w > 1 reason, some psychoacoustic factors, such as roughness, fluctuation strength, tonality, and sharpness were investigated.17 NB: The region yielding Lnor,w = 1 needs: Lnor = 1 ∧ 0 0 The results led to the conclusion that roughness, tonality, and w = 1 and that requires: L2,m = L2,0 ∧ F lsm = F ls0. This sharpness are not suitable predictors for a rating procedure condition is impossible in real buildings and in real situations concerning sound insulation. From reasons discussed above in-situ. and because the specific fluctuation strength, F ls0 (vacil), relates to the temporal structure of the sounds,30, 32 this measure 4. MODEL IMPLEMENTATION is preferred to be an appropriate weighting. To differentiate The filtered sound signal, i.e., the level of interest, is as- the signal in terms of psychoacoustic measures, investigations sumed to be a measure of perception, which means that a dip were focused on specific fluctuation strength. This is in accor- in the frequency dependent airborne sound insulation should dance with investigations concerning indoor acoustic comfort be included in the model. This will be demonstrated explic- by Jeon et al.33 The weighting (w) is the proportion of the itly for single frequency dips in the frequency dependent air- frequency depending specific fluctuation strength of the sig- borne sound insulation. The frequency dependent value en- nal being transmitted through an idealized (i.e., hypothetical ables judgment of the frequency range of that dip. The overall or calculated) partition, F ls0 , and the specific fluctuation (f),0 performance of the model is then demonstrated by means of strength of the signal being transmitted through an actual (i.e., a single value representing the measure of deviation from the measured or simulated) partition, F ls0 . The weighting (w) (f),m ideal value. The single value enables, therefore, the judgment is given by Eq. (6): 0 of an Rw-value to be “real” or “untrue.” This means if the 0 judged airborne sound insulation is “real,” it is likely that the F lsm(f) w(f) = . (6) “heard” sound is subjectively equally assessed as calculated. F ls0 (f) 0 If the airborne sound insulation is “untrue,” it is likely that the The total specific fluctuation strength is calculated as the airborne sound insulation is considered subjectively different sum of all partial fluctuation strength yielding F ls0. The single from a certain expected airborne sound insulation. number quantity of the weighting (w) is then the following: 4.1. Frequency Dependence F ls0 w = m . (7) A calculated airborne sound insulation of 40 dB having a 6 F ls0 0 dB frequency single dip at each 1/3rd octave band frequency from 160 Hz up to 5 kHz is used as an example. In Fig. 7, 3.3. The Weighted Normalized Loudness the investigated idealised airborne sound insulation is shown Level Difference exemplarily. The loudness model describes the frequency-dependent air- Using pink noise as source signal and having a sound pres- borne sound insulation yielding the weighted normalized loud- sure level of 85 dB, the calculated frequency dependent nor- ness level difference. For third-octave band values expressed malized loudness level differences (Lnor(f))) according to Eq. as the product of the frequency-dependent normalized loud- (4) is shown in Fig. 8. A 6 dB dip is evident. It is noted that the ness level difference and a frequency-dependent psychoacous- influence of the single frequency dip on the normalized loud- tic weighting factor, the corresponding formula is given in Eq. ness level difference is observed to be at least one-third octave (8): band below and above the dip. This means that the dip en- larges the influence of a single frequency dip to the normalized loudness level difference. Lnor,w(f) = Lnor(f) ∗ w(f). (8) The envelope depicted in Fig. 8 (represented by the dot- Combining Eq. (5) and Eq. (7) yields the single number ted line) reveals that with increasing frequency, the minimum quantity for the weighted normalized loudness level difference value of the normalized loudness level difference decreases. (Lnor,w) and is written as: That means that the frequency dip reduces the airborne sound insulation. The maximum value (Lnor > 1) increases for low frequencies was very steep; up to about 160 Hz. From 160 Hz Lnor,w = Lnor ∗ w. (9) up to 5 kHz, a nearly constant value is virtually observed. The Equation (9) is case sensitive, i.e., Lnor,w depends on the calculated mean and standard deviation for the linear range individual results of the level differences and the weighting, as (160 Hz – 5 kHz) is Lnor = 1.024±0.003. From the envelope is seen from Eq. (5) and Eq. (6). The following regions occur it is observed that, for minimum values, the low frequencies depending on the six conditions: are about constant up to 250 Hz. Above that frequency, the

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 205 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION

Figure 7. Idealized airborne sound insulation exemplarily for Rw = 40 dB without (left panel) and with a dip of 6 dB at the exemplarily depicted frequency of 1 kHz (middle panel) and 2 kHz (right panel). The solid line is the reference curve given in ISO 717-1.

Figure 8. Normalized loudness level difference over various frequencies, rd Rw = 40 dB with a single dip of 6 dB at each 1/3 octave band frequency Figure 9. Weighted normalized loudness level difference over various fre- from 160 Hz up to 5 kHz. The source signal is pink noise having a SPL of rd quencies, Rw = 40 dB with a single dip of 6 dB at each 1/3 octave band 85 dB. Each solid line shows Lnor with a dip at one frequency and the dotted frequency from 160 Hz up to 5 kHz. The source signal is pink noise having a line shows the envelope. SPL of 85 dB. Each solid line shows Lnor,w with a dip at one frequency and the dotted line indicates the envelope. values decreased linearly until a frequency of approximately 2.5 kHz. Above that frequency, some deviation was observed. 4.2. Specific Fluctuation Strength Overall, the investigated influence of the dip yield in the case of the normalized loudness level difference for all the investi- The calculated specific fluctuation strength (F ls0) for dif- gated sound signals demonstrated that the filtered or processed ferent sound signals and different Rw-values reveal that tran- sound was assumed to be louder when a frequency dip is in- sient sound signals, i.e., the music type signals (Eminem and troduced in the airborne sound insulation. Although the single Beethoven), have greater values than the broadband noise sig- value of the airborne sound insulation is not altered much, the nals (pink noise and white noise). The results are shown in dip is thought to cause a sensation that result in an increased Fig. 10. loudness. Introducing the weighting (w), as defined in Eq. (6), From Fig. 10 it is seen that with increasing Rw-value the results in the weighted normalized loudness level difference fluctuation strength decreases. Broadband noise shows little or (Lnor,w(f)), as defined in Eq. (4). See the results illustrated in almost no fluctuation, whereas music type signals show high Fig. 9 for an example; pink noise was used as a source signal. fluctuation. This is expected, is in line with the literature,34 and The weighted normalized loudness level difference shows confirms the results presented earlier.35 Broadband noise sig- the event of a frequency dip in a manner similar to the normal- nals (pink noise or white noise) do not change much in specific ized loudness level difference. It is observed, however, that a fluctuation strength with increasing sound insulation, which is reversed picture is drawn comparing the normalized loudness expected, but this result could be an indication that transient level difference and the weighted normalized level difference. signals, i.e., non-steady-state signals, can be more influenced This is an indication that the specific fluctuation strength does by appropriate sound insulation in the sense of subjective judg- significantly influence the weighted normalized loudness level ments to rate the annoyance of the receiving sound between a difference. dividing partition.17, 33

206 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION

Figure 10. Speciﬁc ﬂuctuation strength (Fls’) for different sound signals and different Rw-values. Figure 12. Weighted normalized loudness level difference (Lnor,w) for different Rw-values over frequency where a dip of 6 dB is applied in the frequency curve of the sound insulation.

lated ﬁnding that a predicted airborne sound insulation is over- estimated compared to a “biased” sound insulation.

4.4. Weighted Normalized Loudness Level Differences From the explored single number quantities of the normalized loudness level differences, the same airborne sound insulations of 20, 40, and 60 dB with frequency dips at 100, 500, 800, 1k, 2k, and 3.15 kHz at 6 dB are used to compute the weighted normalized loudness level difference (Lnor,w) according to Eq. (9). The results are shown in Fig. 12. The influence of the type of signal used to compute the weighted normalized loudness level difference is clearly ob- Figure 11. Normalized loudness level difference (Lnor) for different Rw- served. However, at 100 Hz, no significant difference between values over frequency where a dip of 6 dB is applied in the frequency curve of the used signal types was observed. A clear variation of the the sound insulation. weighted normalized loudness level difference was observed for frequencies greater than 100 Hz. Obviously the weighting 4.3. Normalized Loudness Level Differences had a key influence on the result in calculating the weighted The result from computing the single number quantities of normalized loudness level difference. This is in agreement the normalized loudness level differences (Lnor) as defined in with the theory used to describe an auditory judgment us- Eq. (5) for different Rw-values of 20, 40, and 60 dB with ing a psychoacoustic measure, such as the specific fluctuation frequency dips of 6 dB at 100, 500, 800, 1k, 2k, and 3.15 kHz strength.31 The single numerical value of the weighted nor- in the airborne sound insulation are shown in Fig. 11. malized loudness level difference is sensitive to the type of test signal and to an introduced frequency dip. In Fig. 11, each symbol shows the calculated result of Lnor for different Rw-values that have a frequency dip of 6 dB at 100, 500, 800, 1k, 2k, and 3.15 kHz in the airborne sound in- 5. MODEL VERIFICATION sulation. From this comparison, it is observed that the level To verify the validity of the model, the results obtained by difference does not show much deviation with respect to dif- the listening tests shown in Figs. 2 to 5 are compared. ferent types of sound signals. At low frequencies (100 Hz), the difference between the investigated signals is smallest. In- 5.1. Calculated Frequency Depending specting results at a single value reveal that at a frequency of 100 Hz the normalized loudness level difference is less than or Values equal to unity. This result means that the dip, i.e., the “biased” The normalized loudness level difference and the weighted airborne sound insulation, is less than or equal to the “ideal”, normalized loudness level difference are calculated for the two i.e., predicted airborne sound insulation. For greater frequen- cases shown in Fig. 5. The idealised airborne sound insulation cies, the normalized loudness level difference is greater than to obtain (LN2,0) was chosen to be the shifted reference curve unity. Hence, the “biased” airborne sound insulation is sup- having a sound reduction index of Rw = 50 (-2; -6) dB. The posed to be greater than the predicted value. This result might results are shown in Fig. 13, which provides a direct visual indicate that a dip at low frequencies yields a subjectively re- comparison of the results. The sound reduction index Rw is

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 207 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION

Figure 13. Weighted normalized loudness level difference (Lnor,w) for different Rw-values over frequency where a dip of 6 dB is applied in the frequency curve of the sound insulation. also depicted separately below Fig. 13. It is seen that the fre- noise signal, which is in line with the results found in the ﬁrst quency dependent normalized loudness level difference does experiment. not discriminate between different sound signals, whereas the weighted normalized loudness level difference does. Further- 5.2. Calculated Single Numerical Values more, the frequency dip is observed in all calculation results. A comparison of the data shows that the frequency dips in case Computing the weighted normalized loudness level differ- “I” at 2.5 kHz and in case “II” at 800 Hz, it is accurately pic- ence for the two cases I and II as single values yields results tured. Results from the listening test have shown that case “I” that are shown in Fig. 14. The results outlined in Fig. 14 (left panel in Fig. 12) was judged to be louder than case “II.” show that the weighted normalized loudness level difference This is also pictured in Fig. 12, where the weighted normalized (Lnor,w) diverges in some way. The results, however, reveal loudness level difference for case “I” is, which is at frequen- that case “I” results in values close to unity, which means cies greater than 1.250 Hz that clearly beyond unity and thus, that the expected sound insulation is achieved. The calculated indicates that the sound insulation is different from the calcu- mean and standard deviation are 1.003 ± 0.025. Furthermore, lated or “expected” value. Moreover, in case “I” (left panel in it is observed that the white noise results are greater than unity Fig. 13), the music type signal is greater than the broadband in both cases “I” and “II,” which means that the sound insulation is supposed to be greater as calculated. The interpretation

208 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION

dependencies on different signal characteristics and different airborne sound insulations. The model allows sound insulation to be evaluated in a more psychoacoustic manner, rather than only by looking at the sound pressure level differences. This approach could be a prime tool for investigating airborne sound insulation having identical single number ratings but different spectra. The most interesting question is, however, which airborne sound insulation spectrum yields best damping to prevent annoyance or disturbance. This determination is a task for further investigations in the ﬁeld of subjective assessment tests.

REFERENCES 1 Langdon, F. J., Buller, I. B., and Scholes, W. E. Noise from Neighbours and the Sound Insulation of Party Walls in Houses, Journal of Sound and Vibration 79 (2), 205–228, Figure 14. Weighted normalized loudness level difference (Lnor,w) for cases “I” and “II” and for different sound samples. The airborne sound (1981). https://dx.doi.org/10.1016/0022-460x(81)90369-2 R C insulation is w = 50 dB with varying spectral adaptation terms and 2 Ctr. Bradley, J. S. Subjective Rating of the Sound Insulation of Party Walls, Canadian Acoustics, 11 (4), 37-45, (1983). is that the “biased” airborne sound insulation is thought to be, 3 Ryu, J. K., Jeon, J. Y. Influence of noise sensitiv- in a subjective sense, greater than the “unbiased” or predicted ity on annoyance of indoor and outdoor noises in res- value. From the subjective test, it was found that white noise is idential buildings, Appl. Acoust., 72, 336-340, (2011). judged to be quieter than the other sound samples (see Fig. 4). https://dx.doi.org/10.1016/j.apacoust.2010.12.005 Thus, it is concluded that, white noise as a test signal causes a fictive increase in the airborne sound insulation. It is further 4 Joiko, K., Bormann, V. Kraak, W., Durchhoren¨ von Sprache concluded from the results in Fig. 14 that the sound insula- bei Leichtbauwanden,¨ Z. Larmbek¨ ampf.¨ 49 (3), 79–85, tion of case “I” is, for all sound samples, “reliable” because (2002). the single numerical value is close to unity whereas case “II” 5 deviates significantly between different sound samples. This Vorlander,¨ M., Thaden, R. Auralisation of Airborne Sound result indicates that the sound insulation of case “II” differs Insulation in Buildings, Acta Acustica united with Acustica, with different sound signals, and hence, the expected airborne 86 (1), 70-76, (2000). http://www.ingentaconnect.com/ sound insulation is not supposed to be consistent with different 6 Tonin, R. The BCA 2004 - A Plan for the future, Proc. applied sound samples. Acoustics 2004, Australia, 141-149, (2004)

7 6. CONCLUSIONS Muellner, H., Humer, C. Stani, M. M. Lightweight Build- ing Elements with Improved Sound Insulation Considering rd The experimental results have revealed that white noise was the Low Frequency Range, Proc. 3 Congress of the Alps assessed to be highly significant (p < 0.001) quieter, and the Adria Acoustics Association, 27–28, Graz, Austria, (2007). rap music sample “Eminem” was judged to be louder com- 8 Rasmussen, B., Lang, J. How much protection do the sound pared with the other sound samples. Furthermore, the exper- insulation standards give and is this enough?, Proc. 8th Eu- iments have shown that, in general, noise samples (i.e., white ropean Conference on Noise Control (EuroNoise), Edin- noise and pink noise) are judged to be not as loud as music burgh, Scotland, UK, 26-28, (2009). sound samples, which also result in lower specific fluctuation strength values. Conversely, non-steady-state signals yield the 9 Ljunggren, F., Simmons, C. Hagberg, K., Correla- greatest specific fluctuation strength values and were judged to tion between sound insulation and occupants’ percep- be louder than the broadband noise samples. In other words, tion — Proposal of alternative single number rating sound insulation is judged differently for different sound sam- of impact sound, Appl. Acoust. 85, 57–68, (2014). ples, as expected. The model was shown to correctly depict the https://dx.doi.org/10.1016/j.apacoust.2014.04.003 experimental results of the loudest and quietest sound samples 10 Neubauer, R. O., Kang, J. Airborne sound insulation based as well as the individual frequency dips in the airborne sound on a model of loudness, Proc. 21st International Congress insulation. on Sound and Vibration (ICSV), Beijing, China, 13-17, The implementation of the model showed that the calcula- (2014). tion scheme is able to capture details in the frequency range as well as in the case of single numerical values. As shown by the 11 Neubauer, R. O., Kang, J. Airborne sound insulation in comparison of calculated values with experimental results, the terms of a loudness model, Appl. Acoust., 85, 34–45, weighted normalized loudness level difference demonstrates (2014). https://dx.doi.org/10.1016/j.apacoust.2014.03.024

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 209 R. O. Neubauer, et al.: A SUBJECTIVE RELATED MEASURE OF AIRBORNE SOUND INSULATION

12 Neubauer, R.O., Kang, J. A Model Based on Loudness 25 ISO 717-1. Acoustics — Rating of sound insulation in Level to Describe Airborne Sound Insulation, Proceedings buildings and of building elements, Part 1: Airborne sound of the 43rd International Congress on Noise Control Engi- insulation, International Organization for Standardization, neering (InterNoise), Melbourne, Australia, (2014). Geneva, (2013).

13 Grimwood, C. Complaints about Poor Sound Insulation be- 26 Hongisto, V., Makil¨ ab,¨ M., Suokas, M. Satisfaction with tween Dwellings in England and Wales, Appl. Acoust., 52 sound insulation in residential dwellings — The effect of (3/4), 211–223, (1997). https://dx.doi.org/10.1016/s0003- wall construction. Building and Environment, 85, 309–320, 682x(97)00027-3 (2015). https://dx.doi.org/10.1016/j.buildenv.2014.12.010

14 Park, H. K., Bradley, J. S. Evaluating standard airborne 27 EN 12354-1. Building acoustics. Estimation of acoustic sound insulation measures in terms of annoyance, loudness, performance in buildings from the performance of elements and audibility ratings, J. Acoust. Soc. Am., 126 (1), 208– (Part 1: Airborne sound insulation between rooms). Euro- 219, (2009). https://dx.doi.org/10.1121/1.3147499 pean Committee for Standardization (CEN), Brussels, Bel- gium, (2000). 15 Masovic, D., Mijic, M. Pavlovic, D. S., Adnadevic, M., Noise in dwellings generated in normal home activities — 28 ISO 532/R. Acoustics- Method for Calculating Loud- Spectral approach, Proc. 6th Forum Acusticum, Aalborg, ness Level. International Organization for Standardization, Denmark, 1383–1388, (2011). Geneva. (Standard conﬁrmed in 2012), (1975).

16 Neubauer R. O, Kang J. What Describes the Airborne 29 DIN 45631/A1. Berechnung des Lautstarkepegels¨ und der Sound Insulation in Technical and Subjective Regard? Lautheit aus dem Gerausch-spektrum¨ - Verfahren nach E. Proc. 6th Forum Acusticum, Aalborg, Denmark, 1783– Zwicker - Anderung¨ 1: Berechnung der Lautheit zeit- 1787, (2011). varianter Gerausche,¨ (in German), Beuth Verlag, Berlin, (2010). 17 Neubauer, R. O., Kang, J. Time Structure Of The Signal In Airborne Sound Insulation, Proc. 9th European Confer- 30 Fastl, H., Zwicker, E. Psychoacoustics: Facts ence on Noise Control (EuroNoise), Prague, Czech Repub- and Models, 3rd ed., Springer, Berlin, (2007). lic, (2012). https://dx.doi.org/10.1007/978-3-540-68888-4

18 Neubauer, R. O., Kang, J. Airborne Sound Insulation as 31 Kitamura T, Shimokura R, Sato S, Ando Y. Measurement a Measure for Noise Annoyance, Proc. 21st International of Temporal and Spatial Factors of a Flushing Toilet Noise Congress on Acoustics (ICA), Montral, Canada, (2013). in a Downstairs Bedroom Journal of Temporal Design in https://dx.doi.org/10.1121/1.4798969 Architecture and the Environment, 2 (1), 13-19, (2002). http://www.jtdweb.org/ 19 Zwicker, E., Feldtkeller, R. Uber¨ die Lautstarke¨ von Gle- ichformigen¨ Gerauschen.¨ (On the loudness of stationary 32 Schone,¨ P. Messungen zur Schwankungsstarke¨ von am- noises), (in German), Acustica 5 (6), 303–316, (1955). plitudenmodulierten Sinustonen,¨ (Fluctuation Strength of http://www.ingentaconnect.com Amplitude-Modulated Tones), (in German), Acustica, 41 (4), 252–257, (1979). http://www.ingentaconnect.com 20 Takeshima, H., Suzuki, Y., Fujii, H., Kumagai, M., Ashihara, K., Fujimori, T., and Sone, T. Equal-loudness 33 Jeon, J. Y., You, J., Jeong C. I., Kim, S. Y., contours measured by the randomized maximum likelihood Jho, M. J. Varying the spectral envelope of air- sequential procedure, Acust. Acta Acust. 87 (3), 389–399, conditioning sounds to enhance indoor acoustic com- (2001). http://www.ingentaconnect.com fort, Building and Environ, 46, 739–746, (2011). https://dx.doi.org/10.1016/j.buildenv.2010.10.005 21 Soeta, Y., Maruo, T., and Ando, Y. Annoyance of bandpass- ﬁltered noises in relation to the factor extracted from auto- 34 Aures, W. Ein Berechnungsverfahren der Rauigkeit, correlation function (L), J. Acoust. Soc. Am., 116 (6), 3275- (in German), ACUSTICA, 58 (5), 268–281, (1985). 3278, (2004). https://dx.doi.org/10.1121/1.1782931 http://www.ingentaconnect.com/

22 Jeon, J. Y., Jeong, J. H., Ando, Y., Objective and Subjec- 35 Neubauer, R.O., Kang, J. Rating Airborne Sound Insulation tive Evaluation of Floor Impact Noise, J. Temp. Des. Arch. in Terms of Time Structure of the Signal, Proc. 42nd Inter- Environm., 2 (1), 20-28, (2002). http://www.jtdweb.org/ national Congress and Exposition on Noise Control Engi- neering (InterNoise), New York, USA (2012). 23 ISO 226. Acoustics - Normal equal-loudness-level contours, International Organization for Standardization, Geneva, (2003).

24 Suzuki, Y., Takeshima, H., Equal-loudness-level contours for pure tones, J. Acoust. Soc. Am., 116 (2), 918-933, (2004). https://dx.doi.org/10.1121/1.1763601

210 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Mechanical Fault Diagnosis Method Based on LMD Shannon Entropy and Improved Fuzzy C-means Clustering

Shaojiang Dong School of Automation, Chongqing University, Chongqing, 400044, Peoples Republic of China School of Mechatronics and Automotive Engineering, Chongqing Jiaotong University, Chongqing 400074, Peoples Republic of China

Xiangyang Xu Chongqing University, College of Optoelectronic Engineering, 400044, Peoples Republic of China School of Mechatronics and Automotive Engineering, Chongqing Jiaotong University, Chongqing 400074, Peoples Republic of China

Jiayuan Luo School of Mechatronics and Automotive Engineering, Chongqing Jiaotong University, Chongqing 400074, Peoples Republic of China

(Received 31 August 2015; accepted 26 August 2016) Aiming to diagnosis the mechanical fault precisely, the current research is proposing a method based on the local mean decomposition (LMD) Shannon entropy and improved fuzzy f-means flustering (IFCM). The features are first extracted by using the time-frequency domain method LMD and the Shannon entropy is used to process the original separated product functions (PF) so as to extract the original features. However, the extracted original features are still with high dimensional and include superfluous information. The nonlinear multi-features fusion technique LTSA is used to merge the features and reduce the dimension. Then, based on the extracted features, the IFCM model is used to achieve the mechanical fault diagnosis. In this model, a feature cluster determine function is established to achieve the weighted factor to achieve a better category effect. Case of a bearing test was analyzed and the results proved the effectiveness of the methodology.

1. INTRODUCTION do not provide a wholesome measure of the bearing health status. Hence, the time-frequency method is used rather fre- Today, diagnosis is a very important research area in indus- quently. Wavelet transformation is the most commonly used try. Traditional concepts of preventive and corrective main- time-frequency analysis method and is widely used in feature tenance are gradually supplemented by diagnosis form. The extraction.4 However, the selection of the wavelet mother co- main objective of this maintenance type is to ensure the de- efficient is difficult, and for different researchers the selected pendability of industrial systems and increase their availability coefficient is not unique. For this reason, it is necessary to find with a lower cost. However, fault diagnosis is not an easy task an effective method to extract the fault-related features hid- since it is essentially a problem of pattern recognition. A more den in the complex and non-linear bearing vibration signals. effective feature extraction and accurate classifier are needed The EMD5 is other tool usually used in time-frequency domain to obtain a higher diagnostic accuracy.1 analysis which is suitable for analyzing non-stationary signals. Feature extraction is the process of transforming the raw vi- However, the EMD method has the problems of endpoint leak, bration data collected from running equipment to relevant in- and the modal aliasing. In this article, the local mean decom- 6 formation of machine working condition. There are three types position (LMD) is used to processing the vibration signals, of methods that deal with the raw vibration data: time domain and the LMD Shannon entropy is used to extract the original analysis, frequency domain analysis, and time-frequency do- features from the signal. main analysis. These three methods are often chosen to ex- Although the original features can be extracted, they are still tract a feature. For example, Fan2 chose the time-frequency high dimensional and include superfluous information, which domain transform ensemble empirical mode decomposition to is why the original features fusion and dimensional reduction describe the characteristics of the vibration signals. Qin3 chose method should be used to deal with the original features so the wavelet transform to extract the feature of machinery wear as to select the typical features. The most commonly used information. The frequency features from FFT analysis re- features fusion and dimensional reduction method is principal sults often tend to average out transient vibrations and thus component analysis (PCA).7 However, the PCA is mainly used

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 211–217) https://doi.org/10.20855/ijav.2017.22.2466 211 S. Dong, et al.: MECHANICAL FAULT DIAGNOSIS METHOD BASED ON LMD SHANNON ENTROPY AND IMPROVED FUZZY. . . for dealing with the linear data set while the bearing vibration To de-modulate h11(t), divide it by the envelope estimation features are usually suppressed by the nonlinear characteristic function a11(t) to obtain s11(t): features so the PCA cannot work effectively. Therefore, it is a challenge to find an effective nonlinear features fusion and h11(t) s11(t) = . (4) dimensional reduction method. In this research, a new feature a11(t) extraction method local tangent space alignment (LTSA)8 is chosen. The LTSA is an efficient manifold-learning algorithm, (3) Ideally, s11(t) should be a pure frequency-modulated which can be used as a preprocessing method to transform the (FM) signal. However, this condition is not always fully sat- high dimensional data into more easily handled low dimen- isfied in reality. Hence, it is necessary to treat s11(t) as the sional data. This method has been used in many fields. In this source data and repeat the above process iteratively until s1n(t) paper, the LTSA is used to achieve extracting the more sensi- is a pure FM signal. Considering the effects of decomposition, tive features. the number of iterations, speed, and other factors, the selected Recognizing condition, in order to identify the work con- conditions for the termination of the iteration process are given a (t) = 1 s (t) dition of machinery further, the SVM and BPNN models are by 1n . When 1n satisfied the condition of being typical used model served as a classifier.9 However, the SVM a pure FM signal, all the envelope estimation functions gener- is not sensitive to the nonlinear feature classification. The ated during the iterative process were multiplied to derive the a (t) BPNN model also carries some disadvantages i.e., the slow envelope signal 1 (i.e., the instantaneous amplitude of the PF (t) rate of learning and getting trapped in local minima. The FCM function) of the first PF component 1 : method based on learning analogy, it is very effective for statis- n Y tical pattern recognition, and can achieve higher classification a1(t) = a11(t)a12(t) ··· a1n(t) = a1q(t); (5) 10, 11 accuracy for unknown and non-normal distribution data. q=1 In this research, the FCM method is revised in order to select the features in order to reduce the classifier complex and where q represents the number of iteration loops. The single computationally intensive, so we need to compress the features component of the AM-FM signal PF1(t) is given by: and supervise them so as to reduce the recognize error rate. Towards this end, a feature cluster determination function is PF1(t) = a1(t)s1n(t). (6) established to achieve a better category effect. (4) The first PF component PF1(t) is separated from the signal x(t) to obtain a new signal ui(t). Repeat the above step 2. METHODS OF SIGNAL PROCESSING using as the source data for k cycles until uk(t) becomes a AND DIMENSIONAL REDUCTION monotonic function. In this way, x(t) can be decomposed into the sum of k PF components and uk(t): 2.1. Original Vibration Process k The nature of LMD is to demodulate AM-FM signals. By X x(t) = PFp(t) + uk(t). (7) using LMD, a complicated signal can be decomposed into a p=1 set of product functions, each of which is the product of an envelope signal and a pure frequency-modulated signal. Further- The use of Eq. (7) ensures that, following LMD, data from more, the completed time-frequency distribution of the original the original signal are better retained and is less likely to get signal can be derived. For any signal x(t), it can be decom- lost. posed as follows:6 (1) Determine all the local extreme points ni (including the 2.2. Feature Extraction maxima and minima) of the non-stationary signal, then calcu- Once the k PFs and a residue uk(t) are obtained, where the late the mean mi of the two adjacent extreme pointsand ni and energy of the k PFs is E1; E2; ...; Ek can be calculated re- ni+1, as well as the envelope estimation function ai: spectively then, due to the orthogonality of the LMD decom- ni + ni+1 position, the sum of the energy of the k PFs should be equal to mi = ; (1) 2 the total energy of the original signal when the residue uk(t) is ignored. As the PFs c1(t); c2(t); ...; ck(t) include different |ni − ni+1| frequency components, E = {E1,E2, ..., Ek} forms an energy ai = . (2) 2 distribution in the frequency domain of roller bearing vibration Use straight lines to connect the two points adjacent to each signal, and then the corresponding LMD energy entropy is des- mean mi and envelope the estimate ai, respectively. Next, use ignated as: the moving average method to obtain the local mean function k m11(t) and envelope the estimation function a11(t). X Hentropy(t) = − pi log pi; (8) (2) Separate the local mean function m11(t) from the signal i=1 x(t) to obtain h11(t): where pi = Ei/E is the percent of the energy of ci(t) in the Pk h11(t) = x(t)m11(t). (3) whole signal energy (E = i=1 Ei). 212 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 S. Dong, et al.: MECHANICAL FAULT DIAGNOSIS METHOD BASED ON LMD SHANNON ENTROPY AND IMPROVED FUZZY. . . 2.3. Basic Concepts of Local Tangent Space 3. THE IMPROVED FUZZY C-MEANS Alignment CLUSTERING METHOD MODEL

The basic idea of LTSA is to use the tangent space of sam- Traditional cluster analysis requires every point of the data ple points to represent the geometry of the local character. set to be assigned into a cluster precisely, so we call it hard Then these local manifold structures of space are lined up clustering. But in fact, most things exist ambiguity in the at- to construct the global coordinates. Given a data set X = tribute, there are no explicit boundaries among the things, and m [x1, x2, ..., xk], xi ∈ R , a mainstream shape of d-dimension no the nature of either-or. So the theory of fuzzy clustering is (m > d) is extracted. The LTSA feature extraction algorithm more suitable for the nature of things and it can a more objec- is as follows:8 tive reﬂection of reality. Currently, the fuzzy c-means cluster- 11 1) Extract local information: for each xi, i = 1, 2, ..., k, ing (FCM) algorithm is the most widely used. used the Euclidean distance to determine a set xi = FCM partitions set of n objects X = (x1, x2, ..., xn) into

[xi,1, xi,2, ..., xi,fi ] of its neighborhood adjacent points (e.g. K fuzzy clusters with C = (c1, c2, ..., cn) cluster centers. In th f, nearest neighbors). fuzzy matrix U = (uij), uij is the membership degree of i th 2) Local linear ﬁtting: In the neighborhood of data points the object with the j cluster. The characters of uij are as xi, a set of orthogonal basis Qi can be selected to construct follows the d-dimension neighborhood space of xi and the orthogonal u ∈ [0, 1] i ∈ 1, 2, ··· , n , j ∈ 1, 2, ··· ,K; projection of each point xi,j(j = 1, 2, ..., k) can be calculated ij (i) T to the tangent space of θj = Qi (xi,j − x¯i). x¯i is the mean data for the neighborhood. The orthogonal projection in the K X tangent space of neighborhood data of xi is composed of lo- uij = 1 i ∈ 1, 2, ··· , n; cal coordinate Θi = [θ(i),1, θ(i),2, ..., θ(i),fi ] that describes the j=1 x most important information of the geometry of the i. n X 3) Global order of the Local coordinates: supposing the 0 < uij < n j ∈ 1, 2, ··· ,K. global coordinates of xi converted by the Θi is Ti = i=1

[ti1, ti2, ..., tifi ], then the error is: Update uij according to Eq. (16): T ) Ei = Ti[I − (1/f)ee ] − LiΘi; (9)  " 2 #−1 K d b−1  P ij  2 , dik 6= 0 where the I is the identity matrix; the e is the unit vector; the k=1 d b−1 uij = ik (16) f is the points number of the neighborhood; and the Li is the  0, dik = 0(k = j)  transformation matrix. In order to minimize the error, the Ti 1, dij = 0(k 6= 0) and Li should be found, then: where: b > 1 is the fuzziness exponentand, and cj (j ∈ [1,K]) T ) ∗ Li = Ti(I − (1/f)ee )Θi ; (10) is the clustering center. dij = ||xi − cj||. Update cj according to Eq. (17): E = T (I − (1/f)eeT ))(I − Θ∗Θ ); i i i i (11) n P ub x where Θ∗ is the Moor-Penrose generalized inverse of Θ . Sup- ij i i i c = i=1 . (17) posing the j n P ub T T ij B = PWW P . (12) i=1

Let P = [P1,P2, ..., Pk]; TPi = Ti; Pi is a selected matrix The objective function is the Eq. (18): from 0-1; and the T are global coordinates, their weight matrix n K X X b 2 J(U, C) = uijdij. (18) W = diag(W1,W2, ..., Wk); (13) i=1 j=1

T ) ∗ Wi = (I − (1/f)ee )(I − Θi Θi). (14) FCM minimizes the objective function when Uij meets the conditions. The U obtained from the algorithm is a fuzzy par- The constraint is: tition matrix, and it corresponds to the fuzzy partition of X. T TT = Id. (15) We apply the method of maximum subjection principle to get the certainty of partition: 4) Extract of the low-dimensional manifolds feature: Since th In the j column of U, if uij0 = max (uij), xi merges e the is the eigenvalue of matrix B, so the corresponding min- 1

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 213 S. Dong, et al.: MECHANICAL FAULT DIAGNOSIS METHOD BASED ON LMD SHANNON ENTROPY AND IMPROVED FUZZY. . . Table 1. A group of LMD energy entropies of different running states of the actual signal.

Running states H1 H2 H3 H4 H5 Normal state 1.2657 1.2316 1.1091 1.1012 1.1316 0.18 mm fault depth 1.0684 1. 0201 1.2452 1.2566 1.3910 0.36 mm fault depth 0.9506 0.9896 0.9597 1.1166 0.9259 0.53 mm fault depth 0.8923 0.8369 0.8949 0.8283 0.8675 0.71 mm fault depth 0.6031 0.5108 0.6392 0.6967 0.7042

Step 3. Construct the FCM model. The determine function is used to improve the FCM. The IFCM model is used to fault diagnosis.

5. VALIDATION AND APPLICATION

5.1. Case 1

Figure 1. The flowchart of the proposed method. The proposed method was applied to bearing fault signals and the used data set in this paper is obtained from Case West- 12 Define the distance of the different category is Dab = ern Reserve University. The bearing type in the experiments Na Nb is SKF 6205-2RS JEM. Experiments were conducted by using 1 P P D(xa, xb). NaNb i j i=1 i=1 a 2 hp reliance electric motor. Bearings were seeded with faults Define the distance of the same category is Da = by using electro-discharge machining. The test is to simulate 2 min kxi − xk . the bearing normal running state and fault running states, with So as the enlarge the distance the different category and fault depths of 0.18 mm, 0.36 mm, 0.53 mm, and 0.71 mm reduce the distance of the same category, then the determine at the inner raceway, outer raceway and the ball to reflect the function is defined as: deteriorating state of the bearing; the outer- raceway fault signals were chosen in this case. Data was collected at the rate of Na Nb 1 P P D(xa, xb) 12,000 samples per second. Approximately 4096 data points NaNb i j dab i=1 i=1 were selected for analysis. There were 50 groups of test data s = = 2 ; (19) daa min kxi − xk for each fault states that were selected with 20 groups for training and the other 30 groups for testing. where D and D represent the different category and D rep- a b ab The collected signal is first decomposed by LMD; each sig- resents the distance of different category. x¯ represents the av- nal can be decomposed into a number of PF components (in erage value of one category. this research, the number is 6), and then the first five PF com- The weight ratio of all the different categories were calcu- ponents that may contain main fault information are selected to λ = s / Pc s lated, and then the weight factor i i i=1 i was used to extract the features. The Shannon entropy was used to extract improve Eq. (18) the features. The collected 4096 data points of 0.18 mm fault

n K signal and its FFT result is shown in Fig. 2, the PFs (decom- X X b 2 posed into 6 PFs ) decomposed by LMD as shown in Fig. 3: J(U, C) = λiuijdij. (20) i=1 j=1 The Shannon entropy is used to extract the features of the 4. PROPOSED METHOD vibration signal. A group of Shannon entropy is obtained as shown in Table 1 (not normalized before). The ﬂowchart of the proposed method is shown in Fig. 1. Because there are 20 groups features for training and the The method consists of three procedures sequentially: data groups of entropy values are still in high dimension, the LTSA processing and features extraction, merge of the original fea- method (the input data dimension is set to 3, and the number tures, constructing, and training IFCM model. The role of each of neighborhood is set to 12) is used to pre-process the entropy procedure is explained as follows: features; the result will work as the input of the improved FCM fault diagnosis model. Step 1. Data processing and features extraction. The time- After the feature pre-process using the LTSA, the extracted frequency domain signal processing method LMD is used features are inputted into the improved FCM model to train the to extract the original features from the collected mass vi- model so as to recognize the states (the category number k is bration data. set to 5). In order to compare the identifying effect of different Step 2. Merge of the original features. The LTSA method is methods, the following comparisons were done: used to extract the typical features and reduce the dimen- (1) Use LMD Shannon entropy to extract the features and sion of the features. The extracted features are used for directly input the extract features into the IFCM without the training the FCM model. LTSA dimension reduction process.

214 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 S. Dong, et al.: MECHANICAL FAULT DIAGNOSIS METHOD BASED ON LMD SHANNON ENTROPY AND IMPROVED FUZZY. . .

Figure 2. The collected 4096 data points of 0.18 mm fault signal and its FFT.

Figure 3. The PFs decomposed by the collected signal.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 215 S. Dong, et al.: MECHANICAL FAULT DIAGNOSIS METHOD BASED ON LMD SHANNON ENTROPY AND IMPROVED FUZZY. . .

Table 2. The recognition rate of different methods. Recognition rate % 0.18 0.36 0.53 0.71 mm mm mm mm Model type Normal fault fault fault fault depth depth depth depth Without the LTSA 63 60 77 65 89 Pre-processed by the PCA 80 92 85 89 98 Pre-processed by the KPCA 90 91 92 96 97 The fault diagnosis FCM model that is not improved 92 100 93 95 96 The proposed method 100 100 100 100 100

(2) Use LMD Shannon entropy to extract the features and Figure 4. The test rig of Case 2. process the extracted features by PCA7 to reduce the dimension, then input the features into the IFCM. (3) Use LMD Shannon entropy to extract the features and process the extracted features by KPCA13 to reduce the dimension, then input the features into the IFCM. (4) Use LMD Shannon entropy to extract the features and process the extracted features by LTSA to reduce the dimension, then input the features into the FCM. (5) The method proposed in this article. The comparison results are shown in Table 2. From the Ta- ble 2 it can be seen that method 1 omitted the process of using the LTSA to extract the features. Hence, the features with a Figure 5. The collected vibration signal. high dimension and the IFCM model cannot deal with the features effectively; the resulting recognition rate is relatively low for one year. Then, a set of data from each of the two months than other methods. Method 2 works without the LTSA by us- is selected and the data sets are used to test whether or not the ing the PCA and the results are inputted into the IFCM model, proposed method can identify the bearing running state. There if the result is not good, the PCA model can achieve some fault were 8192 data points that were selected to be analyzed, 60 identify such as the running state of normal and 0.71 mm seri- groups of collected data of different faults were obtained with ously fault depth, however, the model cannot identify the bear- 30 groups for training and the other 30 groups for testing. ing fault of other depths effectively, that is because the PCA Next, the LMD method is used to decompose each group of model is not easy to deal with the nonlinear feature. Method 3 signals, get the the PFs and the Shannon entropy of the calcu- works without the LTSA by using the KPCA. The features ex- lated PFs. A group of features of different fault conditions are tracted by the KPCA were inputted into the IFCM. The result obtained, as shown in Table 3 (not normalized beforehand). was also not satisfactory because the KPCA model can prelim- Then, the 30 groups’ entropy values are normalized and in- inarily distinguish the features of different fault states, which put into the LTSA in order to pre-processe the typical feature- can have a better effect than the PCA. However, in many states sand the extracted features are input into the IFCM. The rec- the KPCA is still not working very well for complex data, so ognized results are shown in Table 4. Table 4 shows that the the IFCM cannot deal with the features effectively. Method 4 proposed method yields a high recognize accuracy even though proposed that the FCM is not improved; the results show some the actual bearing running state is very complex. The running recognize error because the FCM method may fall into local state of the normal state is about 95% accurate rate, which is recognize error when the data is complex. Hence, an improve- because the bearing in running in safe state, the state is not ment of the FCM model is required and the weight factor can clearly as the bearing running for 2 months, where the bear- help improve the recognize rate of the FCM. So the results of different methods proved the effect of the proposed method. Table 3. A group of LMD energy entropies of different running states of the actual signal. 5.2. Case 2 Running states H1 H2 H3 H4 H5 Normal state 1.2657 1.2316 1.1091 1.1012 1.1316 The bearings are hosted on the shaft; the shaft is driven by Running for 2 month 1.0684 1. 0201 1.2452 1.2566 1.3910 AC motor. The rotation speed is kept at 1000 rpm and a ra- Running for 4 month 0.9506 0.9896 0.9597 1.1166 0.9259 dial load of 3 kg is added to the bearing. The test rig is shown Running for 6 month 0.8923 0.8369 0.8949 0.8283 0.8675 in Fig. 4. The data sampling rate is 25600 Hz and the data Running for 8 month 0.6031 0.5108 0.6392 0.6967 0.7042 length is 102400 collected points, as shown in Fig. 5. Every Running for 10 month 0.7663 0.7095 0.7953 0.7325 0.8034 Running for 12 month 0.7509 0.7993 0.8331 0.7408 0.8296 two hours, the vibration data is collected. The bearing is run

216 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 S. Dong, et al.: MECHANICAL FAULT DIAGNOSIS METHOD BASED ON LMD SHANNON ENTROPY AND IMPROVED FUZZY. . .

Table 4. The states of recognition rate of different states based on the proposed 2 Fan, J. and Zhencai Z., and Wei L. Fault identifica- method (recognition rate η %). tion of rotor-bearing system based on ensemble empirical Running states recognition rate η % mode decomposition and self-zero space projection anal- Normal state 94 ysis, Journal of Sound and Vibration, 333, 3321–3331, Running for 2 month 97 Running for 4 month 95 (2014). Running for 6 month 96 3 Yi, Q. and BaoPing T.and JiaXu W. Higher-density dyadic Running for 8 month 95 wavelet transform and its application, Mechanical Systems Running for 10 month 100 Running for 12 month 100 and Signal Processing, 24 , 823–834, (2010). 4 Shaojiang, D., Baoping, T., and Renxiang, C. Bearing run- ing state was changing, the Shannon entropy were different ning state recognition based on non-extensive wavelet fea- with the normal state. The running state of 10 months and ture scale entropy and support vector machine, Measure- 12 months are recognized clearly, this is because the bearing ment, 46, 4189–4199, (2013) . state was degenerating. The results confirm that the proposed method can recognize the bearing running states effectively. 5 Baoping, T. and Shaojiang, D. and Tao S. Method for eliminating mode mixing of empirical mode decomposition 6. CONCLUSIONS based on the revised blind source separation, Signal Pro- cessing, 92 ,248–258, (2012). Firstly, this research used the LMD Shannon entropy method to extract the original features from the vibration sig- 6 Jiedi, S., Qiyang, X., and Jiangtao, W. Natural gas pipeline nals. The LTSA was used to reduce the dimension and data small leakage feature extraction and recognition based on redundancy of the entropy features. Using these methods, the LMD envelope spectrum entropy and SVM, Measurement, typical features could be extracted effectively. 55, 434–443, (2014). Then, in order to more accurately identify the bearing run- 7 Shaojiang, D. and Tianhong, L. Bearing degradation pro- ning state, the FCM model is improved and the weight factor cess prediction based on the PCA and optimized LS-SVM is used so as to improve the recognition accuracy of FCM ef- model, Measurement, 46, 3143–3152, (2013). fectively. Thirdly, through different comparisons we can see that the 8 Shichang D. Minimal Euclidean distance chart based on proposed method makes good use of the advantage of all parts support vector regression for monitoring mean shifts of and together to obtain better recognition accuracy and effi- auto-correlated processes, International Journal of Produc- ciency. tion Economics, 141, 377–387, (2013). Finally, through the tested signals in the research, the results 9 show the significant efficacy of the proposed method in identi- Yunluo, Y. and Wei, L. A novel sensor fault diagnosis fying the bearing faults. method based on Modified Ensemble Empirical Mode De- composition and Probabilistic Neural Network, Measure- ACKNOWLEDGEMENTS ment, 68, 328–336, (2015) . 10 Telmo, M.,Silva, F., and Bruno, A. P., Hybrid methods for This research was supported by the National Natural Science fuzzy clustering based on fuzzy c-means and improved par- Foundation of China (No. 51405047, 51405048), the Scientific ticle swarm optimization Expert Systems with Applications, and Technological Research Program of Chongqing Munici- 42, 6315–6328, (2015). pal Education CommissionNo.KJ1500529 , KJ1500516), the China Postdoctoral Science Foundation (No.2016M590861), 11 Benaichouche, A. N., Oulhadj, H., and Siarry P. Improved Chongqing Postdoctoral Science Foundation funded project spatial fuzzy c-means clustering for image segmentation (No. xm2015001, xm2015011), and the Natural Science Foun- using PSO initialization, Mahalanobis distance and post- dation Project of CQ cstc2015jcyjA70012. Natural Science segmentation correction Digital Signal Processing, 23, Foundation Project of CQ cstc2014jcyjA1381.China Postdoc- 1390–1400, (2013). toral Science Foundation funded this research, Project no. 12 2014M552316. The authors are grateful to the anonymous re- Case Western Reserve University bear- viewers for their helpful comments and constructive sugges- ing data center, 2009. Available from: tions. http://www.eecs.cwru.edu/laboratory/bearing. 13 Qi W. Product demand forecasts using wavelet kernel sup- REFERENCES port vector machine and particle swarm optimization in 1 Jaouher, B. A. and Nader, F. and Lotfi S. Application of manufacture system, Journal of Computational and Ap- empirical mode decomposition and artificial neural network plied Mathematics, 233, 2481–2491, (2010). for automatic bearing fault diagnosis based on vibration signals, Applied Acoustics, 89,16–27, (2015).

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 217 Analysis of Transverse Vibration Acceleration for a High-speed Elevator with Random Parameter Based on Perturbation Theory

Chen Wang, Ruijun Zhang and Qing Zhang College of Mechanical and Electronic Engineering, Shangdong Jianzhu University, Jinan(250101), China.

(Received 24 September 2015; accepted 4 September 2016) The problem of randomness of design parameters objectively exists in the high-speed elevator cabin system. The definite and random part of the acceleration response expressions were derived according to perturbation theory and the transverse vibration acceleration response of the observation point was analyzed. The sensitivity expressions of various random parameters to the acceleration response were deduced by solving the random response of expression and comparing the coefficients of various parameters of the random part. The different impacts of various parameters on the acceleration response were analyzed according to the expression of acceleration response sensitivity. Based on the displacement response covariance matrix and random parameter covariance matrix, the standard deviation characteristic of the acceleration response was obtained and analyzed. The results showed that the random parameters made the acceleration response to be more discrete. Additionally, the randomness of the geometrical parameters had the greatest influence on transverse acceleration. The results can provide a reference for anti-vibration design of high-speed elevators and safety assessment.

1. INTRODUCTION to the structure of elevator car vibration damping, reliability sensitivity analysis, and safety assessment. As a means of transport in high-rise buildings, the elevator’s Zhang et al. studied the frequency response function of a speed was rapidly increased with the increase of the height statistical feature of one and two degrees of freedom random of buildings.1 Currently, the elevator whose running speed structure using the Monte-Carlo method.6 Zhang et al. used is more than 2.5 m/s is known as the high-speed elevator. the stochastic finite element method to study reliability char- With the increase of elevator’s speed, the transverse vibration acteristics of a mechanical structure component under com- problem due to rail harshness has become more and more se- plex loading conditions.7 The Monte-Carlo numerical mod- rious. In recent years, many domestic and foreign scholars eling method is highly accuracy, but for large and complex have studied the vibration of the cabin system. Noguchi et al. structures, it is time consuming. Thus, it is suitable for the used frequency-domain analysis based on principal component comparison of methods but not suitable for practical engineer- analysis to perform operational modal analysis of an elevator ing structural analysis. The stochastic finite element method car.2 Yin et al. transformed many important factors such as needs to set up all kinds of random parameters correspond- rail harshness, flexion of the rail, and the guide shoe’s defects ing to the stochastic finite element characteristic matrix, which into the contact force of rail to guide shoe, and established a causes much inconvenience to its computer program design. multi-DOF transverse vibration model of a high-speed eleva- Therefore, it is necessary to find a random perturbation method tor car.3 Xia and Shi established a transverse vibration model to avoid the establishment of a random finite element charac- of an elevator car by studying real-time interface stiffness be- teristic matrix which can be realized easily via computation. tween the guide rollers and the guide shoe.4 They took rail harshness as the input excitation to study the transverse vibration of an elevator car. In the product design stage, all of the 2. CONSTRUCTION OF A RANDOM parameters are definite. However, due to the influence of ac- PARAMETER CAR VIBRATION MODEL tual installation circumstance, live debugging, manufacturing errors, installation errors, the influence of temperature and un- The establishment of the car model is the basis for solving certainty of physical parameters of materials, etc., the parame- the differential equations of the motion of car system. The ters of the same batch of elevator products are different. For an model should not be too simple, for it could not accurately re- actual product, its parameters are uncertain and present some flect the actual situation.8 On the other hand, the model cannot randomness. For example, Li and Liao analyzed a shear wall’s be too complicated because the amount of calculations will be vibration response.5 In the case of the variation, the coefficient too large or even impossible to solve. In Fig. 1, a car vibration of elasticity modulus was 0.3 and the vibration power spectral model is established. A rectangular coordinate system oxyz is density of the top of the shear wall was increasing exponen- established with the barycenter of the car as the origin. The tially. However, most of the existing literature about the ele- roller guide shoe system is simplified into a spring damping vator car’s vibration does not consider the random parameters system.9, 10 Each system provides the x and y directions of of an elevator car or approximate them as definite parameters. the two forces, and so the whole system of the car has 5 DOF In fact, these random parameters not only affect the eigenval- along the x and y directions and around the x, y and z-axis ues and eigenvectors of the system in each mode, but also have rotations.11 an effect on the statistical properties of the response. There- For convenient calculation, suppose that the horizontal dis- fore, studying the response to random parameters is significant tances from guide wheel 1, 2 and guide wheel 3, 4, to the cen-

218 https://doi.org/10.20855/ijav.2017.22.2467 (pp. 218–223) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Ch. Wang, et al.: ANALYSIS OF TRANSVERSE VIBRATION ACCELERATION FOR A HIGH-SPEED ELEVATOR WITH RANDOM PARAMETER. . . 3. DYNAMIC RESPONSE ANALYSIS OF THE RANDOM PARAMETERS OF THE ELEVATOR CAR SYSTEM In this section, the dynamic response of the car system was discussed by using the random perturbation method.13 Since M, C, K, and P (t) have random properties, the following transformations are needed:

M = Md + εMr; (2)

C = Cd + εCr; (3)

K = Kd + εKr; (4)

X = Xd + εXr; (5)

P (t) = Pd (t) + εPr (t); (6) where ε is a small parameter. As an abstract value obtained Figure 1. Parameters of elevator cabin system. by the perturbation method will not affect the derivation of the entire equation.14 The subscript d and subscript r denote the troid are respectively equal to l1 and l2. While the vertical determining parts and random parts of random parameters, and distances from guide wheel 1, 3 and guide wheel 2, 4, to the the mean of the random parts is zero. By combining these ε centroid are equal to l3 and l4. As described above, the sys- equations with Eq. (1), comparing them with factor at the 2 tem has 10 random parameters: the mass of the car m, the same power, and omitting higher-order terms above O(ε ), the 15 x-axis rotational moment of inertia Jox, the y-axis rotational following equations were obtained: moment of inertia Joy, the z-axis rotational moment of inertia 0 ε : MdX¨d + CdX˙ d + KdXd = Pd (t); (7) Joz, stiffness and damping in all directions of the guide wheel k, c (assume that the guide wheels’ stiffness and damping at all directions are equal), l1, l2, l3, and l4. Additionally, they 1 ¨ ˙ are assumed to be independent with each other. And they have ε : MdXr + CdXr + KdXr = their own mean value and standard deviation. In the following, P (t) − M X¨ + C X˙ + K X . (8) the dynamic response of the point of observation point O0 at r r d r d r d the bottom of the car was studied. In order to calculate the random response generated due to The elevator car system dynamics equation can be expressed random external excitation and random parameters, the ran- 12 as: dom response was written as: MX¨ + CX˙ + KX = P (t); (1) Xr = Xr1 + Xr2; (9) where M, C, and K are the mass matrix, damping ma- where Xr1 and Xr2 satisfy the following equations: trix, and stiffness matrix respectively, X is the generalized coordinate array, and P (t) is an external stimulus. MdXr1 + CdXr1 + KdXr1 = Pr (t); (10)  m 0 0 0 0   0 m 0 0 0  M =  0 0 Jox 0 0 ,   MdX¨r2 + CdX˙ r2 + KdXr2 =  0 0 0 Joy 0  0 0 0 0 J oz − MrX¨d + CrX˙ d + KrXd . (11)  cxi 0 0 cxilzi −cxilyi  0 c −c l 0 0 4  yi yi zi  Equation (10) and Eq. (11) respectively denote the random P  0 −c l c l2 0 0  C =  yi zi yi zi , response due to the randomness of the excitation and of the i=1  c l 0 0 c l2 −c l l  xi zi xi zi xi yi zi random parameters. For Eq. (11), random variables were ex- −c l 0 0 −c l l c l2 xi yi xi yi zi xi yi panded into the Taylor series in the vicinity of determining part  k 0 0 k l −k l  xi xi zi xi yi bdj(j = 1, 2, . . . , m) of the random parameters, and substi- 0 k −k l 0 0 4  yi yi zi  tuted into Eq. (11), and comparing the brj coefﬁcients the fol- P  0 −k l k l2 0 0  16, 17 K =  yi zi yi zi , lowing equation was obtained: i=1 2  kxilzi 0 0 kxilzi −kxilyilzi  2 −kxilyi 0 0 −kxilyilzi kxilyi ∂X¨d ∂X˙ d ∂Xd Md + Cd + Kd =  x  ∂bj ∂bj ∂bj y   ∂Md ¨ ∂Cd ˙ ∂Kd X =  θ . − Xd + Xd + Xd ;  ox  ∂bj ∂bj ∂bj  θ  oy (j = 1, 2, . . . , m) . (12) θoz International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 219 Ch. Wang, et al.: ANALYSIS OF TRANSVERSE VIBRATION ACCELERATION FOR A HIGH-SPEED ELEVATOR WITH RANDOM PARAMETER. . .

Equation (12) was solved by using the Wilson-θ method.18 Table 1. Parameters of elevator cabin system. ¨ And the system response sensitivities were obtained as: ∂Xd , Mean Standard Para- Mean Standard ∂bj Parameter: value: deviation: meter: value: deviation: ˙ ∂Xd ∂Xd ¨ ˙ , , Xr2 Xr2 Xr2. They were expanded into the Taylor bj Dbj σbj bj Dbj σbj ∂bj ∂bj (CV=0.05) (CV=0.05) series in the vicinity of determining part bdj(j = 1, 2, . . . , m) m[kg] 1000 50 c[kg/s] 1000 50 of the random parameters and the following equations were 2 Jox[kg·m ] 2300 115 l1[m] 1 0.05 obtained: 2 m Joy[kg·m ] 2000 100 l2[m] 1 0.05 ¨ 2 ¨ X ∂Xd Joz[kg·m ] 820 41 l3[m] 1.6 0.08 Xr2 = · brj; (13) −1 ∂b k[N·m ] 100000 5000 l4[m] 1.4 0.07 j=1 j (k) m ˙ Var X represents the variance of the kth element in X ∂Xd X˙ = · b ; (14) vector X, and Cov represents covariance. The following equa- r2 ∂b rj j=1 j tion was obtained: T m ∂Xd ∂Xd X ∂Xd N = N . (20) X = · b . (15) x ∂b b ∂b r2 ∂b rj j=1 j Based on Eq. (20), the standard deviation of the displace- Thus, the random responses of the random parameter structure ment response was obtained 0 X¨r2, X˙ r2, and Xr2 were obtained. The observation point O 1/2 in the x-direction and y-direction acceleration of the matrix is:  m m  X X ∂Xi ∂Xi σi = d d σ σ ρ ; (21) x  bj bk jk x¨ 0 ∂bj ∂bk o = T X¨; (16) j=1 k=1 yö0 i (i)1/2 where T is the transformation matrix: T = where σx is the standard deviation Var x of the ith 1 0 0 l 0 l 0 element in vector X, ρjk is the correlation coefficient of bj zo yo . 0 1 lzo0 0 0 and bk, and σbj is the standard deviation of bj. Similarly, the standard deviation of the velocity response and acceleration response were obtained: 4. STANDARD DEVIATION ANALYSIS FOR  1/2 THE DYNAMIC RESPONSE OF THE m m ˙ i ˙ i i X X ∂Xd ∂Xd ELEVATOR CAR σx˙ =  σbjσbkρjk ; (22) ∂bj ∂bk j=1 k=1 In consideration of the complexity and difficulty of the randomness of the external excitation of the car, only the standard  1/2 m m ¨ i ¨ i deviation of the random response due to random Xr2 parame- i X X ∂Xd ∂Xd σx¨ =  σbjσbkρjk . (23) ter was studied to facilitate random vibration system optimiza- ∂bj ∂bk j=1 k=1 tion.

Firstly, the displacement response covariance matrix Nx, It can be seen that as long as the standard deviation of the random parameter covariance matrix Nb, and displacement re- system architecture’s random parameters and correlation coef- ∂Xd ficients are given, the standard deviation of the response can sponse sensitivity matrix ∂b were defined as: be obtained without involving the statistical properties of other random parameters. This makes it easy to be applied it to engineering practice. Besides, only the required degree of freedom Nx =   is used to calculate the response sensitivity to avoid a large Var X(1) Cov X(2),X(1) ··· Cov X(k),X(1) amount of computation.    Cov X(2),X(1) Var X(1) ··· Cov X(k),X(2)      ;  . . . .  5. CASE ANALYSIS OF AN ELEVATOR CAR  . . .. .    SYSTEM Cov X(k),X(1) Cov X(k),X(2) ··· Var X(k) (17) 5.1. Calculation of the Acceleration Response of a Random Parameters Elevator Car System   Var (b1) Cov (b2, b1) ··· Cov (bm, b1) According to Table 1, the elevator car system vibration  Cov (b2, b1) Var (b2) ··· Cov (bm, b2)  N =   ; model includes the mass parameters m, Jox, Joy, and Joz; the b  . . .. .   . . . .  dynamical parameters k and c; and the geometrical parameters Cov (bm, b1) Cov (bm, b2) ··· Var (bm) l1, l2, l3, and l4. It is assumed that the random parameters are (18) independent and subject to standard normal distribution. For a type of high-speed traction elevator with a running speed of 5 m/s, the parameters are shown in Table 1. ∂Xd h i It is considered that the parameters are independent and sub- = ∂Xd ∂Xd ··· ∂Xd . (19) ∂b ∂b1 ∂b2 ∂bm ject to the standard normal distribution and the coefficient of 220 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Ch. Wang, et al.: ANALYSIS OF TRANSVERSE VIBRATION ACCELERATION FOR A HIGH-SPEED ELEVATOR WITH RANDOM PARAMETER. . .

Figure 2. Lead rail’s excitation in the x-direction. Figure 4. Certain part’s acceleration of the observation point in the x- direction.

Figure 3. Lead rail’s excitation in the y-direction. Figure 5. Certain part’s acceleration of the observation point in the y- direction. variation CV is fixed at 0.05. For convenient calculation, the step pulse excitation was applied to the x and y directions in Fig. 7. the left side of the rail joints and the right side of the rail was Taking the random parameters into consideration, it can be regarded as the ideal rail. A single rail length was 5 m. The seen that the absolute value of the maximum accelerations in excitation was shown in Fig. 2 and Fig. 3. the x direction and the y direction were increased by about The response expression was constructed by perturbation 50% and 40% by comparing acceleration response determining theory and the response acceleration X¨d was obtained by using parts. Furthermore, the response value of each moment had a the Wilson-θ Method to solve Eq. (7). Then, the x-direction different degree of dispersion by comparing the determining and the y-direction accelerations of any point were obtained by part. Especially at 6.5 s and 16.5 s, the y direction always letting the acceleration response pre-multiply the transforma- showed more abrupt acceleration response characteristics. By tion matrix T . Images of response acceleration of determining considering the randomness of the structural parameters, the parts in the x-direction and y-direction were obtained from the discrete degree of acceleration response was indeed increased. center point of the bottom of the elevator car, as shown in Fig. 4 and Fig. 5. As can be seen from the figures, the transverse vi- 5.2. Acceleration Response Sensitivity bration in the x direction tended to be relatively unstable at any Analysis excitation, while the vibration intensity in the y direction was generally greater than that in the x direction. The stochastic parameter system response sensitivity Eq. (12) was solved by the Wilson-θ method, and the accel- The random part of the acceleration response was obtained ¨ eration response sensitivity for each random parameter in the by letting the brj equal to ±σbj together with the Xd substitut- ¨ x direction and y direction at the observation point (the ∂Xx ing it into Eq. (13) and solving it. Then, the total acceleration bj ¨ ¨ response X was obtained by combining the random part and and ∂Xy ) can be obtained. The acceleration response sensi- the definite part of the acceleration response and multiplying bj ¨ ∂X¨ ¨ tivities ∂Xx and y were constantly changing in the whole X by transformation matrix T and then obtained the observa- bj bj tion point total acceleration response, as shown in Fig. 6 and time period. In order to facilitate the comparison, the mean

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 221 Ch. Wang, et al.: ANALYSIS OF TRANSVERSE VIBRATION ACCELERATION FOR A HIGH-SPEED ELEVATOR WITH RANDOM PARAMETER. . .

bation theory to obtain the coefficient of variation (CV). The results were shown in Table 3. From the table, the variation coefficient of the acceleration response were generally around 1 in the case of the random parameters of the variation coefficient CV equal to 0.05. More- over, after comparing the determine part of the acceleration and the total acceleration image at the observation point, it can be shown that the actual response of the discrete degree was large. In conclusion, the randomness of the design parameters of the elevator car system had an obvious effect on the acceleration of the center of the elevator car.

6. CONCLUSIONS In this paper, an elevator car vibration model with random parameters was established, and the dynamic response expressions for determining random parts of an arbitrary point were Figure 6. General acceleration of the observation point in the x-direction. established by using stochastic perturbation theory. The calculation results showed that the randomness of the structural parameters had a certain effect on the response. Response sensitivity expression can be applied in solving the transverse acceleration response’s sensitivity of each random parameter to the observation point. The acceleration sensitivity of geometrical parameters was much higher than the mass and dynamic parameters. For convenient calculation, the random parameters with a relative lower acceleration response sensitivity were simplified as the determining parameters. In the manufacturing and installation processes the parameters with higher sensitivity should be strictly controlled. Based on the analysis of the response characteristics of the acceleration, the random parameters caused the coefficient of variation of the acceleration response becoming larger, which further confirms that the random parameters have a greater impact on the response. The random parameters cannot be simplified to the determining parameters.

Figure 7. General acceleration of the observation point in the y-direction. ACKNOWLEDGEMENTS

¨ ∂X¨ acceleration response sensitivities E ∂Xx and E y were Authors are grateful for the financial support by Shandong bj bj Jianzhu University Doctor Foundation (No.XNBS1514). ¨ obtained by calculating the mean of absolute value of the ∂Xx bj ¨ and ∂Xy . The results were shown in Table 2. REFERENCES bj As can be seen from the table, the acceleration response sen- 1 Wang, W. and Qian, J. Transverse vibration analy- sitivity in the y direction of the observation point was gener- sis for an elevator suspended system subjected to dis- ally larger than that in the x direction, which indicates that the placement excitation due to structure sway, Jour- influence of the change of the random parameters on the accel- nal of Vibration And Shock, 32 (7), 70–73, (2013). eration response in the y direction was larger. In addition, the https://dx.doi.org/10.13465/j.cnki.jvs.2013.07.033 effects from the mess parameters m, J , J , and J , and dy- ox oy oz 2 namical parameters k and c on the sensitivity of the response Naoaki, N., Atsushi, A., Koichi, M.,et al. Study on ac- were relative low. Thus, they can be used as the determina- tive vibration control for high-speed elevators, Journal of System Design and Dynamics, 5 (1), 164-179, (2011). tion of the parameters, while the geometrical parameters l1, l2, https://dx.doi.org/10.1299/jsdd.5.164 l3, and l4 had a higher impact on the response sensitivity, and should be treated as random parameter. 3 Yin Ji-cai, Bing Yan-nian, Jiang Li-ming, et al. Research on high-speed elevator MDOF horizon- 5.3. Mean and Standard Deviation Analysis tal dynamic characteristics and simulation, Jour- of Observation Point nal of Machine Design, 28 (10), 70–73, (2011). https://dx.doi.org/10.13841/j.cnki.jxsj.2011.10.022 In this case, the guide rail excitation is deterministic. The acceleration response’s standard deviations (SD) of the obser- 4 Xia Bing-hu and Shi Xi. Horizontal vibrations of vation point in the x direction and y direction at the first second high-speed elevator with guide rail excitation, Ma- were calculated, then SDs of the acceleration response were chine Building and Automation, 41 (5), 161–165, (2012). divided by the acceleration mean value (MV) based on pertur- https://dx.doi.org/10.3969/j.issn.1671-5276.2012.05.052

222 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Ch. Wang, et al.: ANALYSIS OF TRANSVERSE VIBRATION ACCELERATION FOR A HIGH-SPEED ELEVATOR WITH RANDOM PARAMETER. . .

Table 2. Acceleration response sensitivity of random parameters. Parameters Sensitivity b1(m) b2(Jox) b3(Joy) b4(Joz) b5(k) ¨ E ∂Xx 1.01×10−4 0 4.38×10−5 0 1.94×10−5 bj

¨ ∂Xy E 1.11×10−4 5.42×10−5 0 0 2.84×10−5 bj

b6(c) b7(l1) b8(l2) b9(l3) b10(l4) ¨ E ∂Xx 2.67×10−6 1.17×10−1 1.17×10−1 2.36×10−1 3.40×10−1 bj

¨ ∂Xy E 3.72×10−6 0 0 2.44×10−1 4.49×10−1 bj

Table 3. Acceleration response sensitivity of random parameters. Time [s] 0.2 0.4 0.6 0.8 1.0 0.0037 -0.0064 0.0380 -0.0905 0.0106 X Direction (1.5135) (1.3125) (0.7052) (0.7292) (4.5094) MV (CV) 0.0124 -0.0190 0.1481 -0.3555 0.2438 Y Direction (0.7661) (1.8684) (0.7353) (0.2472) (0.5947) X Direction 0.0056 0.0084 0.0268 0.0660 0.0478 SD Y Direction 0.0095 0.0355 0.1089 0.0879 0.1450

5 Jie Li and Songtao Liao. Response analysis of stochastic vehicle suspension system based on perturbation parameter structures under non-stationary random excita- method, Journal of Central South University (Sci- tion, Computational Mechanics, 27 (1), 61–68, (2001). ence and Technology), 43 (4), 1320–1324, (2012). https://dx.doi.org/10.1007/s004660000214 http://www.zndxzk.com.cn/down/upfile/soft/20120419/20- p1320-91851.pdf 6 Zhang Qiang, Shi Feng-ying, and Liu Jian-xin. Analy- sis of frequency response function of structures with ran- 13 Wang Shi-long, Tian Bo, Zhao Yu, et al. Im- dom parameters. Journal of Vibration and Shock, 25 proved shock load model of stranded wires heli- (2), 64–66, (2006). https://dx.doi.org/10.3969/j.issn.1000- cal springs based on perturbation method, Journal 3835.2006.02.017 of Mechanical Engineering, 51 (7), 85–90, (2015). https://dx.doi.org/10.3901/JME.2015.07.085 7 Zhang Hong-qi, Shao Xiao-dong, Hu Xiang-tao, et al. Reliability assessment for mechanical structure based on 14 Hou Gong-yu, LI Jing-jing, Zhao Wei-wei, et al. Pertur- stochastic finite element method, Journal of Southeast Uni- bation solutions for elasto-plastic problems of circular tun- versity (Natural Science Edition), 44 (1), 76–81, (2014). nel under unequal compression, Chinese Journal of Rock https://dx.doi.org/10.3969/j.issn.1001-0505.2014.01.014 Mechanics and Engineering, 33 (2), 3639–3647, (2014). https://dx.doi.org/10.13722/j.cnki.jrme.2014.s2.031 8 Arrasate, X., Kaczmarczyk, S., Almandoz, G., et al. The modelling, simulation and experimental testing of the dy- 15 Guo Zhi-gang and Sun Zhi. Modal analysis of Euler- namic responses of an elevator system, Mechanical Sys- Bernoulli beam with multiple open cracks based on per- tems and Signal Processing, 42 (1), 258–282, (2014). turbation method, Journal of Vibration And Shock, 32 https://dx.doi.org/10.1016/j.ymssp.2013.05.021 (10), 1–6, (2013). https://dx.doi.org/10.3969/j.issn.1000- 3835.2013.10.001 9 Mei De-qing, Du Xiao-qiang, and Chen Zi-chen. 16 Vibration analysis of high-speed traction elevator Niu Ming-taom Li Chang-sheng, and Chen Li-yuan. based on guide roller-railcontact model, Journal of Perturbation methods for structural-acoustic cou- Mechanical Engineering, 45 (5), 264–270, (2009). pled system with interval parameters, Journal of https://dx.doi.org/10.3901/JME.2009.05.264 Vibration And Shock, 34 (10), 194–198, (2015). https://dx.doi.org/10.13465/j.cnki.jvs.2015.10.034 10 Du Xiao-qiang, Mei De-qing, and Chen Zi-chen. Time- 17 varying element model of high-speed tractiong elevator Zhang Yi-min Liu Qiao-ling, and Wen Bang-chun. Re- and its horizontal vibration response analysis, Jour- sponse analysis of multi-degree-of-freedom nonlinear vi- nal of Zhejiang University(Engineering Science), 43 (1), bration systems with random parameters of probabil- 148–152, (2009). https://dx.doi.org/10.3785/j.issn.1008- ity perturbation finite element method, Chinese Jour- 973X.2009.01.02 nal of Computational Mechancs, 20 (1), 8–11, (2003). https://dx.doi.org/10.3969/j.issn.1007-4708.2003.01.003 11 Feng Yong-hui and Zhang Jian-wu. The modeling and 18 simulation of horizontal vibrations for high-speed eleva- Fang De-ping and Wang Quan-feng. Accuracy analysis tor, Journal of Shanghai Jiaotong University, 41 (4), of modified Wilson-theta method of acceleration, Jour- 557–560, (2007). https://dx.doi.org/10.3321/j.issn:1006- nal of Vibration And Shock, 29 (6), 216–218, (2010). 2467.2007.04.010 https://dx.doi.org/10.3969/j.issn.1000-3835.2010.06.048

12 Jia Ai-qin, Chen Jian-jun, and Xu Ya-lan. Convex model analysis of vibration control eigenvalues of

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 223 Model-Based Algebraic Approach to Robust Parameter Estimation in Uncertain Dynamics Rotating Machinery

Gerardo Pelaez, Pablo Izquierdo, Higinio Rubio, Manuel P. Donsion and Juan Carlos Garcia-Prada Department of Mechanical Engineering, University of Vigo (Pontevedra), Spain Deptartment of Mechanical Engineering, University Carlos III, Leganes (Madrid), Spain

(Received 29 January 2016; accepted 28 October 2016) This work presents a model-based algebraic approach to robust parameter estimation in uncertain dynamics rotating machinery. The approach evades some mathematical intricacies of the traditional stochastic methods, proposing an affordable Jeffcott-model-based scheme with several easy-to-implement computational advantages for processing a real-world rotor frequency response or orbit. Therefore, it takes out the dynamic parameters from one of the orbit’s resonant humps when the multistage rotor orbit shape behaves closely to the Jeffcott-model orbit. This occurs for a valuable array of cases. The approach applies the spatial `2-norm looking forward to the correlation between the analytical Jeffcott-orbit model and the experimental rotor’s orbit hump, handling the normalized frequency ratio. Experimental results are also included to face this method with real-world rotating machinery orbits.

1. INTRODUCTION dom model proposal for an unbalanced rotating machine dates back to 1919, when Henry Hoffman Jeffcott developed a sim- Rotating machinery vibration response reflects the comple jet representative model based on a uniform flexible shaft bined interaction of dynamic and stationary machine elements, over rigid bearings at its ends.4 The shaft mass plus the stage in fact the history of rotodynamics consist in a collection of mass is concentrated at the shaft center. The Jeffcott rotor trade off between theory and practice dealing with the vibra- (named after Henry Homan Jeffcott), also known as the Gustaf tion response. There are situations where the experimental data de Laval rotor in Europe, is a simplified lumped parameter of a case study rotor are unavailable due to some constraints. model. By this time 1920, the critical speeds of shafts with For instance, the machine running speed cannot be increased distributed masses were examined by Grammel.5 In the same to investigate the detrimental effects of a resonant mode due period a discussion of shaft stability conditioned by internal to hazardous conditions. If this is the case, then modelling be- friction forces was given by Kimball and Newkirk.6, 7 Also, comes the most appealing option. Modelling solutions include: the turbine expert A. Stodola made an experimental analysis Transfer Matrix Methods, Finite Element Analysis (FEA), and dealing with the gyroscopic effect of a disc mounted on a shaft affordable analytical approachs. For the first and second cases, and the secondary resonance phenomena.8 Nevertheless, the there are quite valuable computational plus time-cost penalties. presence of disturbing forces is not restricted to unbalancing Alternatively, there is a wide array of the cases where there nature forces, in plain words a keyed shaft generates variable is a chance to develop rotor vibration field measurements un- stiffness disturbing forces that excite resonance at angular ve- der variable running speed. As an illustrative example, con- locities under the criticals.9 sider the pre-programmed shut downs and re-starts of a refin- During the decades of 1940’s and 1950’s other similar cases ery unit in spite of the economic impact. If this is the case were pointed out by Natanzon, Bogdanoff, Green and Dick.10 from a problem solving standpoint, then an investigation needs However, it was not until the next decade of 1960’s when ex- to be by comparing the predicted machine behaviour based on haustive analysis of problems relating to determination of nat- an affordable analytical model. The experimental measured ural vibration frequencies was developed. This analysis was behaviour then becomes the most appealing option. Thus, it is given by Dimentber as a result of experiments carried out at not surprising that a large amount of research and development the ’Elektrosila’ Lab and the Leningrado Foundry.11 This is a has gone to deal with this subject as will be shortly disclosed more general problem than the determination of critical speeds in the following. because the latter are special frequency values of natural vi- To understand the fundamental behaviour of a rotor, per- brations, which match the shaft speed of rotation at a given op- haps the first step should include determining the frequency eration time. The valuable contribution of his hypotheses and of the system’s critical speeds. Historically, one of the ear- propositions about shaft stability, verified experimentally, can liest procedures for critical speed calculations was developed be hardly emphasized at this point. By this time another no- by Julius Frith and Ernest Lamb (1901) dealing with torsional table contribution on the field thanks to Den Hartog, a mechan- vibrations involved by brake of shaft with different inertial ical engineer at Westhinghouse company, must be mentioned.9 loads.1 However, in the beginning of the 1870th, transverse Anyway, in all of these cases the shaft vibrations analyses is cracks occurred in the heavy horizontal rotors of some tur- developed under a varying rotation speed. bosets. This scenario stimulated the research on transverse During the 1980s, progress in analytical rotor modeling has vibrations in shafts. Rankine (1869) and Laval (1889) devel- been closely associated with the process of the transfer ma- oped the primary theory corresponding to the determination of trix calculations. On the other side, during 1980s, the rotating critical speeds by modeling flexible shafts with concentrated machinery monitoring, which was based on the signals sup- and distributed masses.2, 3 A benchmark one degree of free- pield by sharp analog probes and digital keyphasors plus on-

224 https://doi.org/10.20855/ijav.2017.22.2468 (pp. 224–232) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 P. Izquierdo, et al.: MODEL-BASED ALGEBRAIC APPROACH TO ROBUST PARAMETER ESTIMATION IN UNCERTAIN DYNAMICS. . .

line digital signal processing, facilitated the progress of pre- ONE HUMP THEORETICAL ROTOR ORBIT ventive fault detection and field balancing. To this goal, the rotor orbit, i.e., the log of transversal amplitudes of vibration 90 plus phase delay of the deflective response, depicted in polar -5 120 60 7 10 y=1/2 ζ ω ωn -5 = coordinates, reveals as a quite valuable tool as demonstrates 6 10 y’= ζ 12 the popular work of Donald Bently. Also, the primary works 5 10 -5 13 RADIUS -5 ρ of Eisenman reinforce this notion. He is a prolific author on 4 10 n 30 PHASE the field of rotating machinery fault detection and balancing. 3 10 -5 -5 At the 1990th, Eisenman’s son got an application to work as 2 10 ρ manager of rotating equipments at a refinery. He came back 1 10 -5 -1 home almost every day carrying a question on the subject for 180 0 ω/ ωn =0 his father Eisenman Sr. The answers to the questions became PHASE the main argument of a new textbook written by both.14 Any- way, in most of the aforementioned works the rotor orbit shape becomes paramount to examine the shaft vibrations nature resulting form varying rotation speed. Between others the orbit Figure 1. Jeffcott-model rotor orbit. Conditions: ωn=23.56 Hz, ζ=0.1. shape allows to characterize cracked or bowed shafts, mechanical looses, i.e., machinery malfunctions. world multi-stage rotor plus continuos and keyed shafts, are However, the estimation of the frequency corresponding to given. Also, a discussion on the aforementioned parameter es- system critical speeds from the orbit requires a more detailed timation good performance. Finally, section-5 includes some analysis than the only consideration of the orbit shape referred modest conclusions. above. In fact, the exact frequency at which it occurs has a rather valuable uncertainty associated due to the fast phase 2. THE MODEL variation at such resonant region. Overcome this drawback become one of the arguments for the present work. Dealing with the wide array of machines with rolling element bearings without resilience at the shaft journals, usually By the 21st century, a wide array of works have been pub- synchronous whirling takes place. In plain words, the flexi- lished which deal with the stability analysis and control of un- ble shaft and the inertial stages whirls with the same angular balanced shafts vibrations. The stability analysis of the Laval’s velocity, so that rotate together as a rigid body. Also, if the rotor with a transverse crack was developed.15 Vibrations iso- inertial stages are close to the shaft center and the customary lation with a semi-active pneumatic damping was also pre- range of rotating velocities behaves moderate, the gyroscopic sented.16 Auto-balancing using speed dependent vibration ab- effects can be assumed moderate as well. If this the case, the sorbers was also investigated.17 Radial magnetic force was resonant hump shape of a multi-stage rotating machine orbit, used for shaft vibration suppression to go through the first crit- behaves closely to the orbit shape of a Jeffcott-model equiva- ical speed.18 Under a periodic axial force, a rotating Timo- lent machine.22 shenko shaft with a rigid unsymmetrical disc was modelled as a parametrically excited system using the finite-element method Regarding the conditions stated above, the stationary fre- and a harmonic balance method.19 Undesired whirling mo- quency response of a flexible shaft without resilience at the journals and the inertial stage close to the shaft center has been tions of rotating machines are efficiently reduced by support- 11 22 ing journal boxes elastically and controlling their movement by exhaustively addressed in the the literature , and is given by viscous dampers or by dry friction surfaces normal to the shaft 20 2 axis. Damping lateral vibrations using rotor speed modula- ω eu eu 21 n tion also become a recent alternative presented. The method X(iω) = 2 2 = ω 2 ω . ω − ωn + 2ζωωni (( ) − 1)) + i2ζ is based on the generation of an harmonic additive to the con- ωn ωn (1) stant rotation speed, that provides significant damping at criticals. This suggests the notion that the influence of the electric Nonetheless, an interesting geometric translation can be ex- drive-converter can not be neglected dealing with real-world tracted from this expression given in terms of the normalized rotating machines powered by electric motors plus converters. frequency ratio. To this goal, let it be X(iω) = x − iy a customary complex number, q = ω the normalized frequency This issue is taking into account in the present work, by in- ωm stalling a well suited passive LC filter behind a three phase a non-dimensional speed of rotation that states the frequency PWM driver. This set up guarantees that a sinusoidal modu- ratio between the actual rotating frequency and the assumed or modeled natural frequency. This normalized value q, is usu- lation feed the asynchronous electrical motor minimizing har- 23 monics detrimental effects in the experimental orbits. ally adopted for robust filters design. Note that the value anticipated by the referred benchmark model for the natural The present work is organized as follows: in section-2 sym- q k 3 bolic computations, involving the normalized frequency ra- frequency would be ωn ' ωm = m where k = 48EIz/` tio, over the analytical Jeffcott-Orbit expression, are devel- and m states the mass of the inertia disk. Finally, without lost oped. These will facilitate in section-3 synthesize an advan- of generality eu is allowed to be the unit, then Eq. (1) becomes tageous analytical formulation, regarding the orbit extreme points properties. Thus, the estimation of the actual critical 1 x − iy = . (2) frequency on the basis of the correlation between the experi- (q2 − 1) + 2ζqi mental multi-stage rotor orbit-hump to the equivalent Jeffcott- The use of algebraic manipulations to expand Eq. (2) as a func- model-rotor orbit by Least Squares Fitting algebraic method. tion of the normalized frequency ration q gives Later, the synthesized analytical formulation to estimate the corresponding Jeffcott-model-rotor parameters was applied: x(q2 − 1) + 2ζyq = 1; (3) stiffness of the shaft, damping ratio and the unbalance force phase. In section 4, experimental results dealing with a real- 2ζxq − (q2 − 1)y = 0. (4)

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 225 P. Izquierdo, et al.: MODEL-BASED ALGEBRAIC APPROACH TO ROBUST PARAMETER ESTIMATION IN UNCERTAIN DYNAMICS. . .

Table 1. Jeffcott Model Concerned Parameters.

Machine model parameter Assumed value 90 E Young module, 206,000 MPa 120 60 y=1/2ζ Rotor Diameter, d 0.015 m w=wn 4 −9 4 y’= ζ Second moment of the cross section, Iz (πd /64)2.48 · 10 m r Rotor length, l 0.500 m n 30 PHASE Unbalanced distance, eu 0.010 m

Asynchronous motor nominal velocity 1500 rpm r Disk mass, m 9 Kgr. 180 -1 w w 0 PHASE /n =0

(maximum value) R0 R 1 w0 Eliminating the parameter q between Eq. (3) and Eq. (4) can Ri R2 be done by firstly rewriting them as w wi 1 w2 1 − 2ζyq (q2 − 1) = ; (5) x Figure 2. Sketch for comparison of the actual orbit and the Jeffcott-model orbit. 2ζxq (q2 − 1) = ; (6) y Probe Keyphasor Keyphasor signal it follows that Probe 2 2 Dynamic signal y − 2ζy q = 2ζx q; (7) 45 therefore y q = . (8) ω 2ζ(x2 + y2) ϕ :Unbalanced force phase ϕ Substituting the q parameter value given by Eq. (8) into Eq(3) yields Weighted point (Dynamic Signal 2 2 2 2 2 2 2 2 xy − x(2ζ(x + y )) + 2ζy + 2ζy 2ζ(x + y ) High point) Unbalance inertial force = (2ζ(x2 + y2))2; (9) reordering terms and rejecting the null solution, Eq. (9) be- Figure 3. Analog Probe plus digital Keyphasor layout at the measurement comes bridge. y2 − 4ζ2x4 − 8ζ2x2y2 − 4ζ2y4 − 4ζ2y2x − 4ζ2x3 = 0. (10) 1. The parameters ρ and q involved by the Jeffcott-model orbit are dimensionless while the real world hump values: At this point, acknoledging that x2 << y2, rejecting second R the amplitude, and ω the frequency have dimensions. order terms closed to zero Eq. (10) yields y2 − 4ζ2y4 − 4ζ2y2x ' 0. (11) 2. With reference to the initial and final points of the Jeffcott-model orbit, the actual orbit lacks such points or This a quadratic equation in y, again rejecting the trivial solu- regions corresponding to the cases when ω equals zero tion it simply becomes and ω equals ∞, which can not be reached. 1 ' 4ζ2(y2 − x). (12) 3. In fact, each hump of the orbit is not placed at the theoretical orientation, its origin is arbitrary and it is a function of other humps that the actual orbit might have. The Eq. (12) expresses a parabola curve whose hump inter- 4. Finally, the shape of an orbit’s hump would differ from 1 section with the y-axis is at the point y = 2ζ . Differentiating the Jeffcott-model orbit. Eq. (12), the slope value at such point is y˙ = ζ, depicted in Fig. 1. Thus, using this property, it is feasible to extract the Anyway, this work attempts to overcome these drawbacks. Un- damping ratio from the theoretical rotor orbit. Also accord- less otherwise stated the availability of experimental reliable ingly, the orbit hump size is a function of the damping ratio. In measurements corresponding to the rotor deflection Ri at the general terms, as the damping increases the orbit size reduces. frequency ωi, are quite valuable. The maximum deflection Anyway this is only one of the outcomes that can be obtained R0 of the orbit hump is a reliable value, however the exact from the orbit geometric properties. In fact, the orbit hump frequency at which it occurs ωn versus ω0 have a quite valu- provides quite valuables tracks in order to analyse the dynamic able uncertainty associated due to the fast phase variation at behaviour of the rotor. such resonant region. Actually, the most affordable experimental approach consist in measure the inertial unbalanced force 3. METHODS phase depicted in Fig. 3. Two signal are involved: the periodic signal supplied by the keyphasor plus the analog signal sup- Actually, dealing with a real world rotating machine orbit, plied by the probe. For a given rotor angular velocity such an- several humps might take place. Each one corresponds to a res- gular phase, keeping unchanged the angular velocity, remains onant situation. All of them lumped together build the whole proportional to the time delay between the analog signal high rotating machine orbit. In addition, each hump has its own ori- point, corresponding to inertial stage weighted point, and the entation in the complex plane. Thus, between each hump of a keyphasor flank as shows the Fig.4 real world rotating machine orbit and the Jeffcott-model orbit, The experimental measurement outcomes resulting from the depicted in Fig. 2, the following differences can be found: just described set up reinforces the notion of the fast phase

226 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 P. Izquierdo, et al.: MODEL-BASED ALGEBRAIC APPROACH TO ROBUST PARAMETER ESTIMATION IN UNCERTAIN DYNAMICS. . .

Keyphasor periodic signal dimensionless amplitude of vibration ρ can be expressed in Probe dynamic periodic signal terms of its corresponding normalized frequency ratio q = ω ωm ω0 tt2 - tt1 th1- tt1 plus q0 = , without involving the damping ratio ζ parame- = ωm 360 ϕ ter. 1 ρ = . (17) q 2 2 1 2q0 q q4 − q2 + 1 Time tt1th1 tt2 Next, this result was used to solve the correlation between an experimental rotor orbit and the theoretical one synthesized above on the basis of the assumed or modeled natural fre- Figure 4. Time Domain Output signals: Keyphasor plus probe dynamic periodic signals. quency ωm and damping ratio ζm.

Estimation of the actual critical frequency on the basis of the experimental orbit correlation with the Jeffcott-model orbit. At this point, the experimental rotor orbit will be used to extract the actual critical frequency, that coincides with a natural frequency under the resonant situation. To this end, as shown in Fig. 2, let it be the ratio between any experimentally measured amplitude of vibration Ri and the maximum measured amplitude R0. Note that R0 is a reliable value and able to be easily obtained from the experimental measurements

2 Ri 1 2 = ; (18) R0 Zi

where the dimensionless parameter Zi > 1 states the inverse of such ratio. And for each frequency ωi corresponding to the amplitude Ri, let it be

2 2 ωi − ω0 bi = 2 ; (19) ω0 Figure 5. The phase variation at the resonant region (experimental data). 2 the squared normalized ` -norm. Recalling that q = ω/ωm variation corresponding to the inertial force when the rotor tra- states again the normalized frequency. Thus, dividing the sec- 2 verses the critical, as demonstrated overall in Fig. 5. In view ond’s term, num and den, of Eq. (19) by ωm is equivalent to of this, to estimate the actual critical frequency ω of a multi- n 2 2 stage rotating machine from an experimental orbit-hump at a qi = q0(1 + bi); (20) resonant zone, the following analysis is developed. where ω states the actual global machine critical frequency. First, recall from Eq. (2) that the theoretical amplitude of 0 Looking forward, the correlation between the experimental and vibration ρ can be expressed by the theoretical orbit yields 1 ρ = . (13) R1 R2 Ri R0 p(q2 − 1)2 + 4ζ2q2 = = ··· = = ··· = ; (21) ρ1 ρ2 ρi ρ0

Let it be ρ0 the maximum dimensionless rotor deﬂection am- thus, according to Eq. (18) and (21) it can be written plitude that takes place at the normalized frequency ratio q0. If this the case the condition that occurs, at this extreme, is R2 ρ2 1 i = i = ; (22) R2 ρ2 Z dρ 0 0 i = 0. (14) 2 dq R0 0 where Zi = 2 is a dimensionless scalar. At this point, substi- Ri From this point, Maple 13 was used to carry out the values of tuting Eq. (17) into Eq. (22) yields,

ρ0 and correspondingly q0 plus other symbolic computations. 2 4 2 Deriving Eq. (13) according Eq. (14) it is ﬁrst obtained ρi 1 − 2q0 + q0 2 = 2 2 2 ; (23) ρ0 1 − 2q0qi + qi p 2 q0 = ± 1 − 2ζ ; (15) combining Eq. (23), (22), and (20) gives, by simply substituting this value of q0 as a function of ζ, versus 1 1 − q4 Eq. (15) into Eq. (13) yields, = 0 ; (24) Z 1 − q4 + q4b2 1 i 0 0 i ρ0 = . (16) q 2 2 1 2 2ζ rearranging terms, Eq. (24) becomes (1 − 2ζ ) (1 − 1−2ζ2 ) + 1−2ζ2 1 − q4 b2 = (Z − 1) · 0 ; (25) In addition and inversely, by using symbolic computation any i i 4 q0 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 227 P. Izquierdo, et al.: MODEL-BASED ALGEBRAIC APPROACH TO ROBUST PARAMETER ESTIMATION IN UNCERTAIN DYNAMICS. . . Table 2. Experimental Measurements Involved Ratios. 5

2 2 2 2 ωi −ω0 2 R0 i-input Ri ωi bi = ( 2 ) Zi = 2 ω0 Ri 1 10.13 26.4061 0.0315 1 2 10.13 26.441 0.0342 1 3 10.0789 26.476 0.037 1.0102 4 10.0782 26.5252 0.0408 1.0103 5 10.0747 26.5393 0.0419 1.011 6 10.0785 26.5887 0.0458 1.0102 7 10.0805 26.6241 0.0486 1.0099 8 10.0779 26.6667 0.0519 1.0104 9 10.0769 26.7094 0.0553 1.0106 10 10.0776 26.7237 0.0564 1.0104 11 10.0798 26.7666 0.0598 1.01

2 Figure 6. The Least Square polynomial ﬁtting for the set of points (bi , Zi). ﬁnally, letting 4 1 − q0 Q = 4 ; (26) q0

ω0 where q0 = . Thus the Eq. (25) can be simply rewritten as ωm

2 bi = Q · (Zi − 1). (27)

2 A linear relationship between bi and Zi has been found. Such linear polynomial traverses the point (1,0) of the Zi−bi2 plane. In addition, according to Eq. (26), its slope Q links the q k prime concern modelled frequency ωm = m of the Jeffcott model ( ωm = 23.46[Hz] according to table 1), and the actual global machine critical frequency, ω0. However, given a 2 set of experimental points (Zi, bi ) coming from the measured Figure 7. Convergence of the LQE linear polynomial fit procedure. Constrain: response in the frequency domain configuring the rotating ma- traverse the point (1,0) of the Zi − bi2 plane. chine orbit, there is a mismatch between this set of experimen- 2 tal points (Zi, bi ) and the theoretical linear polynomial found. In view of this scenario, the most appealing option comprises a Least Square polynomial fitting procedure shown in Fig. 6. The first step consists in building a table of the type 2 shown in Table 2. Given the dataset (Zi, bi ) the method of 2 least squares minimizes the error associated to saying bi = Q · (Zi − 1) by

3 X 2 2 min → E(Q) = (bi − Q · (Zi − 1)) . (28) i=1 To this goal, it is necessary the availability of an initial modelled or guess frequency ωm. Recall that Q is a function ω0 of q0 and q0 is a function of ωm because q0 = . However, ωm for such a guess frequency, the outcoming line obtained from the ﬁtting process, must traverse the point (1,0). If this Figure 8. Experimental set up. constraint is not veriﬁed, the value of the assumed modeled frequency ωm must be shifted upwards or downwards by Shortly, the procedure just described provides an affordable the algebraic algorithm until the linear polynomial traverses method for evaluating the corresponding critical frequency of the constraint point as shown in Fig. 7 and Table 3. Once a an orbit hump. The relevance of this procedure can be stated by well suited ωm value has been reached from the least squares recalling that the exact frequency at which it occurs ω versus method, the slope Q of the corresponding linear polynomial n ω0 have a quite valuable uncertainty associated due to the fast allows to carry out ω0 using phase variation at such resonant hump-region. 1 However, this is only the primary result. In addition, the ω0 = ωm √ . (29) 4 1 + Q presented method might be helpful to identify others dynamics parameters. To that goal let assume that the mass value of the inertial stage is known, this is a quite reasonable supposition.

228 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 P. Izquierdo, et al.: MODEL-BASED ALGEBRAIC APPROACH TO ROBUST PARAMETER ESTIMATION IN UNCERTAIN DYNAMICS. . .

Table 3. Convergence of the LQE linear polynomial ﬁt procedure.

2 ωm linear polynomial Zi bi 2 26.5 bi = 0.006684Zi + 0.5853 1 0.591984 2 33 bi = 0.004616Zi + 0.0002295 1 0.0048455 2 34 bi = 0.004348Zi − 0.05774 1 -0.0533952

Thus, from Eq. (26) the shaft stiffness can be obtained as,

2 p k = ω0m 1 + Q. (30) The damping ratio can be easily obtained from Eq. (15) r 1 − q2 ζ = 0 . (31) 2 Or equivalently identify the damping of the benchmark model q 2 c = km(1 − q0) (32)

Dealing with the phase delay of the shaft response to the unbalance force F at the system critical ω0, it is given by 2ζq Figure 9. Unbalanced rotor experimental orbit. Conditions: linear response. tan(ϕ ) = 0 . (33) 0 1 − q2 0 the actual critical frequency estimated according Eq. (28) was −4 The direction of ρ0 can be found into the orbit, in view of this, ω0 = 32.962 ± 7 · 10 Hz, carrying a low level of uncertainty from Eq. (33) the direction of the unbalance excitation force associated, as shows the figure 11. F , that causes the rotor orbit, can be also estimated as shown in Fig. 9. Finally, the module of F can be found because 1 R0k = F ρ0. (34) ω0 = 33 √ 4 1 + 0.004616 ± 8.35 · 10−5 Thus, from Eq. (17) and Eq. (33) it is obtained = 32.962 ± 7 · 10−4 Hz. (36) q F = kR 1 − q4. (35) 0 0 To verify this result, a finite element model of the corresponding Jeffcott-model rotor plus the electrical motor was At this point, the procedure just described above allows built, as shown in Fig. 12, giving a first natural frequency of to estimate the corresponding Jeffcott-model parameters from around 44 [Hz]. Regarding that, in general terms, the FEM one of the orbit’s resonant humps, when the multi-stage rotor method is an extension of the Rayleigh’s energy approxima- orbit shape behaves closely to the Jeffcott-model orbit. If, as tion method,24 it is customary to obtain for the lowest natu- m it is customary, the mass is the parameter whose value is ral frequency a value which is somewhat quite high.9 This available then the a simple look up method was given above to result reinforces this notion, when compared with the exper- estimate the addressed model parameters. imental value obtained above, extracted from the actual real world multi-stage rotor orbit. 4. MATERIALS Assuming that the mass m is the parameter whose value is 4.1. Continuous Shaft Rotor known given in table 1, the corresponding Jefcott-model shaft stiffness can be estimated according to Eq. (29), To demonstrate these assessments, a multi-stage rotating machine experimental response to a varying velocity was col- √ 2 p 2 lected. The sketch of the machine is shown in Fig. 8. The k = ω0m 1 + Q = (32.962 ∗ 2π) · 9 · 1 + 0.004616 machine is composed by two inertial stages with associated = 3.87 · 105 N/m. (37) measurement planes consisting of a bridge where two analog probes plus a digital keyphasor have been installed. Table 1 48EIz includes the theoretical parameters corresponding to the mod- The modeled value for the shaft stiffness were km = `3 = eled shaft stiffness plus individual disk inertia, when the two 1.96·105 N/m according to Table 1. The estimated value found stages are coupling to the asynchronous motor. Finally, the fre- for the stiffness differs largely from the beam-model one, actu- quency response bellow the second critical frequency is shown ally it is nearly twice the initially modelled value. The proce- 2 in Fig. 9. This orbit was translated to the Zi −bi plane depicted dure also allows to estimate the damping ratio value. Accord- in Fig. 10. This dataset underwent the Least Square fitting pro- ing to Eq. (30), it is given by cedure with the help of MATLAB R Toolstrip. The MATLAB- script procedure converges to a solution for ω = 33 Hz as m r s demonstrates the Fig. 7. Also, the slope of the optimal lin- 1 − q2 1 − ( 32.96 )2 ζ = 0 = 33 = 0.0335; (38) ear polynomial found was Q=0.004616 ±8.3510−5. Thus, 2 2 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 229 P. Izquierdo, et al.: MODEL-BASED ALGEBRAIC APPROACH TO ROBUST PARAMETER ESTIMATION IN UNCERTAIN DYNAMICS. . .

2 Figure 10. Transformation of the orbit into the Zi − bi plane. Figure 11. Estimation of the actual critical frequency. Conditions: slope Q=0.004616 ±8.35 · 10−5. or equivalently the damping s q 32.962 c = km(1 − q2) = 3.87 · 105 · 9 · (1 − ) 0 33 s = 88.34 · N . (39) m At this point, the whole set of fundamental dynamic parameters of the Jeffcott-model corresponding to the real world multi-stage rotor has been identiﬁed. However, the question of the unbalance force direction remains still unanswered. Such direction can be speciﬁed by Eq. (32) according to

32.96 2ζq0 2 · 0.335 · 33 4 tan(ϕ0) = = = 1.0069 · 10 ; (40) 1 − q2 32.96 2 0 1 − 33 0 this equation leads to a value for ϕ0 of around 88 . Thus, to produce a well suited balancing force at such resonant rotating velocity it should nearly have the opposite direction as shows Figure 12. Estimation of the assumed modeled frequency by the Finite Ele- Fig. 9. Anyway, the balancing procedure has been exhaustively ment Method. Result: ωm ' 44 Hz. addressed in textbooks on the subject.14 The customary ap- ω0 proach establishes the balancing force direction by setting up a critical speed 2 , the impulses of this force will occur at the ro- vector from the beginning to the end of the resonant hump, as tating machine natural frequency. If this occurs, the case a res- shown in Fig. 13 (keeping in mind the angular reference of the onant situation occurs delivering a hump into the whole orbit at ω0 physical rotor). As can be see both directions, the one depicted such frequency: ω = 2 . Thus, a two humps orbit is expected. in Fig. 9 and the second one shown in Fig. 13 agree. Also, this The first hump occurs closed to the half critical speed plus the univocal direction has demonstrated to be effective regarding second one at the first critical. Both appear in the experimental the balancing purposes as corroborates the Fig. 14. This fact orbit depicted in Fig. 15. The main hump with constant radius reinforces the notion that the results obtained by the presented seems to be a circle. The reason can be found in the saturation method agree with the previously mentioned literature on the of the analog probes as a consequence of the vibration severity. subject. This is a weak point of the proposed method. As the pollution of the Ri experimental measurements increases, the reliability 4.2. Keyed shaft rotor of the method obviously decreases. furthermore, the transla- 2 tion of the two humps orbit to the Zi − bi plane is given in For the case of a keyed rotor referring to Fig. 15, the orbit Fig. 17 reinforcing the notion of a complex dynamic behavior. has two humps showing a more complex dynamic response. Also, Table 4 was built using the same experimental measured During the whole revolution, the keyed shaft stiffness passes data. through two cycles, as shows the Fig. 16. Let it be the min- After the fitting procedure, the guess frequency correspond- imum value k - ∆k , and the maximum k + ∆k, with and ing to the linear polynomial, traversing the constraint point average value of k. Then for uniform rotating speed the in- (1,0) of the Z − b2 plane, is ω = 26.5, such polynomial stantaneous stiffness can be expressed by i i m have a slope Q of 0.1772 . Finally, the actual natural frequency k(t) = k + ∆k · sin(2ωt). (41) found was The variable component of the spring force becomes a disturb- 1 ω0 = 26.5 √ = 25.44 Hz. (42) ing force of frequency 2ω. If the shaft is running at its half 4 1 + 0.1772 230 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 P. Izquierdo, et al.: MODEL-BASED ALGEBRAIC APPROACH TO ROBUST PARAMETER ESTIMATION IN UNCERTAIN DYNAMICS. . .

Figure 13. Balancing direction estimation, by setting up a vector from the Figure 15. Experimental keyed rotor orbit that was polluted by probes satura- beginning to the end of the orbit hump. Conditions: clock wise rotation. tion, including one hump of constant radius.

Figure 16. Variable stiffness of the rotor induced by the slot along its length. Conditions: b=5 mm,t=2.65 mm,d=15 mm. Iz = 2485−331.25·10−12 m4.

Table 4. Convergence of the LQE linear polynomial ﬁt procedure for the keyed-rotor.

2 ωm linear polynomial Zi bi 2 ωm = 25 bi = 0.1991Zi − 0.07652 1 0.12258 2 ωm = 26 bi = 0.1841Zi − 0.1462 1 0.0379 2 ωm = 26.5 bi = 0.1772Zi − 0.1781 1 0.0009 Figure 14. Time-based dynamic signals and polar plot: (a) unbalanced rotor and (b) balanced rotor (two correction weights added). Conditions: mechanical frequency 20 Hz, discrete correction weight 8gr at the inertial disk surface. to estimate the most descriptive parameters for uncertain The uncertainty is not important here due to the polluted dynamics rotating machinery has been developed. Thus, the measurement data. However, in spite of something corrupted critical frequency, the stiffness of the shaft, the damping ratio data, as expected from a qualitative view point, the global ro- plus the unbalance force phase, are estimated in a straight tating machine natural frequency is shifted down this is caused forward manner on the basis of the correlation between any by the stiffness loose of the keyed shaft if compared with the experimental real-world rotating machine orbit’s hump to the continuous shaft without the slot along it. This statement can Jeffcott-orbit. This comprises an affordable Least Square be proved by applying the Eq. (29) to the keyed shaft to esti- polynomial ﬁtting procedure applied to the log of to the ex- mate the corresponding Jeffcot-model stiffness perimental orbit. The results show clearly the conformity with the physical evidence and accuracy for the aforementioned √ 2 p 2 parameters estimation. k = ω0m 1 + Q = (25.44 · 2π) · 9 · 1 + 0.1772 = 2.49 · 105 N/m. (43) This is a lower value than the prior case of the non-keyed shaft, verifying the expected theoretical result, and reinforces ACKNOWLEDGEMENTS the presented method conformity with the experimental facts. The authors wish to acknowledge the ﬁnancial support of 5. CONCLUSIONS the Spanish government through Ministerio de Ciencia e Inno- vacion´ and FEDER Funds. Research project ENE2007-6803- At this point, a Jeffcott-model algebraic method that makes C04-01 plus research project MAQ-STATUS DPI2015-69325- affordable analytical expressions available for engineers C2-1-R (MINECO/FEDER, UE).

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 231 P. Izquierdo, et al.: MODEL-BASED ALGEBRAIC APPROACH TO ROBUST PARAMETER ESTIMATION IN UNCERTAIN DYNAMICS. . .

12 Bently, D. E. Polar plotting Applications for Rotating Ma- chinery, Proceedings of the Vibration Institute Machinery Vibrations IV Seminar, Cherry Hill New Jersey, (1980). 13 Eisenmann, R. C. I-Machinery Monitoring, II-Machinery Vibration., Prentice Hall Inc, (1970). 14 Eisenmann, R. C. Sr. and Eisenmann, R. C. Jr. Machinery Malfunction Diagnosis and Correction, Prentice Hall Inc, (1998). 15 Gasch, R. Dynamic behaviour of the Laval rotor with a transverse crack, Mechanical Sys- tems and Signal Processing, 22, 790–804, (2008). https://dx.doi.org/10.1016/j.ymssp.2007.11.023 16 Nieto, A. J., Morales, A. L., Chicharro, J. M., and Pintado, P. Unbalanced machinery vibration iso- lation with a semi-active pneumatic suspension. Jour- nal of Sound and Vibration, 329, 3–12, (2010). https://dx.doi.org/10.1016/j.jsv.2009.09.001 2 Figure 17. Location of the main resonant zone in the Zi − bi plane for the keyed rotor. 17 Cheng, C. C., Wu, F. T., Hsu, K. S., and Ho, K. L. Design and analysis of auto-balancer of an opti- REFERENCES cal disk drive using speed-dependent vibration absorbers. 1 Journal of Sound and Vibration, 311, 200–211, (2008). Frith, J. and Lamb, E. H. The breaking of shafts in direct- https://dx.doi.org/10.1016/j.jsv.2007.08.038 coupled units, due to oscillations set up at critical speeds, Journal of the Institution of Electrical Engineers, 31(155), 18 Chiba, A., Fukao, T., and Rahman, M. A. Vibra- 646–667, (1902). https://dx.doi.org/10.1016/b978-0- tion Suppression of a Flexible Shaft With a Simpli- 85461-076-1.50025-1 ﬁed Bearingless Induction Motor Drive. IEEE Transac- tions on Industry Applications, 44(3), 745–752, (2008). 2 Rankine, W. J. Centrifugal whirling of shafts, Engineer, 27, https://dx.doi.org/10.1109/tia.2008.921401 249, (1869). 19 Yong-Chen, P., Lu, H., and Chatwin, C. Dynam- 3 Smil, V. Creating the Twentieth Century: Technical Innova- ics of a rotating shaft-disc under a periodic ax- tions of 1867-1914 and Their Lasting Impact, Oxford Uni- ial force. Proc. Inst. Mech. Eng. Part K Jour- versity Press, (2005). nal of Multi-body Dynamics., 2, 211–219, (2010). https://dx.doi.org/10.1243/14644193jmbd209 4 Jeffcott, H. H. The lateral Vibration of Loaded Shafts in the Neighbourhood of a Whirling Speed. The effect of want 20 Sorge, F. and Cammalleri, M. Control of hysteretic of balance, Phil. Mag. series 6, 37(219), 304–314, (1919). instability in rotating machinery by elastic suspension https://dx.doi.org/10.1080/14786440308635889 systems subject to dry and viscous friction. Journal of Sound and Vibration, 329(10), 1686–1701, (2010). 5 Grammel, R. Kritische Drehzahl und Kreiselwirkung. (Crit- https://dx.doi.org/10.1016/j.jsv.2009.12.007 ical speed and gyroscope effect.), Z.Ver.dtsch. Ing., (44), (1920). 21 Verichev, N. N., Verichev, S. N., and Erofeyev, V. I. Damp- ing lateral vibrations in rotary machinery using motor speed 6 Kimball, A. L. Internal friction theory of shaft whipping, modulation. Journal of Sound and Vibration, 329, 13–20, Gen. Elec. Rev., 200–244, (1924). (2010). https://dx.doi.org/10.1016/j.jsv.2009.09.018 7 Newkirk, B. L. Shaft whipping, Gen. Elec. Rev., 27, 169, 22 Meirovitch, L. Fundamentals of Vibrations. (1924). McGraw-Hill International Edition, (2001). https://dx.doi.org/10.1115/1.1421112 8 Stodola, A. Dampf und Gasturbinen. (Steam and gas turbines.), Julius Springer, Berlin, (1924). 23 Pelaez´ G., Caparrini N., and Garc´ıa-Prada J. C. Two impulse sequence input shaper design for link 9 Den Hartog, J. P. Mechanical Vibrations, mechanism with non-linearities. Proceedings of the MacGraw-Hill 4th ed. New York, (1956). American Control Conference, New York (2007). https://dx.doi.org/10.1017/s0368393100131049 https://dx.doi.org/10.1109/acc.2007.4282326

10 Dick, J. The whirling of shafts having sections with un- 24 Graham Kelly S. Mechanical Vibrations. MacGraw-Hill equal principal bending moduli, Phil. Mag., 299, (1948). Shaum’s OutLines, (1996). https://dx.doi.org/10.1080/14786444808521703

11 Dimentberg, F. M. Flexural Vibrations of Rotating Shafts., Production Engineering Research Association Dept of Scientiﬁc and Industrial Research, (1961). https://dx.doi.org/10.1017/s0368393100076331

232 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Optimal Vibration Control for Structural Quasi- Hamiltonian Systems with Noised Observations

Zu-guang Ying Department of Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

Rong-chun Hu Department of Civil Engineering, College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China

Rong-hua Huan Department of Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, China

(Received 19 March 2016; accepted 1 March 2017) A structural control system with smart sensors and actuators is considered and its basic dynamic equations are given. The controlled, stochastically excited, and dissipative Hamiltonian system with a noised observation is obtained by discretizing the nonlinear stochastic smart structure system. The estimated nonlinear stochastic system with control is obtained, in which the optimally estimated state is determined by the observation based on the extended Kalman ﬁlter. Then the dynamical programming equation for the estimated system is obtained based on the stochastic dynamical programming principle. From this, the optimal control law dependent on the estimated state is determined. The proposed optimal control strategy is applied to two nonlinear stochastic systems with controls and noised observations. The control efﬁcacy for stochastic vibration response reductions of the systems is illustrated with numerical results. The proposed optimal control strategy is applicable to general nonlinear stochastic structural systems with smart sensors, smart actuators and noised observations.

1. INTRODUCTION presented.29–46 However, the stochastic optimal control for a nonlinear system with a noised observation was only consid- The vibration control for engineering structures subjected to ered in several studies.43 Under a specified condition, the sep- strong random excitations or micro disturbances is a signifi- aration theorem was applied to convert the nonlinear stochastic cant research subject.1–3 Smart materials have been applied to system with a noised observation into a completely observable the structural vibration suppression. For example, magneto- linear system for determining optimal control, but the applica- rheological materials are used for actuators and piezoelectric tion is strongly limited. Thus, an approximate estimation and materials are used for sensors.4–20 A structure system with dis- separation strategy is the alternative in practice. The extended tributed smart sensors, actuators, and controller is called smart Kalman filter is an optimum feasible approximate filter24, 28 structure, which can sense structural response to external exci- and can be applied to the nonlinear stochastic control system tations and produce action to control structural response.21–23 with a noised observation. Based on the extended Kalman fil- The dynamics and controls of structures with piezoelectric and ter, the nonlinear stochastic control system with a noised ob- magneto-rheological materials have been studied extensively. servation is converted into another completely observable non- However, the control performance of the smart structure sys- linear stochastic system, and then the optimal control law is de- tem depends strongly on the used control strategy. The smart termined according to the stochastic dynamical programming structure system control includes two parts: state estimation principle. However, the stochastic optimal control for the par- based on sensing data and response control based on an es- tially observable nonlinear stochastic smart structure system timated state, which are coupled with each other.24–28 The (or quasi-Hamiltonian system) has not been studied based on stochastic system control with a noised observation is called the extended Kalman filter. a partially observable control. Only linear control strategies In the present paper, the stochastic optimal control for the for partially observable smart structures have been proposed vibration response reduction of structural quasi-Hamiltonian presently. systems with a noised observations is studied. A new opti- The smart structure, including multi-degree-of-freedom sys- mal control law expressed by the estimated system state is ob- tems, rods, beams, plates, and shells in modal vibration can tained based on the extended Kalman filter and stochastic dy- be modeled as a controlled, excited, and dissipative Hamilto- namical programming principle. Firstly, the differential equa- nian system (or quasi-Hamiltonian system) with observation.29 tions for the structure system with smart sensors and actuators The stochastic optimal controls for linear and nonlinear sys- are given. The equations are simplified to ordinary differential tems have been studied and many control strategies have been equations of the controlled, stochastically-excited, and dissipa-

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 (pp. 233–241) https://doi.org/10.20855/ijav.2017.22.2469 233 Z. Ying, et al.: OPTIMAL VIBRATION CONTROL FOR STRUCTURAL QUASI-HAMILTONIAN SYSTEMS WITH NOISED OBSERVATIONS tive Hamiltonian system with a noised observation. Secondly, Dominant modes of the main structure with sensors and ac- the extended Kalman filter is applied in order to convert the tuators are used to discretize the structure system in Eqs. (1) controlled quasi-Hamiltonian system with a noised observation to (8), and the Galerkin method is used to eliminate the spa- into the nonlinear stochastic control system for the optimally tial variables generally. The simplified main structure with estimated state. Thirdly, the stochastic dynamical program- actuators is converted into a controlled, excited, and dissipa- ming principle is applied to derive the dynamical program- tive Hamiltonian system and the sensor structure is converted ming equation. The optimal control law based on the estimated into an observation system. The inertias of the sensors are state is determined by the programming equation. Finally, the small and neglected. The differential equation of the con- proposed optimal control strategy is applied to two nonlinear trolled Hamiltonian system with n degree-of-freedoms can be stochastic systems with controls and noised observations. Nu- expressed as44 merical results are given to illustrate the control efficacy. ∂Ha X˙ = J + CX + BΦa + FW (t); (9) 2. VIBRATION EQUATIONS OF A ∂X

QUASI-HAMILTONIAN SYSTEM where Ha is the Hamiltonian, X is the state vector, J is the unit symplectic matrix, C is the damping coefficient matrix, B is Smart materials have been applied recently to the vibration the control coefficient matrix, Φ is the n -dimensional control control of engineering structures under strong random excita- a a vector, F is the excitation coefficient matrix, and W is the m- tions. For example, magneto-rheological materials are used dimensional unit intensity excitation vector that is considered for smart actuators whereas piezoelectric materials are used to be Gaussian white noise. The equation of the observation for smart sensors. A structure that has smart devices is a smart system can be expressed as system that has the ability to sense structural responses to external excitations and produce actions to control structural re- Φs = DX + EWs(t); (10) sponse. System state estimations and control strategies are two key problems for effective implementation. In general, where Φs is the ns-dimensional observation vector, Ws is the the nonlinear dynamic, constitutive, and geometric equations ms-dimensional Gaussian white noise vector with unit inten- 47 of the main structure can be expressed as sity, and D and E are coefficient matrices. The observation noise is independent of the white noise excitation. σij,j + fi−ρuï − cdu˙ i = 0; (1) Equations (9) and (10) describe a nonlinear stochastic quasi- σ = C γ ; (2) ij ijkl kl Hamiltonian control system with a noised observation, which 1 γij = (ui,j + uj,i + uk,iuk,j); (3) is derived from the smart structure Eqs. (1) to (8). The system 2 state is estimated by using Eq. (10) and the estimated state is where σ is the stress tensor, f is the body force vector, u is the used to determine the feedback control for the system in Eq. displacement vector, ρ is the mass density, cd is the damping (9). The optimal estimation and control need to be considered coefficient, γ is the strain tensor, C is the elastic constant ten- for the nonlinear stochastic system in Eqs. (9) and (10). sor, and i, j, k, and l denote Cartesian coordinates. The linear dynamic, constitutive, and geometric equations of the sensor 3. STOCHASTIC OPTIMAL ESTIMATION structure can be expressed as48 AND CONTROL σse + f se − ρseu¨se − cseu˙ se = 0; (4) ij,j i i d i The optimal control of the nonlinear stochastic system (9) se se se se se σij = Cijklγkl + ekijϕ,k; (5) and (10) includes the optimal state estimation and the optimal 1 γse = (use + use ); (6) control based on the estimated state, which is called the par- ij 2 i,j j,i tially observable optimal control.27 The optimal estimation is where the superscript “se” denotes sensor, for example, piezo- to find an accurate state by minimizing the estimated error. The electric sensor, φ is the generalized electric potential and e is error index for the observation in Eq. (10) is given by the piezoelectric stress constant. The differential equations for ˆ ˆ the electric displacement vector D are JF (X) = E[lF (X − X) |Φs ]; (11)

se se ˆ Di,i =ρem; (7) where X is the estimated state, E[·] is the expectation opera- Dse = ese γse − εse ϕse; (8) tor, and lF (·) ≥ 0 is a continuous differentiable function (e.g., i ikl kl ik ,k a quadratic function). For the nonlinear stochastic system (9) where ρem is the free charge density and ε is the dielectric and observation (10) with index (11), the optimally estimated constant. The mechanical and electrical differential equations state probability density with infinite dimensions is generally for the actuator structure (e.g., piezoelectric actuator) can be difficult to be obtained exactly, and then an approximate esti- expressed as similar to Eqs. (4) to (8), where the superscript mation is alternative. The extended Kalman filter is an opti- se is replaced by ac. Equations (1)–(8) give a basic description mum feasible approximate filter for the nonlinear stochastic of the smart structure system. However, the direct use of the system28 and is applied to the stochastic optimal control of equations for optimal estimation and control is impossible and quasi-Hamiltonian systems with noised observations. Based then their simplification is necessary. on the extended Kalman filter, the differential equation for the

234 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Z. Ying, et al.: OPTIMAL VIBRATION CONTROL FOR STRUCTURAL QUASI-HAMILTONIAN SYSTEMS WITH NOISED OBSERVATIONS estimated state is obtained from Eq. (9) with observation (10) on the estimated state Xˆ. The optimally controlled system is 28 ∗ and index (11) as determined by substituting control Φa into Eq. (12), and the controlled state Xˆ is obtained by solving the equation. The ˆ ˆ˙ ∂Ha(X) ˆ T −1 statistics of state X can be calculated by using the state Xˆ X = J + CX + BΦa + RD Rs WI (t); (12) ∂Xˆ and covariance R, which are used for evaluating the control effectiveness. where WI is the Gaussian white noise vector with covariance T matrix Rs = EE , and R is the covariance matrix of the estimated error X˜ = X − Xˆ , which is determined by 4. EXAMPLES AND NUMERICAL RESULTS

∂2H (Xˆ) ∂2H (Xˆ) 4.1. Example 1: single-degree-of-freedom R˙ = {J a + C}R + R{J a + C}T ∂Xˆ 2 ∂Xˆ 2 nonlinear stochastic control system T T −1 + FRW F − RD Rs DR; (13) To illustrate the application and effectiveness of the proposed optimal control strategy, we ﬁrst considered the control where R = I is the identity matrix. Thus, the optimal con- W for a main mode vibration of geometric nonlinear beams with trol of the nonlinear stochastic system in Eqs. (9) and (10) is a piezoelectric sensor and actuator under stochastic excitation. converted into that of the estimated system in Eq. (12). The nonlinear stochastic system with a control and noised ob- The optimal control is to ﬁnd a control law by minimizing servation can be expressed as certain performance index. The performance index for the sys- 3 tem in Eq. (12) is given by q¨ + cq˙ + k1q+k3q = bϕa + e0W (t); (19)

Z tf ϕs =dq + e1Ws(t); (20) JC (Φa) = E[ LC (Xˆ(t), Φa(t))dt + Ψ(Xˆ(tf ))]; (14) 0 where q is the generalized displacement, ϕa is the control, c, where LC (·) ≥ 0 is a continuous differentiable function, k1, and k3 are respectively the damping, linear stiffness, and Ψ(tf ) is a terminal cost and tf is the terminal time. Equations nonlinear stiffness coefficients, b is the control coefficient, e0 (12) and (14) describe the optimal control problem of the non- is the excitation amplitude, W is the Gaussian white noise with linear stochastic system with the estimated state. Based on the unit intensity, ϕs is the observation, d is the observation coeffi- stochastic dynamical programming principle,24, 25 the dynam- cient, e1 is the observation noise amplitude, and Ws is the unit ical programming equation for system (12) with index (14) is Gaussian white noise. Equations (19) and (20) are rewritten as obtained as Eqs. (9) and (10), respectively, where Φa = ϕa, Φs = ϕs, p =q ˙, and ∂V 1 ∂2V + min{ tr(RDTR−1DR ) s 2 1 2 1 2 1 4 ∂t Φa 2 ∂Xˆ H = p + k q + k q a 2 2 1 4 3 ∂Ha ˆ T ∂V ˆ + [J + CX + BΦa] + LC (X, Φa)} = 0; (15) q ∂Xˆ ∂Xˆ X = p where V is the value function, and tr(·) is the trace operator. 0 0 The minimization of the left side of Eq. (15) yields the alge- C = ∗ 0 −c braic equation for the optimal control Φa 0 ˆ ∗ B = ∂LC (X, Φa) T ∂V b ∗ + B = 0. (16) ∂Φa ∂Xˆ 0 F = e The optimal control law is obtained by Eq. (16). For ex- 0 ample, for the function with quadratic control LC = g(X˜) + D = d 0 ΦTR Φ R a C a, where C is a positive definite symmetric constant E = e1 (21) matrix and g(Xˆ) ≥ 0, it is Applying the extended Kalman filter yields the differential 1 ∂V Φ∗ = − R−1BT . (17) Eq. (12) for the estimated state with the covariance Eq. (13). a 2 C ˆ ∂X Equation (12) is converted into the Itoˆ differential equation for By substituting Eq. (17) into Eq. (15), the value function equa- the averaged Hamiltonian29 by using the stochastic averaging tion is obtained as method,

2 ˆ ∂V 1 T −1 ∂ V ∂Ha + tr(RD R DR ) dHâ = [mh(Hâ)+ < bϕa >]dt + σh(Hâ)dξ(t); (22) ∂t 2 s ∂Xˆ 2 ∂p ∂Ha ˆ ∗ T ∂V ˆ ∗ + [J + CX + BΦ ] + LC (X, Φ ) = 0. (18) where m and σ are the drift and diffusion coefficients, re- ∂Xˆ a ∂Xˆ a h h spectively, ξ is the unit Wiener process, and < · > is the Equation (18) can be solved to obtain V and then the optimal averaging operator. According to the stochastic dynamical ∗ control Φa can be determined by Eq. (17), which is based programming principle, the dynamical programming equation International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 235 Z. Ying, et al.: OPTIMAL VIBRATION CONTROL FOR STRUCTURAL QUASI-HAMILTONIAN SYSTEMS WITH NOISED OBSERVATIONS

(a) Figure 2. Controlled and uncontrolled RMS responses q for various observation noise amplitudes e1 (solid line: analytical; dot: simulated).

b = 1.0, d = 100.0, e1 = 0.03 and the quadratic control coefficient Sc2 = 1.0 (i.e., coefficient of linear term of function g) are obtained and shown in Figs. 1 to 10. The probability densities of the controlled and uncontrolled generalized displacement q and velocity p responses are shown in Figs. 1(a) and 1(b), respectively. The controlled response probability density near zero is larger than the uncontrolled probability density so that the controlled response is reduced. Figure 2 illustrates that the controlled Root-Mean-Square (RMS) displacement response is smaller than the uncontrolled response and the controlled RMS response decreases slightly as the observation noise amplitude e1 increases. The corresponding RMS ∗ optimal control ϕa varying with e1 is shown in Fig. 10. Figure 3 illustrates the relative RMS response reduction K > 80% (b) and the relative reduction per unit RMS control µ. The relative RMS response reduction K is the ratio of absolute differ- Figure 1. Controlled and uncontrolled probability densities (solid line: analytical; dot: simulated). The (a) Probability densities of displacement q and ence of controlled and uncontrolled RMS responses to uncon- (b) Probability densities of velocity p. trolled RMS response, and the relative reduction per unit RMS control µ is the ratio of the relative RMS response reduction similar to Eq. (15) is obtained. The optimal control law (17) to RMS control. The stochastic system response is reduced leads to largely by using the proposed optimal control for various ob- ∗ bp dV ϕa = − . (23) servation noise amplitudes. Figure 4 illustrates that the con- 2RC dHâ trolled RMS displacement response is smaller than the uncon- The corresponding stationary value function equation is trolled response for various observation coefficients d. Figure 2 5 illustrates the large relative RMS response reduction K and 1 2 d V dV σ (Hâ) + mh(Hâ) h ˆ 2 ˆ the small relative reduction per unit RMS control µ. 2 dHa dHa Figure 6 illustrates that the controlled RMS displacement b2 dV − < p2 > ( )2 + g(Hˆ ) = γ ; (24) response is smaller than the uncontrolled response, and the 4R ˆ a 0 C dHa controlled RMS response has small increment relative to the where γ0 is a constant. The optimal control is determined fi- uncontrolled response with the excitation amplitude e0. Figure nally by Eq. (23) with (24). Then the controlled response and 7 illustrates the large relative RMS response reduction K and its statistics can be obtained by solving Eq. (12) with Eq. (20) the small relative reduction per unit RMS control µ for various numerically. The analytical system responses can be evaluated excitation amplitudes. Figure 8 illustrates that the controlled by using the probability density, which is obtained by solving RMS displacement response decreases as the nonlinear stiff- the Fokker-Planck-Kolmogorov equation associated with the ness coefficient k3 increases and the controlled RMS response Itoˆ in Eq. (22).29 The proposed optimal control efficacy for is smaller than the uncontrolled response. The corresponding ∗ the nonlinear stochastic system with a noised observation is RMS optimal control ϕa varying with k3 is shown in Fig. 10. evaluated based on the response statistics. Figure 9 illustrates the large relative RMS response reduction The numerical results on the control system (19) and obser- K for various nonlinear stiffness coefficients. Therefore, the vation (20) with c = 1.0, k1 = 10.0, k3 = 3.0, e0 = 1.0, proposed optimal control can reduce largely the stochastic vi-

236 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Z. Ying, et al.: OPTIMAL VIBRATION CONTROL FOR STRUCTURAL QUASI-HAMILTONIAN SYSTEMS WITH NOISED OBSERVATIONS

Figure 3. Relative response reduction K and its ratio to control µ for various Figure 6. Controlled and uncontrolled RMS responses q for various excitation observation noise amplitudes e1. amplitudes e0.

Figure 4. Controlled and uncontrolled RMS responses q for various observa- Figure 7. Relative response reduction K and its ratio to control µ for various tion coefﬁcients d. excitation amplitudes e0.

Figure 5. Relative response reduction K and its ratio to control µ for various Figure 8. Controlled and uncontrolled RMS responses q for various nonlinear observation coefﬁcients d. stiffness coefﬁcients k3.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 237 Z. Ying, et al.: OPTIMAL VIBRATION CONTROL FOR STRUCTURAL QUASI-HAMILTONIAN SYSTEMS WITH NOISED OBSERVATIONS

Figure 9. Relative response reduction K and its ratio to control µ for various (a) nonlinear stiffness coefﬁcients k3.

(b)

∗ Figure 11. Controlled and uncontrolled probability densities (solid line: ana- Figure 10. RMS controls ϕa for various observation noise amplitudes e1 and lytical; dot: simulated). The (a) Probability densities of displacement q1 and nonlinear stiffness coefficients k3 (note that k3 × 100 indicates that the value (b) Probability densities of displacement q2. of k3 is the product of horizontal coordinate and 100). T Φs = [ϕs1, ϕs2] , bration response of the nonlinear system (19) with a noised observation (20), and the control effectiveness has good ro- C = diag[0, 0, −c1, −c2], T bustness for various system parameters. F = [ 0 0 e01 e02 ] , T 0 0 b1 0 4.2. Example 2: two-degree-of-freedom non- B = , 0 0 0 b2 linear stochastic control system d d 0 0 D = 11 12 , To further illustrate the application and effectiveness of the d12 d22 0 0 proposed optimal control strategy, the control for a two-main- e11 mode coupling vibration of geometric nonlinear beams with E = ; (25) e12 piezoelectric sensor and actuator under stochastic excitation was considered. The nonlinear stochastic system with a con- where qi and pi (i = 1, 2) are respectively the generalized dis- trol and noised observation can be described by Eqs. (9) and placement and momentum, ϕai (i = 1, 2) is the control and (10), in which ϕsi (i = 1, 2) is the observation, ci, k1i, k2, and k3 (i = 1, 2) are the damping, linear stiffness, and nonlinear stiffness coef- 1 1 2 2 2 2 ficients, respectively. On the other hand, bi (i = 1, 2) is the Ha = (p1 + p2) + (k11q1 + k12q2) + k2q1q2 2 2 control coefficient, e0i (i = 1, 2) is the excitation amplitude, 1 4 + k3(q1 − q2) , dij (i, j = 1, 2) is the observation coefficient, and e1i (i = 1, 2) 4 is the observation noise amplitude. T X = [ q1 q2 p1 p2 ] , According to the above procedure, using the extended T Φa = [ϕa1, ϕa2] , Kalman filter yields the differential Eq. (12) for the estimated

238 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Z. Ying, et al.: OPTIMAL VIBRATION CONTROL FOR STRUCTURAL QUASI-HAMILTONIAN SYSTEMS WITH NOISED OBSERVATIONS state. Using the stochastic averaging method converts Eq. (12) into the Itoˆ differential Eq. (22) for the averaged Hamiltonian. Using the stochastic dynamical programming principle yields the dynamical programming equation similar to that in example 1. The optimal control law (17) leads to

∗ 1 −1 dV ϕai = − RCijbjpj . (26) 2 dHˆa The corresponding stationary value function equation is

2 1 2 d V dV σ (Hâ) + mh(Hâ) h ˆ 2 ˆ 2 dHa dHa 1 −1 dV 2 ˆ − < bipiRCijbjpj > ( ) + g(Ha) = γ0. (27) 4 dHâ Figure 12. Relative response reduction K and its ratio to control µ for various The optimal control is determined finally by Eq. (26) observation noise amplitudes e11. with (27). Then the controlled response and its statistics can be obtained by solving Eq. (12) with (20) or solving the Fokker-Planck-Kolmogorov equation associated with the Itoˆ Eq. (22). The proposed optimal control efficacy for the nonlinear stochastic system with a noised observation is evaluated based on the response statistics. Numerical results on the control system and observation with c1 = 1.0, c2 = 5.0, k11 = 1.0, k12 = 4.0, k2 = 0.01, k3 = 1.0, e01 = 1.0, e02 = 0.3, b1 = 1.0, b2 = 0.5, d11 = 100.0, d12 = 10.0, d22 = 80.0, e11 = 0.03, e12 = 0.03 and the quadratic control coefficient Sc2 = 1.0 (i.e., coefficient of linear term of function g) are obtained and shown in Figs. 11 to 13. The probability densities of the controlled and uncontrolled generalized displacement responses (q1 and q2) are shown in Figs. 11(a) and 11(b), respectively. The controlled response probability density near zero is larger than the uncontrolled probability density. Figure 12 illustrates that the relative reduction K of the controlled RMS displacement response Figure 13. Relative response reduction K and its ratio to control µ for various observation coefficients d11. q1 compared with the uncontrolled response and the relative reduction per unit RMS control µ vary with the observation noise observations. Numerical results have illustrated that the pro- amplitude e . Figure 13 illustrates that the relative reduction 11 posed optimal control can reduce largely the stochastic vibra- K of the controlled RMS displacement response q and the 1 tion response of nonlinear systems with noised observations, relative reduction per unit RMS control µ vary with the ob- and the control effectiveness is insensitive to varying system servation coefficients d . It is seen again that the stochastic 11 parameters such as observation and nonlinear coefficients. The vibration response of the two-mode coupling nonlinear system proposed optimal control strategy is feasible and effective for with a noised observation is reduced largely by using the pro- the vibration response reduction of nonlinear stochastic smart posed optimal control. structure systems with noised observations. 5. CONCLUSIONS ACKNOWLEDGEMENTS The basic dynamic equations for a nonlinear stochastic control structure system with smart sensors and actuators have This work was supported by the National Natural Science been given and simplified to the controlled, stochastically- Foundation of China under grant nos. 11572279, 11432012 excited, and dissipative Hamiltonian system with a noised ob- and 11602216, and the Zhejiang Provincial Natural Science servation. The optimally estimated nonlinear system with con- Foundation of China under grant no. LY15A020001. trol and stochastic excitation has been determined based on the extended Kalman filter. The dynamical programming equa- REFERENCES tion for the estimated system has been obtained based on the stochastic dynamical programming principle, and the optimal 1 Housner, G. W., Bergman, L. A., Caughey, T. K., Chas- control law has been determined by the programming equa- siakos, A. G., Claus, R. O., Masri, S. F., Skelton, tion. The proposed optimal control strategy has been applied R. E., Soong, T. T., Spencer, B. F., and Yao, J. T. to two nonlinear stochastic systems with controls and noised P. Structural control: Past, present, and future, ASCE

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 239 Z. Ying, et al.: OPTIMAL VIBRATION CONTROL FOR STRUCTURAL QUASI-HAMILTONIAN SYSTEMS WITH NOISED OBSERVATIONS

Journal of Engineering Mechanic, 123, 897-971, (1997). 13 Ying, Z. G., Ni, Y. Q., and Duan, Y. F. Stochas- https://doi.org/10.1061/(asce)0733-9399(1997)123:9(897) tic micro-vibration response characteristics of a sandwich plate with MR visco-elastomer core and mass, 2 Soong, T. T. and Spencer, B. F. Supplemental en- Smart Structures and Systems, 16, 141-162, (2015). ergy dissipation: state-of-the-art and state-of-the- https://doi.org/10.12989/sss.2015.16.1.141 practice, Engineering Structures, 24, 243-259, (2002). https://doi.org/10.1016/S0141-0296(01)00092-X 14 Rao, S. S. and Sunar, M. Piezoelectricity and its use in disturbance sensing and control of flexible structures: A 3 Soong, T. T. and Cimellaro, G. P. Future directions in struc- survey, ASME Applied Mechanics Reviews, 47, 113-123, Structural Control and Health Monitoring tural control, , (1994). https://doi.org/10.1115/1.3111074 16, 7-16, (2009). https://doi.org/10.1002/stc.291 15 Narayanan, S. and Balamurugan, V. Finite element mod- 4 Casciati, F., Rodellar, J., and Yildirim, U. Active and elling of piezolaminated smart structures for active vi- semi-active control of structures theory and application: bration control with distributed sensors and actuators, A review of recent advances, Journal of Intelligent Ma- Journal of Sound and Vibration, 262, 529-562, (2003). terial Systems and Structures, 23, 1181-1195, (2012). https://doi.org/10.1016/S0022-460X(03)00110-X https://doi.org/10.1177/1045389X12445029 16 Ray, M. C. Optimal control of laminated shells using piezo- 5 Spencer, B. F. and Nagarajaiah, S. State of the art of electric sensor and actuator layers, AIAA Journal, 41, 1151- structural control, ASCE Journal of Structural Engineering, 1157, (2003). https://doi.org/10.2514/2.2058 129, 845-856, (2003). https://doi.org/10.1061/(asce)0733- 9445(2003)129:7(845) 17 Gupta, V. K., Seshu, P., and Issac, K. K. Finite element and experimental investigation of piezoelectric actu- 6 Dyke, S. J., Spencer, B. F., Sain, M. K., and Carlson, J. ated smart shells, AIAA Journal, 42, 2112-2123, (2004). D. Modeling and control of magnetorheological dampers https://doi.org/10.2514/1.2902 for seismic response reduction, Smart Materials and Struc- tures, 5, 565-575, (1996). https://doi.org/10.1088/0964- 18 To, C. W. S. and Chen, T. Optimal control of ran- 1726/5/5/006 dom vibration in plate and shell structures with distributed piezoelectric components, International Jour- 7 York, D., Wang, X. and Gordaninejad, F. A new MR nal of Mechanical Sciences, 49, 1389-1398, (2007). fluid-elastomer vibration isolator, Journal of Intelligent https://doi.org/10.1016/j.ijmecsci.2007.03.015 Material Systems and Structures, 18, 1221-1225, (2007). https://doi.org/10.1177/1045389X07083622 19 Jin, Z. L., Yang, Y. W., and Soh, C. K. Semi- analytical solutions for optimal distributions of sen- 8 Zhou, G. Y. and Wang, Q. Study on the adjustable sors and actuators in smart structure vibration con- rigidity of magnetorheological-elastomer-based sandwich trol, Smart Structures and Systems, 6, 767-792, (2010). beams, Smart Materials and Structures, 15, 59-74, (2006). https://doi.org/10.12989/sss.2010.6.7.767 https://doi.org/10.1088/0964-1726/15/1/035 20 Ying, Z. G., Feng, J., Zhu, W. Q., and Ni, Y. 9 Choi, W. J., Xiong, Y. P. and Shenoi, R. A. Vibration char- Q. Stochastic optimal control analysis of a piezoelec- acteristics of sandwich beam with steel skins and magne- tric shell subjected to stochastic boundary perturba- torheological elastomer cores, Advances in Structural Engi- tions, Smart Structures and Systems, 9, 231-251, (2012). neering, 13, 837-847, (2010). https://doi.org/10.1260/1369- https://doi.org/10.12989/sss.2012.9.3.231 4332.13.5.837 21 Vepa, R. Dynamics of Smart Structures, 10 Rajamohan, V., Rakheja, S., and Sedaghati, R. Vi- John Wiley & Sons, Chichester, (2010). bration analysis of a partially treated multi-layer https://doi.org/10.1002/9780470710623 beam with magnetorheological fluid, Journal of Sound and Vibration, 329, 3451-3469, (2010). 22 Fisco, N. R. and Adeli, H. Smart structures: Part I -active https://doi.org/10.1016/j.jsv.2010.03.010 and semi-active control, Scientia Iranica A, 18, 275-284, 11 Ying, Z. G., Ni, Y. Q., and Huan, R. H. Stochastic mi- (2011). https://doi.org/10.1016/j.scient.2011.05.034 crovibration response analysis of a magnetorheological vis- 23 Fisco, N. R. and Adeli, H. Smart structures: Part II -hybrid coelastomer based sandwich beam under localized mag- control systems and control, Scientia Iranica A, 18, 285- netic fields, Applied Mathematical Modelling, 39, 5559- 295, (2011). https://doi.org/10.1016/j.scient.2011.05.035 5566, (2015). https://doi.org/10.1016/j.apm.2015.01.028 24 Stengel, R. F. Optimal Control and Estimation, John Wiley 12 Nayak, B., Dwivedy, S. K., and Murthy, K. S. R. K. & Sons, New York, (1994). Dynamic analysis of magnetorheological elastomer-based sandwich beam with conductive skins under various bound- 25 Yong, J. M. and Zhou, X. Y. Stochastic Controls, Hamil- ary conditions, Journal of Sound and Vibration, 330, 1837- tonian Systems and HJB Equations, Springer-Verlag, New 1859, (2011). https://doi.org/10.1016/j.jsv.2010.10.041 York, (1999). https://doi.org/10.1007/978-1-4612-1466-3

240 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Z. Ying, et al.: OPTIMAL VIBRATION CONTROL FOR STRUCTURAL QUASI-HAMILTONIAN SYSTEMS WITH NOISED OBSERVATIONS

26 Fleming, W. H. and Soner, H. M. Controlled Markov Pro- 39 Dimentberg, M. F. and Bratus, A. S. Bounded para- cesses and Viscosity Solutions, Springer, New York, (2006). metric control of random vibrations, Proceedings of https://doi.org/10.1007/0-387-31071-1 Royal Society London A, 456, 2351-2363, (2000). https://doi.org/10.1098/rspa.2000.0615 27 Bensoussan, A. Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, (1992). 40 Crespo, L. G. and Sun, J. Q. Stochastic optimal con- https://doi.org/10.1017/cbo9780511526503 trol of nonlinear dynamical systems via Bellmans principle and cell mapping, Automatica, 39, 2109-2114, (2003). 28 Anderson, B. D. O. and Moore, J. R. Optimal Filtering, https://doi.org/10.1016/S0005-1098(03)00238-3 Dover Publication, New York, (2005). 41 Park, J. H. and Min, K. W. Bounded nonlinear stochastic 29 Zhu, W. Q. Nonlinear stochastic dynamics and control based the probability distribution for sdof oscilla- control in Hamiltonian formulation, ASME Ap- tor, Journal of Sound and Vibration, 281, 141-153, (2005). plied Mechanics Reviews, 59, 230-248, (2006). https://doi.org/10.1016/j.jsv.2004.01.008 https://doi.org/10.1115/1.2193137 42 Scruggs, J. T., Taﬂanidis, A. A., and Iwan, W. D. Non- 30 Socha, L. Linearization in analysis of nonlinear stochas- linear stochastic controllers for semiactive and regen- tic systems: Recent results -part II: application, ASME erative systems with guaranteed quadratic performance Applied Mechanics Reviews, 58, 303-315, (2005). bounds -part 1: State feedback control, Structural Con- https://doi.org/10.1115/1.1995715 trol and Health Monitoring, 14, 1101-1120, (2007). 31 Datta, T. K. A brief review of stochastic control of https://doi.org/10.1002/stc.196 structures, Proc. International Symposium on Engineer- 43 Ying, Z. G. and Zhu, W. Q. A stochastic optimal con- ing under Uncertainty, Springer, India, 119-139, (2013). trol strategy for partially observable nonlinear quasi- https://doi.org/10.1007/978-81-322-0757-3 6 Hamiltonian systems, Journal of Sound and Vibration, 310, 32 Agrawal, A. K. and Yang, J. N. Optimal polynomial 184-196, (2008). https://doi.org/10.1016/j.jsv.2007.07.065 control of seismic-excited linear structures, ASCE Jour- 44 Ying, Z. G., Luo, Y. M., Zhu, W. Q., Ni, Y. Q., and nal of Engineering Mechanics, 122, 753-761, (1996). Ko, J. M. A semi-analytical direct optimal control https://doi.org/10.1061/(asce)0733-9399(1996)122:8(753) solution for strongly excited and dissipative Hamilto- 33 Nagarajaiah, S. Adaptive passive, semiactive, smart tuned nian systems, Communications in Nonlinear Science mass dampers: identiﬁcation and control using empiri- and Numerical Simulation, 17, 1956-1964, (2012). cal mode decomposition, Hilbert transform, and short-term https://doi.org/10.1016/j.cnsns.2011.09.010 Fourier transform, Structural Control and Health Monitor- ing, 16, 800-841, (2009). https://doi.org/10.1002/stc.349 45 Hu, R. C., Ying, Z. G., and Zhu, W. Q. Stochastic minimax 34 Li, J., Peng, Y. B., and Chen, J. B. A physical ap- optimal control strategy for uncertain quasi-Hamiltonian proach to structural stochastic optimal controls, Prob- systems using stochastic maximum principle, Structural abilistic Engineering Mechanics, 25, 127-141, (2010). and Multidisciplinary Optimization, 49, 69-80, (2014). https://doi.org/10.1016/j.probengmech.2009.08.006 https://doi.org/10.1007/s00158-013-0958-x 46 35 Aldemir, U. Predictive suboptimal semiactive control of Ying, Z. G., Ni, Y. Q., and Duan, Y. F. Parametric optimal earthquake response, Structural Control and Health Moni- bounded feedback control for smart parameter-controllable toring, 17, 654-674, (2010). https://doi.org/10.1002/stc.342 composite structures, Journal of Sound and Vibration, 339, 38-55, (2015). https://doi.org/10.1016/j.jsv.2014.11.018 36 Ying, Z. G. and Ni, Y. Q. Optimal control for vibra- 47 tion peak reduction via minimizing large responses, Struc- Mase, G. E. Continuum Mechanics, McGraw-Hill, New tural Control and Health Monitoring, 22, 826-846, (2015). York, (1970). https://doi.org/10.1002/stc.1722 48 Ying, Z. G. and Ni, Y. Q. Dynamic asymme- 37 Socha, L. A. Application of true linearization in try of piezoelectric shell structures, Journal of stochastic quasi-optimal control problems, Structural Sound and Vibration, 332, 3706-3723, (2013). Control and Health Monitoring, 7, 219-230, (2000). https://doi.org/10.1016/j.jsv.2013.03.002 https://doi.org/10.1002/stc.4300070207

38 Dimentberg, M. F., Iourtchenko, A. S., and Brautus, A. S. Optimal bounded control of steady-state random vibrations, Probabilistic Engineering Mechanics, 15, 381-386, (2000). https://doi.org/10.1016/S0266-8920(00)00008-4

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 241 Detection of Damage in Spot Welded Joints Using a Statistical Energy Analysis-like Approach

Achuthan C. Pankaj CSIR-National Aerospace Laboratories, Bangalore, 560017, Karnataka, India Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, Mangalore, 575025, India

Marandahalli V. Shivaprasad CSIR-National Aerospace Laboratories, Bangalore, 560017, Karnataka, India

S. M. Murigendrappa Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, Mangalore, 575025, India

(Received 16 August 2016; accepted 12 October 2016) Vibration-based damage detection has been frequently used for low frequency problems. However, there are situations where the damage, like connections of structures with spot welds, mainly affects the highest modes. Energy- based approaches such as Statistical Energy Analysis (SEA) is one of the most widely used methods for high frequency analysis that is well-suited for periodic structures. The present work studies the damage detection of joints based on statistical energy analysis-like principles using apparent coupling factors to predict velocity/acceleration responses and detect damage in the spot welds located at various positions on a sub-system (spot-welded plate configurations). Apparent coupling factors have been derived for four cases of spot-welded plates and used further to predict the velocity/acceleration responses using the statistical energy analysis like (SEAL) approach for an assembly of three subsystems (three plates lap joined by spot-welds) for all the possible combinations. The results are discussed, compared, and validated by experimentation and finite element simulations for a healthy and damaged configuration. A database of the predicted values using the SEAL approach for the remaining combinations has been compared with values obtained from finite element simulations. The proposed SEAL-based approach can be effectively applied as a simulation tool to locate the damaged joint in an assembly of subsystems for future use.

1. INTRODUCTION models can be used to generate data for the damaged class.

Vibration-based health monitoring has been frequently used New advanced instrumentation like the scanning laser vi- for low frequency problems.1 However, there are situations brometer and motion magnification approaches has motivated and problems wherein the damage affects mainly the highest the development of damage identification methods based on modes, rather than the lowest, such as connections of struc- higher-frequency vibrational responses. Nevertheless, unlike tures with cleats/spot welds etc.2 Structural health monitor- finite element methods used at low-frequencies, energy-based ing techniques based on identifying the damaged samples (de- approaches like SEA is one of the widely used methods in the lamination) by shifts in the frequencies of some of the vibra- high frequency regime that is well-suited to periodic structures. tion modes has been carried out by exciting the samples at The application of SEAL approaches and parameters like ap- high frequencies using lead zirconate titanate (PZT) piezoelec- parent/effective coupling loss factors/energy influence coeffi- tric elements acting as sensors and actuators.3 Similarly, the cients between sub-elements can be used to detect and localize use of the electro-mechanical impedance signatures of piezo- the damage. electric wafer active sensors (PWAS), which are permanently The present paper presents the studies of damage detection mounted to the structure along with a probabilistic neural net- of spot-welded joints based on SEAL principles and effective work (PNN), to classify the spectral data and identify the sever- coupling factors derived from experiments and finite element ity of damage has been used in the health monitoring of thin simulations. Apparent coupling factors has been derived for plates and aerospace structures.4 Pattern recognition methods four cases of two spot-welded plates and used further to predict based on neural networks and support vector machines have the velocity/acceleration responses using the SEAL approach also been used for damage detection.5, 6 The main challenge for an assembly of three subsystems (three plates lap joined by with these methods is that if the structure is considered undam- spot-welds). The outcomes acquired have been examined and aged in its current state, data from the damaged class is not validated by experimentation and finite element simulations available unless some simulated finite element/experimental for a healthy and damaged configuration. Further, a database

242 https://doi.org/10.20855/ijav.2017.22.2470 (pp. 242–251) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

  N   X  η1 + η1i n1 −η12n1 ... −η1N n1     i6=1        hE1i    N  n1 Pi,1  X   hE2i   −η21n2 η2 + η2i n2 ... −η2N n2     Pi,2  ω   ×  n2  =   . (1)  i6=2   .   .     .   .   . . .. .     . . . .  hEN i Pi,N     n  N−1  N  X   −ηN1nN ...... ηN + ηNi nN  i6=N

of the predicted values using the SEAL approach for remaining combinations has been compared with values obtained from finite element simulations. The proposed SEAL-based prediction approach can thus be used to generate a database of velocity/acceleration responses for various possibilities of sequentially coupled combinations of the basic cases, which can be effectively applied to locate the damaged joint in an assembly of subsystems for future use. The authors propose the use of the SEAL approach as a simulation tool for prior prediction and generation of a database of the joint damage scenarios for a structural assembly of sub-systems. The data generated can also be further utilized for optimal location of sensors and Figure 1. Geometric dimensions of plates with spot-welds (case-1). pattern recognition methods for damage detection. terial damping and the coupling factors can then be found out Mild steel plates have been used in the present work that by the matrix inversion approach and vice versa. Equation (1), have low values of material damping. The coupling loss fac- which is a SEA equation, relates ensemble average powers and tor (CLF) computed by the wave approach is independent of energies, whereas the simulated FE results are based on a sin- the internal loss factor as compared to the values computed gle estimate of frequency average quantities. In the present pa- using finite element method, wherein CLF increases linearly per, the focus is on using the energy flow balance for the indi- as the internal loss factor varies from a zero value, followed vidual cases, based on the principles of SEA and not applying by a transition region and converges to the values obtained its underlying assumptions and procedural rules of analysis. 7, 8 by the analytical wave approach at higher frequencies. The The coupling loss factor estimated in this manner has been re- finite element modeling of spot-welded joints requires easy- ferred to as an apparent coupling loss-factor (ACLF) or Energy to-connect congruent as well as non-congruent meshes and to Influence Coefficient (EIC) and the term SEA-like (SEAL) ap- locate weld nuggets anywhere in the meshes. For this pur- proach has been adopted to distinguish between SEA and the pose, Area Contact Model 2 (ACM2) and CWELD are widely use of the energy flow balance in SEA as applied in the energy 9 used finite element models in the automotive industry. In the flow model.11 The variance of the ACLF / EICs generally de- present study, a finite element model of spot welds has been creases as modal overlap or the number of modes in the band modelled as an ACM2 model using ANSYS software. The increase.12 modelling of discrete joints like the spot welded joints in computation of the coupling factors is significant in terms of its 2. TEST SPECIMENS further use in computation of energies and velocity responses 10 using statistical energy approach. SEA involves predicting The dimensions of each plate considered in the studies are the vibration response of a complex structure by dividing it 500 mm×500 mm×1.2148 mm, made of mild steel. The plates into a number of subsystems. In predictive SEA, coupling and have been joined by spot welds with an overlap length between damping loss factors are estimated through experiments, an- two sheets of 50 mm. Three spot welds are lined up on the cen- alytical, and numerical approaches by solving the power bal- tre of the overlap in a longitudinal direction and in a transverse ance equations Eq. (1) (see top of the page), where hEi/nii are direction, one of the three spot welds is centrally located and the modal energy (energy per mode) of subsystem i and ni is the others are located 50 mm away from the edge of the plate the modal density of subsystem i. The power injected in sub- shown in Fig. 1. The diameters of the spot weld nuggets are system N is PiN , frequency averaging has been denoted by a approximately 6 mm. The density of the mild steel plates has ¯. hEii is the spatially averaged energy in subsystem i and ω - been assumed to be 7850 kg/m3 with a modulus of elasticity central band frequency, ηi -internal damping loss factor in sub- of 210 GPa. The following cases of two plates joined by spot- system i and ηij is coupling loss factor (CLF) from subsystem welds have been considered to obtain the apparent coupling i to subsystem j. factors using the SEAL approach. The energies in all the subsystems are computed for the corresponding power to the respective sub-system, with known ma- • Case-1: Two plates with lap joint having three spot welds

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 243 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

(Healthy Conﬁguration)

• Case-2: Two plates with lap joint having two spot welds (Centre weld assumed to be damaged)

• Case-3: Two plates with lap joint having two spot welds (Bottom most weld assumed to be damaged)

• Case-4: Two plates with lap joint having one spot weld at the Centre (Two extreme spot welds assumed to be damaged)

3. EXPERIMENTAL SETUP The experimental setup consists of spot-welded plates to be tested, data acquisition hardware, sensors, shakers, and computer with modal analysis software as shown in Fig. 2. SCADAS III is a multichannel 24 bit data acquisition system with inbuilt ADC and signal conditioners for ICP type of Figure 2. The experimental set-up. accelerometers. Communication between the data acquisition hardware and computer system is established through SCASI card. Laptop with advanced modal analysis software LMS Test lab is used for data acquisition, analysis, and extracting the modal parameters. ICP based PCB make accelerometers with sensitivity of 100 mV/g (Model No. 333B30) has been used for acceleration response measurement. Electrodynamic shakers of Bruel and Kjaer with a force rating of 200 N sine (Type 4825) with an excitation frequency limitation beyond 3000 Hz and a compatible power ampliﬁer has been used to excite the plate and the excitation force is measured using force transducer with a sensitivity of 500 mV/N and force range of 44 N. The maximum excitation frequency for the experiments has been limited to 3000 Hz. The spot welded plates have been hung by strings to simulate a free-free boundary condition. The geometry as per the test points chosen for measurements have been created in the LMS test lab. Each plate has been divided equally into nine regions and the responses have been Figure 3. The geometry of the two-spot welded plates created in Test-lab. obtained using nine accelerometers placed centrally for each of the region. The nodes i.e., the accelerometer positions and power input into every single subsystem and its corresponding wire frame geometry created has been shown in Fig. 3. The measurement. Subsequently the SEA parameters, like internal channels setup was carried out with setting type of sensor, ex- loss and coupling loss factor, can be obtained by inversion of citation voltage, units, reference point, measurement point IDs, the measured energy matrix.13 The time averaged power input and gain settings. To compute the natural frequencies, scope into a subsystem14 is given as: settings with maximum frequency was set to 512 Hz and spec- 1 tral lines of 2622 Hz, which gave a frequency resolution of P = Im(Saf (ω)). (2) ω 0.19 Hz. Uniform windows were chosen for both excitation and response signals. A 50% burst random signal has been Power input measurements (P ) requires the evaluation of the used for exciting the plate. Same scope settings are imported imaginary part of the cross spectrum Saf . PIM is based on a to test setup and required functions such as time domain data, comparison of the dissipated energy of a system to the total en- cross power spectrums, peak spectra, FRF, and auto power ergy of the system (ET otal) under steady state vibration. Since spectra etc. are selected to store into the computer database. the power input (P ) into a single system is dissipated by the In case of harmonic sinusoidal excitation and acceleration re- system under steady state conditions, the dissipated power can sponse measurement 32768 Hz sampling frequency and maxi- be replaced with the power input (Eq. (3)) to obtain the internal 15 mum bandwidth of 10000 Hz has been set. loss factor at a particular excitation frequency. P η = . (3) 4. POWER INJECTION METHOD ωET otal In coupled sub-systems, the power injection method (PIM) Excitation was carried out at discrete frequencies of 1000 to involves the measurement of energy for each subsystem for the 3000 Hz in steps of 500 Hz. The test was repeated for 3 to 4

244 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

Figure 4. A diagram of ACM2 (Spot welded joint) model. trials in each case. The input excitation force was applied on the centre (0.25 m, 0.25 m) of one plate (Pt-5, Fig. 3). The Figure 5. The finite element model (case-1). plate was excited for flexural vibratory modes. has been applied in the range of frequencies of 1000-3000 Hz at steps of 500 Hz. A unit force of 1 N was applied on the first 5. FINITE ELEMENT ANALYSIS plate and the velocity responses on both the plates were com- Simplified models of spot welds in FEA have been cre- puted. The internal loss factors estimated by the experimental ated using shell elements for the plates and using a single power injection method on a single plate for different frequen- solid element or beam element for the weld nuggets. The cies have been used for the finite element simulation at the re- most commonly used finite element models in the industry spective frequencies. The mass of the accelerometers (5 gms) are the CWELD and the area contact model (ACM2) mod- has been modelled as a concentrated mass element (mass 21) els.16, 17 The ACM2 model consists of a single solid element at the locations wherein the accelerometers have been placed (shaded cuboid) connecting the upper and lower shell elements during the experiments. The additional mass of the force trans- with constraint elements. Each node of the solid element is ducer (10 gms) has also been included in the concentrated mass connected to four nodes of one shell element via constraint element (mass 21 element) at the point of excitation force. equation elements shown in Fig. 4. The patch for the ACM2 The velocity responses at the nodes of the plates, wherein re- model consists of four shell elements for each of the two sponses have been obtained in the experiments, including the sheets. The rigid body element (RBE3) is an interpolation el- power input location has been determined. Macros have been ement that forms constraint equations and is used to distribute developed in ANSYS Parametric Design language (APDL) for force/moment from master node to slave nodes proportional to post-processing and computation of power input and veloci- the weighting factors. ties. Finally the energy (Ei) of each region of the plate esti- The spot weld nuggets have been modelled using solid el- mated by using Eq. (4), wherein (Mi) is the mass of one of ements (SOLID185 with 6 mm square and 1.2148 mm high), the 9 regions on each plate and the representative subsystem that is the ACM2 model. The natural frequencies and the mode velocity (Vi) corresponding to that region. shapes have been computed under the free-free boundary con- M V 2 E = i i . (4) ditions. The upper and lower plates have been meshed using i 2 SHELL63 elements with a mean mesh size of 6 mm (Fig. 5). The total energy of each plate has been computed by the sum- Shell 63 has both bending and membrane capabilities. The mation of the energies obtained from individual regions of that finite element method calculations were carried out using ele- plate (E = E + E ...E ). The apparent coupling factors ment dimensions smaller than one-sixth of the bending wave- 1 2 9 have been computed by the matrix inversion approach from length. The patches have been meshed with 6 mm square shell Eq. (1) after computation of power inputs and corresponding elements, while areas along the edges are meshed with less energies of each plate. The maximum average velocity re- than 6 mm non-square elements. In addition, for the ACM2 sponse of each plate has been obtained directly from the post- model, the centre of the patch area is coincident with the centre processing of the output results. The codes for post-processing of solid element. The damaged spot welds have not been mod- the responses, energies have been developed in the parametric elled representing the stiffness loss for a spot welded joint at design language available in ANSYS and MATLAB software. that particular location. Dynamic analysis of the FE model of the spot-welded plates has been carried out for a free-free con- 6. RESULTS AND DISCUSSION dition and the dynamic frequency spectrum has been obtained by invoking the lanczomethod in ANSYS, with unit mass cri- The comparison of the natural frequencies obtained from the teria for normalizing mode shapes. finite element analysis and experiments for few of the elastic The full method available in ANSYS has been used for the modes of case-1 (healthy configuration) has been shown in Ta- harmonic analysis. A harmonic force with unit load intensity ble 1. It can be observed that the maximum error is within an

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 245 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

(a)

(b) (b)

Figure 6. Comparison of the Experimental and FEM mode shapes (Mode. No-7). Figure 7. Comparison of the Experimental and FEM mode shapes (Mode. No-10). acceptable range of 5%. Figures 6 and 7 shows the comparison of two typical mode shapes obtained from the ﬁnite element analysis and experiments for case-1. Similar trends have been observed for the other cases. The acceleration responses have been obtained from the de- ﬁned locations by experimentation on the plates. The last col- Table 1. umn of Table 2 shows the sinusoidal force excitation values at Natural frequencies comparison for elastic modes (Case-1). the excitation point on the plate. The responses and the ex- Mode No. FEM (Hz) Experimental (Hz) % Error 7 6.1166 5.98 2.28 citation force being sinusoidal in nature, the maximum value 8 8.198 8.01 2.34 obtained are computed by post-processing the values in MAT- 9 17.403 16.8 3.58 LAB software. The maximum acceleration responses obtained 10 18.744 18.5 1.31 at all the points on both the plates is divided by the maximum 11 27.474 28.9 4.93 excitation force to obtain the acceleration responses for unit 12 31.456 31.25 0.65 13 31.867 31.5 1.16 excitation loading of 1 N as shown in Table 3. 14 35.07 36.7 4.44 The acceleration responses have been divided by the excita- 15 44.546 45.5 2.09 tion frequency to obtain the velocity responses. The vibratory 16 47.516 49.02 3.06 17 60.62 59.5 1.88 energies in the plates have been computed based on Eq. (4). 18 62.625 62.5 0.2 The power injected at the excitation location has been obtained 19 72.231 71.29 1.31 from the cross-spectrum of force and acceleration by Eq. (2). 20 76.313 76.23 0.10 Finally the apparent coupling factors have been computed by the inversion of power balance equation as given in Eq. (1). The apparent/effective coupling factors (η12 = η21) for all the

246 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

Table 2. Sample acceleration responses (Case-1). Time Acceleration Response (g) Force (a) (secs) . (N) Node-1 Node-2 . Node-18 . 4.9999762 9.74 -24.1 . 3.28 -8.73 . 5.0000067 -13.1 -24.5 . -10.8 -8.04 . 5.0000372 -31.2 -17.2 . -21.1 -4.76 ...... 5.0999823 5.32 -24.8 . 0.363 -8.81 . 5.1000128 -17.2 -23.5 . -13.4 -7.55 (b) Table 3. Acceleration responses for unit force (1 N) (Case -1). Acceleration Response (g) Force Excitation Node-1 Node-2 Node-18 (N) Location 4.5011 2.8571 ...... 2.7098 1 cases of the spot-welded plates obtained from ﬁnite element analysis and experiments has been listed out in Table 4 and Ta- ble 5 respectively. Similarly, the velocity responses obtained by both the methods for all the cases under consideration have been compared in Figs. 8 to 11. The internal loss factor (η12) for two coupled plates of the same size and material with an internal loss factor (η) can be also directly obtained from Eq. (5), wherein E11 is the energy in plate 1 and E12 is the energy in Figure 8. Velocity responses for case 1 (a) Plate 1 and (b) Plate 2. plate 2, when plate 1 is excited. E22 is the energy in plate 2 and E21 is the energy in plate 1, when plate 2 is excited.

ηE21(E22 + E12) (a) η12 = . (5) (E11E22 + E12E21)

The graphs reveal that the variation in the velocity responses obtained from experimental and FE simulations reduces with the increase in excitation frequencies. At low frequencies, due to reduction in modal overlap, there can be inaccurate assumptions in the energy balance equations in SEA approach. At higher frequencies, the excitation frequency selected, would have equivalent energies as generated by its accompanying modes representing a uniform energy density in a particular bandwidth due to higher modal overlap. Excitation frequencies that generate sudden peak responses as observed near 1500 Hz is undesirable for the present studies, as small perturbations (b) around it may lead to a large variation in output responses. The theoretical value of modal density of each plate turns out to be 0.0207. The modal overlap factor (M = ωηn) preferred for application of the SEA principle should be greater than 1. The response signals obtained from experiments has been found to be better at higher frequencies. It had also been observed that the

Table 4. Apparent coupling factors (experimental). Frequency (Hz) Case- no. 1000 1500 2000 2500 3000 1 1.6e-3 1.46e-3 3.0e-4 1.2e-3 1.8e-3 2 3.4e-4 8.7e-4 1.2e-3 1.0e-3 8.5e-4 3 3.1e-4 9.0e-4 1.1e-3 1.6e-3 6.0e-4 Figure 9. The velocity responses for case-2 (a) Plate 1 and (b) Plate 2. 4 1.6e-3 1.6e-3 1.0e-4 3.0e-4 4.0e-4

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 247 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

Table 5. Apparent coupling factors (FEM). (a) Frequency (Hz) Case- no. 1000 1500 2000 2500 3000 1 1.4e-3 1.4e-3 5.0e-4 3.9e-3 1.0e-3 2 2.0e-4 2.0e-4 7.0e-3 2.0e-3 5.0e-4 3 6.3e-4 1.3e-4 2.0e-3 9.8e-3 4.0e-4 4 -1.8e-3 4.9e-3 2.5e-4 3.2e-4 1.0e-4

Table 6. Ratio of velocity responses of Plate 1 and Plate 2 (3000 Hz). Case 1 2 3 4 Plate1 (P1) (m/s) 1.92e-3 1.9e-3 2.06e-3 8.16e-4 EXP Plate2 (P2) (m/s) 8.75e-4 6.2e-4 5.62e-4 1.98e-4 (P1/P2) 2.19 3.06 3.66 4.12 Plate1 (b) (P1) (m/s) 1.14 e-3 1.2 e-3 1.18 e-3 1.12 e-3 FEM Plate2 (P2) (m/s) 4.10 e-4 3.2 e-4 2.80 e-4 1.40 e-4 (P1/P2) 2.77 3.85 4.19 8.02

variation in the derived coupling factors due to the variation in the excitation force location on the excited plate also reduces with the increase in the excitation frequencies. Table 6 shows the ratio of velocities obtained in plate-1 and plate-2 at an excitation frequency of 3000 Hz for all the cases of two plates joined by spot-welds. The deterministic frequency of 3000 Hz with an internal modal loss factor of 0.007 has been selected for further damage detection studies in the present work. It can be observed that as the number of spot welds decreases the energies and consequently the velocities in the second plate Figure 10. Velocity responses for case 3 (a) Plate 1 and (b) Plate 2. is reduced considerably and therefore the ratio of velocity responses of plate-1 to plate-2 (P1/P2) increases.

(a) 6.1. Joint Damage Detection The studies for damage detection consists of three identical plates coupled in sequence by a simple lap spot welded conﬁg- uration has been considered for all the possible combinations of the cases as explained in section 2. The statistical energy analysis-like approach has been applied by using the apparent coupling factors in Tables 4 and 5 to predict the velocity responses in the three plates for all the possible combinations, as listed out in Table 7. The velocity responses for two such combinations, viz.

Table 7. List of possible combinations for three spot-welded plates. Plate-1 Plate-2 (b) Combination and 2 and 3 1 case-1 case-1 2 case-1 case-2 3 case-1 case-3 4 case-1 case-4 5 case-2 case-1 6 case-2 case-2 7 case-2 case-3 8 case-2 case-4 9 case-3 case-1 10 case-3 case-2 11 case-3 case-3 12 case-3 case-4 13 case-4 case-1 14 case-4 case-2 15 case-4 case-3 Figure 11. The velocity responses for case-4 in (a) Plate 1 and (b) Plate 2. 16 case-4 case-4

248 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

Table 8. Comparison of acceleration responses (3000 Hz). (a) Acceleration Ratio Plate no. Approach (m/s2) (a/b) Healthy Damaged (a) (b) Experimental 22.09 24.12 0.915 SEAL (Exp) 20.34 21.41 0.950 1 FEM 21.73 21.41 1.014 SEAL (FEM) 22.44 21.39 1.049 Experimental 8.80 8.01 1.098 SEAL (Exp) 8.54 8.85 0.965 2 FEM 5.54 7.72 0.717 SEAL (FEM) 7.55 7.37 1.025 Experimental 4.17 2.83 1.473 SEAL (Exp) 3.86 2.74 1.409 3 FEM 4.07 2.80 1.449 SEAL (FEM) 2.67 1.90 1.405 (b)

(c)

Figure 12. The experimental set-up (Combination-1). combination 1 (three plates with all the spot-welds intact) and combination 2 (three plates with the last plate having one centre-weld damaged), with an excitation at the central point of the first plate has been predicted using statistical energy analysis-like approach. The codes required for carrying out the computations have been developed using MATLAB software. The velocity responses have been predicted for the combi- Figure 13. The velocity responses for Combination-1 (a) Plate 1, (b) Plate 2, nation 1 shown in Fig. 12 and combination 2 shown in Fig. 14 and (c) Plate 3. by using the effective coupling factors obtained from the experimentation and finite element simulation, respectively. The predicted value for combination 1 has been compared with the results obtained from actual experimentation and finite element simulation in Figs. 13(a) to 13(c). The predicted value for combination 2 has been compared with the results obtained from actual experimentation and finite element simulation in Figs. 15(a) to 15(c).The excitation frequency of 3000 Hz has been selected as the deterministic frequency for detection of damage for the combinations as listed out in Table 7 Table 8 shows the acceleration responses obtained for combination 1 (healthy) and combination 2 (damaged) in plate-1, plate-2 and plate-3 at an excitation frequency of 3000 Hz. In practice, the acceleration responses would be measured. Acceleration response for each plate can be computed by multiplying the plate velocity with the frequency of excitation. It has been observed from Table 8 that for combination- Figure 14. The experimental set-up (Combination-2).

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 249 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

(a) ues. In such cases, the ends of the plates away from the spot- weld have large amplitudes vibrations that are limited in practical conditions during the experimental test, as compared to the ﬁnite element simulations. The SEAL predicted values obtained for the other combinations are in close agreement with the ones obtained from actual FE simulation. Such a simulated database bank could be used to identify the joint damage in a particular sub-system for further use.

7. CONCLUSIONS The ease of computing responses and energies through power balance equations using an SEA-like approach as a sim- (b) ulation tool has been used to generate acceleration, velocity responses, and energies for an assembly of sub-systems with a known damage in one of the sub-systems. The reduction in the number of spot welds for sequentially coupled plates lead to reduction in the values of apparent coupling factors, energies, velocities, and an acceleration response of the subsequent plate following the damaged spot-weld joint. Proper FE modeling of spot-welded joints and incorporation of damping from experiments plays a critical role in the estimation of coupling factors using FEA. In case of highly complex conﬁgu- rations, experimental determination of the coupling factors for the basic sub-system may become inevitable. The excitation (c) frequency selected, should represent a uniform energy density in a particular bandwidth fulﬁlling the assumptions in the energy balance equations in an SEA-like approach, which is obtained at higher frequencies due to higher modal density and modal overlap. The requirement of online response measurements and the need to obtain responses in a spatially averaged sense may increase the cost of set-up. The proposed SEAL based prediction approach can be used to generate a database of velocity/acceleration responses for various possibilities of sequentially coupled combinations of the basic cases, which can be effectively applied to locate the damaged joint in an assembly of subsystems for future use. The data generated can Figure 15. The velocity responses for Combination-2 (a) Plate 1, (b) Plate 2, also be further utilized for optimal location of sensors and use and (c) Plate 3. in pattern recognition methods for damage detection.

2 (Third plate connected to the second plate with the centre REFERENCES weld assumed to be damaged) the energy, velocity and conse- 1 Doebling, S.W., Farrar, C.R., and Prime, M.B. A summary quently the acceleration response in Plate-3 (combination-2) is review of vibration-based damage identification methods, reduced in comparison to the values of the same in Plate-3 for The Shock and Vibration Digest, 30 (2), 91-105, (1998). combination-1. https://dx.doi.org/10.1177/058310249803000201 The values predicted using the SEAL approach is in close 2 agreement with other approaches and has been further used to Lopez-D´ ´ıez, J.,Torrealba, M.,Guemes,¨ A., and Cuerno, create a database bank of the expected acceleration responses C. Application of Statistical Energy Analysis for for the remaining combinations of available sub-system cases damage detection in spacecraft structures, Key using the apparent coupling factors as given in Tables 4 and 5. Engineering Materials, 293-294, 525-532, (2005). The ratio of acceleration responses on each plate of combi- https://dx.doi.org/10.4028/www.scientific.net/kem.293- nation 1 (healthy configuration) with the values obtained for 294.525 the remaining combinations respectively has been used as the 3 Whittingham, B., Herszberg, C., and Chiu, W.K. damage indicator as listed out in Table 9. Disbond detection in adhesively bonded compos- It has been seen that the values obtained from the actual FE ite structures using vibration signatures. Jour- simulation for plates joined by a single weld combinations (13, nal of Composite Structures, 75, 351363, (2006). 14, 15, 16) are higher as compared to the SEAL predicted val- https://dx.doi.org/10.1016/j.compstruct.2006.04.055

250 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 A. C. Pankaj, et al.: DETECTION OF DAMAGE IN SPOT WELDED JOINTS USING A STATISTICAL ENERGY ANALYSIS-LIKE APPROACH

Table 9. Ratio of acceleration responses (Excitation Frequency-3000 Hz). Seal Predicted Seal Predicted (FEM) FEM Combination (Exp) Plate-1 Plate-2 Plate-3 Plate-1 Plate-2 Plate-3 Plate-1 Plate-2 Plate-3 3 0.99 0.95 1.54 0.99 0.97 1.47 0.99 1.02 1.23 4 0.99 0.95 1.84 0.99 0.95 2.84 0.99 0.92 2.08 5 0.96 1.33 1.33 0.97 1.34 1.34 0.93 1.21 1.21 6 0.96 1.28 1.76 0.97 1.30 1.78 0.95 1.35 1.84 7 0.96 1.26 2.03 0.97 1.29 1.97 0.90 1.26 2.67 8 0.96 1.25 2.43 0.97 1.27 3.79 0.88 0.83 2.09 9 0.94 1.54 1.54 0.96 1.47 1.47 0.99 1.47 1.65 10 0.94 1.48 2.03 0.96 1.43 1.97 1.00 1.29 2.48 11 0.94 1.46 2.35 0.96 1.43 2.17 0.99 1.54 3.25 12 0.94 1.44 2.81 0.96 1.40 4.18 0.99 2.2 6.53 13 0.93 1.84 1.84 0.94 2.84 2.84 1.03 8.12 5.79 14 0.93 1.77 2.43 0.94 2.77 3.79 1.03 8.48 5.43 15 0.93 1.75 2.81 0.94 2.75 4.18 1.03 5.39 6.52 16 0.93 1.73 3.36 0.94 2.70 8.04 1.03 5.79 14.8

4 Giurgiutiu, V., and Zagrai, A. Damage detec- 11 Fredo, C. R, SEA-like approach for the derivation of en- tion in thin plates and aerospace structures with ergy flow coefficients with a finite element model, Jour- the electro-mechanical impedance method, Struc- nal of Sound and Vibration, 199 (4), 645-666, (2014). tural Health Monitoring, 4, 99-118, (2005). https://dx.doi.org/10.1006/jsvi.1996.0634 https://dx.doi.org/10.1177/1475921705049752 12 Thite, A. N., and Mace, B. R. Robust estimation 5 Samanta, B., Al-Balushi, K. R., Al-Araimi,S. A. Artifi- of coupling loss factors from finite element analysis, cial neural networks and support vector machines with ge- Journal of Sound and Vibration, 303, 814-831, (2007). netic algorithm for bearing fault detection, Engineering https://dx.doi.org/10.1016/j.jsv.2007.02.004 Applications of Artificial Intelligence, 16, 657-665, (2003). 13 https://dx.doi.org/10.1016/j.engappai.2003.09.006 Bies, D. A., and Hamid, S. In situ determination of loss and coupling loss factors by the power injection 6 Worden, K., and Lane, A. J. Damage identification us- method, Journal of Sound and Vibration, 70,187-204, ing support vector machines, Smart Materials and Struc- (1980). https://dx.doi.org/10.1016/0022-460x(80)90595-7 tures, 10, 540547, (2001). https://dx.doi.org/10.1088/0964- 14 1726/10/3/317 De-Langhe K., and Sas, P. Statistical analysis of the power injection method, Journal of the 7 Pankaj, Achuthan. C., Sastry, S., and Murigendrappa, S.M. Acoustical Society of America, 100, 291-303, (1996). A comparison of different methods for determination of https://dx.doi.org/10.1121/1.415915 coupling factor and velocity response of coupled plates, 15 Journal of Vibro-engineering, 15 (4),1885-1897, (2013). Bloss, B., and Rao, M. D. Measurement of damping in structures by the power input method, 8 Yap, F. F., and Woodhouse, J. Investigation of damping Experimental Techniques, 26, 30-33, (2002). effects on statistical energy analysis of coupled structures, https://dx.doi.org/10.1111/j.1747-1567.2002.tb00066.x Journal of Sound and Vibration, 197 (3), 351-371, (1996). 16 https://dx.doi.org/10.1006/jsvi.1996.0536 Palmonella, M., Friswell, M.I., Mottershead, J.E., Lees, A.W. Finite element models of spot welds 9 Kuratani, F.,Matsubara, K., and Yamauchi, T. Finite Ele- in structural dynamics: review and updating, ment Model for Spot Welds Using Multi-Point Constraints Computers and Structures, 83, 648661, (2005). and its dynamic characteristics, SAE International Journal https://dx.doi.org/10.1016/j.compstruc.2004.11.003 of Passenger Cars - Mechanical Systems, 4 (2),1311-1319, 17 (2011). https://dx.doi.org/10.4271/2011-01-1697 Palmonella, M., Friswell, M.I., Mottershead, J.E., Lees, A.W. Guidelines for the implementation of the CWELD 10 Pankaj, Achuthan. C., and Murigendrappa,S.M. Determi- and ACM2 spot weld models in structural dynamics, Fi- nation of coupling factors for spot welded plates, Interna- nite Elements in Analysis and Design, 41, 193210, (2004). tional Conference on Theoretical, Applied, Computational https://dx.doi.org/10.1016/j.finel.2004.04.003 and Experimental Mechanics, IIT-Kharagpur, (2014).

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 251 Probes Design and Experimental Measurement of Acoustic Radiation Resistance Xiaoqing Wang Mechanical Engineering Department, The University of Alabama, Tuscaloosa, AL 35487, USA. Energy and Power Engineering Department, Wuhan University of Technology, Wuhan 430063, China.

Yang Xiang Energy and Power Engineering Department, Wuhan University of Technology, Wuhan 430063, China

(Received 5 September 2016; accepted 31 January 2017) An experimental method used to measure radiation resistance was developed, which was based on a lumped parameter model. Three probes (devices for measuring acoustical radiation resistance) were designed, and the test system was built. To verify the accuracy of the experimental results and to ﬁnd the application frequencies range for each of the probes, self-resistance and cross-resistance measurements of bafﬂed circular pistons were conducted. The results show that this method is a practical way to obtain the acoustical resistance and that the probe with a 57 mm speaker can obtain resistance values in the frequency range from 260 Hz to 1700 Hz. The probe with a 50 mm speaker, on the other hand, presents resistance in the frequency range from 460 Hz to 1900 Hz; the range of 700 Hz to 2600 Hz is for the probe with a 40 mm loudspeaker. To verify the actual application effects, experiments measuring the resistance of a horn were performed. The results show that the measuring system with three probes can be used to predict acoustical resistance of various structures with high accuracy in a convenient and simple way.

1. INTRODUCTION with the volume velocities of the surface elements. Each of the surface elements were assumed to be vibrating as pistons. The 1–3 Active or passive noise control methods are common equation of the power output was determined as ways to lower the level of sound power radiated from vari- 1 T ∗ ous products. Designing a quiet structure and considering the Wav = U RU ; (1) mechanical vibration and acoustic characteristics in the design 2 T stage of the product is the fundamental method to optimize where, U = [U1 ··· UN ] is the volume velocity of the sur-   the radiation resistance of a structure based on the theory of R11 ··· R1N a lumped parameter model, which was developed by Koop- . . . face elements and R =  . .. .  is the acous- mann et al.4–6 The resistance matrix of the structure can be  . .  obtained by analytical and numerical methods, which are suit- RN1 ··· RNN tic surface radiation resistance matrix. The resistance matrix able for products with simple boundary conditions.2, 7 There could be determined by the numerical method. However, due are still accounts of acoustic boundary value problems that are to the huge amount of preparation work and time-consuming difficult to compute with the numerical method for the com- calculations, the experimental method has some advantages in plexity of the geometrical shape.8, 9 The singularity problem determining the values of the resistance matrix. has not been totally solved, although some progress had been 10–13 made. However, the experimental method was not be af- 2.2. Resistance Matrix Measurement Theory fected by this, but it also had the potential to give precise values of the radiation resistance in a frequency ranges. Arenas et In a lumped parameter model, the single term of resistance al.14–16 has studied the experimental method of measuring the matrix is defined as acoustic radiation resistance with good results obtained. Even p(~rm) though some studies have been conducted,14–19 such as the Ra,mn = R ; (2) u(~rn) there is no other research work on the accuracy of the experimental results and the optimal size of the experimental device where p(~rm) is the sound pressure of a point m on the struc- with its applicable frequency range. The research presented in tures surface, and u(~rn) is the volume velocity of a point n on this paper is aimed to solve these two problems. the structure surface. When m = n, it is called self- resistance and when m 6= n, it is called cross-resistance. In this experimental method, a small source with a known 2. THEORY volume velocity is needed to approximate a point simple 2.1. Lumped Parameter Model source, and it must meet two conditions. First, the size of the source should be small enough so that the influence of its In a lumped parameter model4, 20 the acoustic power output shape can be ignored when it is installed on the structures sur- of a vibrating product can be written in the resistance matrix face. Second, it must be able to radiate a sound pressure that

252 https://doi.org/10.20855/ijav.2017.22.2471 (pp. 252–259) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 X. Wang, et al.: PROBES DESIGN AND EXPERIMENTAL MEASUREMENT OF ACOUSTICAL RADIATION RESISTANCE

2 is strong enough to be measured in a certain frequency range, where k is the free-field wavenumber. Since V2 = πld 4, so so that every point on the structures surface can be measured. Eq. (6) can be simplified as These two requirements are conflicting, because a source with ρ0c0u kl a smaller size means it will has lower power output at low fre- pˆ2 = ; (7) quency. To simulate a point simple source,21 the loudspeaker jωV2 sin(kl) used must be small compared to the acoustic wavelength, so a where pˆ is the sound pressure on the top of the cavity 2(z = flat panel loudspeaker with a small diameter was used in the 2 l). Using V and V to represent the volume of the sound cavity present study. The loudspeaker was enclosed in a designed s c and the calibration cavity respectively, and pˆ to represent the cavity to guarantee it is working as a point simple source, s sound pressure in the source cavity, gives which is essential because the loudspeaker will be more representative of a point dipole source if not enclosed. A point ωVc dipole source has a very weak radiation output ability at low uˆ = j 2 pˆsFc. (8) ρ0c0 frequencies.22 According to the acoustic theory, the sound pressure inside a The volume velocity of the arbitrary point k on the surface small volume cavity is determined by the harmonic oscillation of the structure can be calculated with Eq. (9) velocity of the cavity wall and it is defined as ωV uˆ(~r ) = j c pˆ (~r )F . (9) 2 k 2 s k c jρ c ρ0c pˆ = 0 0 uˆ; (3) 0 ωV When the sound pressure of point i on the surface of the where ρ0 is the density of the medium, c0 is sound velocity in structure is measured, then the acoustic radiation resistance be- the medium, ω is the circular frequency (2πf), V is the cavity tween two arbitrary points can be calculated with volume, and uˆ is the harmonic oscillation velocity of the cavity 2 wall. p(~ri) ρ0c0 p(~ri) −1 Za,ik = R = R Fc . (10) In the analysis, ignoring the influence of a standing wave in u(~rk) jωVc ps(~rk) the cavity and given the condition that the volume of the cavity When i = k, it is called self-resistance, and if i 6= k, it is and the response of the microphone measuring sound pressure called cross-resistance. have no phase error, the volume velocity of the chamber wall can be calculated from Eq. (3). In fact, the shape of the loud- Then Eq. (10) can be further simplified as speaker is not regular, so it is difficult to measure the volume 2 ρ0c0 p(~ri) of the cavity accurately. The microphone responses generally za,ik = Im . (11) ωV p (~r ) have some phase error. Therefore, it is necessary to calibrate c s k the measurement devices prior to the experiment. Thus, the acoustic radiation resistance can be calculated The calibration is conducted with a cavity (called calibration with Eq. (11) by amending the phase error of the microphone cavity) whose volume is known and fixed on the source cavity. −1 with Fc which was calculated by the calibration measure- The microphone fixed on the top of the calibration cavity is ment, and by multiplying the sound pressure of arbitrary points used to measure the sound pressure in it. Figure 2 below is the on the surface of the structure and the pressure inside the schematic diagram for the position of the calibration cavity and source cavity with the designed device. the source cavity. The transfer function Fc, referred to as the sound pressure ratio between the two cavities, is determined 3. THE MEASUREMENT SET-UP by 2 2 pˆ2 jρ0c0uˆ2 jρ0c0uˆ1 A schematic of the instrumentation is shown in Fig. 1. It in- Fc = = / ; (4) pˆ1 ωV2 ωV1 cludes one probe, which is the designed experimental device to measure acoustic radiation resistance and the NI cDAQ-9172 Since the air compression in the source cavity corresponds chassis, NI 9263 Analog voltage output module, NI 9234 dy- to the air expansion in the calibration cavity, this means uˆ1 = namic signal acquisition modules, and the computer for data −uˆ2. So the Eq. (4) can be simplified as acquisition and post-processing program. The loudspeakers are flat panel types and their rated power output is 2 W. In pˆ2 V1 Fc = = − . (5) this study, three sets of experimental devices (probes) were de- pˆ V 1 2 signed with the diameters of the three loudspeakers used are 57 mm, 50 mm, and 40 mm, respectively and their projected Thus, V1 can be calculated by measuring the sound pressure of the two cavities. diaphragm diameters are 52 mm, 46 mm, and 36 mm, respectively. The name of the microphone is DGO9767CD, and the Assuming that the diameter of cavity V2 is d, the length 2 sensitivities used are -26 dB, -30 dB, -32 dB, and -50 dB. along the X axis is l (V2 = πld 4), and the transmission of acoustic wave inside of the cavity is in the form of the plane The data acquisition program, which was developed by Lab- wave, the sound pressure along the cavity axis (z axis) can thus VIEW, can gather data automatically after the initial parame- 23 be determined by ters are set up. The data post-processing program was designed by MATLAB. In each of the experiments, the input 4ρ c2u cos(k(l − z)) volume of the loudspeaker was kept constant the loudspeaker pˆ = 0 0 ; (6) z jπd2 sin(kl) was driven with sinusoidal signals, and the frequency range International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 253 X. Wang, et al.: PROBES DESIGN AND EXPERIMENTAL MEASUREMENT OF ACOUSTICAL RADIATION RESISTANCE

Figure 1. Schematic of experimental set-up. b)

Figure 2. Schematic diagram of the calibration and in-situ experimental set- up. c) was from 20 to 4000 Hz with frequency increments of 20 Hz. The sampling rate was 25.6 KHz and the sampling number is 20000. The acoustical radiation resistance was calculated with the actual experimental data and Fc calculated by the calibration measurement. Before the formal measurements, calibration of the experimental devices is needed. As illustrated in Fig. 2, the experimental device was placed on the floor of a hemi-anechoic chamber, which has an acoustical treatment on the walls and ceiling only as well as feature hard floors with no acoustical treatment.24 The design of the calibration cavity is based on the calibration principle developed above. Its length is deter- Figure 3. Results of calibration for the three devices: Loudspeaker, P = 2 W, mined by the attenuation of sound pressure along the calibra- (a) D = 57 mm, (b) D = 50 mm, and (c) D = 40 mm. tion cavity axis within 1dB. Thus, the loudspeaker stimulates acoustic fields synchronously in the source cavity and the cal- tronics, such as capacitors and resistors. ibration cavity. The microphone fixed on the top of the calibration cavity measures the sound pressure in the calibration Even though the imaginary parts of transfer functions show cavity. Based on former studies,25–27 after reducing the effect some fluctuations, they stay close to zero, tending to be linear of the major factors influencing the accuracy of the measure- variation. The real parts approximation show straight lines in ment, the results obtained showed greater improvements. Each the range of the interested frequencies, which suggests that the of the three devices‘ transfer function between the calibration loudspeakers have a good frequency characteristic. The stable cavity and the source cavity calculated by calibration measure- working range of frequencies for the loudspeakers will become ments is illustrated as in Fig. 3. wider with the decreasing of the loudspeaker‘s diameter. Recalling Eq. (5), the transfer function between the pressure in the calibrator and the speaker cavities should be constant as a function of frequency with its imaginary part identically zero. However, it is known that the actual physical systems 4. APPLICABLE FREQUENCY RANGE OF always have some difference from their idealized models and THE PROBE the difference will lead to the systems having some limitations in their actual application. It can be seen from Fig. 3 that the transfer function will vary with the frequency in the actual Two kinds of tests were conducted to verify the ability of measuring conditions, which is caused by the different phase the experimental method determining the acoustical resistance: response between the two microphones together with other fac- verification of the self- resistance measurements and verifica- tors brought by the loudspeaker and the other associated elec- tion of cross-resistance measurements.

254 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 X. Wang, et al.: PROBES DESIGN AND EXPERIMENTAL MEASUREMENT OF ACOUSTICAL RADIATION RESISTANCE

Figure 4. Self-resistance measurement of baffled circular piston. 4.1. Self-resistance of Baffled Circular Piston b) To simulate measuring the resistance of a piston in an infinite baffled plate, the experimental device was placed on the floor of the hemi-anechoic chamber, as illustrated in Fig. 4. The influences of the device‘s height were neglected in this study, as they have a relatively low height compared to the size of the hemi-anechoic chamber. Referring back to Eq. (11), the radiation resistance can be calculated with the experimental data and Fc. For the convenience of comparison, the experimental solutions were changed to a dimensionless quantity by multiplying the area c) of the speaker diaphragm Si (below) and then divided by the characteristic impedance of the acoustic medium ρc. By assuming the diaphragm of the loudspeaker moves as a piston, the self-resistance Rii of the baffled circular piston was calculated using the Eq. (12)28

J (2kr) R = ρ cS 1 − 1 ; (12) ii 0 i kr where J1 is the first order Bessel function and r is the radius of the loudspeaker‘s diagram. The measured self-resistances are compared with the analytical values respectively, as shown in Fig. 5. Figure 5. Results for the normalized self-resistance of baffled circular piston: It can be seen from the comparison in Fig. 5 that the exper- Loudspeaker, P = 2 W, (a) D = 57 mm, (b) D = 50 mm, and (c) D = imental values agree well with the analytical values in a wide 40 mm. frequency range, which states that this experimental method is should not be viewed as a point simple source. This is not able to obtain the acoustical resistance. However, there are sig- in correspondence with the condition of the lumped parameter nificant differences at the low frequencies. The experimental model and thus the performance of the measuring probe will results are far from the analytical values and also show some be poor. natural random fluctuations, which will lighten with the diaphragm of the loudspeakers increasing from 36 mm to 52 mm. 4.2. Cross-resistance of Baffled Circular This phenomenon only occurred at the low frequencies be- Piston cause of the lower SNR (signal-to-noise ratio). At certain frequencies and currents, the smaller the diameter of the loud- As stated in the Eq. (1), it is known that the cross-resistance speaker‘s diaphragm, the lower the power radiated from the terms take a large proportion of the resistance matrix in diaphragm. Thus, the radiation resistance measurement is lim- computing the average sound power of the vibration struc- ited by the noise floor of this hemi-anechoic chamber, which is ture. Using the approach of the discrete calculation method, at approximately RSρc = 1×10-2. Therefore, the experimen- Hashimoto (2001) presented the approximated values of the tal results will not show good agreement with the analytical cross-resistance between the ith element and the kth element,29 values below 0.01. That is why the degree of agreement be- 2ρ ck2S S J (kr ) J (kr ) sin kd tween the experimental and the analytical values declines. The R = 0 i k 1 i 1 k ik ; (13) ik π kr kr kd low frequency limitations will ascend at the same time. i k ik In addition, the difference between them becomes bigger at where Si and Sk√are the areas of√ the ith and kth elements re- high frequencies, for example, above 2000 Hz, because the spectively, ri = Siπ and rk = Skπ are the equivalent radii condition of the lumped parameter model will not be satisfied of the ith and kth elements respectively, and dik is the distance there. When the wavelength is shorter, the ratio of it to the between the centers of the two circular vibrating piston ele- size of the probe is not big enough and the measuring probe ments.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 255 X. Wang, et al.: PROBES DESIGN AND EXPERIMENTAL MEASUREMENT OF ACOUSTICAL RADIATION RESISTANCE

Figure 6. Results of the cross-resistance for two circular elements (r = ri = rk = 0.026 m; d = dik = nD (n = 1, 2, 3, 4; D = 0.057 m)).

Figure 7. Cross-resistance measurement of bafﬂed circular piston c) (d = 0.057 m).

Thus, the two circular elements are taken as an example, ri = rk = 0.026 m, dik = nD (n = 1,2,3,4; D = 0.057 m), and the absolute value of cross-resistance (C-R) for the ith and kth elements are presented in Fig. 6. It can be seen that the results of the cross-resistance decrease when the distance between the two elements increase. There are more valleys with the increasing of the distance, which will seriously reduce the power radiated from the elements. Thus, only the first 3 steps off- axis cross-resistance of the devices were measured. Similar to the self-resistance measurements, the device was Figure 8. Results of Cross-Resistance Verification (Loudspeaker D = 57 mm, dik = 57 mm): (a) One Step Off-Axis (n = 1), (b) Two steps off-axis (n = placed on the floor of the hemi-chamber, and the field point 2), and (c) Three steps off-axis (n = 3). was measured with the microphone at one, two and three dik (57 cm, 50 cm and 40 cm) from the center of the loudspeaker. linear variation frequency range of the transfer function. The The in-situ cross-resistance measurement of a baffled circular device whose diameter of the loudspeaker is 50mm presents a piston is shown in Fig. 7 and the comparison of the experimen- precise resistance value in the frequency range from 460 Hz to tal and analytical results are shown in Fig. 8. 1900 Hz, and, 700 Hz to 2600 Hz for the device with a 40 mm The results show very good agreement between the exper- diameter loudspeaker. imental and analytical results. The degree of the agreement between the experimental and the analytical results goes down 5. APPLICATION OF THE PROBES IN THE with the diameter of the loudspeaker decreasing from 60 mm PIPE AND HORN to 40 mm, which resulted from the same reasons mentioned above. The first one is the lower SNR due to the powder ra- The pipe is a critical application scenario for sound trans- diated from the diaphragm, especially for the speakers with a mission. One of its important functions is that it can be used small diaphragms as the smaller the diaphragm, the lower the to amplify, silence or even eliminate sound. Take a horn with powder radiated from it. Another reason is the frequency lim- high power output for daily use as an example. As is shown in itation from the lumped parameter model as the requirements Fig. 11, the horn is circular symmetry with its radius varying for it will not be satisfied at high frequencies, which lead to the as a complex function of the distance along the centerline. It is big errors. made of 2 mm aluminum alloy with length of 355 mm and the Based on the self-resistance and cross-resistance tests, the radius at the throat and mouth are 128 mm and 500 mm, re- applicable frequency range can be determined respectively. spectively. There is a connection tube at the throat of the horn The device with a loudspeaker of 60 mm can obtain resistance with the same radius as the throat and its length is 40 mm. values that agree well with the analytical solutions in the fre- During the test, the horn was placed on the floor of a hemi- quency range from 260 to 1700 Hz, which is almost the same anechoic chamber and the gap between it and the floor was

256 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 X. Wang, et al.: PROBES DESIGN AND EXPERIMENTAL MEASUREMENT OF ACOUSTICAL RADIATION RESISTANCE

a) a)

b) b)

c) c)

D = Figure 9. Results of Cross-Resistance Verification, (Loudspeaker D = Figure 10. Results of Cross-Resistance Verification, (Loudspeaker d = n = 50 mm, d = 50 mm): (a) One Step Off-Axis (n = 1), (b) Two steps off-axis 40 mm, ik 40 mm): (a) One Step Off-Axis ( 1), (b) Two steps off-axis ik n = n = (n = 2), and (c) Three steps off-axis (n = 3). ( 2), and (c) Three steps off-axis ( 3). sealed by the BLU Tack. The measuring probe was placed at are some differences at the low frequencies and high frequen- the center of the throat of the horn, as illustrated in Fig. 11. cies, which has the same reasons as discussed above. In ad- To verify the accuracy of the measured results, they were dition, the inaccuracy of the horn model might also results in compared to numerical values because the cross-section of the lead to the difference between the test and numerical values. horn changes continuously and therefore cannot be analyzed There would be some error between the model used in calcu- using Webster‘s horn equation which is applicable to exponen- lation and the actual size of the horn, because the detail size of tial horns, conical horns and so on.15, 16 The numerical values the horn is not easily measured accurately. were calculated with the boundary element method (BEM) us- Through this application, it can be seen that the measuring ing Virtual.Lab.Acoustics, in which the air is set as a homo- system can present the resistance results with high accuracy. geneous medium with density and velocity of the sound. The This will save a lot of time in building the model and mak- boundary condition of velocity is exerted at the center of the ing calculations, especially when the structures have complex tube‘s bottom in a circular area with radius same as the di- shapes. The system is convenient and very easy to operate, and aphragm of the loudspeaker used in the tests. In acoustical thus can be used to determine the acoustical resistance matrix grid model, the field point and boundary condition of velocity of various structures. is shown in Fig. 12. The comparison between the measured results and numerical values of the radiation resistance as a 6. CONCLUSIONS function of frequency is shown in Fig. 13. The frequency increment in both case is 2 Hz. In this paper, a method has been presented to measure the As can be seen from Fig. 13, the test results agree well with acoustic radiation resistance. With the three sets of probes de- the numerical results which indicates that the designed devices signed, the self-resistance and cross-resistance of the infinite can obtain the resistance of the horn with high accuracy. There baffled piston source had been conducted. Comparing the ex-

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 257 X. Wang, et al.: PROBES DESIGN AND EXPERIMENTAL MEASUREMENT OF ACOUSTICAL RADIATION RESISTANCE

Figure 11. In-situ measurement of horn. b)

Figure 12. Acoustic grid model of horn. perimental and analytical results showed that this method is applicable to obtain the acoustical resistance and the applicable frequency ranges for the three devices tested. The frequency ranges 260 ∼ 1700 Hz, 460 ∼ 1900 Hz, and 700 ∼ 2600 Hz for the devices with 57 mm, 50 mm, and 40 mm loudspeakers, respectively were obtained. Then the resistance at the throat of a horn was tested to verify actual application effects. The results showed that the measuring system with three probes can be used to measure vari- Figure 13. Comparison between the tests results and numerical values of the ous structures‘ acoustical resistance with a high accuracy in a radiation resistance of the horn: Loudspeaker, P = 2 W, (a) D = 57 mm, (b) D = 50 mm, and (c) D = 40 mm. convenient and simple manner. The qualities of the loudspeakers, such as harmonic distor- spherical cap, Applied Acoustics, 115, 23–31, (2017). tion and characteristic sensitivity, have signiﬁcant inﬂuence on https://dx.doi.org/10.1016/j.apacoust.2016.08.010 the experimental results. The results would be better at low frequencies if the loudspeaker has a high quality with high power 3 Wise, S., and Leventhall, G. Active noise control as output. a solution to low frequency noise problems, Journal of Low Frequency Noise, Vibration and Active Control, ACKNOWLEDGEMENTS 29 (2), 129–137, (2010). https://dx.doi.org/10.1260/0263- 0923.29.2.129 This work was supported by the National Natural Science Foundation of China (Grant No. 5107911851279148). Xiao- 4 Fahnline, J. B., and Koopmann, G. H. A lumped qing Wang also gratefully acknowledges the help from Xuebao parameter model for the acoustic power output from Xia, Yuxiao Shi, Rui Li, Sichong Qian, Lei Shi, and Peng Xue. a vibrating structure, The Journal of the Acousti- cal Society of America, 100 (6), 3539–3547, (1996). REFERENCES https://dx.doi.org/10.1121/1.417330 1 Marburg, S. Developments in structural-acoustic opti- 5 Xiang, Y., Guo, Z., and Wang, X. Research on design of the mization for passive noise control, Archives of computa- resistance probe and experiments using the probe, Journal tional methods in engineering, 9 (4), 291–370, (2002). of Wuhan University of Technology (Transportation Science https://dx.doi.org/10.1007/bf03041465 & Engineering), 36 (6), 1140–1142, (2012).

2 Liu, Z., and Maury, C. Numerical and experimental study 6 Xiang, Y., Guo, Z., and Wang, X. Research on the Measure- of Near-Field Acoustic Radiation Modes of a bafﬂed ment Principle Design and Calibration of the Resistance

258 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 X. Wang, et al.: PROBES DESIGN AND EXPERIMENTAL MEASUREMENT OF ACOUSTICAL RADIATION RESISTANCE

Probe, Journal of Wuhan University of Technology (Trans- 18 Driesch, P. L., Iwata, H., Koopmann, G. H., and Dosch, portation Science & Engineering), 36 (5), 899–906, (2012). J. Development and evaluation of a surface acoustic intensity probe, Review of Scientific Instruments, 71 (10), 3947– 7 Faverjon, B., and Soize, C. Equivalent acoustic impedance 3952, (2000). https://dx.doi.org/10.1063/1.1311944 model. Part 2: analytical approximation, Journal of Sound and Vibration, 276 (3–5), 593–613, (2004). 19 Aewnas, J. P., (2001). Analysis of the Acoustic Radiation https://dx.doi.org/10.1016/j.jsv.2003.08.054 Resistance and Its Applications to Vibro-Acoustic Prob- lems, Doctor PhD Thesis, Auburn University. 8 Robjohns, H. A brief history of microphones, First published Microphone Data Book, (2001). 20 Arenas, J. P., Ramis, J., and Alba, J. Far-Field Sound Ra- diation from a Baffled Elliptical Piston Using a Lumped- 9 Crocker, M. J., Arenas, J. P., and Dyamannavar, R. Parameter Model and Singular Value Decomposition, 15th E. Identification of noise sources on a residential split- International Congress on Sound and Vibration, Daejeon, system air-conditioner using sound intensity measure- Korea, 1633–1640, (2008). ments, Applied Acoustics, 65 (5), 545–558, (2004). https://dx.doi.org/10.1016/j.apacoust.2003.10.008 21 Bruneau, M. Fundamentals of acoustics, John Wiley & Sons, (2013). 10 Zellers, B., Koopman, G., Hwang, Y., and Naghshineh, K. Element Free Structural Acoustics for Efficient 22 Oskooi, A. F., Roundy, D., Ibanescu, M., Bermel, Shape Optimization, Proc. ASME International P., Joannopoulos, J. D., and Johnson, S. G. MEEP: Mechanical Engineering Congress and Exposition. A flexible free-software package for electromag- https://dx.doi.org/10.1115/imece2005-82958 netic simulations by the FDTD method, Computer Physics Communications, 181 (3), 687–702, (2010). 11 Wu, S., Xiang, Y., Xia, X., and Wang, X. Research on a https://dx.doi.org/10.1016/j.cpc.2009.11.008 method for acoustic superposition in element free spatial discrete domains, Journal of Huazhong University of Sci- 23 Wang, X., (2013). Research on the Experimental Measure- ence and Technology (Natural Science Edition), 41 (11), ment and Feasible Technical Solutions of Acoustic Radia- 22–25, (2014). tion Resistance, Master Master, Wuhan University of Tech- nology Wuhan. 12 Xia, X., Xiang, Y., Wang, X., and Wu, S. Research on the acoustic radiation resistance calculation based on wave 24 Davidson, M. A. Hemi-anechoic chamber qualification and superposition method, Journal of Ship Mechanics, 19 (1), comparison of room qualification standards, The Journal 206–214, (2015). of the Acoustical Society of America, 135 (4),2379–2380, (2014). https://dx.doi.org/10.1121/1.4877859 13 Qian, S., Xiang, Y., Xiao, X., and Wang, X. Application of Gammatone Filter Bank to Vibration Characteristics Analy- 25 Wang, X., Xiang, Y., Guo, Z., Xia, X., Shi, Y., Xue, sis of Engine Cylinder Head, Chinese Internal Combustion P., and Wu, S. Research on experimental measurement of Engine Engineering, 34 (6), 36–42, (2013). acoustic resistance and major accuracy influencing factors analysis, J Mech Sci Technol, 28 (4), 1219–1227, (2014). 14 Arenas, J. P., and Crocker, M. J. Numerical solu- https://dx.doi.org/10.1007/s12206-014-0302-1 tion for the transmission loss in pipes and reactive expansion chambers of gradual area change, Noise Con- 26 Wang, X., Xiang, Y., Shi, Y., Wu, S., and Xue, P. Exper- trol Engineering Journal, 49 (5), 224–230, (2001). imental Research on the Measurement of Acoustic Radi- https://dx.doi.org/10.3397/1.2839664 ation Impedance of Several Typical Acoustic Structures, Journal of Wuhan University of Technology, 35 (1), 151– 15 Arenas, J. P., and Crocker, M. J. Approximate Results 156, (2013). of Acoustic Impedance for a Cosine-Shaped Horn, Jour- nal of Sound and Vibration, 239 (2), 369–378, (2001). 27 Wang, X., Xiang, Y., Shi, L., Qian, S., and Guo, Z. Design https://dx.doi.org/10.1006/jsvi.2000.3062 of acoustic radiation resistance measuring system and analysis of factors influencing measuring accuracy, Journal of 16 Arenas, J. P., and Crocker, M. J. A note on a WKB Applied Acoustics, 33 (3), 228–237, (2014). application to a duct of varying cross-section, Ap- plied Mathematics Letters, 14 (6), 667–671, (2001). 28 Pierce, A. D. Acoustics: an introduction to its physical prin- https://dx.doi.org/10.1016/s0893-9659(01)80024-0 ciples and applications, McGraw-Hill New York, (1981).

17 Paddock, E. H., and Koopmann, H. G. The Use of Radiation 29 Hashimoto, N. Measurement of sound radiation efﬁciency Resistance Measurements to Assess the Noise Characteris- by the discrete calculation method, Applied Acoustics, tics of Machines, Journal of American Society of Mechani- 62, 429–446, (2001). https://dx.doi.org/10.1016/s0003- cal Engineers, 3 (B), 655–662, (1995). 682x(00)00025-6

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 259 Thermoviscoelastic Vibrations of a Micro-Scale Beam Subjected to Sinusoidal Pulse Heating

D. S. Mashat Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

A. M. Zenkour Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt

A. E. Abouelregal Department of Mathematics, College of Science and Arts, Aljouf University, Al-Qurayat, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

(Received 11 September 2016; accepted 1 March 2017) In this article, we studied the vibrational effects of a viscoelastic micro-scale beam induced by sinusoidal pulse heating based on Euler-Bernoulli beam theory. The formulation of the present problem is applied to the generalized thermoelasticity with phase lags. The deﬂection, temperature, axial displacement, and bending moment of the micro-scale beam are determined by using the Laplace transform method. The system of governing equations is reduced to a novel six-order thermoelastic differential equation in either deﬂection or temperature. The numerical results in case of silicon material are presented with the help of Mathematica programming software. Some plots are illustrated to investigate the effects of the phase-lags and the width of the sinusoidal pulse parameters. The effect of the viscosity on the micro-scale beam resonator is also investigated.

1. INTRODUCTION variable warming. Extremely fast thermal procedures, under the activity of a ultra-short laser heartbeat, are intriguing from The linear viscoelasticity remains an important area of re- the outlook of thermoelasticity, since they require an examina- search since most of the solids and polymer-like materials are tion of the coupled temperature and strain fields. This implies subjected to a dynamic loading exhibit viscous effect. The the absorption of the laser beat energy results in a limited tem- stress-strain law for many materials, such as polycrystalline perature expand, which thusly causes thermal expansion and metals and high polymers, can be approximated by the linear creates fast movements in the structure components, in this viscoelasticity theory. In using different generalized thermoe- manner bringing about the ascent of vibrations. lasticity theories, many authors have considered various linear thermoviscoelastic problems.1–9 Micro-scale mechanical resonators have high affectability as The classical coupled theory of thermoelasticity is based on well as a fast response and are widely used as sensors and mod- a parabolic type of heat conduction equation while the gen- ulators. Micro- and nano-mechanical resonators have recently eralized thermoelasticity theory was postulated by Lord and pulled in impressive consideration due to their many signifi- Shulman10 as well as Green and Lindsay,11 who used a hyper- cant industrial applications. Perfect analysis of many effects bolic type of heat conduction equation admitting finite speed on the features of resonators, such as resonant frequencies and for thermal disturbances. The theory developed by Lord and quality influences, is crucial for designing high-performance Shulman10 takes into account only one parameter of relax- components. Many authors have considered the vibration and 22–28 ation time and has been extended by Dhaliwal and Sherief12 heat exchange procedure of micro-scale beams. in anisotropic media. Another generalization to the theory of The objective of the present work is to determine the com- thermoelasticity is the dual-phase-lag (DPL) model, which is ponents of displacements, stress, temperature, and strain dis- proposed by Tzou.13, 14 tributions in an isotropic homogeneous thermoviscoelastic thin Laser ultrasound is another innovation; it uses a laser to pro- beam subjected to sinusoidal pulse heating.29–31 The problem duce and test ultrasound.15–21 It is another branch of ultrasound is solved in context of the theory of generalized thermovis- and includes optics, acoustics, calorifics, electrics, material, coelastic with phase lags. An exact solution of the problem physics and so on. Laser ultrasound has numerous points of is first obtained in Laplace transform space. The inversions of interest. For example, it is noncontact, nondestructive, quick, Laplace transforms have been carried out numerically. Numer- precise, and requires minimal effort. In this way, it is typi- ical results predict finite speeds of propagation for thermoelas- cally utilized to distinguish and portray absconds as a part of tic and diffusive waves. To investigate the viscosity and phase aviation materials or an aerocraft. Thermally prompted vibra- lags effects, a comparison is made with the results obtained tion of bars has practical significance in space vehicles, reac- in the thermoelastic problem. Finally, in taking an appropri- tor vessels, turbines, and other machine parts are subjected to ate material, the results are presented in a graphical form to

260 https://doi.org/10.20855/ijav.2017.22.2472 (pp. 260–269) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING illustrate the impotency of this problem.

2. MATHEMATICAL MODELING AND THE FUNDAMENTAL EQUATIONS The Kelvin-Voigt model is one of the macroscopic mechanical models commonly used to describe the viscoelastic manner of a material. The model represents the delayed flexible reaction subjected to stress when the deformation is time- dependent but recoverable. The dynamic association of the mechanical and thermal fields in solids has pronounced practi- Figure 1. Schematic illustration of the beam set-up. cal applications in current aeronautics, astronautics, atomic re- dynamical thermoviscoelasticity theory, the generalized ther- actors, and high-energy element accelerators, for instance we moelasticity theory proposed by Lord and Shulman, the gen- refer to Mishra et al.32 eralized thermoelasticity theory with two relaxation times de- We consider a homogenous isotropic thermally conducting veloped by Green and Lindsay, and Green and Naghdi theory Kelvin-Voigt type thermoviscoelastic solid initially at uniform without energy dissipation and dual-phase-lag model for dif- temperature T and undeformed. The governing equations for 0 ferent sets of values of the parameters τ and τ . a linear isotropic and homogeneous thermoviscoelastic solid in θ q In Eqs. (1) to (5): the absence of body forces take the following forms.33 The equation of motion has the form 1) If we put τθ = τq = 0, it reduces to the equations of the classical theory of thermoviscoelasticity (CTE). ∂2u µ∗u + (µ∗ + λ∗) u − β∗θ = ρ i ; (1) i,jj j,ij ,i ∂t2 2) When τθ = 0, τq ≥ 0 and δ = 1, it reduces to the equations of the generalized thermoviscoelasticity theory with where ui are the components of the displacement vector, θ = one relaxation time (LS). T − T0, T is the absolute temperature of the medium, T0 is the reference uniform temperature of the body chosen such that 3) Putting τθ = 0, τq = 1 and δ = 0, reduces to the equa- ∗ tions of the generalized thermoviscoelasticity theory with |θ/T0| 1, t is time and ρ is the density. The parameters, µ , λ∗, and β∗, are defined as two relaxation time (GL). ∗ ∗ ∂ ∂ 4) Putting τθ = 0, τq = 1, δ = 0, and K = K ,(K is a λ∗ = λ 1 + α ; µ∗ = µ 1 + α ; material constant characteristic of Green and Naghdi the- 1 ∂t 2 ∂t ory), reduces to the equations of the generalized thermo- ∂ β∗ = β 1 + β ; β = (3λ + 2µ) α ; viscoelasticity theory without energy dissipation (GN). 1 ∂t t 5) If 0 ≤ τθ < τq and δ = 1, it reduces to the equations (3λα1 + 2µα2)αt β1 = ; (2) of the generalized thermoviscoelasticity theory with dual- β phase-lags (DPL). where λ and µ are Lames´ constants, α1, α2 are the thermovis- In absence of viscous effect the viscoelastic relaxations times coelastic relaxation times, and αt is the coefficients of linear are zero (i.e., α1 = α2 = 0). thermal expansion. The strain-displacement relations are given by 3. FORMULATION OF THE PROBLEM 1 We consider a homogeneous, thermally conducting, Kelvin- eij = (ui,j + uj,i) . (3) 2 Voigt type visco-thermoelastic micro-scale beam (Fig. 1) with length L, thickness h, and width b. A Cartesian coordinate The heat conduction equation in the context of generalized system is attached to the beam such that the x-coordinate is thermoelastic with phase lags, is modified to be of the form parallel to the beam axis. The thickness and width directions are parallel to the y- and z-axes, respectively, and then struc- ∂ ∂ ∂θ ∗ ∂e K 1 + τθ θ,ii = δ + τq ρCE + β T0 − ρQ ; ture occupies the domain defined by 0 ≤ x ≤ L, −b ≤ 2y ≤ b ∂t ∂t ∂t ∂t and −h ≤ 2z ≤ h. (4) The beam undergoes bending vibrations of small amplitude where e is the strain dilatation, K is the thermal conductivity, about the x-axis, such that the deformation is consistent with the linear Euler-Bernoulli theory. The usual Euler-Bernoulli CE is the specific heat at constant strain, τq is the phase-lag assumption34 is made so that any plane cross-section, initially of the heat flux, τθ is phase-lag of the temperature gradient perpendicular to the axis of the beam, remains plane and per- 0 ≤ τθ < τq, and Q is the heat source per unit volume. The dummy index implies summation. pendicular to the neutral surface during bending. Thus, the The constitutive equations have the form displacements can be given by ∂w σ = 2µ∗e + (λ∗e − β∗θ) δ ; (5) u = −z ; v = 0; w = w(x, z, t); (6) ij ij kk ij ∂x where σij is the stress tensor and δij is Kronecker’s delta func- where u is the axial displacement and w is the transverse dis- tion. Equations (1) to (5) respectively describe the coupled placement in the z direction (the lateral deflection). The elastic

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 261 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING stress field in the beam is assumed to be uniaxial; only σxx can beam can be modeled as a one-dimensional problem with an attained non-zero values. This assumption is valid for slender energy source Q(z, t) near the surface, i.e., Euler-Bernoulli beams and neglects any shear stresses in the 34 vicinity of clamped boundaries. The constitutive equation Ra 2z−h L0Ra 2z−h − t Q(z, t) = I(t)e 2δ0 = te 2δ0 tp ; (17) reduces to one-dimensional equation yields 2 δ0 δ0tp ∂u σ = (2µ∗ + λ∗) − β∗θ. (7) xx ∂x where δ0 the absorptive depth of heating energy and Ra the absorptivity of the irradiated surface. Substituting the Euler- The flexure moment of the cross-section is given as Bernoulli assumption, namely Eq. (6), and (17) into Eq. (4) gives the thermal conduction equation for the beam. Z h/2 M(x, t) = −b zσxxdz. (8) The heat conduction in Eq. (4), for the five theories CTE, −h/2 LS, GL, GN, and DPL are described by the following system of partial differential equations for isotropic materials: Using Eq. (7) into Eq. (8) to get the flexure moment of beam as follows: ∂ ∂2 ∂2 2 K 1 + τθ + θ = ∗ ∗ ∂ w ∗ ∂t ∂x2 ∂z2 M(x, t) = I (2µ + λ ) z 2 + β MT ; (9) ∂x ∂ ∂θ ∂3w δ + τ ρC − β∗T z − ρQ . (18) where I = bh3/12 is moment of inertia of the cross-section q ∂t E ∂t 0 ∂x2∂t and h/2 Z Using Eqs. (11) and (17) into Eq. (18), we get MT = b zθ(x, z, t)dz; (10) −h/2 2 is the thermal moment of the beam. ∂ ∂ 2 ∂ sin(pz) 1 + τθ − p φ = δ + τq · There is no heat flow across the upper and lower surfaces of ∂t ∂x2 ∂t ∂θ h ∗ 3 the beam, hence = 0 at z = ± . For a very thin beam, it ρCE ∂φ β T0 ∂ w ρL0Ra 2z−h − t ∂x 2 sin(pz) − z − te 2δ0 tp . is assumed that the temperature increment varies in terms of a 2 2 K ∂t K ∂x ∂t Kδ0tp sin function along the thickness direction. That is (19) π θ(x, z, t) = φ(x, t) sin(pz); p = . (11) h Multiplying Eq. (19) by z and integrating it with respect to z from −h/2 to h/2, yields Substituting Eq. (11) into Eq. (10), gives

2 ∂ ∂2 MT = 2ap φ(x, t). (12) 1 + τ − p2 φ = θ ∂t ∂x2 Substituting from Eq. (12) into Eq. (9), gives 3 ∂ ∂φ ∗ ∂ w δ + τq η0 − β η1 − η2f(t) ; (20) ∂2w ∂t ∂t ∂x2∂t M(x, t) = I (2µ∗ + λ∗) z + 2ap2β∗φ(x, t). (13) ∂x2 where The equation of motion for free ﬂexural vibrations of the beam is ρC T h3 ρL R δ h ∂2M ∂2w η = E ; η = 0 ; η = 0 a 0 − + ρA = 0; (14) 0 K 1 2p2K 2 2p2Kt2 2δ ∂x2 ∂t2 p 0 h −h − t where A = bh, the cross-section area. By substituting Eq. (13) 1 + + 1 e δ0 ; f(t) = te tp . (21) into Eq. (14), we can get the motion equation of the beam as 2δ0 follows: ∂4w ∂2w ∂2φ I (2µ∗ + λ∗) + ρA + 2ap2β∗ = 0. (15) 4. DIMENSIONLESS QUANTITIES ∂x4 ∂t2 ∂x2 The initial temperature distribution in the beam T (x, t, 0) = For convenience, we shall use the following non- dimensional quantities T0, i.e., θ(x, z, t) = 0. Also, for t = 0, the upper surface (z = h/2) of the beam is heated uniformly by a laser pulse 0 0 0 0 0 0 with non-Gaussian form temporal proﬁle as follows: {x , w , u , z ,L , b } = c1η0{x, w, u, z, L, b};

0 0 0 2 0 β L0 − t {t , τq, τθ} = c1η0{t, τq, τθ}; θ = θ; tp ρc2 I(t) = 2 te ; (16) 1 tp 0 1 0 1 2 2µ + λ σxx = 2 σxx; Q = 2 2 Q; c1 = ; (22) where tp is a characteristic time of the laser-pulse, L0 is the ρc1 Kc1η2T0 ρ laser intensity, which is deﬁned as the total energy carried by a laser pulse per unit cross-section of the laser beam. It is clear where c1 is the speed of propagation of isothermal elastic that the maximum laser intensity decreases as tp increases. In waves. For simplicity, we drop the dashes of all variables and accordance with Royer,18 the conduction heat transfer in the parameters. The resulting non-dimensional Eqs. (15) and (20)

262 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING are The characteristic equation of Eq. (32) is

∂ ∂4w ∂2w ∂ ∂2φ k6 − a k4 + a k2 − a = 0; (34) 1 + c2 + η + η 1 + β = 0; 1 2 3 2 ∂t ∂x4 3 ∂t2 4 1 ∂t ∂x2 2 2 2 (23) where k1, k2, and k3 are the roots of the characteristic equation. These roots are given by21 2 ∂ ∂ 2 1 + τθ − p φ = ∂t ∂x2 2 1 h 3 1/3 i ki = ψi 2a1 + 27a3 − 9a1a2 + a2 , i = 1, 2, 3; ∂ ∂φ ∂ ∂3w 3 δ + τ − εη 1 + β − η f(t) ; (35) q ∂t ∂t 5 1 ∂t ∂x2∂t 6 (24) where 2 1 √ √ 2 ∂ ∂ w ∂ σ = −z 1 + η c − sin(pz) 1 + β φ; {ψ1, ψ2, ψ3} = − {2R0, −R0 + i 3R1, −R0 − i 3R1}; xx 0 2 ∂t ∂x2 1 ∂t 2 2 2 (25) ξ − R0 ξ + R2 R = ; R = ; 0 ξ 1 ξ where 3a − a2 R = 2 1 ; 2 5 2 3 2/3 2 α1λ + 2α2µ 12 24p c1η0 (2a1 + 27a3 − 9a1a2) c2 = ; η3 = 2 ; η4 = 3 ; ρ h h 1 q 1/3 3 2 ξ = 4 + 4 4R3 + 1 . (36) h βη2 β T0 2 2 η5 = 2 6 4 ; η6 = 4 3 ; ε = 2 2 . (26) 2p c1η0 ρc1η0 ρ CEc1 Using Eq. (28) in Eq. (29), we get 5. SOLUTION OF THE PROBLEM IN −1 d4w d2w LAPLACE TRANSFORM DOMAIN φ = 4 − A2A4 2 + A1w − A2A5g(s) . A2A3 dx dx (37) In order to solve the problem, both the initial and boundary Then we can express the general solutions of Eqs. (32) and (37) conditions should be considered. The initial conditions of the by the linear combination of the fundamental solutions. That problem are taken as is ∂w(x, t) w(x, t) = = 0; 3 t=0 X −kj x kj x ∂t t=0 w = Cje + Cj+3e ; (38)

∂φ(x, t) j=1 φ(x, t) = = 0. (27) t=0 ∂t 3 t=0 X −kj x kj x φ = Hj Cje + Cj+3e + H4; (39) By applying Laplace transform to Eqs. (23) to (25), we get the j=1 ﬁeld equations in the Laplace transform space as where d4w d2φ −1 4 2 A5g(s) 4 + A1w + A2 2 = 0; (28) Hj = kj − A2A4kj + A1 ; H4 = ; (40) dx dx A2A3 A3 d2 d2w − A φ = −A − A g(s); (29) are parameters depending on s. Thus, Eqs. (38) and (39) con- dx2 3 4 dx2 5 stitute a complete set of governing equations for the homoge- d2w σ = − 1 + η c2s z − sin(pz) (1 + β s) φ; (30) nous isotropic Kelvin-Voigt type thermoviscoelastic beam. In xx 0 2 dx2 1 addition, these equations can also be supplemented with appropriate initial and boundary conditions of the relevant problem where to be modelled. The displacement after using Eq. (38) takes s2η η (1 + β s) the form A = 3 ; A = 4 1 ; A = p2 + q; 1 1 + sc2 2 1 + sc2 3 2 2 dw 3 X −kj x kj x s(δ + τ s) u = −z = z kj Cje − Cj+3e ; (41) q dx q = ; A4 = qη5ε (1 + β1s); A5 = qη6; j=1 1 + τθs t 2 and the strain will be g(s) = p ; (31) 1 + tps 3 d2w X e = −z = −z k2 C e−kj x − C ekj x . (42) and an over bar symbol denotes its Laplace transform, s dx2 j j j+3 j=1 denotes the Laplace transform parameter. The solution of Eqs. (28) and (29) leads to the following differential equation 6. APPLICATIONS for w d6w d4w d2w − a1 + a2 − a3w = 0; (32) When the two ends of the micro-scale beam are clamped, dx6 dx4 dx2 then the boundary conditions are given by where the coefﬁcients a1, a2, and a3 are given by ∂w(x, t) w(x, t) x=0,L = = 0. (43) a1 = A2A4 + A3; a2 = A1; a3 = A1A3. (33) ∂x x=0,L International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 263 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING

      1 1 1 1 1 1 C1  0  −k1L −k2L −k3L k1L k2L k3L      e e e e e e  C2  0   −k −k −k k k k  C   0   1 2 3 1 2 3  3 = ; (48)  −k1L −k2L −k3L k1L k2L k3L  0  −k1e −k2e −k3e k1e k2e k3e  C4     C     H1 H2 H3 H1 H2 H3   5 G(s) −k1L −k2L −k3L k1L k2L k3L     −k1H1e −k2H2e −k3H3e k1H1e k2H2e k3H3e C6 −H4

Additionally, we consider the beam is loaded thermally on the in which ε1 is a persecuted small positive number that corre- boundary x = 0 as sponds to the degree of accuracy to be achieved. The parameter c is a positive free parameter that must be larger than the θ(x, z, t) = φ(x, t) sin(pz) = θ f(x, t) sin(pz); (44) 0 real parts of all singularities of Ψ(s). The optimal choice of c on x = 0, where θ0 is a constant. Additionally, let us consider was obtained according to the criteria described in Honig and that f(x, t) is varying sinusoidal pulse function with time de- Hirdes.35 In order to find the displacement distribution u(t), scribed mathematically as take after we use the expression that appeared in Eq. (48) with replacing Ψ(t) and Ψ(s) respectively. This procedure is repeated for the ( π sin t , 0 ≤ t ≤ t0 other functions. f(x, t) = f(t) = t0 ; (45) 0 , t > t0, t < 0 where t0 is a non-negative constant called pulse width param- 8. DISCUSSION OF THE NUMERICAL eter. Moreover, the temperature at the end boundary should RESULTS fulfill the accompanying equation Numerical results with the help of Mathematica program- ∂θ(x, t) = 0. (46) ming software in the case of silicon nitride have been pre- ∂x x=L sented. For this purpose, we take the following values of the After using the dimensionless parameters and Laplace trans- different physical constants of the thermoviscoelastic material 3 3 form, the boundary conditions in Eqs. (43), (44), and (46) take ρ = 8.954 × 10 kg/m , CE = 383.1 J/(kgK), T0 = 296 K, −5 −1 10 3 the forms αT = 1.78 × 10 K , λ = 7.76 × 10 kg/m , µ = 10 3 10 3 3.86×10 kg/m , K = 386 W/(mK), E = 8.4×10 kg/m , dw(x, s) α1 = 0.6 s, t = 0.1 s, α2 = 0.9 s, τq = 0.05 s, τθ = 0.01 s, w(x, s) x=0,L = = 0; dx x=0,L ν = 0.33. πt dφ(x, s) The aspect ratios of the beam are fixed as L/h = 10 and φ(x, s) = 0 ; = 0. (47) x=0 2 2 2 b/h = 0.5. When h is varied, L and b change accordingly with π + t0s dx x=L h. The figures were prepared by using the non-dimensional Substituting Eqs. (38), and (39) into the boundary conditions variables which are defined in Eq. (22) for a wide range of of Eq. (47) yields Eq. (48) (see the top of the page), where beam length when L = 1 and z = h/6. Generally speaking, πt0 G(s) = 2 2 2 − H4. Solution of the above system of linear π +t0s the value of Ra depends on the material of the resonator and equations gives the unknown parameters Cj, j = 1, 6. This the wavelength. For a resonator made of silicon, Ra is about completes the solution of the problem in the Laplace transform 0.17 without coating. With an energy absorbing coating, Ra domain. can be larger than 0.99. Without loss of generality, we take Ra = 0.5 in the following studies. The energy intensity of 11 2 7. INVERSION OF THE LAPLACE the laser pulse is L0 = 10 J/m . For different values of L0, TRANSFORMS (NUMERICAL which is related to the absorption depth, the beam absorbed INVERSION) different quantity of energy and it vibrates in different man- ners. To get a solution of the problem in the physical field, the Numerical calculations are carried out for non-dimensional transforms in Eqs. (38), (39), (41), and (42) are inverted. In lateral vibration, temperature, displacement and stress along order to invert the Laplace transform in the above equations, the x-direction for various values of the phase lags, differ- we adopt a numerical inversion technique based on a Fourier ent theories of thermoelasticity, pulse width, viscosity, and the 35 series expansion. In this technique, the inverse Ψ(t) of the time of the laser-pulse and the laser intensity parameters, re- Laplace transform Ψ(s) is come close to the relation spectively. Then computations were performed for one value " ( n )# of time, namely for t = 0.1. The solutions displayed in dif- ect 1 X ikπ Ψ(t) = Ψ(c) + Re Ψ c + ; ferent figures are observed to be completely in agreement with t1 2 t1 k=1 the boundary conditions under four different cases as follows:

0 ≤ t ≤ t1; (48) Case I: Discussing the behavior of the field quantities through where n is a sufficiently large integer representing the num- the axial direction of the micro-scale beam with differ- n ber of terms in the truncated infinite Fourier series, may be ent values of the phase-lags τq and τθ for fixed values of chosen such that the pulse width parameter t0 = 0.2 and the time of the ct inπt ikπ laser-pulse tp, and the laser intensity L0 parameters re- e Re e t1 Ψ c + ≤ ε1; (49) t1 main constant (see Fig. 2). 264 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING

Figure 2. Distributions of the micro-scale beam’s: (a) transverse deﬂection w, (b) temperature θ, (c) displacement u, and (d) thermal stress σxx versus x for different values of the phase-lags τq and τθ.

Case II: Investigating how the dimensionless lateral vibra- out in the coupled theory (CTE) by setting τq = τθ = 0, in tion, thermodynamic temperature, displacement, and Lord-Shulman theory (LS) putting τq > 0, τθ = 0, and in the stress vary with different values of the dimensionless generalized theory of thermoelasticity proposed by Tzou when pulse width parameter t0 (see Fig. 3). τq ≥ τθ > 0. Figure 2a depicts the distribution of the lateral vibration w Case III: Introducing the study of the effects of viscosity α 1 which always begins from the zero values (i.e., vanishes) and and α on the dimensionless lateral vibration, thermody- 2 satisfies the boundary condition at x = 0. Figure 2b shows the namic temperature, displacement, and stress (see Fig. 4). variant of temperature θ with x and it indicates that tempera- Case IV: Studying how the dimensionless lateral vibration, ture field has maximum value at the boundary x = 0 of the thermodynamic temperature, displacement, and stress beam and then falls to zero in the region 0.3 ≤ x ≤ 1. The vary with different values of the time of the laser-pulse temperature distribution θ decays along the direction of the tp and the laser intensity L0 parameters (see Fig. 5). transmitted wave propagation for the effects of diffusion. Fig- ure 2c is plotted to show the variation of thermal displacement Figures 2a to 2d are drawn to give comparison of the re- u against x for theories of thermoelasticity. It is observed from sults obtained for displacements, temperature, and stress of the this figure some difference in values of displacement is noticed beam against x for different values of phase-lags τq and τθ for different values of the parameters τq and τθ. As shown in when the pulse width parameter t0, the time of the laser-pulse Fig. 2c, displacement u increases near the micro-scale beam tp, and the laser intensity L0 parameters remain constant. The edge x = 0 (always starts from a negative value), then smooth graphs in Fig. 2 represent four curves predicted by three dif- decreases again to reach its minimum magnitude just at about ferent theories of thermoelasticity obtained as a special case of the beam end. Figure 2d displays thermal stress σxx. We can the dual-phase-lag model. The computations were performed see that stress starts from a negative value and terminates at the for one value of time, namely for t = 0.12 and various values zero value. It reaches the minimum value nearby the end of the of the parameters τq and τθ. These computations were carried beam and converges to zero with the increasing distance x.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 265 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING

Figure 3. Distributions of the micro-scale beam’s: (a) transverse deﬂection w, (b) temperature θ, (c) displacement u, and (d) thermal stress σxx versus x for different values of the pulse width parameter t0.

It is known that the phase-lag time τq can dictate the man- the studied fields. The field quantities are very sensitive to the ners of thermal wave propagation, slow down the propagation variation of the pulse width parameter t0. speed of thermal indication, and manifest the feature of a ther- In order to study the effects of viscosity on displacement, mal wave. The influence of τθ can assist heat energy diffuse temperature, stress we now present our results of the numeri- and create the characters of thermal wave decline in DPL heat cal evaluation in the form of graphs (Figs. 4a to 4d). It is ob- transmission. We found that the phase lags τq and τθ param- served that due to the existence of viscosity term in the phase eters have significant effects on all the considered fields. The lag model the amplitude of the thermoelastic fields has sig- value of the studied fields for CTE model are larger compared nificantly decreased for viscous case in comparison with non- to those for other theories. The distribution in LS theory is viscous case. Due to the presence of viscosity, the magnitude close to that in dual phase lag theory, whereas the distributions of temperature decreases for all models, and in this case, the in the coupled theory CTE is different from that in DPL theory. magnitude of temperature also approaches zero value with in- The phenomenon of a finite speed of propagation is manifested crease of x. Additionally, it is observed that the magnitude of in all figures. This is different from the cases in both the uncou- displacement and stress distributions is large for viscous case pled and coupled theories of thermoelasticity where an infinite in comparison with non-viscous case. speed of propagation is inherent, and hence all the considered The last case is studying how the non-dimensional lateral functions have non-zero values for any point in the medium. vibration, temperature, displacement and stress vary with dif- The second case is investigating how the non-dimensional ferent the time of the laser-pulse tp and the laser intensity L0 lateral vibration, temperature, displacement, and thermal stress parameters when the phase-lags τq and τθ remain constant. vary with pulse width parameter t0 when the phas lags τq and The numerical results are obtained and presented graphically τθ, the the time of the laser-pulse tp, and the laser intensity in Figs. 5a to 5d. Here, we observe that the significant effect of L0 parameters remain constants. The numerical results are ob- the time of the laser-pulse and the laser intensity parameters on tained and presented graphically in Figs. 3a to 3d. We can all the studied fields. The increasing in the value of the laser- see the significant effect of the pulse width parameter t0 on all pulse parameter tp and the laser intensity L0 causes decreasing

266 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING

Figure 4. Distributions of: (a) transverse deflection w, (b) temperature θ, (c) displacement u, and (d) thermal stress σxx versus x for the viscous and nonviscous micro-scale beams. in the values of all the fields, which is very obvious in the peek influence plays a significant character on all distributions. points of the curves. • The difference between coupled theory and dual-phase- 9. CONCLUSIONS lag (DPL) model is very clear. In the present study, a generalized thermoviscoelastic prob- • Due to the existence of viscosity term in the model of ther- lem of Kelvin-Voigt type micro-scale beams subjected to a si- moelasticity with phase lags, the amplitude of the ther- nusoidal varying thermal load is considered. The upper sur- moelastic fields has significantly decreased for viscous face of the beam is also excited regularly by a laser pulse with case in comparison with non-viscous case. a temporal shape in a non-Gaussian form. The phase lag ther- • Also we found that, the laser-pulse and the laser inten- moelastic model is applied to study the mechanical relaxations sity parameters have significant effects on all the studied (viscous effects) and thermomechanical coupling (thermal ef- fields. fects). The effects of the phase-lags (τq and τθ) and the pulse width parameters on the field variables were studied. Addi- • Lord and Shulman theory (LS) and Green and Naghdi tionally, the effects of the laser-pulse and the laser intensity model (GN) as well as the classical thermoelasticity the- parameters of thermal vibration have been studied. ory (CTE) were obtained as special cases. It was shown The analysis of the results permits some concluding re- that the present DPL model results are close to the LS marks: results. • Visco-thermoelastic resonators provide a useful contribution to thermoelastic coupling in solids in micro- ACKNOWLEDGEMENTS electromechanical systems (MEMS) applications. This project was funded by the Deanship of Scientific Re- • It can be observed from the discussion that the phase lags search (DSR), King Abdulaziz University, Jeddah, under grant

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 267 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING

Figure 5. Distributions of the micro-scale beam’s: (a) transverse deflection w, (b) temperature θ, (c) displacement u, and (d) thermal stress σxx versus x for different values of the laser-pulse tp and the laser intensity L0 parameters. no. G-1436-130-195. The authors, therefore, acknowledge conducting thermodiffusive medium with a spherical cav- with thanks DSR technical and financial support. ity, Journal of Earth System Science, 124 (8), 1709–1719, (2015). https://dx.doi.org/10.1007/s12040-015-0628-z REFERENCES 5 Bucur, A. Rayleigh surface waves problem in linear thermoviscoelasticity with voids, Acta Mechanica, 227 (4), 1 Grover, D. Viscothermoelastic vibrations in micro-scale 1199–1212, (2016). https://dx.doi.org/10.1007/s00707- beam resonators with linearly varying thickness, Cana- 015-1527-8 dian Journal of Physics, 90 (5), 487–496, (2012). https://dx.doi.org/10.1139/p2012-044 6 Ezzat, M. A. and El-Bary, A. A. Unified fractional derivative models of magneto-thermo-viscoelasticity the- 2 Grover, D. Transverse vibrations in micro-scale ory, Archives of Mechanics, 68 (4), 285–308, (2016).

viscothermoelastic beam resonators, Archive of 7 Applied Mechanics, 83 (2), 303–314, (2013). Dogus¸can˘ Akbas¸, S¸. Forced vibration responses of func- https://dx.doi.org/10.1007/s00419-012-0656-y tionally graded viscoelastic beams under thermal environment, International Journal of Innovative Research in Sci- 3 Pal, P., Das, P., and Kanoria, M. Magneto-thermoelastic ence, Engineering and Technology, 5 (12), 36–46, (2016). response in a functionally graded rotating medium due to a 8 Ezzat, M. A. and El-Bary, A. A. On thermo-viscoelastic periodically varying heat source, Acta Mechanica, 226 (7), inﬁnitely long hollow cylinder with variable thermal con- 2103–2120, (2015). https://dx.doi.org/10.1007/s00707- ductivity, Microsystem Technologies, in press, (2017). 015-1301-y https://dx.doi.org/10.1007/s00542-016-3101-2 4 Zenkour, A. M., Alzahrani, E. O., and Abouelregal, A. E. 9 Zenkour, A. M. and Abouelregal, A. E. Effects of phase- Generalized magneto-thermoviscoelasticity in a perfectly lags in a thermoviscoelastic orthotropic continuum with a

268 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 Mashat, D. S., et al.: THERMOVISCOELASTIC VIBRATIONS OF A MICRO-SCALE BEAM SUBJECTED TO SINUSOIDAL PULSE HEATING

cylindrical hole and variable thermal conductivity, Archives 24 Carrera, E., Abouelregal, A. E., Abbas, I. A., and of Mechanics, 67 (6), 457–475, (2015). Zenkour, A. M. Vibrational analysis for an axially

10 moving micro-scale beam with two temperatures, Lord, H. W. and Shulman, Y. A generalized dynam- Journal of Thermal Stresses, 38, 569–590, (2015). Journal of the Me- ical theory of thermoelasticity, https://dx.doi.org/10.1080/01495739.2015.1015837 chanics and Physics of Solids, 15, 299–307, (1967). https://dx.doi.org/10.1016/0022-5096(67)90024-5 25 Abouelregal, A. E. and Zenkour, A. M. Thermoelastic problem of an axially moving micro-scale beam subjected 11 Green, A. E. and Lindsay, K. A. Thermoe- to an external transverse excitation, Journal of Theoret- lasticity, Journal of Elasticity, 2, 1–7, (1972). ical and Applied Mechanics, 53 (1), 167–178, (2015). https://dx.doi.org/10.1007/BF00045689 https://dx.doi.org/10.15632/jtam-pl.53.1.167 12 Dhaliwal, R. S. and Sherief, H. H. Generalized 26 Abouelregal, A. E. and Zenkour, A. M. Effect of phase lags thermoelasticity for anisotropic media, Quarterly on thermoelastic functionally graded micro-scale beams of Applied Mathematics, 38 (1), 1–8, (1980). subjected to ramp-type heating, IJST, Transactions of Me- https://dx.doi.org/10.1090/qam/575828 chanical Engineering, 38 (M2), 321–335, (2014). 13 Tzou, D. Y. A uniﬁed approach for heat conduction from 27 Abouelregal, A. E. and Zenkour, A. M. Generalized ther- macro-to micro-scales, Journal of Heat Transfer, 117, 8– moelastic vibration of a micro-scale beam with an ax- 16, (1995). https://dx.doi.org/10.1115/1.2822329 ial force, Microsystem Technologies, 21 (7), 1427–1435, 14 Tzou, D. Y. Macro- to microscale heat transfer: The (2015). https://dx.doi.org/10.1007/s00542-014-2220-x lagging behavior, series in chemical and mechanical en- 28 Zenkour, A. M. Free vibration of a micro-scale beam rest- gineering, Taylor & Francis, Washington, DC, (1997). ing on Pasternak’s foundation via the GN thermoelasticity https://dx.doi.org/10.1002/9781118818275 theory without energy dissipation, Low Frequency Noise, Vibration and Active Control, 35 (4), 303–311, (2016). 15 Sun, H. X. and Zhang, S. Y. Thermoviscoelastic ﬁnite element modeling of laser-generated ultrasound in viscoelas- 29 Zenkour, A. M. and Abouelregal, A. E. Thermoelastic vi- tic plates, Journal of Applied Physics, 108, 123101, (2010). bration of an axially moving micro-scale beam subjected https://dx.doi.org/10.1063/1.3520675 to sinusoidal pulse heating, International Journal of Struc- tural Stability and Dynamics, 15 (6), 1450081, (2015). 16 Kumar, R., Kumar, A., and Singh, D. Interaction of laser https://dx.doi.org/10.1142/S0219455414500813 beam with micropolar thermoelastic solid, Advances in Physics Theories and Applications, 40, 10–15, (2015). 30 Zenkour, A. M. and Abouelregal, A. E. The ef-

17 fect of two temperatures on a FG nanobeam in- Scruby, C. and Drain, L. Laser ultrasonics: Techniques and duced by a sinusoidal pulse heating, Structural En- applications, Adam Hilger, New York, (1990). gineering and Mechanics, 51 (2), 199–214, (2014). 18 Royer, D. Mixed matrix formulation for the analy- https://dx.doi.org/10.12989/sem.2014.51.2.019 sis of laser-generated acoustic waves by a thermoe- 31 Zenkour, A. M. and Abouelregal, A. E. Nonlocal ther- lastic line source, Ultrasonics, 39, 345–354, (2001). moelastic nanobeam subjected to a sinusoidal pulse heat- https://dx.doi.org/10.1016/S0041-624X(01)00066-X ing and temperature-dependent physical properties, Mi- 19 Wang, X. and Xu, X. Thermoelastic wave induced by crosystem Technologies, 21 (8), 1767–1776, (2015). pulsed laser heating, Applied Physics A, 73, 107–114, https://dx.doi.org/10.1007/s00542-014-2294-5 (2001). https://dx.doi.org/10.1007/s003390000593 32 Mishra, J. C., Samanta, S. C., and Chakrabarty, A. 20 Wang, X. and Xu, X. Thermoelastic wave in K. Magneto-thermo-mechanical interaction in an ae- metal induced by ultrafast laser pulses, Jour- olotropic viscoelastic cylinder permeated by magnetic nal of Thermal Stresses, 25, 457–473, (2002). ﬁeld subjected to a periodic loading, International Jour- https://dx.doi.org/10.1080/01495730252890186 nal of Engineering Science, 29, 1209–1216, (1991). https://dx.doi.org/10.1016/0020-7225(91)90025-X 21 Allam, M. N. M. and Abouelregal, A. E. The ther- 33 Mukhopadhyay, S. Effects of thermal relaxations on thermoelastic waves induced by pulsed laser and vary- moviscoelastic interactions in an unbounded body with ing heat of inhomogeneous microscale beam resonators, a spherical cavity subjected to a periodic loading on Journal of Thermal Stresses, 37 (4), 455–470, (2014). the boundary, Journal of Thermal Stresses, 23, 675–684, https://dx.doi.org/10.1080/01495739.2013.870858 (2000). https://dx.doi.org/10.1080/01495730050130057 22 Fu, Y. and Zhang, J. Free vibration analysis of a micro-scale 34 Boley, B. A. Approximate analyses of thermally beam subjected to a symmetric electrostatic ﬁeld, Mechan- induced vibrations of beams and plates, Jour- ics of Advanced Materials and Structures, 20 (4), 257–263, nal of Applied Mechanics, 39, 212–216, (1972). (2013). https://dx.doi.org/10.1080/15376494.2011.584264 https://dx.doi.org/10.1115/1.3422615 23 Zenkour, A. M. and Abouelregal, A. E. Effect of harmon- 35 Honig, G. and Hirdes, U. A method for the numerical ically varying heat on FG nanobeams in the context of inversion of the Laplace transform, Journal of Compu- a nonlocal two-temperature thermoelasticity theory, Euro- tational and Applied Mathematics, 10, 113–132, (1984). pean Journal of Computational Mechanics, 23 (1–2), 1–14, https://dx.doi.org/10.1016/0377-0427(84)90075-X (2014). https://dx.doi.org/10.1080/17797179.2014.882141

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 269 Noise and Vibration Analysis of a Heat Exchanger: a Case Study

Thiago A. Fiorentin and Alexandre Mikowski Department of Mobility Engineering, Federal University of Santa Catarina, Joinville, Brazil 88040-900

Olavo M. Silva and Arcanjo Lenzi Federal University of Santa Catarina, Florianopolis, Brazil 88040-900

(Received 15 November 2016; accepted: 26 April 2017) Flow-induced vibration of heat-exchanger tube bundles often causes serious damage, resulting in reduced efficiency and high maintenance costs. The excitation mechanism of flow-induced vibration is classified as vortex shedding, acoustical resonance, turbulent buffeting, or fluid-elastic instability. This paper aims to identify the mechanism that causes flow-induced vibration in a specific heat-exchanger tube bundle with cross-flow and pro- poses a solution to this problem. This case is investigated through acceleration and sound pressure level measurements. Moreover, finite- element models are developed to view the acoustic models of the cavity and vibration modes of the tubes and plates. The layout pattern of the tube array, the spacing ratio, the Strouhal number, and the flow characteristics are used to determine the excitation mechanism.

1. INTRODUCTION In cogeneration mills, the excess of energy generated can be transferred to an electrical system.1 In an attempt to improve the efficiency of these systems, some manufacturers have developed new heat exchangers in which the speed of the fluid flowing around the tubes has been increased; they have also changed their tube arrangements. The production cost of major heat-exchanger equipment is high. Also, since maintenance is necessary, the cost of the entire operation is quite high. Ac- cording to the literature, many incidents of failure due to apparent flow-induced vibration in heat exchangers have been re- Figure 1. Lock-in condition. ported.2 Flow-induced vibration and acoustical resonance have cause flow-induced vibration in heat exchangers. The follow- caused serious damage to the system integrity of heat exchang- ing four main mechanisms of flow-induced vibration in heat ers.3 The four principal sources of vibration in cross-flow tube exchangers have been investigated. banks are vortex shedding, acoustical resonance, turbulent buffeting and fluid-elastic instability.4, 5 All these mechanisms 2.1. Vortex Shedding arise because of the various forces that act on a tube due to the shell-side cross-flow. Flow across a tube produces vortices in the wake generated from the opposite sides of the tube. The oscillation frequency During the last decades, researchers were successful (in of a wake is proportional to the flow velocity and results in varying degrees) in better understanding the main sources of an oscillatory force on the tube. This phenomenon may excite noise and vibration in different kind of heat exchangers.6, 9 vibration in a liquid flow or acoustical resonance in a gas flow. Distinct solutions have been proposed to control the noise and vibration problem in heat exchangers. In order to suppress When the frequency of the vortex shedding coincides with acoustical resonances, a rigid baffle is normally placed inside the natural frequency of the tube vibration, higher levels of vi- a container and is parallel to the flow stream. It modifies the brations and rapid tube damage may occur, especially for liq- acoustical field and inhibits the instability.10 To avoid the vi- uid flows. The velocity range over which the tubes exhibit bration of the equipment, it is sometimes necessary to reduce high-amplitude vibrations is referred to as the ”lock-in” range, the shell side flow rate, remove the tubes in the window area to as shown in Fig. 1. The lock-in conditions can be estimated form a bypass, and redesign and reinstall a new bundle.11 from the following equation: However, it is difficult to find case histories presenting ex- 0.8fn < fv < 1.2fn; (1) perimental analyses, finite element models, and a solution to the problem. Normally, the investigations are focused on ex- with citation mechanisms, instabilities criteria, and methods to pre- U f = St ; (2) dict the problem. v v D

where fn is the tube’s natural frequency, fv is the frequency 2. EXCITATION MECHANISMS of vortex shedding, Stv is the Strouhal number, U is the gap velocity, and D is the tube diameter Recently, a signiﬁcant amount of research has been con- Recent studies have shown that this excitation results from ducted to predict, understand, and resolve the mechanisms that vortex shedding around the tubes. It occurs in the beginning of

270 https://doi.org/10.20855/ijav.2017.22.2473 (pp. 270–275) International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 T. A. Fiorentin, et al.: NOISE AND VIBRATION ANALYSIS OF A HEAT EXCHANGER: A CASE STUDY

Figure 2. Acoustial modes inside the heat exchanger. Figure 3. Fluid-elastic instability, turbulent buffeting, and vortex shedding. the section tubes, mainly in heat exchangers with little space between tubes.12, 13 " 2 # ftbD D 2.2. Acoustical Resonance Sttb = XlXt = 3.05 1 − + 0.28 ; (4) U Lt Acoustical resonance excited by a flow can occur in different types of heat exchangers. Normally, they have tubes with where Sttb is the Strouhal number considering turbulent buf- fluid flow inside (tube-side fluid) and a shell that wraps around feting, ftb is the turbulent buffeting frequency, U is the cross- the tubes (shell-side fluid). The gas in the cavity of the heat flow velocity, D is the outer diameter of the tube, Xl is the exchanger generates natural acoustic frequencies that depend longitudinal pitch ratio ( Ll ), X is the transverse pitch ratio on the speed of sound and the shell geometry. These acoustic D t Lt L L modes can be excited by either vortex shedding or turbulent ( D ), l is the longitudinal pitch and t is the transverse pitch. buffeting. This resonance, if and when excited, is established within 2.4. Fluid-Elastic Instability the tube bundle in a direction perpendicular to the flow and the Fluid-elastic instability can be defined as a mechanism tube axis, as shown in Fig.2. For a cylindrical heat exchanger, where fluid forces cause tube movement in the heat exchanger. the natural acoustic frequencies can be predicted by the fol- At a critical flow velocity, tube banks subjected to increas- lowing equation: ing flow velocity begin to vibrate with a large amplitude, as shown in Fig. 3. The resultant tube failure can occur in a rela- nUs tively short period due to fluid-elastic instability, which can be fa = ; (3) 2d avoided by controlling the cross-flow velocity. where fa is the acoustic frequency, n is the mode number, d is The flow field around a tube bundle causes displacement of the shell diameter, and Us is the velocity of sound in shell-side a tube. This displacement modifies the flow field, changing the fluid.14 fluid forces acting on the tube. Damping and stiffness forces of Acoustical resonance requires two conditions: (i) coinci- the tube act to restore the tube to its initial position. The com- dence of vortex shedding and acoustic frequency and (ii) suffi- petition between the energy input from the fluid flow across the ciently high acoustic energy or sufficiently low acoustic damp- tube and the energy dissipated by damping and stiffening will ing to allow sustained acoustic standing-wave resonance.15 An determine the amplitude of the vibration. If the energy dissi- acoustical resonance can be broken by inserting acoustic baf- pated by damping and stiffening is less than the energy input fles into the shell. These baffles can change the natural acoustic from the fluid flow, then fluid-elastic vibrations will establish frequency and the mode shape. themselves. Connor’s equation is used as a parameter to verify the possi- 2.3. Turbulent Buffeting bility of developing fluid-elastic instability in a tube array. The equation below calculates the critical flow velocity based on High flow rates produce turbulent flow within a fluid, which tube parameters: can be a cause of structural excitation. In any flow, there exists a range of oscillatory flow components spread over a broad U mδ a cr = K ; range of frequencies. Tubes respond randomly to flow tur- 2 (5) fnD ρD bulence. Turbulent buffeting is a low-amplitude vibration re- 16 sponse of the tube bundle below a certain critical velocity. where Ucr is the critical pitch flow velocity, D is the diameter Although the vibration amplitudes are small, the phe- of the tube, fn is the natural structural frequency, m is the nomenon persists over the entire life of the heat exchanger, tube mass per unit length, ρ is the density, δ is the logarithmic 18 because some wear is inevitable. Even when other vibration decrement, and K and a are Connor’s constants. mechanisms are successfully mitigated, this turbulent buffeting may eventually cause tube damage. For the heat exchanger 3. EXPERIMENTAL AND NUMERICAL to remain in operation for many years, the potential for wear ANALYSIS due to turbulent buffeting needs to be considered in the design stage of the heat exchanger. The heat exchanger under analysis is a cross-flow type with Flow turbulence is a significant excitation mechanism in a a geometric tube distribution, as shown in Fig. 4. The tubes heat exchanger with cross-flow. The dominant frequency of have an external diameter of 63 mm, length of 3.80 m and a 4- excitation can be determined by Owen’s equation:17 mm-thick wall. They are distributed over 26 rows by 54 rows.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 271 T. A. Fiorentin, et al.: NOISE AND VIBRATION ANALYSIS OF A HEAT EXCHANGER: A CASE STUDY

Figure 4. Cross-section of tube geometric distribution. Figure 6. Sound pressure level measured in front of the fan inlet.

Figure 5. Accelerometer and microphone positions in the heat exchanger.

The heat exchanger is 3.80 m in height, 3.50 m in width, and 6.70 m in length. A 350 hp fan blower insufflates an airflow through the heat Figure 7. Acceleration measured on the lateral plate. exchanger. To identify the cause of excessive noise and vibration, the rotational speed of the fan is varied from 450 to 750 ated as approximately 53 Hz along with its resultant harmon- rpm, which corresponds to an upstream air velocity of 2.90 to ics. 5.00 m/s, respectively. The inlet air temperature is 25◦C and the air temperature at the end of the heat exchanger is 240◦C. 3.2. Numerical Analysis According to the configuration of the heat exchanger anal- The correct identification of the mechanism that generates Lp ysed in this paper, the space ratio Xp D = 1.44, the noise and vibration requires a precise knowledge of the natural acoustic frequencies of the tubes and their respective vibration Strouhal number from the literature is St = 1.0 19 and the modes. Numerical models are developed to analyse the vibra- Strouhal number considering turbulent buffeting St = 1.14, tb tions of the tubes, the side plate, and the acoustical field in the as calculated by Eq. (4). heat exchanger.

3.1. Experimental Analysis 3.2.1. Tube Vibrations To help with the problem identification, noise and vibration A bundle of tubes is modelled by a finite-element method measurements are performed at various speeds of the primary using a shell element. These tubes are coupled to 20-mm-thick fan blower, resulting in a range of mean flow velocities in the plates at the ends. When the tubes are welded to the plate, it heat exchanger. This procedure is used to analyse situations is considered to be a rigid connection. In this model, a larger when the excitation frequency coincides with the acoustic or number of tubes produce a slow computational solution. It is structural modes of vibration. assumed that the model can reproduce the true behaviour of the The vibration is measured on the side wall of the heat ex- tubes in the heat exchanger. Due to the fixation of the tubes in a changer inlet duct. The sound pressure level (SPL) is measured rigid structure, the behaviour of this system is associated with in front of the blower inlet nozzle, distant 1.0 meter of fan at- the modes located on each tube. The steel damping coefficient tenuator. The vibration is measured on the side wall of the ex- is defined as 0.05%. changer inlet duct. The sound and vibration measurements are The objective of the model is to obtain the frequency ranges made simultaneously. These quantities are measured for dif- of the natural vibration modes. Figure 8 demonstrates that it is ferent fan rotation values. The microphone and accelerometer possible to identify a typical frequency response for each tube position, as well as the main components of the heat exchanger excited by a point force. The first three modes are plotted in can be identified at Fig. 5. Figs. 8 and 9. Analysing the results shown in Figs. 6 and 7 enables the There is no peak at 53 Hz; however, the second mode of identification of the highest amplitude of the frequency gener- vibration occurs between 62 and 68 Hz. Experimentally, the

272 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 T. A. Fiorentin, et al.: NOISE AND VIBRATION ANALYSIS OF A HEAT EXCHANGER: A CASE STUDY

Figure 11. Acoustical modes of the heat exchanger.

Figure 8. Frequency response function of tubes.

Figure 12. Frequency response function of the lateral duct plate. Figure 9. Modes of vibration.

Figure 13. Vibration modes of a lateral duct plate.

reinforcements. A typical frequency response function for this model can be observed in Fig. 12. The presence of a global mode resonance of 51 Hz can be observed, which is very close Figure 10. Frequency response function of the acoustical field. to the acoustic mode and the high response frequency (53 Hz) modes occur at 21 Hz, 58 Hz, and 118 Hz. The excitation is observed during the measurements. applied at one side of the tube e the response measured at the The three vibration modes with frequencies of around 53 Hz other side of the tube. can be observed in Fig. 13. 3.2.2. Acoustic Modes 4. RESULTS AND DISCUSSION This simulation considers air with a variable temperature The noise and vibration measurements show a high level distribution over the length of the exchanger, from 30◦C at ◦ of responses at 53 Hz using a blower fan speed of 550 rpm. the blower exit up to 225 C at the end of the exchanger. The Through the use of numerical models, an acoustic mode of the finite-element method can be used to consider the temperature heat exchanger, and a mode of plate vibrations have been iden- variation in the acoustical field of the exchanger. The speed of tified within this frequency band. Beyond this, however, it is sound in the longitudinal and transversal directions is consid- important to identify the excitation mechanism. ered to be 95% of that in air due to the porosity caused by the Eq. (5) is used to calculate the critical speed for fluid-elastic tubes. In the model, there is no coupling between the acous- instability to occur, using the data of the heat exchanger under tic and structural components. A typical frequency response study. The value is found to be 73 m/s, which is well above the function can be observed in Fig. 10. operating speed of the equipment. This shows that fluid-elastic The boundary condition defined at fan inlet is a panel veloc- instability is not the excitation mechanism responsible for the ity. At heater outlet is applied an admittance. The damping is flow-induced vibration. considered as the imaginary part of the velocity and the value Another possible cause cited in the literature is turbulent used is 0.1% of the real part. buffeting. Eq. (4) is used to obtain values for the frequency The modes around 53 Hz are 51 Hz, 55 Hz, and 59 Hz, at which this phenomenon would occur. Using the data from as seen in Fig. 11. This indicates the presence of acoustical the heat exchanger under study, these values are found to be resonances around 53 Hz, which can be easily excited. above 53 Hz. Thus, turbulent buffeting is not the excitation mechanism for flow-induced vibration. 3.2.3. Plate Vibrations The last mechanism to be analysed is vortex shedding. Eqs. High vibration levels at the lateral duct plates are observed (1) and (2) are used to calculate the frequency of vortex, which during operation. Severe fatigue problems require permanent is shedding in this case. Using the flow rate data generated by welding repairs due to strong excitation levels. the fan, the frequency values are found near 53 Hz, indicat- A finite-element model of a detailed and full-length lateral ing that vortex shedding is the likely excitation mechanism for duct plate is developed using shell and beam elements for the flow-induced vibration.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 273 T. A. Fiorentin, et al.: NOISE AND VIBRATION ANALYSIS OF A HEAT EXCHANGER: A CASE STUDY

Figure 15. Acoustical bafﬂes in the heat exchanger.

(a)

(b)

Figure 14. Vortex shedding frequency (a) and the vibration response at 53 Hz, Figure 16. Vibration of lateral plate with acoustical baffles installed inside the as a function of flow velocity (b). heat exchanger. 4.1. Problem Identification Using Eqs. (1) and (2), it is possible to create a graph that correlates the vortex shedding frequency with flow velocity. Figure 14(a) plots the vortex shedding frequency and the natural frequencies of the structure and the acoustical field. The velocity values on the x-axis are indicated by the corresponding blower fan rotation, in revolutions per minute. At the air velocity of 3.5 m/s, the vortex shedding frequency coincides with the transverse acoustic mode of the heat exchanger and with a global mode of the side plate, Figure 14(a). According to Fig. 14(b), it is possible to identify the vibration response measured at lateral duct plate at 53 Hz for the various fan rotation values. A large response is initiated at an air velocity value of approximately 3.25 m/s. Around this frequency there is an acoustics mode and the lateral duct plate mode. These modes can be excited by vortex shedding. Tube Figure 17. SPL at the inlet of the fan with acoustical baffles installed inside vibrations do not play a role in this excitation mechanism. the heat exchanger. Further evidence of the vortex shedding acoustic mode excitation can be seen from the noise spectra measured at 1 m To verify the efficacy of the proposed solution, vibration and from the fan inlet in Fig. 6. After a specific value of fan ro- noise measurements are taken after the installation of the baf- tation, the sound pressure level measured at 53 Hz increases fles; the results can be seen in Figs. 16 and 17. The fan rota- the value. According to Fig. 14(a) there is an acoustic near tional speed is varied again from 450 to 750 rpm, generating this frequency. Also, according to Eq. (2) the value of vortex typical air flows inside the heat exchanger. The accelerometer shedding frequency increases with fan rotation. and microphone are placed at the same position as before. Initially, the vortex shedding mechanism was exciting the 4.2. Proposed Solution acoustic mode at the frequency of 53 Hz. Also, the acoustical To avoid excitation of the acoustic modes in the heat ex- resonance excited the lateral duct plate at the same frequency. changer, the exchanger width needs to be shortened. For this After installation of the baffle, the acoustical mode is not ex- reason, two longitudinal rows of tubes are removed, and in the cited by vortex shedding (see Fig. 17) and the acoustical reso- empty space, 4-mm-thick plates are placed and welded at the nance does not exist to excite the lateral duct plate Fig. 16. top and bottom borders of the heat exchanger. In Fig. 15, it is Overall, noise levels at the sugar-refining plant are now con- possible to identify the location of these plates, also known as sidered perfectly acceptable and as expected from this type of acoustic baffles. This procedure does not interfere significantly industry. The SPL measured at 1 m from the fan inlet indicates with the efficiency of the heat exchanger. that acoustical resonance disturbances have been completely

274 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 T. A. Fiorentin, et al.: NOISE AND VIBRATION ANALYSIS OF A HEAT EXCHANGER: A CASE STUDY avoided by this modification of the heat exchanger. Some re- 5 Gorman, D. J. Experimental development of design crite- searchers have developed models and methodologies to simu- ria to limit liquid cross-flow-induced vibration in nuclear late and optimize this kind of solution.10 reactor heat exchanger equipment, Nucl. Sci. Eng., 61 (3), 324-336, (1976). 5. CONCLUSIONS 6 Weaver, D. S., Lian, H. Y., and Huang, X. Y. Vortex shed- This paper presents some experimental results enabling the ding in rotated square tube arrays, J. Fluids Struc., 7, 107- identification of a specific frequency of 53 Hz, which is asso- 121, (1993). https://doi.org/10.1006/jfls.1993.1009 ciated with sound and vibration measurements that cause flow- 7 Goyder, H. G. D. Acoustic modes in heat exchangers, induced vibrations in a heat exchanger. The value of accelera- ASME PVP, 420-2 (107), 114, (2001). tion measured at the lateral plate of the heat exchanger is also high at this frequency, as is the noise measured in front of the 8 Owen, P. R. Buffeting excitation of boiler tube vi- fan, which demonstrates the same behaviour. This frequency is bration, J. Mech. Eng. Sci., 4-2, 431-439, (1965). correlated with a fan speed of between 500 and 650 rpm. This https://doi.org/10.1243/jmes jour 1965 007 065 02 indicates problems caused by flow-induced vibrations at this fan speed. 9 Lever, J. H. and Weaver, D. S. A theoretical model for According to the literature, the main sources of vibration in a the fluidelastic instability in heat exchanger yube bun- heat exchanger are vortex shedding, acoustical resonance, tur- dles, J. Press. Vess. Technol., 104, 147-158, (1982). bulent buffeting and fluid-elastic instability. Using equations https://doi.org/10.1115/1.3264196 from the literature to characterize each source, vortex shed- 10 Eisinger, F. L. and Sullivan, R. E. Suppression of acous- ding is identified as the main mechanism for flow-induced vi- tic waves in steam generator and heat exchanger Tube brations. Banks, Journal of Pressure Vessel Technology, 125, 221- Thus, it is necessary to identify which natural frequencies 227, (2003). https://doi.org/10.1115/1.1565079 are excited by this mechanism. Models are developed based on the finite-element method. First, a bundle of tubes is mod- 11 Goyder, H. G. D. Flow-induced vibration in heat ex- elled to enable the identification of vibration modes and nat- changers, Trans IChemE, 80 (A), 226-232, (2002). ural frequencies of the heat-exchanger tubes. Following this, https://doi.org/10.1205/026387602753581971 the acoustical field inside the heat exchanger is also modelled. 12 Analysis of these results indicates that there are some acous- Ziada, S., Oengoerem, A., and Buehlman, E.T. On acousti- tical modes where the maximum frequency value is close to cal resonance in tube arrays Part I: Experiments, J. Fluids 53 Hz. A lateral duct plate is also modelled, and through the Struc., 3, 293-314, (1989). https://doi.org/10.1016/s0889- results, it is possible to identify some vibration modes with 9746(89)90083-2 natural frequencies close to the frequency of interest. 13 Ziada, S., and Oengoerem, A. Vorticity shedding and Using the results of finite-element models and the data on acoustic resonance in an in-line tube bundle-part I: Vor- the excitation mechanism, it is possible to assume that vorticity shedding, J. Fluids Struc., 6, 271-292, (1992). tex shedding excites the acoustic mode of the cavity and the https://doi.org/10.1016/0889-9746(92)90010-z acoustical resonance excites the vibration mode of the plate. The solution to this problem is to insert an acoustic baffle in 14 Chenoweth, J. M., Chisholm, D., Cowie, R. C., Harris, D., the cavity and break the acoustic mode. The proposed solu- Illingworth, A., Loncaster, J. F., Morris, M., Murray, I., tion resolves the presented problem, and the subsequent noise North, C., Ruiz, C., Saunders, E. A. D., Shipes, K.V., Den- and vibration measurements after the installation of the baffles nis, U., and Webb, R. L. Heat exchanger design handbook confirm this. HEDH. Hemisphere Publishing Corporation, (1993).

15 ACKNOWLEDGEMENTS Pettigrew, M.J., and Taylor, C.E. Vibration analysis of shell-and-tube heat exchangers: An overview- The generous financial support of Conselho Nacional de Part 2: vibration response, fretting wear, guide- Desenvolvimento Cientco e Tecnolgico (CNPq) and Comisso lines, J. Fluids Struc., 18, 485-500, (2003). de Aperfeioamento de Pessoal do Nvel Superior (CAPES) is https://doi.org/10.1016/j.jfluidstructs.2003.08.008 highly appreciated. 16 Weaver, D. S., and Grover, L. K. Cross flow induced vibrations in a tube bank - turbulent buffeting and fluidelas- REFERENCES tic instability. J. Sound and Vibration, 59, 277-294, (1978). 1 Cortez, L. A. B. Sugarcane bioethanol - R&D for https://doi.org/10.1016/0022-460x(78)90506-0 productivity and sustainability, Saö Paulo, Blucher. 17 Owen P. R. Buffeting excitation of boiler tube vi- https://doi.org/10.5151/blucheroa-sugarcane bration, J. Mech. Eng. Sci., 7, 431-439, (1965). 2 Wootton, L. R. Industrial implications of flow induced https://doi.org/10.1243/jmes jour 1965 007 065 02 vibrations, Proc. Institution of Mechanical Engineers,C 18 416/111, Brighton, England, (1991). Price, S. J. An investigation on the Connors equation to predict fluid elastic instability in cylinder ar- 3 Gelbe, H., Jahr, M., and Schrder, K. Flow-induced vi- rays. J. Press. Vess. Technol., 123, 448-453, (2001). brations in heat exchanger tube bundles, Chem. Eng. https://doi.org/10.1115/1.1403445 Proc., 34, 289-298, (1995). https://doi.org/10.1016/0255- 19 2701(94)04016-8 Ziada, S. Vorticity shedding and acoustic resonance in tube bundles, J. Brazilian Society of Mechanical 4 Padoussis, M. P. A review of flow-induced vibrations in re- Science and Engineering, 28 (2), 186-199, (2006). actor and reactor components, Nucl. Eng. Des., 74, 31-60, https://doi.org/10.1590/s1678-58782006000200008 (1983). https://doi.org/10.1016/0029-5493(83)90138-3

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 275 About the Authors

Milad Azimi obtained his MS in 2010 from the Department of Mechanical Engineering at Iran University of Science and Technology (IUST) and his PhD in aerospace engineering from the Space Research Center, Tehran, Iran in 2015. His research interests include spacecraft dynamics and control, multi-body dynamics, active vibration control, and space system design and analysis. He has also held visiting appointments as an assistant professor with the Islamic Azad Universities North and East Tehran Branches.

Morteza Shahravi received his PhD from the Department of Mechanical Engineering at Amirkabir University of Technology (AUT), Tehran, Iran in 2006. He is currently a PhD supervisor in the Space Research Center. His research interests include structural dynamics, vibration, control and space system design, and analysis.

Keramat Malekzadeh Fard received his PhD from the Department of Mechanical Engineer- ing at Khajeh Nasir Toosi University of Technology (KNTU), Tehran, Iran in 2005. He is now a professor in the Department of Mechanical/Aerospace engineering at the Space Research Center where he currently teaches and performs research in the ﬁelds of vibration analysis, smart structures, sandwich panel theories, and applications.

Masoud Asgari is assistant professor of mechanical engineering at K.N. Toosi University of Technology. He received his PhD and BS from the Department of Mechanical Engineering of the Amirkabir University of Technology (Tehran Polytechnic). He received his MS degree in mechanical engineering from Sharif University of Technology. His research interests include structural vibration analysis, thermo-mechanical analysis of smart materials, and human body modeling for vehicle crash and impacts.

Jiashi Yang received his BE and ME in engineering mechanics in 1982 and 1985 from Ts- inghua University, China, and PhD in civil engineering in 1993 from Princeton University, USA. He was a Postdoctoral Fellow from 1993 to 1994 at the University of Missouri-Rolla and at Rensselaer Polytechnic Institute from 1994 to 1995. He was employed by Motorola, Inc. from 1995 to 1997. Since 1997, he has been an assistant, associate, and professor at the Department of Mechanical and Materials Engineering at the University of Nebraska-Lincoln. His main research area is theoretical and numerical modeling of piezoelectric structures and devices. He is an Associate Editor of the IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control.

276 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 About the Authors

Jianke Du received his BS in aircraft design from Beijing University of Aeronautics and As- tronautics in 1992. He received his PhD in mechanics from Xian Jiaotong University in 2004. From 1992 to 1999, he worked on structural analysis of composite material vessel at Xiangyang Dynamic Mechanical Corporation in Xian, Shaanxi, China. From 2005 to 2007, he was an associate professor at Ningbo University, Ningbo, China. From 2007, he has been a professor at Ningbo University. From 2008 to 2009, he was a visiting scholar at Rutgers University, USA. His research focuses on the surface acoustic waves in ferroelectric materials and phononic crystal structures.

Ji Wang has been a Qianjiang Fellow of Zhejiang Province at Ningbo University since 2002. He received his PhD and Master degrees from Princeton University in 1996 and 1993 and bachelor from Gansu University of Technology in 1983. Professor Wang has been working on acoustic waves in piezoelectric solids for resonator design and analysis in his research with US and Chinese patents and over 100 journal papers. Professor Ji Wang is the founding director of the Piezoelectric Device Laboratory, which is a designated Key Laboratory of City of Ningbo, and Ningbo University-Dawning Joint High Performance Computing Laboratory. He was employed at SaRonix, Menlo Park, CA, as a senior engineer from 2001 to 2002; NetFront Communications, Sunnyvale, CA, as a senior engineer and manager from 1999 to 2001; and Epson Palo Alto Laboratory, Palo Alto, CA, as a Senior Member of the Technical Staff from 1995 to 1999. Professor Ji Wang also held visiting positions at Chiba University, University of Nebraska-Lincoln, and Argonne National Laboratory. He has been a member of many international conference committees and currently serving the IEEE UFFC Technical Program Committees of the Frequency Control and Ultrasonics Symposia, the IEEE MTT-S, and the IEC TC-49.

Hui Chen received his BE in engineering mechanics in 2012 from Ningbo University, China. He is currently a graduate student in the School of Mechanical Engineering and Mechanics, Ningbo University. His research interests are in the electrical properties of quartz crystal resonators.

Hsueh-Chi Lu was born in Taiwan. He received his MS in aerospace engineering in 2014 from Tamkang University, Taiwan. His dissertation included research on nonlinear system vibration reduction and system stability analysis. He is currently working as a reliability engineer in PEGATRON, RD Center-Environmental Reliability QA Div.-Enterprise Products Testing De- partment. He specializes in mechanism design and the analysis of product reliability test.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 277 About the Authors

Yi-Ren Wang was born in Taipei, Taiwan. He received his PhD in aerospace engineering in 1992 from Georgia Institute of Technology, USA. He was promoted to be an associate professor in the Department of Aerospace Engineering at Tamkang University in 1992. In 1995, he was designated a Distinguished Adviser/Teacher by the University. He was awarded the Excellence in Teaching in the year of 2009. He was also awarded Outstanding Mentorship in the year of 2014. Dr. Wang has also served as Department Chairperson from 2005 to 2009. He was awarded the best paper of the year in 2013 from the Journal of Aeronautics, Astronautics and Aviation. He was also a council member for the Aeronautical and Astronautical Society of the Republic of China. He is currently a full professor in the Department of Aerospace Engineering at Tamkang University. Dr. Wang’s research work includes the ﬁelds of nonlinear vibrations, rotary wing aeroelasticity and structural dynamics. He specializes in the analysis of the internal resonance of nonlinear beams, vibration reductions and the Nonlinear Energy Sink (NES) problems.

Aviral Rajora is currently an MS student in fluid mechanics in the Mechanical, Maritime and Materials Engineering department at Delft University of Technology. He holds his BS in mechanical engineering from the Indian Institute of Technology, Varanasi. He previously worked with Dr. Prabal Talukdar on developing a 3D code for heat transfer in a walking beam type heat furnace. His current areas of interest are computational fluid dynamics, numerical heat transfer, turbulence, and multiphase flows.

Ajit Kumar Dwivedi is currently working as an Assistant Manager in Bhilai Steel Plant (BSP). BSP is the ﬂagship unit of Steel Authority of India Limited (SAIL) manufacturing ﬂat as well as long products. He graduated in mechanical engineering from Indian Institute of Technology, Varanasi. He has two years of experience in blast furnace operation. Presently, he is working in the direction of maintaining hot metal quality with minimal variation of Silicon and Sulphur content. His research interests include optimization of different furnace parameters like hot blast temperature and pressure, RAFT, charging pattern for smooth burden movement.

Ankit Kumar Vyas is currently working as an Assistant Section Ofﬁcer in the Ministry of Health & Family Welfare, Government of India. He has previously worked in an Education Company where he was involved in developing educational tools, content and live tutoring of physics. He graduated in mechanical engineering from Indian Institute of Technology, Varanasi.

Satyam Gupta is working as an Engineering Analyst in Infosys Limited, Mysore. Infosys provides R&D service support for ANSALDO and GE, which holds interests in the electricity generation and rail transport markets. He has 3 years of experience performing CFD, combustion and heat transfer analysis for steam and gas turbine components. He has supported and well acknowledged in the International paper presented by client on 3D Sensitivity for Aerodynamic design guidelines of LP Rear stages. Presently, he is conducting CFD and CHT analysis of gas turbine combustor domains to enhance cooling performance. He received his BS mechanical engineering from Indian Institute of Technology (BHU) Varanasi.

278 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 About the Authors

Amit Tyagi is working as an assistant professor in the Department of Mechanical Engi- neeringat - Indian Institute of Technology, Varanasi. He has 14 years of teaching experience and has taught various undergraduate courses such as kinematics and dynamics of machines, engineering mechanics, and engineering drawing. He has authored many papers on the topic of energy harvesting capabilities of various vibrating structural members. Currently, two research scholars are working with him. Presently, his research group is working on ﬁltering Lamb waves in the process of crack detection in the structures, particularly in extremely noisy conditions. His principle research interests are the vibration of structural members.

Ying-Qing Guo is an associate professor in School of Mechanical and Electronic Engineering, Nanjing Forestry University, China. She received her PhD in control theory and control engineering from Southeast University in China in 2010 and worked as a visiting scholar in North Carolina State University, USA, from September 2008 to September 2009 and again from June 2012 to September 2012. Her research interests include smart materials and structures, intelligent control, and structural vibration control.

Zhao-Dong Xu is a Professor in the Civil Engineering School, Southeast University, China. He received his PhD degree in civil engineering from Xian University of Architecture and Technology in 2000, worked as a senior visiting researcher at North Carolina State University from November 2008 to March 2009 and again from June 2012 to September 2012, as a visiting professor in University of Illinois at Urbana & Champaign from March 2011 to July 2011, as an SVBL researcher at Ibaraki University from November 2004 to April 2006. His research interests include structural vibration control and dynamics, structural health monitoring, smart materials, and structures. He has published 169 international journal papers and four books, and he has seventeen patents. He is currently an editorial board member of the Journal of Dis- aster Advances, the Journal of Control Engineering and Technology, the International Journal of Automation and Control Engineering, and the Journal of Engineering. He is also a Fellow Member of the InternationalCongress of Disaster Management and a member of 11 kinds of Committees in China. He has been a keynote speaker in international conferences twice and a chair or co-chair four times for conferences.

Bing-Bing Chen received his MS in civil engineering from Southeast University, China, in 2015 and BS in civil engineering from Zhengzhou University, China, in 2012. He currently works as a research assistant at Southeast University and as a designer at Henan Xuhui Archi- tectural Engineering Design Co., Ltd. His major research interest is structural vibration control and structural design.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 279 About the Authors

Cheng-Song Ran received his MS in civil engineering from Southeast University in 2010 and BS in civil engineering from Hohai University, China, in 2007. He currently works as a research assistant in Southeast University and as an engineer in Sichuan Provincial Architectural Design and Research Institute. His major research interest is structural vibration control and structural design.

Wei-Yang Guo received his MS in civil engineering from Southeast University, China, in 2013 and BS in civil engineering from Nanjing Tech University, China, in 2010. And he currently works as a research assistant at Southeast University and as a designer at Nanjing Dongrui Damping Control Technology Co, Ltd. His major research interest is structural vibration control and structural design.

Reinhard O. Neubauer completed his masters degree on sound and vibration studies (I.S.V.R.) at the University of Southampton, UK, in 1991. He received his first PhD from the Building Research Institute, Warsaw, Poland, and his second PhD from the University of Sheffield, UK. He possesses a sound working knowledge of a wide variety of laboratory and measurement techniques. His scientific interests mainly focus on the field of building physics in which he has gained over 20 years of practical experience at IBN for various projects. His managerial experience includes assisting the development of standards. One of his key accom- plishments includes representing Germany as a committee member for the EU granted COST Action TU0901: Integrating and Harmonizing Sound Insulation Aspects in Sustainable Urban Housing Constructions.

Jian Kang is a professor of acoustics at the University of Shefﬁeld, UK. He obtained his BS and MS from Tsinghua University, China, and his PhD from the University of Cambridge. He also worked as an A. V. Humboldt Fellow at the Fraunhofer Institute of Building Physics in Germany. His research ﬁeld is architectural/environmental acoustics on which he has written more than 800 publications, performed more than 70 research projects, and conducted more than 80 engineering projects worldwide. He is a Fellow of the IIAV, IOA, and ASA. He chairs the Technical Committee for Noise of the European Acoustics Association and EU COST Action on Soundscape of European Cities and Landscapes. He was awarded the John Connell Award and IOA Tyndall Medal 2008.

Shaojiang Dong received his PhD in mechanical and electronic engineering from Chongqing University, Chongqing, China, in 2012. He now works at Chongqing Jiaotong University. His current research interests include signal processing and fault diagnosis.

280 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 About the Authors

Xiangyang Xu received his PhD - in mechanical and electronic engineering from Chongqing University, Chongqing, China, in 2012. He now works at Chongqing Jiaotong University. His current research interests include gear dynamics research and fault diagnosis.

Jiayuan Luo received his PhD in mechanical and electronic engineering from Chongqing University, Chongqing, China, in 2011. He now works at Chongqing Jiaotong University. His current research interests include control and reduction of aluminum alloy quenching residual stress.

Chen Wang was born in Yantai of Shandong Province, China in 1991. He received his BS from the Science College of Shandong Jianzhu University in Jinan City, China in 2014. He is now studying for an MS in mechanical engineering at College of Mechanical and Electronic Engineering of Shandong Jianzhu University. His main research interests are uncertain mechanical structure dynamics and mechanical reliability design. Along with his mentor, he has published a monograph titles Evaluation with Uncertainty Parameters and Approaches to Re- liability Design. He is currently the main researcher of the SDJZU-FJZY elevator technology research institute.

Ruijun Zhang was born in Heze of Shandong Province, China in 1965. He earned his BS in mechanical engineering from the Shandong Jianzhu University in Jinan, China in 1985. He earned his MS in mechanical engineering from the Harbin Institute of Technology in Harbin, China in 1990. He earned his PhD in mechanical engineering from the Beijing University of Posts and Telecommunications in Beijing, China in 2015. He is currently a professor in the College of Mechanical and Electronic Engineering of Shandong Jianzhu University. His main research interests are engineering mechanical design theory and safety assessment technology.

Qing Zhang was born in Liaocheng of Shandong Province, China in 1968. He earned his BS in mechanical engineering from Shenyang Institute of Technology in Shenyang, China in 1992. He earned his MS in mechanical engineering from Shandong University in Jinan, China in 2010. He is currently studying for his PhD at Tianjin University. He is also a professor in the College of Mechanical and Electronic Engineering of Shandong Jianzhu University. He is also the chief engineer of SDJZU-FJZY elevator technology research institute. His main research interests is structure and design of construction machinery.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 281 About the Authors

Gerardo Pelaez´ is a full professor at the Mechanical Engineering Department of the University of Vigo. He received his MD in electrical engineering from University of Santiago (1987). He worked in CAD-CAM-CNC at the R&D Department of Televes Co (1989). Later, he worked at Repsol Refinery of La Corua in advanced control (1990). He has a pre-doctoral application at the Mechanical Engineering Department of the UPM (1997). He concluded his post-doctoral studies at Georgia Tech in the field of command generation for flexible systems (2003).

Higinio Rubio Alonso is currently professor of mechanical engineering at Carlos III Univer- sity of Madrid (Spain). He received his masters in industrial engineering from UNED (Spain) and PhD from Carlos III University of Madrid, in 1998 and 2003, respectively. From 1999 to 2017, he has been professor and investigator of the Mechanical Engineering Department at the University Carlos III of Madrid. His research has been concerned with vibration and acoustic analysis; machine maintenance and fault detection; analysis, synthesis, design, and simulation of mechanical elements, particularly rolling bearings; and design of mechatronic systems and biped robots.

Manuel Perez-Donsi´ on´ received his MD in industrial engineering from the Polytechnic Uni- versity of Catalonia in 1980 and PhD from the University of Vigo in 1986. In 1987, he became a full professor at the University of Vigo. He was dean of Escuela Tcnica Superior de In- genieros Industriales y Minas from 1995 to 2001. Further, from 2001 to 2004, he was head of the Department of Electrical Engineering. He is a member of the Scientiﬁc Committee of different conferences and journals. He is the president of the European Association for the Development of Renewable Energies, Environment and Power Quality (EA4EPQ), European Association for the Development of Electrical Engineering (EADEE), and Spanish Association for the Development of Electrical Engineering (AEDIE).

Juan Carlos Garc´ıa-Prada is chair professor at the Mechanical Engineering Department of the University Carlos III of Madrid. He received his MD in mechanical engineering from Poly- technique University of Madrid (1984). He concluded his PhD in the ﬁeld of Fourier analysis of rotating machinery response (1991). He has published works about mechatronics, mechanical engineering, gait, ﬁnite element analysis, mechanical design, mechanical vibrations, and mechanical testing plus condition monitoring.

Zu-guang Ying received his BS, MS, and PhD from Zhejiang University in 1984, 1987, and 1995, respectively. He is currently a professor at Zhejiang University in Hangzhou, China. He visited the Hong Kong Polytechnic University and worked as researcher in 1999, 2000, 2002, 2008, 2013, and 2016. His research interests include nonlinear random vibration, optimal vibration control, structural dynamics and stability, periodic and quasi-periodic structures, smart composite structures, structural parameter identiﬁcation and damage detection, system uncertainty, and probability analysis.

282 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 About the Authors

Rong-chun Hu received his BS in applied mathematics from Anhui Normal University in 2007, MS in applied mathematics from Shantou University in 2010, and PhD in engineering mechanics from Zhejiang University in 2014. He is currently a post-doctoral researcher at the Department of Civil Engineering in Zhejiang University. His research interests include nonlinear stochastic dynamics, stochastic optimal control, intelligent structure, and Markovian jump process.

Rong-hua Huan received his BS and PhD from Zhejiang University in 2002 and 2007, respectively. He is currently an associate professor in the Department of Engineering Mechanics, Zhejiang University. He visited the Department of Mechanical Engineering, University of Cal- ifornia at Berkeley from 2012 to 2013 and the School of Engineering, University of California at Merced from 2013 to 2014. His research interests include linear and nonlinear stochastic dynamics, nonlinear stochastic optimal control, stochastic stability, and MEMS dynamics.

Achuthan C. Pankaj received his BE degree in mechanical engineering from the University of Pune, India, in 1997 and his ME degree in mechanical engineering from the National In- stitute of Technology in Allahabad, India in 2000. He joined Central Mechanical Engineering Research Institute, Durgapur India in 2001 as a scientist working in the ﬁelds of design and ﬁnite element analysis. He is currently working as a Principal Scientist in the Structural Tech- nologies Division at CSIR-National Aerospace Laboratories, Bangalore, India in the areas of aircraft structural dynamics, aeroelasticity and reliability of aircraft systems. He has more than 10 journal and 20 conference publications to his credit.

M. V. Shivaprasad received his MS in mechanical engineering with a specialization in machine design from Bangalore University in 2002. He is currently working as a Senior Scientist in the Structural Technologies Division of CSIR-National Aerospace Laboratoriesin Banga- lore, India in the area of experimental structural dynamics and flight flutter testing, a position he has held for the last twelve years. He has carried out system identification of aerospace components and aircraft through ground vibration tests for fixed wing aircraft (fighters and civil aircraft) and helicopters. His research interests include active vibration control using piezo actuators, developing semi-active vibration isolators using magneto rheological fluids and structural health monitoring through fiber brag grating sensors.

S. M. Murigendrappa received his BE and MTech degrees in mechanical engineering from the University of Mysore, India, in 1989 and 1998 respectively. He completed his PhD in mechanical engineering from the Indian Institute of Technology, Mumbai, India in 2004 and is currently working as an associate professor in the Mechanical Engineering Department at the National Institute of Technology Karanataka, Surathkal, India. He has two years of industrial experience and more than 16 years of teaching and research experience in machine dynamics and vibrations, fracture mechanics and fatigue, stress analysis, ﬁnite element method, and advanced materials. He is the recipient of Sir C. V. Raman Award for the best article contributed to the Journal of Acoustical Society of India and the National Award for engineering design for students studying in engineering colleges. He is also the author of a textbook titled Fundamen- tals of Finite Element Method. He has more than 20 journal and 15 conference publications to his credit.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 283 About the Authors

Xiaoqing Wang received his BS and MS degrees from Chongqing Jiaotong University and Wuhan University of Technology, China, in 2010 and 2013. He is currently a PhD candidate in the Department of Mechanical Engineering at the University of Alabama. His research interests include vibration and noise control, signal processing, machine learning, fault diagnosis, and processing-microstructure-mechanical property relationships in metallic alloys, such as magnesium alloys, titanium alloys, and nickel-based super-alloys. The focus of his doctoral dissertation is additive manufacturing (AM) of metals using powder bed fusion (PBF) technologies, including electron beam melting (EBM, also known as EBAM) and selective laser melting (SLM, may also called ‘Laser-PBF’).

Xiang Yang is currently a professor at the School of Energy and Power Engineering, Wuhan University of Technology. Professor Xiangs research interests include vibration and noise control, optimal design, condition monitoring, and fault diagnosis technology, and signal processing technology.

Ahmed E. Abouelregal was born in Dakahlia, Egypt, in 1974. He received his BS in mathematics from Mansoura University, Egypt, in 1997, and his MS and PhD in applied mathematics from the same university in 2002 and 2007, respectively. He is an associate professor of applied mathematics at Mansoura University, Egypt and he is currently an associate professor of applied mathematics at Aljouf University, Saudi Arabia. His current research interests include elasticity, thermoelasticity, wave propagation, beams and solid mechanics. Dr. Abouelregal is a reviewer of many international journals and his research papers have been cited in many articles and textbooks.

Ashraf M. Zenkour graduated from Mansoura University, Egypt in mathematics in 1985 and recieved his MS and PhD degrees from the same university in 1989 and 1995, respectively. He is a professor of applied mathematics at Kafrelsheikh University, Egypt and he is currently a professor of applied mathematics at King Abdulaziz University, Saudi Arabia. His research interests are in the areas of structural stability, vibration, plated structures and shells. He is the author or co-author of over 220 scientiﬁc publications. He is a reviewer of many international journals such as Solid Mechanics and Applied Mathematics, and an editorial member of many others. In addition, he has delivered various lectures at national and international conferences.

Daoud S. Mashat graduated from King Abdul Aziz University, SaudiArabiain mathematics in 1985 and went on to receive his MS. and PhD from Texas A&M university, USA in 1991 and 1997, respectively. He is a professor of numerical analysis at King Abdul Aziz University. Now, he is a vice dean of distant learning deanship at King Abdul Aziz University. His research interests are in solving partial differential equations numerically. Professor Daoud has supervised many MS and PhD thesis across the kingdom of Saudi Arabia.

284 International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 About the Authors

Thiago Antonio Fiorentin graduated in mechanical engineering from Universidade Federal de Santa Catarina with a masters and PhD invibration and acoustics. Visiting researcher at Uni- versity of Southampton, currently professor at Universidade Federal de Santa Catarina, Brazil. His research interests include aircraft noise and vibration, vehicle dynamics, ﬁnite element method and noise control. He developed analytical and numerical models to predict the noise generated by air-core dry-type reactors. Also, he researched noise and vibration generated by aircraft hydraulic system to Embraer. He is currently researching wave propagation in solids, model reduction, and optimization applied to vibroacoustics problems.

Alexandre Mikowski graduated in physics from Universidade Estadual de Ponta Grossa with his masters in 2003. He then graduated with his PhD in 2008 in physics at Universidade Federal do Paran (UFPR). He completed a postdoctoral internship at UFPR with the research line of superﬁcial mechanical properties of lamellar materials and applications in polymer nanocomposites. He has experience on the characterization and development of methods for the analysis of mechanical properties of materials. Has didactic experience in higher education in the areas of physics, mathematics, statistics and engineering.

Olavo Mecias da Silva Junior graduated in mechanical engineering, from Universidade Fed- eral de Santa Catarina and is also finishing his PhD studies along with his work as a researcher. He has experience in dynamic analysis of structures and optimization. Currently, his focus is the topology optimization of components to improve their vibroacoustic behavior. He de- velops structural optimization for some companies such as Whirlpool. Over the course of his research, he developed numerical models to identify the noise sources of electrical equipment. Normally, he uses finite element methods and boundary element methods to do the analysis. He is currently working with vibratory energy, especially identifying the energy flow. This methodology is very applicable to optimize the vibroacoustic behavior of some components.

Arcanjo Lenzi graduated in mechanical engineering from Universidade Federal de Santa Cata- rina with a masters in sound and vibration studies at University of Southampton followed by a PhDin machinery noise from the same university. He is currently professor at Universidade Federal de Santa Catarina, Brazil, and has experience in mechanical engineering with an emphasis in noise and vibration. His research interests include machinery noise, structural damping and noise control. He is currently researching porous materials and structural damping to aircraft applications, the activities are sponsored by Embraer. Additionally, he is working in new technologies to reduce the noise generated by hermetic compressors, Whirlpool group is the main partner.

International Journal of Acoustics and Vibration, Vol. 22, No. 2, 2017 285