The Transcendental Eigenvalue Problem and Its Application In

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LSU Doctoral Dissertations Graduate School

2003 The transcendental eigenvalue problem and its application in system identification Kumar Vikram Singh Louisiana State University and Agricultural and Mechanical College, [email protected]

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THE TRANSCENDENTAL EIGENVALUE PROBLEM AND ITS APPLICATION IN SYSTEM IDENTIFICATION

A Dissertation

Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy

The Department of Mechanical Engineering

by Kumar Vikram Singh B.E., Birla Institute of Technology, Mesra, India, 1997 May 2003

This work is dedicated

to my inspirations,

of the past

“my grandfather Late Shri S.P.Singh who taught me how to read and write”

and

of the future,

“my newly born daughter Shreya”

ii ACKNOWLEDGEMENTS

I would like to express my gratitude to my advisor, Dr. Yitshak Ram, for his encouragement, advice, mentoring, and research support throughout my doctoral studies.

His technical and editorial advice was essential to the completion of this dissertation. His ability to teach, depth of understanding and ability to achieve perfection will always be my inspiration.

I would also like to thank to my doctoral committee members, Dr. George

Voyiadjis, Dr. Su-Seng Pang, Dr. Michael Khonsari, Dr. Michael Murphy and dean’s representative Dr. Joseph Giaime for evaluating my research work and providing valuable comments to improve the presentation and contents of this dissertation.

I highly appreciate my colleagues Akshay Singh and Jayho Shim for their friendship and intellectual input. I would also like to thank them along with Mr. Ed

Harris for their assistance during my experimental studies. I am thankful to my friend

Mayank Tyagi for interesting and good-spirited discussions relating to this research. I am very appreciative to all the faculty members and staffs of mechanical engineering department who assisted me in my research, as well as in my graduate studies.

Finally, I am grateful to my parents Madhuri and Rama Shankar Singh for their love, support and guidance. They have always been supportive of my academic pursuit in the United States as well as in India. I would like to thank my wife Sangeeta for her love and understanding. Her presence, encouragement and support in the last few years made this dissertation possible. Last but not the least, my room-mates Anirudha Yadav, Proyag

iii Datta, Abhinav Bhushan and all my friends at LSU and BIT thank you for your encouragement.

The research work is supported in part by a National Science Foundation research grant CMS-9978786.

iv TABLE OF CONTENTS

DEDICATION………………………………………………………….. ii

ACKNOWLEDGEMENTS.………………………………………….... iii

ABSTRACT………………....…………………………………………. viii

1. INTRODUCTION………………………………………………... 1 1.1 Inverse Problem in Vibration: A Simple Example………………. 2 1.2 Background and Overview………………………………………… 4 1.3 Research Objectives………………………………………………... 9 1.4 Scope of the Dissertation…………………………………………… 9

2. CONTINUOUS MODELS AND THEIR DISCRETE APPROXIMATION: MOTIVATION………………………….. 12 2.1 Analytical Models of Continuous Vibrating Systems……………. 12 2.1.1 Spectrum of an Axially Vibrating Rod………………………. 13 2.1.2 Spectrum of a Transversely Vibrating Beam………………… 16 2.2 Discrete Models of Continuous Vibratory Systems...……………. 18 2.2.1 Finite Difference Model of Axially Vibrating Rod…………... 19 2.2.2 Discrete Model of Vibrating Beam…………………………… 22 2.2.3 Finite Element Model of Vibrating Rod……………………… 25 2.2.4 Finite Element Model of Transversely Vibrating Beam……… 27 2.3 Solution for Inverse Problem of Continuous Systems Using Its Associated Discrete Model…………………………………………. 29 2.4 Motivation of the Research: Continuous Systems vs. Their Discrete Models…………………………………………………….. 33 2.4.1 Comparison of the Spectrum of Fixed-Free Uniform Rod with Its Approximated Model.………………………………... 34 2.4.2 Comparison of the Mode Shapes of a Fixed-Free Uniform Beam with Its Discrete Model………………………………... 36 2.4.3 Parameter Estimation of an Uniform Rod Using Finite Difference Model……………………………………………... 38

3. TRANSCENDENTAL EIGENVALUE METHODS IN CLASSICAL DIRECT PROBLEMS………………..………. 42 3.1 Introduction………………………………………………………… 42 3.2 Transcendental Eigenvalue Problem……………………………… 42 3.2.1 Free Vibration of Non-uniform Rods...………………………. 43 3.2.2 Free Vibration of Non-uniform Beams……………………….. 47 3.3 An Algorithm for the Solution of Transcendental Eigenvalue Problems: Newton’s Eigenvalue Iteration Method………………. 54 3.3.1 Algorithm 1…………………………………………………… 56

v 3.4 Applications of the Transcendental Eigenvalue Problem……….. 57 3.4.1 Example: Spectrum of Piecewise Continuous Rod…..………. 57 3.4.2 Example: Spectrum Comparison of Exponential Rod………... 59 3.4.3 Example: Comparison of the Tapered Beam Eigenvalues…… 63 3.4.4 Example: Buckling of a Column with Composite Loads…….. 67 3.5 Higher Order Approximation in Newton’s Eigenvalue Algorithm…………………………………………………………… 71 3.6 Extension of the Algorithm in Quadratic Eigenvalue Problems… 72 3.6.1 Example: Spectrum of a Discrete Model of a Vibrating System………………………………………………………… 73 3.7 Summary of the Results and Conclusions………………………… 75

4. INVERSE PROBLEMS IN THE CONTINUOUS VIBRATORY SYSTEM…………………………………………. 77 4.1 Mathematical Modeling for Non-uniform Continuous Rod for the Associated Inverse Problems…………………….……………. 78 4.2 An Algorithm for the Solution of Inverse Problems……………... 83 4.2.1 Some Mathematical Preliminaries for the Algorithm...………. 83 4.2.2 Development of an Algorithm for Non-uniform Continuous System………………………………………………………… 87 4.2.3 Algorithm 2…………………………………………………… 89 4.3 Reconstruction of Continuous Vibrating System from Its Spectral Data………………………………………….……………. 90 4.3.1 Example: Reconstruction of the Piecewise Continuous Rod. 91 4.4 Significance of the Multiple Solutions and Uniqueness of the Results………………………………………………………………. 95 4.4.1 Multiplicity of the Solutions………………………………….. 95 4.4.2 Modification towards Obtaining the True Solution: Least Square Method………………………………………………... 100 4.4.3 Modification towards Obtaining the True Solution: Verification of Solution Set…………………………………... 102 4.5 Numerical Example and Conclusions……………………………... 103 4.5.1 Example: Reconstruction of the Shape of the Exponential Rod 103

5. IDENTIFICATION OF THE BEAM FROM SPECTRAL DATA: EXPERIMENT……………...………………………… 107 5.1 Physical Significance of Spectral Data……………...…………... 107 5.1.1 Spectral Data: Resonant and Anti-resonant Frequencies…... 107 5.1.2 Frequency Response Function of an Axially Vibrating Rod. 110 5.1.3 Frequency Response Function of the Transversely Vibrating Beam………………………………………………………... 111 5.1.4 Obtaining Fixed-sliding Frequencies of the Transversely Vibrating Beam…………………………………………….. 114 5.2 Modal Analysis Experiment for Frequency Response Measurement……………………………………………………... 118 5.2.1 Experimental Setup…………………………………………. 118

vi 5.3 Modeling of an Inverse Problem for the Piecewise Beam……... 121 5.4 Identification of the Physical Properties of the Piecewise Beam 128

6. OTHER ENGINEERING APPLICATIONS…………..……... 138 6.1 Introduction………………………………………………………. 138 6.2 Application in Vibration Control…………………..……………. 138 6.3 Application in Structural Health Monitoring…………………... 144 6.4 Possible Future Applications…………………………………….. 150

7. CONCLUSIONS……………………………………………….... 153

REFRENCES………………………………………………………….. 161

APPENDIX…………………………………………………………….. 168

A NATURAL FREQUENCIES OF THE NON-UNIFORM RODS 168

B DYNAMIC ABSORPTION IN CONSERVATIVE 170 VIBRATORY SYSTEM…………………………………………...

C PARAMETERS FOR CRACK PROBLEM……………...……… 174

VITA…………………………………………………………………… 175

vii ABSTRACT

An accurate mathematical model is needed to solve direct and inverse problems related to engineering analysis and design. Inverse problems of identifying the physical parameters of a non-uniform continuous system based on the spectral data are still unsolved.

Traditional methods, for the system identification purpose, describe the continuous structure by a certain discrete model. In dynamic analysis, finite element or finite difference approximation methods are frequently used and they lead to an algebraic eigenvalue problem. The characteristic equation associated with the algebraic eigenvalue problem is a polynomial. Whereas, the spectral characteristic of a continuous system is represented by certain transcendental function, thus it cannot be approximated by the polynomials efficiently. Hence, finite dimensional discrete models are not capable of identifying the physical parameters accurately regardless of the model order used.

In this research, a new low order analytical model is developed, which approximates the dynamic behavior of the continuous system accurately and solves the associated inverse problem. The main idea here is to replace the continuous system with variable physical parameters by another continuous system with piecewise uniform physical properties.

Such approximations lead to transcendental eigenvalue problems with transcendental matrix elements. Numerical methods are developed to solve such eigenvalue problems.

The spectrum of non-uniform rods and beams are approximated with fair accuracy by solving associated transcendental eigenvalue problems. This mathematical model is extended to reconstruct the physical parameters of the non-uniform rods and beams.

viii There is no unique solution for the inverse problem associated with the continuous system. However, based on several observations a conjecture is established by which the solution, that satisfies the given data by its lowest spectrum, is considered the unique solution. Physical parameters of non-uniform rods and beams were identified using the appropriate spectral data. Modal analysis experiments are conducted to obtain the spectrum of the realistic structure. The parameter estimation technique is validated by using the experimental data of a piecewise beam. Besides the applications in system identification of rods and beams, this mathematical model can be used in other areas of engineering such as vibration control and damage detection.

ix 1. INTRODUCTION

Mathematical models are often required for the analysis and design of any engineering problem. These models can be classified into two types of problems in engineering application,

a) direct problems and

b) inverse problems.

Analysis and prediction of the “behavior” of system e.g. forced response, natural frequencies and stresses from the known input and physical parameters, is called a direct problem. In the structural vibration field of study, inverse problems are defined as the determination or estimation of system parameters, such as density, modulus of elasticity and cross sectional area, by knowing the behavior of system (Gladwell, 1996). These definitions are presented in schematic diagram as shown in figure 1.1.

Inverse problems are used in the fields of physics, science and engineering sciences. There is a subclass of inverse problems, which deals with the determination and estimation of mechanical properties based on certain spectral data. This area of knowledge is associated with inverse eigenvalue problems. The mathematical problem utilizes exact and complete spectral data to determined the system uniquely. In engineering problems, the measured data is inaccurate and incomplete thus parameters, in this case, are estimated rather than reconstructed.

Only inverse problems that are related to vibrating structures are presented here.

There are other inverse problems that exist in different engineering and science

1 applications e.g. pole assignment, vibration control, geophysics, quantum mechanics and numerical analysis. These applications are briefly discussed in Chu (2001).

Direct Problems

Known Known ? INPUTPHYSICAL SYSTEM OUTPUT Forces Physical Parameters Natura l Freq ue nc ie s, Mode Shapes, Mass, Stiffness, Density, Rigidity System Response

Inverse Problems ? Known Known

PHYSICAL SYSTEM INPUT OUTPUT Forces Physical Parameters Natura l Freq ue nc ie s, Mode Shapes, Mass, Stiffness, Density, Rigidity System Response

Figure 1.1: Direct Problems and Inverse Problems

1.1 Inverse Problem in Vibration: A Simple Example

Consider a simple inverse problem (Gladwell, 1986-a), associated with a mass spring system shown in figure 1.2(a). An inverse problem is defined as the determination

of the mass m1 and spring constants k1 , k2 based upon certain spectral data such as natural frequencies and mode shapes. It is evident that in order to estimate the above mentioned three quantities, more than one natural frequency is necessary. Thus, it is required to obtain another set of data, which could be available from different boundary

conditions as shown in figure 1.2(b). Note that even if the values of m1 , k1 and k2 are

2 found they will yield the same natural frequency which can be obtained by the system

α α α α having physical parameters m1 , k1 and k2 , where is any positive constant. Thus in order to obtain an unique solution one non-spectral data such as total mass or static deflection is required.

k1 k2 k1

m1 m1

(a) (b)

Figure 1.2: A mass spring system with two different end conditions (a) original system (b) modified system

The eigenvalue of freely vibrating system, as shown in figure 1.2(a), can be evaluated as,

k + k λ~ = 1 2 λ~ = ω~ 2 1 , 1 , (1.1) m1 where, ω~ is the natural frequency of original system. Similarly, for the system, as shown in figure 1.2(b), the eigenvalue can be obtained as,

k λ = 1 λ = ω 2 1 , 1 , (1.2) m1 where, ω is the natural frequency of modified system. The physical parameters of a given mass-spring system are thus determined uniquely by solving (1.1) and (1.2) by using the values of ω~ , ω and one non-spectral data set. This problem can be extended for multi-degree-of-freedom systems and they are developed and discussed in the following chapter.

3 The aim of research work presented here is to study inverse problems in the field of structural dynamics, vibration and control and to develop a method by means of which physical properties of a continuous vibratory system can be estimated using measured spectral data. In this work, inverse problems related to one-dimensional distributed parameter systems i.e. axially vibrating rod and transversely vibrating beam, have been studied. A schematic of the research objective is shown in figure 1.3. It is required to determine the cross-sectional area of the rod when the resonant and anti resonant frequencies are known.

Resonant Frequency response function frequencies Anti-resonant

frequencies

Mathematical model Spectral data from and experiment numerical algorithm

A(x),E(x), ρ(x) x Identified physical parameters Vibrating structure

Figure 1.3: Schematic of problem description

1.2 Background and Overview

Past research in the field of inverse eigenvalue problems related to vibration is summarized in a monograph by Gladwell (1986-a) and in the review papers by Boley and

4 Golub (1987), Friedland et al (1987), Chu (1998) and Gladwell (1986-c, 1996). The problem of reconstruction of distributions of density and rigidity for a continuous vibratory system is well studied. Fundamental research works on inverse problems related to continuous systems by Borg, Levinson, Gel’fand and Hald are summarized in

Gladwell (1986-c) which concludes that in theory, two complete sets of spectral data associated with two different boundary conditions along with the total mass, determine the distributions of density and rigidity along a rod. However, there are no results demonstrating that certain numerical methods exist for actually identifying these functions.

The classical inverse problems in vibration involve reconstruction of mass-spring systems (Gantmakher and Krein, 1961) and mass-rod-spring systems (Gladwell, 1984) from spectral data. There are several publications dealing with the reconstruction of discrete vibrating systems mainly by Gladwell (1986-a, 1995), Movaheddy (1995) and

Ram (1992, 1993, 1994). Hald (1976) and de Boor and Golub (1978) found that identification of the mass and stiffness matrices for a simply connected system requires a reconstruction algorithm for tridiagonal matrices. Some numerical methods for solving inverse eigenvalue problems associated with discrete models have been proposed by

Friedland (1979, 1987), Nocedal and Overton (1983), Chu (1992) and recently by Elhay and Ram (2001). Ram and Caldwell (1992) considered reconstruction of discrete models of block in line structures where all the members, of the system, are interconnected with springs. In addition, it is shown by Ram (1993), that density and rigidity functions may be determined from natural frequencies of a rod together with natural frequencies obtained by attaching a simple oscillator to the free end. Ram and Elhay (1996-a) demonstrated that a simply connected mass-spring-damper system could be reconstructed

5 from a single frequency response function. The reconstruction of a discrete model for axially vibrating rods and transversely vibrating beams based on the spectral sets of data are investigated by Barcilon (1979, 1982), Gladwell (1986-b), Gladwell and Williams

(1988), Ram and Gladwell (1994-c) and Ram and Elhay (1998). The discrete model of a rod can also be reconstructed from one natural frequency and two mode shapes (Ram,

1994-a). Similarly, the discrete model of a Bernoulli-Euler beam can be reconstructed from two natural frequencies and three mode shapes (Ram, 1994-b).

Past research demonstrates that reconstruction of a continuous model, based on its discrete approximation, leads to an inaccurate identification of physical parameters. This is expected because the mathematical formulation for a continuous system differs from its associated discrete model. The motion of a n degree-of-freedom discrete system is governed by n simultaneous ordinary differential equations and its solution is obtained by solving an associated algebraic eigenvalue problem. The solution consists of n eigenvalues and n eigenvectors. In contrast, the motion of a continuous system is governed by partial differential equations. These equations are to be satisfied inside the domain and an appropriate number of boundary conditions are to be satisfied over the boundary. The solution is obtained by solving the associated differential eigenvalue problem, and it consists of infinite sets of eigenvalues and eigenfunctions (see

Meirovitch, 1997).

Consider a fixed-free axially vibrating rod as shown in figure 1.4. The motion of a fixed-free vibrating rod is governed by the following differential equation,

∂  ∂u()x,t  ∂ 2u()x,t  E()()x A x  = ρ()()x A x , (1.3) ∂x  ∂x  ∂t 2 where,

6 u()x,t is the axial displacement of an element dx ,

E()x is the Young’s modulus of elasticity,

A()x is the cross sectional area, and,

ρ()x is the density per unit length.

E()x , A ()x , ρ ()x

u()x,t x = L

Figure 1. 4: A continuous fixed-free axially vibrating rod

Assume that the motion is harmonic,

u()x,t = v ()x sinωt . (1.4)

Hence, substitution of (1.4) in equation (1.3) leads to the associated eigenvalue problem

d  dv()x   E()()x A x  + ω 2 ρ()()()x A x v x = 0 . (1.5) dx  dx 

λ = ω 2 The eigenvalues i i cannot usually be obtained when the area A(x) , the Young’s modulus of elasticity E()x , and the density ρ()x are non-uniform along the length of the rod. Hence, in practice, finite dimensional models are used to approximate the eigenvalues of continuous systems.

7 For example, the rod in figure 1.4 can be divided into n elements as shown in figure 1.5. Inverse problem for estimating physical properties of the rod can be stated as follows:

ω ω ω ω Given: Measured natural frequencies 1 , 2 , 3 , , n of the fixed-free

ω~ ω~ ω~ ω~ configuration, measured natural frequencies 1, 2 , 3 , , n−1 of the fixed-fixed

configuration, and the total mass of system.

Find: Physical parameters of the rod,

l = = l a) the axial rigidity p1, p2 , p3 , , pn , where pi Ei Ai , for i 1,2,3, ,n ; and

= ρ = b) the mass per unit length q1 ,q2 ,q3 , ,qn , where qi i Ai , for i 1,2,3, ,n .

p1 p2 p q q n 1 2 qn

u()x,t x = L

Figure 1. 5: Proposed approximate model of the rod

Reconstruction, based on spectral data, is preferred because natural frequencies can be measured by experiments. Two spectral sets can be obtained by experimentally extracting a single transfer function at the free end of rod and by evaluating poles and zeros corresponding to its resonant and anti-resonant frequencies. Solution of the above stated problem could also be approached by constructing its associated finite dimensional model and by solving the corresponding inverse eigenvalue problem. However, the algebraic eigenvalue problem for a finite dimensional model is relatively simple to model

8 and solve, but they lead to inaccurate physical parameter estimation, as demonstrated in the following chapter.

1.3 Research Objectives

In summary the following issues are addressed in this research work:

• develop a low order analytical model to approximate continuous vibratory

systems effectively.

• study and develop a numerical method to solve transcendental eigenvalue

problems associated with proposed approximation models for non-uniform

axially vibrating rods and transversely vibrating beams.

• develop a method to solve inverse transcendental eigenvalue problems to

estimate physical parameters of vibrating beams and rods from the spectral

data sets.

• conduct laboratory experiments i.e. modal analysis of beams, to verify

these theoretical results.

1.4 Scope of the Dissertation

The discrepancies, between the continuous system and its discrete approximation, are demonstrated in chapter 2. Mathematical modeling of a continuous system based upon their discrete approximation is developed. It is shown that discrete models are not a good approximation of continuous systems. This chapter concludes that a new mathematical model is required for solving the direct and inverse problems for continuous vibrating structures.

To overcome the difficulty stated in chapter 2, a new formulation is introduced for the modeling of continuous system in chapter 3. Proposed mathematical model

9 approximates the given continuous system by another continuous system with piecewise constant physical properties. Such models lead to transcendental eigenvalue problems where the elements of the approximating matrices are transcendental in nature. In this chapter, an effective algorithm based on Newton’s eigenvalue iteration method is developed for solving transcendental eigenvalue problems. Numerical examples involving elastic stability and eigenvalue calculation for non-uniform continuous rods and beams are presented. Development of higher order approximation of the solution for transcendental eigenvalue problem has been discussed. Application of this algorithm in the context of quadratic eigenvalue problems has also been studied for discrete vibrating structures. The effectiveness of the solution for a transcendental eigenvalue problem has been compared with other well-established numerical methods such as finite difference and finite element methods.

In chapter 4, the inverse problem for continuous vibratory system is introduced based on transcendental eigenvalue formulation. A numerical algorithm is developed in order to solve the inverse transcendental eigenvalue problem. An accurate reconstruction procedure for the estimation of physical parameters of non-uniform rods is demonstrated conclusively. Least square estimation and uniqueness of the solutions are also discussed.

Laboratory experiments with piecewise beams have been conducted and the results are discussed in chapter 5. The inverse eigenvalue problem for a piecewise beam has been developed and physical significance of spectral data set is introduced. The objective of this chapter is to validate proposed mathematical model and to authenticate the predictions of the reconstruction algorithm. The issues related to collection of spectral data set are emphasized and conditions on data to ensure successful reconstruction procedure are proposed in this chapter.

10 Applications of the developed mathematical model and algorithms, in the areas of vibration control and damage detection, are shown in chapter 6. Some fundamental problems are addressed and the results are compared with the existing work. The results are summarized in chapter 7 with principal conclusions from each chapter.

Recommendations have been made for the possible future work related to the proposed mathematical model and numerical algorithm.

2. CONTINUOUS MODELS AND THEIR DISCRETE APPROXIMATION: MOTIVATION

Often the vibration analysis of continuous system leads to an eigenvalue problem with no closed form solution. In the absence of closed form solution, it is necessary to approximate the solution. Finite element and finite difference models are widely used to approximate continuous systems. In this chapter, some examples are presented to demonstrate the behavior of continuous vibrating systems and their discrete approximations. Analytical models of axially vibrating rods and transversely vibrating beams along with their finite dimensional models are developed. The behavior of a vibrating structure, i.e. natural frequencies and mode shapes, obtained from analytical models and discrete models are compared.

Solution of the inverse problem associated with a continuous system based upon the discrete approximation is developed and analyzed. Numerical results demonstrate that parameter estimation techniques of a continuous system based upon its discrete model, leads to an inaccurate approximation. In summary, this chapter intends to address the research issues related to inverse problems in vibration of continuous system, in a new direction.

2.1 Analytical Models of Continuous Vibrating Systems

Analytical solutions for axially vibrating rods and transversely vibrating beams have been developed. A few examples involving uniform vibrating rods and beams have been considered for simplicity.

2.1.1 Spectrum of an Axially Vibrating Rod

The equation of motion for a unit length uniform rod, as shown in figure 2.1, can be modeled by the following partial differential equation

∂ 2u()x,t ∂ 2u()x,t EA = ρA , 0 < x < 1, t > 0, (2.1) ∂x 2 ∂t 2 or equivalently,

∂ 2u()x,t ∂ 2u()x,t = c 2 , (2.2) ∂t 2 ∂x2 where,

E c = , (2.3) ρ is the speed of wave propagation in the rod.

x u()x,t

L =1

Figure 2. 1: Uniform cross-section axially vibrating rod

The fixed-free boundary conditions

 =  u(0,t) 0 ∂u(L,t) , (2.4) = 0  ∂x ensure that there is no displacement at x = 0, and that no axial force applies at x = 1.

Assuming harmonic motion of the following form,

u()x,t = v ()x sinωt , (2.5) leads to the frequency equation (equation 7.122, Meirovitch, 1997),

cos λ = 0, (2.6) with the roots

 ()2i −1 cπ 2 λ =   . (2.7) i  2 

Note that the frequency equation (2.6) is a transcendental equation.

p , q p ,q 1 1 2 2 A1 A2

x1 L = 1

Figure 2. 2: Piecewise axially vibrating rod of order n=2

Now consider a unit length piecewise continuous rod as shown in figure 2.2. Here the axial rigidity and material density parameters are defined as

 E A , 0 < x < x p()x =  1 1 1 , (2.8) < < E2 A2 , x1 x 1 and

 ρ A , 0 < x < x q()x =  1 1 1 , (2.9) ρ < <  2 A2 , x1 x 1 respectively.

Let the values of E and ρ be constant along the length of the rod and cross-

sectional area is assumed to be A1 and A2 as shown in figure 2.2. Thus, the eigenvalue problem for the piecewise continuous rod is given by following sets of differential equations

p v′′+ λq v = 0, 0 < x < x  1 1 1 1 1 p v′′ + λq v = 0, x < x < 1  2 2 2 2 1 , (2.10) ()= ′ ()= v1 0 0, v2 1 0  ()= ()′ ()= ′ () v1 x1 v2 x1 , p1v1 x1 p2v2 x1 with λ = ω 2 c2 , and, where the motion of the rod,

()= () λ < < u1 x,t v1 x sin t, 0 x x1 , (2.11) and

()= () λ < < u2 x,t v2 x sin t, x1 x 1, (2.12) is harmonic. Equation (2.10) consists of two differential equations, two boundary conditions and two matching conditions. These two matching conditions ensure the continuity in displacement and force between two portions of the rod having different constant physical properties.

The general solution of (2.10) is written as,

= λ + λ v1 (x) z1 sin x z2 cos x , (2.13) and,

= λ + λ v2 (x) z3 sin x z4 cos x . (2.14)

Applying the boundary conditions and imposing the matching conditions of displacement and force from (2.10), the eigenvalue problem can be written in matrix form as

 0 1 0 0  z   1   λ λ − λ − λ   sin x1 cos x1 sin x1 cos x1  z2    = o . (2.15) A cos λ x − A sin λ x − A cos λ x A sin λ x  z  1 1 1 1 2 1 2 1  3  λ − λ    0 0 cos sin  z4 

Equation (2.15) is termed as transcendental eigenvalue problem of the following form

A()ω z = o , (2.16) which determines non-trivial solutions of ω and z ≠ 0 for (2.16), where the elements of matrix A are transcendental functions in ω , and z is a constant eigenvector. The determinant relation

det()A()ω = 0 (2.17)

= gives the same transcendental frequency equation as (2.6) for the case where A1 A2 and

λ the eigenvalues i can be obtained as in (2.7).

2.1.2 Spectrum of a Transversely Vibrating Beam

Consider a unit length transversely vibrating uniform Bernoulli-Euler beam as shown in figure 2.3. Physical properties of beam are modulus of elasticity E , density ρ , cross sectional area A and moment of inertia I . The equation of motion for a transversely vibrating uniform beam is given by,

∂ 4 w(x,t) ∂ 2 w(x,t) EI + ρA = 0 , (2.18) ∂x 4 ∂t 2 with the following boundary conditions

w(x,t) = 0 no displacement at the fixed end  x=0  ∂w(x,t)  = 0 no slope at the fixed end ∂  x x=0   . (2.19) ∂ 2 w(x,t) EI = 0 no bending moment at the free end  ∂x 2  x=1   ∂ 3 w(x,t) EI = 0 no shear force at the free end  ∂  x x=1

Assuming the solution to be a following harmonic motion of the form,

w(x,t) = v(x)sinωt . (2.20)

w()x,t x

L =1

Figure 2. 3: Uniform cross-section transversely vibrating beam

Substituting (2.20) in (2.18) and (2.19), the following eigenvalue problem for beam can be derived,

v'''' − β 4v = 0, 0 < x < 1, (2.21) where,

ρA β 4 = ω 2 , (2.22) EI with the following boundary conditions,

v(0) = 0  v′(0) = 0  (2.23) v′′(1) = 0 v′′′(1) = 0.

The solution of (2.21) can be written as,

= β + β + β + β v(x) z1 sin x z2 cos x z3 sinh x z4 cosh x , (2.24) and applying the boundary conditions from (2.23) in the solution (2.24) leads to following formulation of transcendental eigenvalue problem in the form of (2.16) where,

 0 1 0 1    β 0 β 0 A =   , (2.25) − β 2 sin β − β 2 cos β β 2 sinh β β 2 cosh β    − β 3 cos β β 3 sin β β 3 cosh β β 3 sinh β 

and,

= ()T z z1 z2 z3 z4 . (2.26)

Determinant of (2.25) leads to following frequency equation,

cos β cosh β +1 = 0 (2.27) with the roots

 1.87510   4.69409  7.85475 β = i  . (2.28)  10.99554  14.92256   (2i −1)π 2, for i > 5

The mode shapes corresponding to these eigenvalues can be evaluated by (see

Blevins, 1979),

 sinh β − sin β  φ(x) = cosh β x + cos β x −  i i ()sinh β x − sin β x . (2.29) i i  β + β  i i  cosh i cos i 

It is evident from both of the formulations of rods and beams that distributed parameter system leads to a transcendental frequency equation and a transcendental eigenvalue problem.

2.2 Discrete Models of Continuous Vibratory Systems

Discrete models are widely used in direct classical problems for approximating the spectrum of continuous systems as well as in estimating physical parameters of the vibratory system.

In general, discretization methods are classified in two major classes,

a) finite difference methods and,

b) finite element methods.

2.2.1 Finite Difference Model of Axially Vibrating Rod

The rod shown in figure 2.1 is divided into n equal subintervals as shown in figure 2.4(a) and length of the finite difference segment is assumed to be h . Eigenvalue problem associated with the fixed-free uniform rod, as shown in figure 2.4, can also be written as,

 ′ ()EAv′(x) + λρAv(x) = 0, λ = ω 2  v(0) = 0 . (2.30)  ′ = v (L) 0

v 0 vi−1 vi vi+1 vn

′ ′ ′ ′ vi−1 vi vi+1 vn

(a)

ki−1 ki ki+1 kn

mi−1 mi mi+1 mn (b)

Figure 2. 4: (a) Finite difference scheme for the axially vibrating rod (b) Discrete model of the rod based on finite difference scheme

By partitioning the rod into n elements of equal length h = L n and denoting the

th displacements of the right and left boundaries of i element by vi−1 and vi respectively as

shown in figure 2.4(a), n eigenvalues and eigenvectors of the above stated differential equation can be evaluated.

Axial rigidity and density per unit length of the mid-section of ith element is

()ρ th defined by(EA)i and A i . Finite difference approximation for the i element gives

 −  ′ vi vi−1 ()EAv = − ≅ (EA)  , (2.31) x (ih h 2)  i h 

and

v − v v − v ′ i+1 i − i i−1  −  (EA)i+1 (EA)i vi vi−1 h h (EA)  x=ih = . (2.32)  i h  h

Substituting the values of (2.31) and (2.32) the finite difference scheme in (2.30) gives

+ (EA)i (EA)i (EA)i+1 (EA)i+1 v − − v + v + + λ(ρA) v = 0 . (2.33) h 2 i 1 h 2 i h 2 i 1 i i

Multiplying (2.33) by − h and defining

()EA p k = i = i , (2.34) i h h and,

= ()ρ = mi h A i hqi , (2.35) the eigenvalue problem (2.30) can be written as,

− + ()+ − − λ = ki vi−1 ki ki+1 vi ki+1vi+1 mi vi 0 . (2.36)

The left boundary condition of (2.30) can be written in terms of finite differences as,

= v0 0 , (2.37) and the right boundary condition implies that,

= vn+1 vn . (2.38)

Applying the boundary conditions (2.37) and (2.38) in (2.36) and for i = 1,2,m,n the generalized eigenvalue problem can be written in matrix form as,

()K − λM v = o , (2.39) where,

 + −  p1 p2 p2  − + −   p2 p2 p3 p3  1 K =   , (2.40) h   − + −  pn−1 pn−1 pn pn   −   pn pn 

q   1   q2  M = h  , (2.41)    qn−1     qn  and,

= () T v v1 v2 vn . (2.42)

It is important to note that finite difference formulation used for approximating a continuous system leads to an algebraic eigenvalue problem (2.39) contrary to a transcendental eigenvalue problem (2.16), which is obtained analytically. The algebraic eigenvalues are simple to evaluate and various numerical methods to solve such problems have been well established.

The finite difference model of an uniform rod can be physically interpreted as an approximation of rod by mass-spring system where the mass of each element is lumped at the element center and all the masses are connected by mass-less springs. Mass-spring approximation of the uniform rod as shown in figure 2.4(b) is also known as lumped parameter model.

2.2.2 Discrete Model of Vibrating Beam

Discrete model for the transversely vibrating Bernoulli-Euler beam can be represented by mass-spring-rod model (Gladwell, 1986-a). Let the beam shown in figure

+ 2.3 be divided into n elements of length hi . It consists of n 2 masses linked by mass-

less rigid rods of length li that are connected by n rotational springs ki as shown in figure 2.5.

kn−1 V− k k 2 1 n VR k2 l k3 l l0 1 n l m − τ m0 2 n 1 τ − m1 R 1 m−1 mn m2

Figure 2. 5: Discrete mass-spring-rod model of the beam

The density, modulus of elasticity, cross sectional area and moment of inertia of

th ρ the i element of the beam is denoted by i , Ei, Ai and I i respectively. The parameters of discrete mass-spring-rod model of beam can be written as,

= li hi , (2.43)

= ρ = mi i Ai hi qi hi , (2.44) and,

E I r = i i = i ki , (2.45) hi hi

where, qi is mass per unit length of each element and ri is the flexural rigidity of the beam element.

th θ Let wi be the displacement of i mass from its static equilibrium, i be the slope

th τ of the i rod, i be the bending moment and Vi be the shearing force. Thus, left boundary (fixed) condition implies no displacement and no slope at the fixed end and they can be written as,

= θ = w−1 0 0, (2.46) which also implies that,

= w0 0. (2.47)

Free body diagram of figure (2.6) shows the configuration of ith mass and its neighboring rods. Force balance equation of the mass is written as,

= − mi wi Vi Vi−1 , (2.48) and the summation of moment about left end of (i+1) th rod gives,

τ −τ = i+1 i Vi . (2.49) li+1

Substituting (2.49) in (2.48) gives,

τ −τ τ −τ       = i+1 i − i i−1 =  1 τ −  1 + 1 τ +  1 τ mi wi   i−1   i   i+1 . (2.50) li+1 li  li   li li+1   li+1 

The moment needed to produce a relative rotation between two rigid rods on either side of the ith mass is given by,

τ = ()θ −θ i ki+1 i+1 i . (2.51)

Therefore substitution of (2.51) in (2.50) gives,

k  k + k k   k k + k  k = − i θ +  i+1 i + i+1 θ −  i+1 + i+1 i+2 θ + i+2 θ mi wi i−1   i   i+1 i+2 . (2.52) li  li li+1   li li+1  li+1

Also, for small displacement, the slope is given by,

w − w θ = i i−1 i , (2.53) li then substitution of (2.53) in (2.52) gives,

 k  k + k k   DD = i ()− +  i+1 i + i+1 ()− mi wi wi−1 wi−2  2  wi wi−1  l l −  l l l +   i i 1 i i i 1 . (2.54a)   k k + k  k −  i+1 + i+1 i+2 ()− + i+2 ()−   2  wi+1 wi wi+2 wi+1   lili+1 li+1  li+1li+2

Rearranging (2.54) gives,

 k  1 1   1 1    DD = − i −  +  +  +   mi wi wi−2  2 ki  2 ki+1 wi−1  lili−1  lili−1 li   li lili+1      1   1 1 1   1    +  k +  + + k +  k w  2  i  2 2  i+1  2  i+2  i . (2.54b)   li   li lili+1 li+1   li+1          1 1 1 1 k + −  + k +  + k w + i 2 w   2  i+1  2  i+2  i+1 i+2   lili+1 li+1   li+1 li+1li+2   li+1li+2

Thus, equation (2.54b) along with the left end boundary condition (2.46), (2.47) and right boundary condition can be written in the matrix form as follows,

MwDD + Kw = o , (2.55)

= with the value of hi h ,

q   1  = q2 M h  , (2.56)    qn 

= −1 T −1 T − −τ −1 K EL EθE L E VRe n R Ln Ee n , (2.57) where,

1 −1   1 −1  =  −  E  1 1  , (2.58)    1 

l   1  = l2 L   , (2.59)    ln 

  r1 1   = r2 Θ   , (2.60) h    rn 

= () T ∈ × en 0 0 1 (n 1) , (2.61) and,

= () T u u1 u2 un . (2.62)

Thus, trying the harmonic motion of the form (2.20), eigenvalue problem (2.39) associated with this discrete model of beam can be obtained as,

()K − λM v = o . (2.63)

This mass-spring-rod approximation of beam is frequently used for dynamic analysis. Natural frequencies and mode shapes of the beam can be obtained by solving the eigenvalue problem (2.63). Mass-spring-rod models can also approximate beams having nonlinear distribution of physical properties and they can be used for approximating their dynamic characteristics.

2.2.3 Finite Element Model of Vibrating Rod

The finite element method is the essence of a discretization procedure as it expresses the displacement of a continuous system in terms of finite number of

displacements at the nodal points. Finite element model for n elements has been shown in figure 2.6 that approximates an uniform rod.

u1 u2 u3 un p , q p ,q p ,q 1 1 2 2 n n h L

Figure 2. 6: Finite element model for an uniform axially vibrating rod

Axial displacement in the first element of vibrating system depicted in figure 2.6, can be written in the form (see Meirovitch, 1967),

= + = T u(x,t) L1 (x)u1 (t) L2 (x)u2 (t) L(x) u(t ), (2.64) where, L(x) is the interpolating function, known as shape function and u(t) is the corresponding vector of nodal displacements. By selecting the appropriate shape function,

 x  1−  L(x) =  h  , (2.65)  x     h  elemental mass and stiffness matrices can be obtained as,

   − x  h 1  x x  q h 2 1 m = q ∫  h 1− dx = 1   , (2.66) e 1  x  h h  6 1 2 0    h  and

p h −1 p  1 −1 k = 1  ()−1 1 dx = 1   . (2.67) e 2 ∫  − h 0  1  h  1 1 

After evaluating the elemental matrices the global mass and stiffness matrices M and K can be obtained by using appropriate continuity conditions between the elements as,

 ()+  2 q1 q2 q2  ()+   q2 2 q2 q3 q3  h M =  r r r  , (2.68) 6   ()+  qn−1 2 qn−1 qn qn     qn 2qn  and

 + −  p1 p2 p2  − + −   p2 p2 p3 p3  1 K =  r r r  . (2.69) h   − + −  pn−1 pn−1 pn pn   −   pn pn 

The dynamics of the vibrating rod can then be represented using finite element method by the following differential equation,

MuDD + Ku = o , (2.70) that can be solved by solving the generalized eigenvalue problem

(K − λM)v = o . (2.71)

2.2.4 Finite Element Model of Transversely Vibrating Beam

Similarly, the finite element approximation for transversely vibrating beam as shown in figure 2.7 can be obtained by expressing the bending displacement of any beam element in the following form as,

= + θ + + θ = T v(x) L1 (x)w1 L2 (x) 1 L3 (x)w2 L4 (x) 2 L(x) w , (2.72)

where, L(x) is cubic interpolating function, known as shape function and w is the corresponding vector of nodal displacements.

w w w 1 w2 3 n+1 θ θ θ θ 1 2 3 n+1

p1,q1 p2 ,q2 pn ,qn h L

Figure 2. 7: Finite element model for the uniform transversely vibrating beam

The shape functions are obtained from the Hermite cubic interpolating function as

(see Meirovitch 1967, Logan 1992),

 3x 2 x 3  1− + 2   h 2 h3   2x 2 x 3   x − +  = h h 2 L(x)  2 3  , (2.73)  3x − 2x   h 2 h3   x 2 x 3   − +   h h 2  and the elemental mass and stiffness matrices for beam element can be obtained as,

T  3x 2 x 3  3x 2 x 3  1− + 2 1− + 2   h 2 h3  h 2 h3   2x 2 x 3  2x 2 x 3  h − + − +  x 2  x 2  m (e) = q h∫  h h  h h  dx , (2.74) i 3x 2 2x 3 3x 2 2x 3 0  −  −   h 2 h3  h 2 h3   x 2 x 3  x 2 x 3   − +  − +   h h 2  h h 2  or,

 156 22h 54 −13h   q h 22h 4h 2 13h − 3h 2 m (e) = i   , (2.75) 420  54 13h 156 − 22h   −13h − 3h 2 − 22h 4h 2  and

  T − + x − + x  3 6  3 6   h  h  − + x − + x  h  2 3  2 3  (e) 4ri h h k = ∫    dx , (2.76) h 4 6x 6x 0  3 −  3 −   h  h  x x  −1+ 3  −1+ 3   h  h  or equivalently,

 12 6h −12 6h    r 6h 4h 2 − 6h 2h 2 k (e) = i   . (2.77) h3 −12 − 6h 12 − 6h    6h 2h 2 − 6h 4h 2 

After evaluating the elemental matrices the global mass and stiffness matrices M and K can be assembled for fixed-free configuration by using appropriate continuity conditions between the elements and the corresponding eigenvalues can be evaluated by solving,

K − λM = 0 . (2.78)

2.3 Solution for Inverse Problem of Continuous Systems Using Its Associated Discrete Model

This section deals with parameter estimation of an uniform rod by using its finite difference model. Our aim is to take the finite difference model of an uniform rod from

the previous section and develop an estimation technique for the reconstruction of its physical parameters based on spectral data.

Inverse problem for a finite difference model of order n of uniform rod thus can be defined as follows:

λ λ λ Given: The spectrum for the fixed-free rod 1 , 2 , , n , the spectrum for

n µ µ µ = fixed- fixed rod 1 , 2 , , n−1 and the total mass of the system mT ∑ mi . i=1

m m m Find: The parameter m1 ,m2 , mn and k1 ,k2 , kn or equivalently p1 , p2 , pn

m and q1 ,q2 , qn .

p(x) = α, q(x) = β p ,q p , q p ,q n n 2 2 1 1 L

(a)

k k − k k n n 1 2 1

mn mn−1 m2 m1

(b)

kn kn−1 k2 k1

m m m n n−1 2 (c) Figure 2. 8: Axially vibrating rod for physical parameter estimation (a) Distributed parameter model, (b) finite difference model of order n (c) finite difference model for constrained model of order n-1

The uniform rod as shown in figure 2.8(a) has been chosen for analysis and its physical parameters are to be estimated. Finite difference models of uniform rod for fixed-free and fixed-fixed configurations are shown in figure 2.8(b) and 2.8(c) respectively. These two sets of eigenvalues hold the interlacing property,

< λ < µ < λ < µ < λ < µ m < µ < λ 0 1 1 2 2 3 3 n−1 n , (2.79) for the discrete systems.

The physical meaning of this inequality is interpreted as follows: “If a linear constraint is applied to a system, each natural frequency increases, but does not exceed the next natural frequency of the original system”, (Gladwell, 1986-a).

The reconstruction of the mass-spring system, from its spectral data, has been proposed by several researchers mainly de Boor and Golub (1978), Friedland et al.

(1979), Gladwell (1986-a) and by Ram (1992). The following is a reconstruction procedure based upon eigenvalues where a tridiagonal symmetric matrix A is defined such that

1 1 A = M 2 KM 2 , (2.80) where,

1  1 1 1  M 2 = diag m . (2.81)    m1 m2 mn 

The symmetric tridiagonal matrix

α β  1 1 β α β   1 2 2  A =  β α β  (2.82)  2 3 3   r r r  β α   n−1 n 

− λ λ λ m λ ˆ is equivalent to K M and it has the eigenvalues 1 , 2 , , n . Furthermore, A is formed as

α β  2 2 β α β  Aˆ =  2 3 3  , (2.83)  r r r   β α  n−1 n 

µ µ m µ which has the eigenvalues 1 , 2 , , n . Denoting the spectral decomposition of A by,

AV = VΛ , (2.84)

= where V [vij ]is orthogonal such that,

V T V = I . (2.85)

The jth element of the first row of V can be determined by the eigenvalue-eigenvector relationship (see, Boley and Golub, 1987),

n−1 ()λ − µ ∏ i i 2 = i=1 = m v1 j n , j 1,2, n . (2.86) ()λ − λ ∏ j i i=1 i≠ j

Using the above stated decomposition for each element of the matrix, the mass and the stiffness matrices can be extracted by reconstructing A as,

= Ay e n , (2.87) where,

m T = n ()m y m1 m2 mn , (2.88) kn

and, where en is defined as the n-th unit vector,

= ()m T en 0 0 1 . (2.89)

The vector y is solved by (2.87) after constructing A , so that it gives

m n m m T = n = n T y y 2 ∑ mi 2 , (2.90) k n i=1 kn which implies that by the transformation,

m y T y n = , (2.91) kn mT

= m the masses mi , i 1,2, ,n , can determined by (2.88) and (2.91).

After evaluating M and A , stiffness matrix is found by

1 1 K = M 2 AM 2 (2.92)

= m and hence, ki , i 1,2, ,n can be calculated for the system. Values of pi (axial rigidity)

and qi (mass per unit length) can be obtained from the identified mi and ki by using

(2.34) and (2.35).

In this way, the physical parameters, of the rod, are obtained by using its finite difference approximation. In the following section, numerical results have been presented in order to demonstrate the results of the numerical modeling discussed in this section. It has been shown that a matrix approximation of the continuous system may not be useful for the purpose of system identification.

2.4 Motivation of the Research: Continuous Systems vs. Their Discrete Models

This section demonstrates some numerical examples involving the uniform rod and beam models developed in the previous sections and the results of exact solution with their associated approximate solutions are analyzed.

2.4.1 Comparison of the Spectrum of Fixed-Free Uniform Rod with Its Approximated Model

In section 2.2.1, an analytical solution for the spectrum of the uniform rod is obtained by equation (2.7). The eigenvalues of the uniform rod can also be obtained by finite difference and finite element methods using (2.39) and (2.71) respectively.

Assuming, the model order n = 10 for finite difference and finite element approximation, the associated mass and stiffness matrices M and K are obtained from (2.40)-(2.41) and

(2.68)-(2.69) respectively. By solving the eigenvalue problems (2.39) and (2.71) for model order n = 10 , the spectrum of the rod is approximated.

The lowest 10 eigenvalues of the rod using finite element and finite difference models are compared with the corresponding exact eigenvalues of rod obtained from

χ = λ 2 2 (2.7). The results are expressed in the non-dimensional form i i L c and plotted together in figure 2.9(a). Exact eigenvalues can be expressed in non-dimensional form as,

 ()2i −1 π  2 χ = λ L2 c 2 =   , i = 1,2,m,10 . (2.93) i i  2 

It is apparent from figure 2.9(a) that only a few lower eigenvalues, of the rod, are approximated correctly. In order to improve the approximation, model order is increased to n = 30 and the eigenvalues are compared in figure 2.9(b). These results demonstrate that about n / 3 eigenvalue are approximated accurately. Increasing the model order

 n  improves the accuracy of low eigenvalues but at the same time, n −  inaccurate  3  eigenvalues are also obtained. Thus, the increase in number of elements in the approximated system does not ensure the spectral consistency between the continuous system and its discrete approximation.

χ χ i i 35 120 Exact Solution Exact Solution 30 Fin it e Element method 100 Fin it e Elem ent m ethod Finite difference method Finite difference method 25 80 20 60 15 40 10

5 20 i 0 i 0 12345678910 0 5 10 15 20 25 30

(a) (b)

Figure 2. 9: Comparison of the natural frequency of the rod using finite element, finite difference and exact solution, (a) model order n=10, (b) model order n=30

The higher eigenvalues obtained from finite difference method are lower than the analytical eigenvalues because the mass of element is lumped into the center of element that increases the effective inertia of system. Moreover, the assumption of mass-less springs lowers the stiffness of system leading to approximated eigenvalues that are lower than the exact ones. However, finite element method overestimates the higher eigenvalues. The exact solution of eigenvalue problem is obtained by minimization of the

Rayleigh quotient. In finite element method, a subset of admissible function is used therefore instead of evaluating the exact minimum finite element scheme evaluates those eigenvalues that are higher than the true values. It can thus be concluded from figure

2.9(a) and 2.9(b) that higher eigenvalues, of a continuous system, are not evaluated accurately by using its discrete model.

2.4.2 Comparison of the Mode Shapes of a Fixed-Free Uniform Beam with its Discrete Model

Consider the approximated discrete models of fixed-free uniform beam as shown in section 2.2.2 and 2.2.4. The uniform beam is divided into 10 equal parts for mass- spring-rod model and finite element model respectively. The eigenvectors or mode shapes, for the discrete beam, are evaluated by solving eigenvalue problem (2.63). Mass and stiffness matrices for the mass-spring-rod model, used in (2.63) are obtained from

β o (2.56) and (2.57). The mode shapes are calculated for the corresponding i that are the eigenvalues for the mass-spring rod model of uniform beam. Similarly, by solving (2.78) for finite element model of beam, the mode shapes are evaluated for the corresponding

β ∆ i (eigenvalues of finite element model).

The non-dimensional mode shapes are evaluated for each discrete model and they are compared with the mode shape of the fixed-free uniform beam evaluated from (2.29) as shown in figure 2.10. It is evident from figure 2.10 that mode shape approximation using discrete model is accurate for only lower mode shapes. Moreover, these discrete models cannot capture strong variation of motion associated with higher mode shapes.

Thus, it can be concluded that the higher mode shape cannot be evaluated accurately by discrete mass-spring rod model for the beam or by finite element model of the beam.

Examples, 2.4.1 and 2.4.2, demonstrate that finite dimensional models of rods and beams cannot approximate higher frequencies and mode shapes accurately. These results are also relevant while solving inverse problem of rods and beams, because they suggest that discrete model of these structures try to approximate motion of the entire element by a single point or node. Such an approximation may be good for the lower frequencies or mode shapes but they are meaningless for the high modes.

Analytical Mass-spring-rod model Finite element method

1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1  β = 1.8751  1 st β = (a) Normalized 1 mode shape corresponding to  1 1.7084 β ∆ =1.8751  1

1 1 0.5 0.5 0 0 -0.5 -0.5

-1 0 0.2 0.4 0.6 0.8 1 -1 0 0.2 0.4 0.6 0.8 1 β = 4.6940  2 nd β = (b) Normalized 2 mode shape corresponding to  2 4.2752 β ∆ =  2 4.6943

1 1 0.5 0.5 0 0 -0.5 -0.5 -10 0.2 0.4 0.6 0.8 1 -1 0 0.2 0.4 0.6 0.8 1 β = 29.8451  10 th β = (c) Normalized 10 mode shape corresponding to  10 19.7676 ∆ β = 30.1150 10

Figure 2. 10: Mode-shapes comparison of the uniform beam with the mode-shapes of their finite approximated models

Furthermore, identification of the physical parameter is based on all frequencies of the continuous system. However, inaccurate higher frequencies and mode shapes of

finite dimensional models deteriorate the identification process. Such a phenomenon is demonstrated in the next section.

2.4.3 Parameter Estimation of an Uniform Rod Using Finite Difference Model

Considering an uniform rod as discussed in section 2.3 and substituting the non- dimensional constants α and β to obtain p = EA = α, q = ρA = β . For numerical simplicity values of α = 1 and β = 1 are taken. Substituting these values, the eigenvalues for the fixed free configuration of rod are obtained as,

(2i −1)2π 2 λ = , i = 1,2,3... . (2.94) i 4 and the eigenvalues for fixed-fixed configuration are obtained as,

µ = π 2 = i (i ) . i 1,2,3... . (2.95)

The axial rigidity, pi and the density per unit length, qi of the uniform, rod are to be obtained by using the spectra (2.94) and (2.95). The uniform rod is approximated by finite difference method of model order n = 6 as shown in figures 2.8(b) and 2.8(c).

λ λ m λ µ µ m µ Using two set of spectra 1 , 2 , , 6 , 1 , 2 , , 5 of the uniform rod and the sum of

the total mass of the system, the physical parameters p1 , p2 , p3 ,..., p6 and q1 ,q2 ,q3 ,...,q6 can then be estimated by using (2.80)-(2.92). The mass matrix is estimated as

M = diag{}0.0702 0.1426 0.1498 0.1644 0.1947 0.2782 , (2.96) and stiffness matrix from (2.92) can be obtained as,

 11.4376 −11.4376    −11.4376 22.5232 −11.0856   −11.0856 21.4322 −10.3466  K =   . (2.97)  −10.3466 19.4759 − 9.1294   − 9.1294 16.3367 − 7.2074    − 7.2074 10.9824 

Density per unit length (q) Axial Rigidity ( p) 1.8 3 1.6 Identified Identified

1.4 Theoretical 2.5 Theoretical 1.2 2 1

0.8 1.5

0.6 1 0.4

0.2 0.5 123 456123456

(a)

3 Identified 1.8 Identified

2.5 Theoretical 1.6 Theoretical 1.4 2 1.2 1.5 1 1 0.8

0.5 0.6

0 0.4 0 2 4 6 8 1012141618200 2 4 6 8 101214161820

(b)

Figure 2. 11: Parameter estimation of uniform rod (a) based on six degree-of-freedom finite difference model (b) based on twenty degree-of-freedom model

The finite difference model leads to a simply connected mass-spring system

model where the individual stiffness ki can then easily be determined as,

= () ki 11.4376 11.0856 10.3466 9.1294 7.2074 3.7753 (2.98)

Based on these values of spring elements ki and the mass elements mi , and the values of

pi and qi can be identified using (2.34) and (2.35). These values of pi and qi are

= = = m compared with theoretical values of pi qi 1, for i 1,2,3, ,6 in figure 2.11(a). The model order is increased to n = 20 and the physical parameters are estimated again and compared with theoretical results as shown in figure 2.11(b). These results clearly indicate that the identification is inaccurate.

It is clear from the examples of uniform rod that the approximation, based upon finite difference scheme is inaccurate, which is true for both direct as well as inverse problems associated with the rod. By inspection of figures (2.9) and (2.11), it can be concluded that no improvements in the results are achieved by increasing the model order. The discrepancies that are presented in approximated models of continuous systems are the main motivation of our work presented here.

In conclusion, it is observed that the behavior of discrete model of vibrating system leads to an algebraic eigenvalue problem where as the associated continuous model leads to a transcendental eigenvalue problem. Spectral characteristics of the continuous systems are completely different from their corresponding discrete model.

Hence, this leads to inaccurate approximation of higher eigenvalues, higher mode shapes and physical parameters.

Boley and Golub (1987) have concluded that “There are many similarities between the matrix problems and the continuous problems, but numerical evidence

demonstrates that the solution to the matrix problem is not a good approximation to the continuous one”. This phenomenon has been demonstrated via the examples in the previous sections. Paine, Hoog and Anderson (1981) also noted this discrepancy in terms of the asymptotic behavior of the continuous system. Therefore, a new mathematical model is required that can realistically model the behavior of continuous vibrating systems accurately while solving associated direct and inverse problems.

3. TRANSCENDENTAL EIGENVALUE METHODS IN CLASSICAL DIRECT PROBLEMS

3.1 Introduction

The limitations of determining analytical solutions for distributed parameter vibratory systems are discussed in chapter 2. It has been concluded that finite difference or finite element methods are not capable of accurate parameter estimation for continuous systems. In order to overcome the difficulties illustrated in previous chapter, a new approximation method, for the continuous systems, is introduced. This method leads to a transcendental eigenvalue problem. The distributed parameter model, which depends on the continuously variable parameters, is approximated by a continuous model with finite number of physical parameters of known distribution.

This chapter introduces such formulations and develops a rapidly converging numerical algorithm for evaluating eigenvalues of the transcendental eigenvalue problems (Singh and Ram, 2002). The algorithm is also used to evaluate eigenvalues of the quadratic eigenvalue problems. This chapter includes certain classical problems in vibration and buckling to demonstrate the effectiveness and practicality of the proposed method and algorithm.

3.2 Transcendental Eigenvalue Problem

The transcendental eigenvalue problem is defined as the problem of determining the non-trivial solutions ω and z ≠ o of

A()ω z = o , (3.1)

where the elements of n × n matrix A , are transcendental functions in ω , and the vector z is a constant eigenvector.

3.2.1 Free Vibration of Non-uniform Rods

Consider a non-uniform axially vibrating rod of length L , axial rigidity

p(x) = E(x)A(x) , and mass per unit length q(x) = ρ(x)A(x) , which is fixed at x = 0 and free to oscillate at x = L , as shown in figure 3.1. The axial motion of the rod is governed by the following differential equation

∂  ∂u()x,t  ∂ 2u()x,t  p(x)  = q(x) , 0 < x < L , t > 0 , (3.2) ∂x  ∂x  ∂t 2 and boundary conditions

∂u(L,t) u(0,t) = 0, = 0 . (3.3) ∂x

Assuming a harmonic motion

u(x,t) = v(x)sinωt , (3.4) separates (3.2) and (3.3) to the following eigenvalue problem,

( pv′)′ + λqv = 0, 0 < x < L, (3.5)

v(0) = v′(L) = 0, (3.6) where λ = ω 2 and prime denotes the derivatives with respect to x .

The non-uniform rod, as shown in figure 3.1, can also be represented by dividing

m the rod into n uniform continuous rods of constant physical parameters p1 , p2 , , pn

m and q1 ,q2 , ,qn as shown in figure 3.2. The length of each uniform rod can be represented as,

= Li xi . (3.7)

p(x), q(x)

x dx

u(x,t)

Figure 3. 1: Non-uniform axially vibrating rod

Thus (3.2) and (3.3) can be expressed as

 ∂ 2u (x,t) ∂ 2u (x,t) 1 = 1 < < >  p1 2 q1 2 , 0 x L1 , t 0  ∂x ∂t  ∂ 2u (x,t) ∂ 2u (x,t) p 2 = q 2 , L < x < L , t > 0  2 ∂x 2 2 ∂t 2 1 2  o , (3.8)   o  ∂ 2 ∂ 2 un (x,t) un (x,t)  p = q , L − < x < L, t > 0  n ∂x 2 n ∂t 2 n 1 and,

∂u (L,t) u (0,t) = 0, n = 0 . (3.9) 1 ∂x

m m m p1 p2 p3 pn−1 pn m m m q1 q2 q3 qn−1 qn

x1 x2 x3 un−2 u3 u − u2 n 1 u1 x − n 1 un xn

Figure 3. 2: Non-uniform rod approximation using piecewise uniform rod of model order n with constant physical properties

The matching conditions representing the continuity in displacement and continuity in

= m axial force at x x1 , x2 , x3 , , xn−1 , can be written as,

 ∂u (L ,t) ∂u (L ,t) u (L ,t) = u (L ,t) p 1 1 = p 2 1  1 1 2 1 1 ∂ 2 ∂  x x ∂u (L ,t) ∂u (L ,t)  u (L ,t) = u (L ,t) p 2 2 = p 3 2  2 2 3 2 2 ∂x 3 ∂x . (3.10)  o o  ∂ ∂ un−1 (Ln−1 ,t) un (Ln−1 ,t) u − (L − ,t) = u (L − ,t) p − = p  n 1 n 1 n n 1 n 1 ∂x n ∂x

By assuming harmonic motion of the following form

u (x,t) = v (x)sinωt  1 1 u (x,t) = v (x)sinωt  2 2 , (3.11)  o  = ω un (x,t) vn (x)sin t equations (3.9) through (3.10) yield a set of ordinary differential equations that can be written as,

 λ ′′ + = < <  v1 2 v1 0, 0 x L1  c1  λ  v′′ + v = 0, L < x < L  2 2 2 1 2 , (3.12) c2  o  λ v′′ + v = 0, L < x < L  n 2 n n−1  cn

λ = ω 2 = = m where , ci pi qi , i 1,2, n , with the following boundary conditions

= ′ = v1 (0,t) 0,vn 0, (3.13) and matching condition

 v (L ) = v (L ) p v′(L ) = p v′ (L )  1 1 2 1 1 1 1 2 2 1  v (L ) = v (L ) p v′ (L ) = p v′ (L )  2 2 3 2 2 2 2 3 3 2 . (3.14)  o o  = ′ = ′ vn−1 (Ln−1 ) vn (Ln−1 ) pn−1vn (Ln−1 ) pn vn (Ln−1 )

The general solution for the system (3.12) through (3.14) is given as,

 λ λ = + < <  v1 (x) Q1 sin x R1 cos x, 0 x L1  c1 c1  λ λ  v (x) = Q sin x + R cos x, L < x < L  2 2 2 1 2 . (3.15) c2 c2  o  λ λ v (x) = Q sin x + R cos x, L < x < L  n n n n−1  cn cn

Imposing the conditions (3.13) and (3.14) on (3.15), the resulting equations can be rearranged to yield a transcendental eigenvalue form equation (3.1),

A()ω z = o . (3.16)

ω ()ω The eigenvalues i , which make the matrix A i singular, are to be determined. The matrix A is a transcendental matrix and is defined as;

− − − − − −pT − − − − − − −    − − − − − −U − − − − − − −  A()ω = , (3.17)  − − − − − −V − − − − − − −    − − − − − −qT − − − − − − − where,

p = ()0 1 0 0 T , (3.18)

α β −α − β  1 1 1 1 0 0 0  α β −α − β   0 0 2 2 2 2 0  U =   , (3.19)   α β −α − β  0 n−2 n−2 n−2 n−2 0 0   α β −α − β   n−1 n−1 n−1 n−1 

γ −σ − γ σ  11 11 12 12  γ −σ − γ σ   0 0 22 22 23 23  V =   , (3.20)   γ − σ − γ σ  n−2,n−2 n−2,n−2 n−2,n−1 n−2,n−1   γ −σ − γ σ   n−1,n−1 n−1,n−1 n−1,n n−1,n 

 λ λ λ λ T   q,= 0 0 cos L − sin L (3.21)  cn cn cn cn 

λ α = i sin Li , (3.22) ci

λ β = i cos Li , (3.23) ci

λ λ γ = ij p j cos Li , (3.24) c j c j

λ λ σ = = − = ij p j sin Li , i 1,2, ,n 1, j 1,2, ,n , (3.25) c j c j and

= () T z Q1 R1 Q2 R2 Qn Rn . (3.26)

ω Once eigenvalues i are determined, the corresponding eigenfunctions

v (x,ω ) 0 < x < L  1 i 1 v (x,ω ) L < x < L φ = 2 i 1 2 i (x)  (3.27)   ω < < vn (x, i ) Ln−1 x Ln can be obtained by solving the linear system (3.16) for z and substituting the values of

= Qi , Ri , i 1,2,3,....,n in (3.15).

3.2.2 Free Vibration of Non-uniform Beams

By using the approximation method described in the previous section, a mathematical model associated with the non-uniform vibrating beams can be developed.

Consider a transversely vibrating non-uniform Bernoulli-Euler beam, as shown in figure

3.3(a), that is approximated by another non-uniform beam with piecewise constant

physical properties as shown in figure 3.3(b). The differential equation of motion for the transversely vibrating non-uniform beam as shown in 3.3(a) is governed by

∂ 2  ∂ 2 w(x,t)  ∂ 2 w(x,t)  EI(x)  + ρA(x) = 0 , (3.28) ∂x 2  ∂x 2  ∂t 2 where, E(x)I(x) is known as the flexural rigidity of beam and ρ(x)A(x) is the mass per unit length of beam.

E(x), A(x), ρ(x), I(x)

x dx w(x,t)

(a)

c c c c 1 2 n−1 n

x1 x2 x3

w1 w2 w3 wn−2 wn−1 wn L

(b) Figure 3. 3: (a) Non-uniform beam (b) approximated model of order n

This beam is then approximated by another beam of order n, such that the modulus of elasticity, density, area and moment of inertia remains constant for each element. The differential equation for such approximated beam can be written as,

 ∂ 4 w (x,t) ∂ 2 w (x,t) 1 + ρ 1 = < <  E1I1 4 1 A1 2 0, 0 x x1  ∂x ∂t  ∂ 4 w (x,t) ∂ 2 w (x,t) E I 2 + ρ A 2 = 0, x < x < x  2 2 ∂x 4 2 2 ∂t 2 1 2 , (3.29)   ∂ 4 w (x,t) ∂ 2 w (x,t) E I n + ρ A n = 0, x < x < L  n n ∂x 4 n n ∂t 2 m where, m is defined as model order (n-1).

The fixed-free boundary condition as shown in figure 3.3(b) represents no displacement and slope at the fixed end and no bending moment and force at the free end, and they can be expressed as,

 ∂w (x,t) w (x,t) = 0, 1 = 0  1 x=0 ∂x  x=0 . (3.30) ∂ 2 w (x,t) ∂ 3 w (x,t) E I n = 0, E I n = 0  n n ∂x 2 n n ∂x 3  x=L x=L

The four matching conditions, representing the continuity of displacement, slope, bending moment and force between the elements, can be expressed as,

 ∂w (x ,t) ∂w (x ,t) w (x ,t) = w (x ,t) 1 1 = 2 1  1 1 2 1 ∂ ∂  x x ∂w (x ,t) ∂w (x ,t)  w (x ,t) = w (x ,t) 2 2 = 3 2  2 2 3 2 ∂x ∂x  o o  ∂w (x ,t) ∂w (x ,t) = m m = n m  wm(xm ,t) wn(xm ,t)  ∂x ∂x  ∂ 2 w (x ,t) ∂ 2 w (x ,t) ∂ 3 w (x ,t) ∂ 3 w (x ,t) . (3.31)  E I 1 1 = E I 2 1 E I 1 1 = E I 2 1  1 1 ∂x 2 2 2 ∂x 2 1 1 ∂x3 2 2 ∂x3  ∂ 2 w (x ,t) ∂ 2 w (x ,t) ∂ 3 w (x ,t) ∂ 3 w (x ,t)  E I 2 2 = E I 3 2 E I 2 2 = E I 3 2 2 2 ∂x 2 3 3 ∂x 2 2 2 ∂x3 3 3 ∂x3  o o  ∂ 2 w (x ,t) ∂ 2 w (x ,t) ∂ 3 w (x ,t) ∂ 3 w (x ,t) E I m m = E I n m E I m m = E I n m  m m ∂x 2 n n ∂x 2 m m ∂x3 n n ∂x3

Assuming harmonic motion of the form,

 w (x,t) = v (x)sinωt  1 1 w (x,t) = v (x)sinωt  2 2 , (3.32)  o  = ω wn (x,t) vn (x)sin t and substituting (3.32) in (3.29) and (3.31), following eigenvalue problem for a beam can be derived,

 β 4 E I v'''' − v = 0 where, β 4 = ω 2 , c 4 = 1 1 , 0 < x < x 1 4 1 1 ρ 1  c1 1 A1  β 4 E I v '''' − v = 0 where, β 4 = ω 2 , c 4 = 2 2 , x < x < x  2 4 2 2 ρ 1 2 , (3.33) c2 2 A2   β 4 E I v'''' − v = 0 where, β 4 = ω 2 , c 4 = n n , x < x < L  n 4 n n ρ m  cn n An with the following boundary conditions,

v (0) = 0, v′(0) = 0  1 1 , (3.34) ′′ = ′′′ = vn (L) 0, vn (L) 0 and matching conditions,

 = ′ = ′ v1 (x1 ) v2 (x1 ) v1 (x1 ) v2 (x1 )  = ′ = ′  v2 (x2 ) v3 (x2 ) v2 (x2 ) v3 (x2 )  o o   v (x ) = v (x ) v′ (x ) = v′ (x )  m m n m n m n m . (3.35) ′′ = ′′ ′′′ = ′′′  E1I1v1 (x1 ) E2 I 2v2 (x1 ) E1I1v1 (x1 ) E2 I 2v2 (x1 )  E I v′′(x ) = E I v′′(x ) E I v′′′(x ) = E I v′′′(x )  2 2 2 2 3 3 3 2 2 2 2 2 3 3 3 2  o o  ′′ = ′′ ′′′ = ′′′ Em I mvm (xm ) En I n vn (xm ) Em I mvm (xm ) En I nvn (xm )

The solution of (3.33) can be written as,

 β β β β = + + +  v1 (x) z11 sin x z12 cos x z13 sinh x z14 cosh x  c1 c1 c1 c1 β β β β v (x) = z sin x + z cos x + z sinh x + z cosh x  2 21 22 23 24 . (3.36) c2 c2 c2 c2   β β β β  v (x) = z sin x + z cos x + z sinh x + z cosh  n n1 n2 n3 n4  cn cn cn cn

Applying the boundary and matching conditions from (3.36) in the solution (3.35) leads to the formulation of transcendental eigenvalue form of (3.16) where,

− − − − − − − − − − − − − − − − A L − − − − − − − − − − − − − − − −  A d  − − − − − − − − A − − − − − − − − A =  s  , (3.37) − − − − − − − − − − − − − − − −  A m  − − − − − − − − A − − − − − − − −  f  − − − − − − − − − − − − − − − −  A R  and constant vector,

= () T z z11 z12 z13 z14 z21 z22 zn4 . (3.38)

The matrix A is of size ()4n × 4n and it contains the matrices associated with the left

()× boundary conditions, A L of size 2 4n , the continuity of displacement, A d of size

()× ()× m 4n , the continuity of slope, A s of size m 4n , the continuity of moment, A m of

()× ()× size m 4n , the continuity of force, A f of size m 4n , and the right boundary

()× conditions A R of size 2 4n .

These matrices are defined as follows:

 0 1 0 1 0 0 0 0 0 0 0 0 0 A =  1 1  , (3.39) L  0 0 0 0 0 0 0 0 0 0 0 c1 c1 

α η δ γ −αˆ −ηˆ − δˆ − γˆ 0 0 0 0 0   1 1 1 1 1 1 1 1   0 0 0 0 α η δ γ −αˆ 0 0 0 0  A = 2 2 2 2 2 , (3.40) d     −α −η − δˆ − γ  0 0 0 0 0 0 0 0 0 ˆ m ˆm m ˆm 

η α γ δ rηˆ rαˆ r γˆ r δˆ   1 − 1 1 1 − 1 1 1 1 − 1 1 − 1 1 0 0 0   c1 c1 c1 c1 c2 c2 c2 c2   η α γ δ r ηˆ  0 0 0 0 2 − 2 2 2 − 2 2 0 0 A =   , (3.41) s  c2 c2 c2 c2 c3     r γˆ r δˆ   0 0 0 0 0 0 0 0 0 − m m − m m   cn cn 

 α η δ γ r αˆ rηˆ r δˆ r γˆ  − 1 − 1 1 1 1 1 1 1 − 1 1 − 1 1  2 2 2 2 2 2 2 2 0 0 0  c1 c1 c1 c1 c2 c2 c2 c2   α η δ γ rαˆ   0 0 0 0 − 2 − 2 2 2 1 2 0 0  A = 2 2 2 2 2 , (3.42) m  c2 c2 c2 c2 c3     r δˆ r γˆ   − m m − m m  0 0 0 0 0 0 0 0 0 2 2  cn cn 

 η α γ δ rηˆ rαˆ r γˆ r δˆ  − 1 1 1 1 1 1 − 1 1 − 1 1 − 1 1  3 3 3 3 3 3 3 3 0 0 0  c1 c1 c1 c1 c2 c2 c2 c2   η α γ δ r αˆ   0 0 0 0 − 2 2 2 2 2 2 0 0  A = 3 3 3 3 2 , (3.43) f  c2 c2 c2 c2 c3     r γˆ r δˆ   − m m − m m  0 0 0 0 0 0 0 0 0 3 3  cn cn  and,

 βL βL βL βL   sin cos sinh cosh  c c c c 0 0 0 0 0 0 0 0 0 − n − n n n   c 2 c 2 c 2 c 2  A = n n n n , (3.44) R  βL βL βL βL   cos sin cosh sinh   c c c c  0 0 0 0 0 0 0 0 0 − n n n n  3 3 3 3   cn cn cn cn  where the elements of these matrices are defined as,

E I = i+1 i+1 ri , (3.45) Ei I i and,

  βx   βx   βx   βx  α =  i  η =  i  δ =  i  γ =  i   i sin  i sin  i sin  i sin    c   c   c   c   i i i i , (3.46)  βx   βx   βx   βx  αˆ =  i  ηˆ =  i  δˆ =  i  γˆ =  i   i sin  i sin  i sin  i sin    ci+1   ci+1   ci+1   ci+1  for i = 1,2,3,,m .

By analyzing the behavior of transcendental eigenvalue problems developed for non-uniform rods and beams, it is found that the elements of the matrices associated with such problems are not algebraic in nature. In order to find the non-trivial solution of such eigenvalue problems it is necessary to evaluate the determinant of matrices containing transcendental elements. The main difficulty here is to evaluate the determinant of these matrices analytically. In theory, the determinant of transcendental matrices can be expanded by fundamental definitions but such expansion involves (n!) floating-point operations for a matrix of dimension n × n . Thus, a relatively small problem of dimension (n>13) may require the computations of almost billions of floating point operations, which may hamper calculations even if, faster computers are used. In addition, determinant expansion using Gaussian elimination (see Lake and Mikulas,

1991) requires symbolic manipulations. In order to overcome these difficulties, a new numerical method is required to solve the transcendental eigenvalue problems that compute desired eigenvalues without the evaluating determinants of these transcendental matrices by circumventing any symbolic manipulations.

In order to solve the transcendental eigenvalue problem (3.16) associated with the non-uniform rods and beams, a rapidly converging algorithm is proposed here. This algorithm is based on Newton’s eigenvalue iteration method and it is developed in the following section.

3.3 An Algorithm for the Solution of Transcendental Eigenvalue Problems: Newton’s Eigenvalue Iteration Method

Using the Taylor’s series expansion (Yang 1983), the matrix A()ω can be expanded in the neighborhood of ω (0) using

() ω = ω 0 + ε , (3.47) as

dA()ω (0) A()()ω (0) + ε = A ω (0) + ε + O(ε 2 ) , (3.48) dω where O(ε 2 ) represents the higher order terms.

Let,

dA()ω B()ω = − . (3.49) dω

Neglecting the second and higher order terms in (3.48) gives

() () () A(ω 0 + ε ) = A(ω 0 )− εB(ω 0 ). (3.50)

Now the determinant of (3.50) gives

() () () det(A(ω 0 + ε )) = det(A(ω 0 )− εB(ω 0 )). (3.51)

Thus, the non-trivial solution of (3.16) can be obtained by

() det(A(ω 0 + ε )) = 0 (3.52) which can be substituted into (3.51) giving

() () det(A(ω 0 )− εB(ω 0 )) = 0 . (3.53)

It thus follows from (3.16) that the above equation can be written as

() () (A(ω 0 )− εB(ω 0 ))z = 0 . (3.54)

Note that in (3.54), the value of ε can be determined by solving an equivalent algebraic eigenvalue problem (3.54).

An iterative procedure has been adopted with an initial guess in the neighborhood

() of ω 0 , and the iterative value for k-th iteration is determined by,

()− ω (k ) = ω k 1 + ε (k ) (3.55) where ε (k ) is an eigenvalue of

()− ()− () (A(ω k 1 )− λB(ω k 1 ))z k = 0 . (3.56)

λ = Note that the solution of (3.56) gives n eigenvalues i , i 1,2,...,n , where n is the dimension of the system. By the definition of Taylor series approximation (3.48), it is

() only valid for small ε , thus ε k is chosen as the smallest eigenvalue of (3.56) in an absolute value sense. For n = 1, equation (3.56) reduces to

()ω ()k −1 λ = − f ()− , (3.57) f ′()ω k 1 where, f ()ω = A ()ω and − f ′()ω = B ()ω ,

Thus, the new value for ω is determined by (3.47) with ε = λ . It follows that the

Newton’s eigenvalue iteration method for evaluating the roots of a function is a degenerated case of the eigenvalue extraction algorithm presented here.

3.3.1 Algorithm 1

Suppose that for a given A()ω , the non-trivial solution of (3.16) is desired. The algorithm for solving such transcendental eigenvalue problems is termed as Newton’s eigenvalue iteration method, and can be summarized as follows.

INPUT

(a) The matrix A()ω and matrix B()ω as defined in (3.49).

(b) The initial estimate of ω (0) .

ITERATION

(a) Choose the initial guess ω (0) and start the iteration.

(b) Compute the eigenvalues of system (3.56) based upon the initial guess ω (0) .

(d) Compute the new estimate ω (1) = ω (0) + ε .

(e) Compute the matrices A()ω and B()ω by substituting ω = ω (1) .

(f) Repeat steps (b)-(e) for kth iteration until the condition ε < δ is satisfied.

(g) Stop the iteration.

ω (h) Store the value of k .

ω ω (0) (i) Repeat steps (a)-(h) to evaluate another k for different starting value .

OUTPUT

This method produces the solution vector ω , which is the non-trivial solution of

eigenvalue problem (3.16) that will make the matrix A()ω singular.

The above stated algorithm has been tested for the problems involving free vibration of non-uniform rods, beams, exponential rods and buckling of column. These examples are presented in the next section to demonstrate the effectiveness of the proposed algorithm.

3.4 Applications of the Transcendental Eigenvalue Problem

3.4.1 Example: Spectrum of Piecewise Continuous Rod

Consider a piecewise uniform axial vibrating rod, as shown in figure 3.3, with axial rigidity

α, 0 < x < 0.5, p()x =  (3.58) β, 0.5 < x < 1, and mass per unit length

γ , 0 < x < 0.5, q(x) =  (3.59) δ , 0.5 < x < 1, where α , β , γ , and δ are constants. Infinitesimal axial oscillations, of this rod, are determined by the Sturm-Liouville system

αu′′ + λγu = 0, 0 < x < 0.5  ′′ βv + λδv = 0, 0.5 < x < 1 ku()0 = αu′ ()0  , (3.60) u()0.5 = v ()0.5 αu′()0.5 = βv′ ()0.5  v′()1 = 0, where primes denote derivatives with respect to x and λ = ω 2 denote the eigenvalues of system (3.60).

x x = 0.5 x =1

Figure 3. 4: Piecewise axially vibrating rod

Applying two boundary and two matching conditions in (3.60), transcendental matrix of the form (3.16) is obtained. Two boundary conditions of the configuration are a spring force that is proportional to the displacement at x = 0, and stress free end at

x = 1. Matching conditions in (3.60) represent continuity in displacement and axial force at x = 0.5. Using the above stated four conditions the solution of the (3.60) can be written as follows,

 αω − k 0 0  Q  0  1     ()ω ()ω − ()ω − ()ω   sin 2 cos 2 sin 2 cos 2  R1  0   =   , (3.61) α cos()ω 2 −α sin()ω 2 − β cos()ω 2 β sin()ω 2  Q 0   2    ω − ω      0 0 cos sin  R2  0 which is the same as developed in (3.16) through (3.26).

ω ()ω The eigenvalues i are determined, which make matrix A i singular and after calculating the eigenvalues, corresponding eigenfunctions

v ()x,ω , 0 < x < 0.5 φ ()x =  1 i (3.62) i ()ω < < v2 x, i , 0.5 x 1 is calculated from (3.27).

Let the value of k = 300, α = 2 and β = 1 is substituted in (3.61) to obtain

 2ω − 300 0 0    sin()ω 2 cos()ω 2 − sin()ω 2 − cos()ω 2 A(ω) =   . (3.63) 2cos()ω 2 − 2sin()ω 2 − cos()ω 2 sin()ω 2     0 0 cosω − sinω 

ω ω The proposed algorithm is applied to find the values of i by evaluating matrix B( ) by using (3.49) as follows.

 − 2 0 0 0    − 0.5cos()ω 2 0.5sin()ω 2 0.5cos()ω 2 − 0.5sin()ω 2 B()ω =   . (3.64)  sin()ω 2 cos()ω 2 − 0.5sin()ω 2 − 0.5cos()ω 2     0 0 sinω cosω 

This problem is solved by algorithm presented in previous section with various initial values and the lowest five natural frequencies of rod have been tabulated with their initial guess in table 3.1. Initial guess, number of iterations applied, and calculated natural frequencies (with precision tolerance of 10−12 ) are shown in table 3.1.

Table 3.1: Five lower natural frequencies of the piecewise continuous rod

ω ()0 ω i i Number of iterations i 1 0.1 14 1.89795 2 3.2 10 4.34375 3 6.4 13 8.13909 4 9.6 6 10.58616 5 13.8 9 14.37982

3.4.2 Example: Spectrum Comparison of Exponential Rod

The main application of the transcendental eigenvalue problem is in estimating the spectrum of continuously varying distributed parameter systems. Consider an example of an axially vibrating rod, as shown in the figure 3.5 (a), of unit length with cross sectional area A(x) = e x , modulus of elasticity E , and density ρ , which is fixed at

x = 0 and free to oscillate at x = 1.

1 2 3 n −1 n

h h

x1 x 2 x 3 xn−1 xn

x x = 0.5 x =1 (a)

1 2 3 n −1 n

h h

x1 x2 x3 xn−1 xn

x x = 0.5 x =1

(b) Figure 3. 5: An axially vibrating exponential rod: (a) discrete model of order n, (b) approximation by finite number of rods of constant physical properties

The system is governed by the following differential equation

 ∂  ∂u()x,t  ∂ 2u()x,t   Ee x  = ρe x , 0 < x < 1,t > 0 ∂x  ∂x  ∂t 2 , (3.65)  u(0,1) = 0,u′(1,t) = 0 which leads to the following eigenvalue problem

ρ  2 v′′ + v′ + λv = 0, λ = ω  E v()0 = 0 . (3.66)  ′()= v 1 0, 

Eigenvalues of the system (3.66), (Ram 1994-a), are the roots of the frequency equation

1 1 tan λ − = 2 λ − . (3.67) 4 4

70 Exact Solution 60 Finite element method 50 Finite difference method 40

30 η 20

0 0 2 4 6 8 101214161820 i (a)

60 Exact Solution Transcendental Eigenvalue 50

η 20

0 0 2 4 6 8 101214161820 i (b)

Figure 3. 6: Comparison between exact natural frequencies of exponential rod and (a) approximate solution obtained with finite difference and finite element model of order n=20, (b) approximated solution obtained with the transcendental eigenvalue model of order n=3

In order to demonstrate the effectiveness of the algorithm, given continuous system is replaced by an equivalent discrete model of order n with uniform element length h = 1/ n as shown in figure 3.5(b). The eigenvalue problem corresponding to the discrete model of the rod is obtained by (2.39). Finite element modeling, using piecewise

constant elements with linear shape function, leads similarly to the generalized eigenvalue problem (2.71) with the same stiffness matrix as given by (2.69), and mass matrix of the form (2.68). Twenty eigenvalues of exponential rod have been estimated using finite difference and finite elements models of order n = 20 . The results are shown in figure 3.6(a). It is apparent from figure 3.6(a) that only about six eigenvalues have been estimated accurately by these methods.

This problem is solved by using a small model order of n = 3 , where the rod is divided into three uniform portions of cross-sectional areas

 ()e1/ 3 + e0 2 i = 1  = ()2 / 3 + 1/ 3 = Ai  e e 2 i 2 (3.68)  1 2 / 3  ()e + e 2 i = 3.

()ω = = = The matrix A in (3.17) with L1 1/ 3 , L2 2 / 3 and L 1, takes the form

0 1 0 0 0 0    sinξ cosξ − sinξ − cosξ 0 0  0 0 sinη cosη − sinη − cosη  A()ω =   , (3.69) ξ − ξ − ξ ξ A1 cos A1 sin A2 cos A2 sin 0 0  0 0 A cosη − A sinη − A cosη A sinη  2 2 3 3  0 0 0 0 cosω − sinω  where ξ = λ 3 and η = 2 λ 3

The twenty lowest roots of det()A()ω = 0 are determined by using the method introduced in section 3.3. The results are shown in figure 3.6(b) together with the exact analytical solution. It is clearly demonstrated that the transcendental model has produced good approximation for all the eigenvalues obtained. Note that these results have been obtained using a matrix of order 6× 6, whereas inferior results are obtained by finite difference and finite element models using matrices of higher dimensions 20 × 20 . Also

note that a small order transcendental model (3.69) can produce unlimited number of eigenvalues, while finite elements and finite difference models of order n can at most produce n eigenvalues, of which about two-third of the eigenvalues are inaccurate.

3.4.3 Example: Comparison of the Tapered Beam Eigenvalues

Consider a tapered beam with fixed-free configuration as shown in figure 3.7(a).

The height of beam varies linearly along the direction of length. Let the height of beam

be hL at the left end and hR at the right end. Thus, the height distribution of beam along the length can be expressed by the following linear function,

 h − h  h(x) = h − L R x . (3.70)  L L 

Let, the thickness be b , Young’s modulus of elasticity be E and density be ρ then the mass per unit length can be expressed as,

 h − h  ρA(x) = ρbh(x) = ρbh − L R x , (3.71)  L L  and flexural rigidity of the beam can be written as,

3 bh(x)3 Eb  h − h  EI(x) = E = h − L R x , (3.72) 12 12  L L  where, A(x) is the cross sectional variation and I(x) is the moment of inertia variation along the length of the beam.

Associated finite element model for the tapered beam using PLANE82 element in

ANSYS is shown in figure 3.7(b). Tapered beam can also be replaced by an equivalent approximated model of order n with piecewise constant physical properties within the elemental length h = 1/ n as shown in figure 3.7(c).

Side View Direction of vibration

hL hR

Plan View b L

(a)

(b)

c1 c2 c3 cn

(c)

Figure 3. 7: (a) Fixed-free configuration of the tapered beam (b) representative finite element model using PLANE82 element in ANSYS, (c) piecewise constant beam of model order n.

= = = = α ρ = γ Let, L 1, hL 0.1, b 0.05, E and . The lower three natural

frequencies (in Hz.) are evaluated by finite element method, f fem , using ANSYS with

PLANE82, SHELL63 elements and by solving transcendental eigenvalue problem associated with non-uniform beam of form (3.16) using (3.37) for model order n=2 and 8.

Table 3.2: Comparison of eigenvalues of fixed-free configuration of the tapered beam for non-dimensional values α = γ = 1

First three non-dimensional eigenvalues

Blevins FEM FEM Transcendental (1979) SHEL63 PLANE82 eigenvalue problem with λ λ λ λ ref fem fem Model order ( trans ) n=2 n=8 h 1.87 1.8691 1.8685 1.8751 1.8751 L = 1 h 4.69 4.6940 4.5927 4.6941 4.6941 R 7.85 7.8546 7.4845 7.8548 7.8548

h 1.95 1.9500 1.9500 1.8903 1.9510 L = 2 h 4.27 4.2257 4.2241 4.1245 4.2690 R 6.90 6.6821 6.6778 6.9417 6.8575

h 2.00 2.0007 2.0007 1.9195 2.0004 L = 3 h 4.10 4.0704 4.0690 3.9366 4.0977 R 6.45 6.3181 6.3146 6.7002 6.4381

h 2.05 2.0373 1.9419 1.9419 2.0359 L = 4 h 4.00 3.9858 3.8569 3.8569 4.0041 R 6.20 6.1073 6.6011 6.6011 6.1977

The natural frequencies for the tapered beam are using ANSYS are,

1 2 λ ()EI ρA 2 f = fem L L , (3.73) fem 2πL2 where,

bh3 A = bh and I = L . (3.74) L L L 12

λ The true eigenvalues of this tapered beam ref , (Blevins, pp. 163, fig 1.8, 1979) are then compared in the table 3.2 with the eigenvalues obtained by higher dimension finite

λ element method using ANSYS, fem using (3.73) and with the eigenvalues obtained by

λ solving transcendental eigenvalue problem (3.16) of model order n=2 and 8, trans . The natural frequencies obtained from transcendental eigenvalue problem can be defined as,

β 2 1 f = ()EI ρA 2 , (3.75) trans 2πL2 L L

λ = β where, trans i are the eigenvalues from (3.37).

β Similarly, the eigenvalues i of the fixed-pinned tapered beam can be obtained by modifying the right boundary condition of (3.37) as,

 βL βL βL βL  0 0 0 0 0 0 0 m 0 0 sin cos sinh cosh   cn cn cn cn  A =  βL βL βL βL  , (3.76) R sin cos sinh cosh  c c c c  m − n − n n n 0 0 0 0 0 0 0 0 0 2 2 2 2   cn cn cn cn  and solving the transcendental eigenvalue problem of the form (3.16) by using algorithm

1 for model order n=2,4 and 8.

Table 3.3: Comparison of eigenvalues for fixed-pinned tapered beam for non- dimensional values α = γ = 1

Lower three eigenvalues Melnikov Transcendental eigenvalue problem with Model (1999) order n=2 n=4 n=8 h 3.9266 3.926602 3.926602 3.926602 L = 1 h 7.0685 7.068582 7.068582 7.068582 R 10.2101 10.210170 10.210176 10.210176 h 3.5741 3.634720 3.593077 3.556341 L = 2 h 6.1231 6.597649 6.273425 6.211665 R 8.8042 9.348211 8.997746 8.901484 h 3.2867 3.634720 3.593077 3.383601 L = 3 h 5.6082 6.597649 6.273425 5.826686 R 8.1227 9.348211 8.997746 8.318701 h 3.1598 3.506586 3.345659 3.278970 L = 4 h 5.3259 6.320536 5.734400 5.599225 R 7.7393 8.966614 8.272348 7.962290

The lower three eigenvalues are then compared with analytical

( )= λ = eigenvalues pk k , for k 1,2,3 , (see, Melnikov, pp. 121, table 3.4, 1999). These results are tabulated here in the table 3.3.

It is evident from tables 3.2 and 3.3 that the eigenvalues of the non-uniform beam can be obtained with fair accuracy by solving low order transcendental eigenvalue problem.

Note that results shown in tables 3.2 and 3.3 are in agreement with results obtained from finite element model of higher dimension and with analytical results (Blevins, 1979 and

Melnikov, 1999).

3.4.4 Example: Buckling of a Column with Composite Loads

Consider the problem of finding the critical load P applied to an uniform column of unit length L = 1 and flexural rigidity EI . Suppose that an intermediate axial load Q is applied to the rod at x = a , as shown in figure 3.8(a). Let y()x be the lateral deflection

= () of the column, and denote ya y a .

The free body diagram of the entire column shown in figure 3.8(b) determines the forces at the supported ends. The free body diagrams of the virtual cross sections shown in figures 3.8(c) and 3.8(d) imply that

− ′′ = ()+ − < < EIy P Q y Qya x , 0 x a , (3.77) and

− ′′ = + ()− < < EIy Py Qya 1,x a x 1. (3.78)

Denote,

P Q u(x) 0 < x < a λ = , µ = , and y(x) =  (3.79) EI EI v(x) a < x < 1.

Then (3.77) and (3.78) yield

′′ + ()λ + µ = µ < < u u ua x , 0 x a , (3.80) and

′′ + λ = −µ − < < v v va (1 x), a x 1, (3.81)

= () = () where, ua u a and va v a .

P P P P Qy Qy Qy a a a

Rx Q Q Q Ry M a ya

R x

Ry x M y Qya

P + Q

(a) (b)(c) (d)

Figure 3. 8: Buckling of column with composite loads

It thus follows that

= ω + ω + u(x) A1 sin 1 x A2 cos 1 x Cx (3.82) and

= ω + ω + ()− v(x) B1 sin 2 x B2 cos 2 x D 1 x (3.83) where

ω = λ + µ 1 , (3.84)

ω = λ 2 , (3.85)

µu C = a , (3.86) λ + µ and

µv D = − a . (3.87) λ

By imposing the boundary condition u(0) = 0 ,

= A2 0 , (3.88) is obtained so that

= ω + u A1 sin 1 x Cx . (3.89)

From the condition v(1) = 0 , the following relationship is obtained,

ω + ω = B1 sin 2 B2 cos 2 0. (3.90)

The continuity of the displacement and the slope at x = a imply

ω + − ω − ω − ()− = A1 sin 1a Ca B1 sin 2 a B2 cos 2 a D 1 a 0 , (3.91) and

ω ω + − ω ω + ω ω + = A1 1 cos 1a C B1 2 cos 2 a B2 2 sin 2 a D 0. (3.92)

10 Q = 0

P 9 Q =1 EI 8 Q = 2

7 Q = 3

6 Q = 4 Graph of P as a function of Q and its position a 5 = Timoshenko and Gere, pp-100, Table 2 -6 Q 5 4 Results for a = 0.5 P = 9.8696 and Q = 0 3 P = 8.7487 and Q = 2.1872 Q = 7 P = 7.9456 and Q = 3.9728 2 P = 7.1200 and Q = 5.3400 1 P = 6.5198 and Q = 6.5198

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x − Position of Q

Figure 3. 9: Critical buckling load as a function of intermediate load and its position

Equations (3.90), (3.91), (3.92), (3.86) and (3.87) can be written in the following matrix

form

 0 sinω cosω 0 0  A  0 2 2  1     ω − ω − ω −   sin 1a sin 2 a cos 2 a a a 1 B1  0     ω cosω a −ω cosω a ω sinω a 1 1  B = 0 .(3.93)  1 1 2 2 2 2  2    ω 2 ω 2      0 0 0 1 2  C 0  µ ω − ()λ + µ − µ      sin 1a 0 0 a 0  D  0

ω ω Note that 1 is determined by 2 via (3.85), hence (3.93) has the transcendental

()ω = eigenvalue form with A 2 z o . The Newton’s eigenvalue method has been applied to

this problem with variable parameters a = 0.1k , k = 0,1,...,10 and q = Q EI = j ,

j = 1,2,...,9 . The critical load P(q,a) obtained is indicated by the graphs in figure 3.9.

The analytical results (Section 3.11, table 3.6 of Timoshenko and Gere, 1961) are compared with our parametric solutions obtained by the algorithm of section 3.3 in figure

3.9 and they indicate good agreement.

3.5 Higher Order Approximation in Newton’s Eigenvalue Algorithm

The proposed algorithm can also be extended to produce a higher order approximation as follows. Expanding Taylor series of order p gives

p () () A()ω 0 + ε = A()ω 0 + ∑ε k A , (3.94) k ω =ω ()0 k =1 where,

1 d k A A = , k =1,2,..., p . (3.95) k k! dω k

Hence

() det(A(ω 0 + ε )) = 0 (3.96) implies

( (ω ()0 )+ ε (ω ()0 )+ + ε p (ω ()0 )) = det A A1 ... A p 0 . (3.97)

Iterative correction for ε can be determined by the least eigenvalue (in the absolute value sense) for the higher order eigenvalue problem

( (ω ()0 )+ λ (ω ()0 )+ + λ p (ω ()0 )) = A A1 ... A p z 0 . (3.98)

A first order realization of (3.98) is given by

()G()ω − λH ()ω y = o (3.99) where

 0 I   n   0  G()ω = , (3.100)  I   n  − ()ω − ()ω − ()ω  A A1 A p−1 

I   n  I ()ω =  n  H   , (3.101)   ()ω  A p  and

   z   λz  y = . (3.102)      −  λ p 1z

Table 3.4: Higher order approximation and the rate of convergence

λ st nd Initial Guess i 1 order 2 order 0.1 1.8980 14 6 3.2 4.3437 10 5 6.4 8.13909 13 6 9.6 10.58616 6 5

In order to demonstrate the use of higher order approximations the axially vibrating rod, example 3.4.1 is resolved by using the approximation of the order p = 3 .

Results obtained are shown in table 3.4. As expected the number of iterations required for the convergence has been reduced.

3.6 Extension of the Algorithm in Quadratic Eigenvalue Problems

The method, developed in the previous sections is quite general and holds for discrete model with matrices that are not necessarily transcendental. Consider the quadratic eigenvalue problem associated with finite dimensional vibrating systems,

A(λ) = λ2M + λC + K , (3.103) which involves three symmetric matrices M , C , and K . Algorithm of Section 3.3 can be applied to the problem using

dA B()λ = − = −2λM − C . (3.104) dλ

Standard method for solving such problems involves transforming the equations into first order form. The transformation requires doubling the dimensions of problem and destroying the symmetry of system. With the method developed here matrix size as well as the symmetry of the system is unaltered, as demonstrated in the following example.

3.6.1 Example: Spectrum of a Discrete Model of a Vibrating System

The equation of motion for the system shown in figure 3.10 is given by (3.106) with

1 0 0  1 0 −1   5 − 2 −1        M = 0 1 0 , C =  0 3 −1  , K = − 2 4 − 2  . (3.105) 0 0 1 −1 −1 2   −1 − 2 3 

2 22 111

2 1

Figure 3. 10: A three degree-of-freedom mass-spring-damper system

Define,

λ2 + λ + 5 − 2 − λ −1    A()λ =  − 2 λ2 + 3λ + 4 − λ − 2  . (3.106)  2   − λ −1 − λ − 2 λ + 2λ + 3

The values of B(λ) can be obtained from (3.104) as,

− 2λ −1 0 1    B()λ =  0 − 2λ − 3 1  . (3.107)  1 1 − 2λ − 2

The Newton’s eigenvalue method has been applied to the problem yielding the eigenvalues shown in table 3.5. Initial guess used, and the number of iterations required for the convergence with a tolerance of 10−12 , are shown in the table. Note that the eigenvalues have been determined by using 3× 3 symmetric matrices, whereas the standard method of solving this problem involves a first order realization with matrices of dimension 6× 6.

Higher order approximation may be also applied to the quadratic eigenvalue problem (3.103) yielding via (3.94)

(λ + ε )= ()λ + ε ()λ + ε 2 ()λ A 0 A 0 A1 0 A 2 0 , (3.108) where

A = λ2M + λC + K , (3.109)

= λ + A1 2 M C, (3.110) and

= A 2 M . (3.111)

However, a first order realization leads via (3.99) through (3.102) to an eigenvalue problem of dimension 2n , which is of the same complexity as the original

problem. It is noted that in general a discrete model gives all the eigenvalues numerically whereas this approach guarantees one eigenvalue at a time.

Table 3.5: The eigenvalues of equations (3.103) and (3.104)

λ Initial guess Number of iteration Calculated i -1-2i 5 -0.7226 - 3.2521i -1+2i 5 -0.7226 +3.2521i -1-i 6 -0.4863 - 0.6063i -1+i 6 -0.4863 + 0.6063i -2-i 5 -1.7911 - 1.2358i -2+i 5 -1.7911 + 1.2358i

3.7 Summary of the Results and Conclusions

In this chapter a low order analytical model has been developed based upon transcendental eigenvalue problem. Unlike generalized eigenvalue problem, the matrices involved in this numerical model involve elements that are transcendental in nature. In order to solve such matrices a fast convergence algorithm has been developed based on

Newton’s eigenvalue iteration method, which uses algebraic eigenvalue solver.

Application of this model and its algorithm has been shown effectively with the examples of spectrum evaluation for non-uniform rods, beams and stability of column. In summary, a new improved mathematical model is now available for the analysis of continuous systems. This formulation can be extended to any problem that leads to the form of a transcendental eigenvalue problem. An effective algorithm is developed which solves these types of models efficiently. Contrary to the approximation of continuous systems based on the finite element and the finite difference methods, proposed mathematical model uses small size matrices and leads to accurate solutions.

This model also compensates for asymptotic corrections of eigenvalues, which were not present in the discrete models of continuous systems. The method developed here predicts the eigenvalues of a continuous system with uniform accuracy. With this method, a large number of eigenvalues can be obtained by solving a transcendental eigenvalue problem of small dimension. For the sake of verification, eigenvalues of three different rods having non-uniform cross sections are evaluated using the new model and the presented algorithm. The results are then compared with the existing results in the appendix A. These examples demonstrate that proposed mathematical model could be effectively used for evaluating the eigenvalues of any non-uniform cross-sectional continuous system. In the following chapter the same model can be extended to solve, inverse problem associated with continuous systems.

76 4. INVERSE PROBLEMS IN THE CONTINUOUS VIBRATORY SYSTEM

In this chapter, a method is presented for the solution of the inverse problem for continuous vibratory systems. This mathematical model for continuous system is more complicated than its associated finite discrete model and it leads to a transcendental eigenvalue problem as discussed in chapter 3. In the previous chapter, a transcendental eigenvalue problem for a single variable has been solved. This chapter deals with solving multivariable transcendental matrices that will lead to the solution of associated inverse problem by extending the same model.

An analytical mathematical model for solving inverse problem of continuous vibratory system is developed here. A numerical method is presented for the solution of multivariable transcendental eigenvalue problems. The algorithm presented here is also based on Newton’s eigenvalue iteration method and it uses an algebraic eigenvalue solver for multivariable system. The algorithm implements a variation of the Newton’s iterative method to determine a differential relation between change in physical parameters of the system and the corresponding change of the eigenvalues.

For simplicity, a piecewise continuous rod is used for the analysis. An example involving reconstruction of the area variation of exponential rod is also illustrated.

Conclusions are drawn from the results and the behavior of the solution is summarized.

4.1 Mathematical Modeling for Non-uniform Continuous Rod for the Associated Inverse Problems

Consider a piecewise uniform rod as shown in the figure 4.1(a). The equation of the motion for vibratory rod is represented by equation (3.2) and their associated

77 eigenvalue problem consisting of two equally spaced portions as shown in figure 4.1(a) is given by the following equations,

 p v′′ + λ 2 q v = 0, where, λ = ω c , c = E ρ , 0 < x < x  1 1 1 1 1 1 1 1 1 1 1 ′′ + 2 = = = < <  p2v2 λ2 q2v2 0, where, λ2 ω c2 , c2 E2 ρ2 , x1 x L . (4.1)  = = ′ = ′ ′ = v1 (0) 0, v1 (x1 ) v2 (x1 ), p1v1 (x1 ) p2v2 (x1 ), v2 (L) 0

p1,q1 p2 , q2

x =1 2 1 L = 1

u()x,t (a)

p1,q1 p2 , q2

= x1 1 2 L =1

u()x,t (b)

Figure 4. 1: Piecewise continuous axially vibrating rod with constant distribution of physical parameters: (a) fixed-free configuration (b) fixed-fixed configuration

Considering the general solution of form,

 ω ω = x + x < <  v1 (x) F1 sin G1 cos , 0 x x1  c c  1 1 , (4.2) ωx ωx v (x) = F sin + G cos , x < x < L  2 2 2 1  c2 c2 and applying boundary and matching conditions from (4.1) gives the transcendental eigenvalue problem as,

78  0 1 0 0   ωx ωx ωx ωx  1 1 − 1 − 1    sin cos sin cos  F1 c c c c    1 1 2 2  G  p ω ωx p ω ωx p ω ωx p ω ωx 1  1 1 − 1 1 − 2 1 2 1   = o . (4.3) cos sin cos sin F  c c c c c c c c  2   1 1 1 1 2 2 2 2   ω ωL ω ωL G   0 0 cos − sin  2    c2 c2 c2 c2 

Similarly, for fixed-fixed configuration of the same rod as shown in figure 4.1(b), the governing eigenvalue problem can be written as,

 p v′′ + λ 2 q v = 0, where, λ = µ c , c = E ρ , 0 < x < x  1 1 1 1 1 1 1 1 1 1 1 ′′ + 2 = = µ = < <  p2v2 λ2 q2v2 0, where, λ2 c2 , c2 E2 ρ2 , x1 x L . (4.4)  = = ′ = ′ = v1 (0) 0, v1 (x1 ) v2 (x1 ), p1v1 (x1 ) p2v2 (x1 ), v2 (L) 0

Considering the general solution of form,

 µ µ = x + x < <  v1 (x) F1 sin G1 cos , 0 x x1  c c  1 1 . (4.5) µx µx v (x) = F sin + G cos , x < x < L  2 2 2 1  c2 c2

Applying boundary and matching conditions from (4.4) gives the transcendental eigenvalue problem as,

 0 1 0 0   µx µx µx µx  1 1 − 1 − 1    sin cos sin cos  F1 c c c c    1 1 2 2  G  p µ µx p µ µx p µ µx p µ µx 1  1 1 − 1 1 − 2 1 2 1   = o (4.6) cos sin cos sin F  c c c c c c c c  2   1 1 1 1 2 2 2 2   µL µL G   0 0 sin cos  2    c2 c2 

p 2 = = = Let, the values r1 , L 1 and x1 1 2 are substituted in (4.3) and (4.6). Simplified p1 matrices are obtained as,

79  ω ω ω   sin − sin − cos  2c1 2c2 2c2     F1 ω ω ω ω ω ω   cos − r cos r sin  F  = o . (4.7) c 2c 1 c 2c 1 c 2c  2  1 1 2 2 2 2   ω ω ω ω G2   0 cos − sin     c2 c2 c2 c2  and

 µ µ µ   sin − sin − cos  2c1 2c2 2c2     F1  µ µ µ µ µ µ   cos − r cos r sin  F  = o . (4.8) c 2c 1 c 2c 1 c 2c  2  1 1 2 2 2 2   µ µ G2   0 sin cos     c2 c2 

Note that the solution of the transcendental eigenvalue problems (4.7) and (4.8)

ω ω ω µ µ µ provides two sets of spectrum 1 , 2 , , n and 1 , 2 , , n for the same rod with different configurations as shown in figure 4.1. In the inverse analysis, it is assumed that these sets of data are available from an experiment.

Now for the above stated system the inverse problem can be defined as follows:

ω ω “Given the two natural frequencies 1 , 2 of the fixed-free configuration, one natural

µ frequency 1 of fixed-fixed configuration and the total mass of the rod. Determine the

physical parameters i.e. axial rigidities p1 , p2 and density q1 ,q2 of the piecewise rod”.

The motivation here is to find the constants c1 ,c2 and r1 in such a way that will lead to the singular solution of (4.7) and (4.8). Singular solution of equations (4.7) and (4.8) for the given natural frequencies can be represented by,

80 ω ω ω sin 1 − sin 1 − cos 1 2c1 2c2 2c2 ω ω ω ω ω ω = 1 1 − 1 1 1 1 = F1 (c1 ,c2 , r1 ) cos r1 cos r1 sin 0 , (4.9) c1 2c1 c2 2c2 c2 2c2 ω ω ω ω 0 1 cos 1 − 1 sin 1 c2 c2 c2 c2

ω ω ω sin 2 − sin 2 − cos 2 2c1 2c2 2c2 ω ω ω ω ω ω = 2 2 − 2 2 2 2 = F2 (c1 ,c2 , r1 ) cos r1 cos r1 sin 0, (4.10) c1 2c1 c2 2c2 c2 2c2 ω ω ω ω 0 2 cos 2 − 2 sin 2 c2 c2 c2 c2 and,

µ µ µ sin 1 − sin 1 − cos 1 2c1 2c2 2c2 µ µ µ µ µ µ = 1 1 − 1 1 1 1 = F3 (c1 ,c2 ,r1 ) cos r1 cos r1 sin 0. (4.11) c1 2c1 c2 2c2 c2 2c2 µ µ 0 sin 1 cos 1 c2 c2

These 3-equations are to be solved for the three unknowns c1 ,c2 and r1 . However, these equations can also be obtained by evaluating its determinant in the form of transcendental frequency equation. The difficulties in evaluating determinant of transcendental matrices are discussed in the previous chapter. It is suggested that the determinant of a transcendental matrix of dimension n requires n! number of addition and multiplication operations, which cannot be obtained numerically for large n . In addition, such a method will require symbolic manipulations. Thus our aim here is to avoid calculating determinant and solve for its roots so that it can be extended to relatively bigger and complex problems.

81 For now, the problem of determining the roots, for the system of transcendental frequency equations, is as follows,

F (c ,c ,r ) = 0  1 1 2 1 = F2 (c1 ,c2 ,r1 ) 0 , (4.12)  = F3 (c1 ,c2 ,r1 ) 0

ω ω µ for the given values of 1 , 2 and 1 .

This problem can be generalized of variable n that can be defined as the n

= functional of variables xi , i 1,2, , n ,

F ()x , x , x ,, x = 0  1 1 2 3 n F ()x , x , x ,, x = 0  2 1 2 3 n , (4.13)   () = Fn x1, x2 , x3 , , xn 0 and, they represent the following nonlinear system of equations,

()()m ϖ = = m det A x1 , x2 , , xn , ϖ =ϖ 0, for i 1,2, ,n , (4.14) i where, A ∈ R m×m and ϖ are given sets of spectrum.

The motivation here is to determine the roots of nonlinear system (4.13).

Conventionally, the system of nonlinear equations (4.12) can be solved by Newton’s method (Hamming, 1987, Press et. al., 1992). However, this method requires evaluation of determinant for obtaining frequency equations (4.9-4.11) and it also requires evaluation of Jacobian matrix. It must be noted that the frequency equations cannot be evaluated analytically for the relatively large system thus the evaluation of Jacobian matrix for Newton’s method by analytical means is not feasible. Thus, in order to solve such system of equations an algorithm based upon Newton’s eigenvalue iteration method is proposed in the following section. This algorithm will then be used to estimate the

82 physical parameters of the piecewise continuous rod circumventing the need of the evaluation of transcendental frequency equations analytically.

4.2 An Algorithm for the Solution of Inverse Problems

In order to avoid the difficulties in obtaining the frequency equation, a new algorithm is proposed here which uses eigenvalue variation of Newton’s eigenvalue iteration method in conjunction with algebraic eigenvalue solver. The algorithm is developed here and the solution methodology is demonstrated effectively with the help of an example. The following algorithm thus can be used for solving inverse problems in continuous vibratory systems e.g. rods and beams.

4.2.1 Some Mathematical Preliminaries for the Algorithm

Let, A be a n× n square matrix and consider the eigenvalue problem,

= λ = Aφi iφi , for i 1,2,3, ,n , (4.15)

λ where, i and φi are the eigenvalues and eigenvectors of A . The eigenvectorsφi are also known as right eigenvectors.

Consider an eigenvalue problem associated with the matrix AT ,

T = λ = A ψ i i ψ i , for i 1,2,3, ,n , (4.16)

λ T where, i and ψ i are the eigenvalues and eigenvectors of A . The eigenvectors ψ i are also known as left eigenvectors.

Define the eigenvalue in the form of spectral matrix as,

λ  1  λ  = 2 Λ   , (4.17)   λ  n  and the modal matrices can be written as,

83 = [] Φ φ1 φ 2 φ n , (4.18) and,

= [] Ψ ψ1 ψ 2 ψ n . (4.19)

Theorem 1: Let, Φ and Ψ are the modal matrices containing the left and right eigenvectors of A , such that,

AΦ = ΦΛ , (4.20)

A T Ψ = ΨΛ . (4.21)

Then the relationship between left and right eigenvectors is,

Ψ = Φ −T . (4.22)

Proof: Substitution of (4.22) in the definition of (4.21) gives,

AT Φ −T = Φ −T Λ . (4.23)

Multiplying (4.23) from left and right by ΦT then gives,

ΦT AT Φ −T ΦT = ΦT Φ −T ΛΦT , (4.24) or,

ΦT AT = ΛΦT . (4.25)

By using (4.22) following relationships can also be obtained,

= −T ψ i φi , (4.26) and,

= −T φi ψ i , (4.27)

th −1 −1 where, φi and ψ i are the i eigenvector of Φ and Ψ respectively, and φi and ψ i are the ith eigenvector of Φ −1 and Ψ −1 respectively.

84 The determinant of A can be obtained analytically from its fundamental definition (Kreyszig, 1999). It can also be evaluated by using the product of eigenvalues

λ i of matrix A that are the n roots of its characteristic polynomials (Golub and Van

Loan, 1996), and they can be written as,

()= λ λ λ mλ det A 1 2 3 n . (4.28)

The Rayleigh quotient is a scalar quantity that is defined as (see, Meirovitch, 1986),

xT Ax R(x) = λ = , (4.30) xT x and if x is a good approximation to an eigenvector of A then R(x) is the good approximation to associated eigenvalue. Based upon the definitions of (4.28) and (4.30) and supposing that A is a symmetric matrix and φ is an eigenvector, then the associated eigenvalue can be defined as,

φT Aφ λ = . (4.31) φT φ

Similarly, when A is a non-symmetric matrix, φ is right eigenvector and ψ is left eigenvector then the associated eigenvalue can be defined as, (Meirovitch, 1997),

ψT Aφ λ = . (4.32) ψ T φ

Theorem 2: Let, A(c) be a symmetric matrix depending on a parameter c , and let

Aφ = λφ , (4.34) so that λ = λ(c) and φ = φ(c) are an eigenvalue and right eigenvector of A(c) .

Then,

85 ∂A φT φ ∂λ ∂ = c . (4.35) ∂c φTφ

Proof: From the definition of Rayleigh’s quotient (4.31), we have

φT Aφ λ = . (4.36) φTφ

Differentiating (4.36) we get,

 ∂φT ∂A   ∂φT  2 Aφ + φT φφTφ − ()φT Aφ 2 φ ∂λ ∂c ∂c ∂c =     2 . (4.37) ∂c ()φTφ

Substituting (4.34) in (4.37) we get,

 ∂ T ∂   ∂ T  φ T A T T φ 2 λφ + φ φφ φ − ()φ λφ 2 φ ∂λ ∂c ∂c ∂c =     2 , (4.38) ∂c ()φTφ which simplifies to the expression (4.35).

Theorem 3: Let, A(c) be a non symmetric matrix depending on a parameter c , and let

Aφ = λφ , (4.39) and,

ATψ = λψ , (4.40) so that λ = λ(c) is an eigenvalue, φ = φ(c) is a right eigenvector and ψ = ψ(c) is a left eigenvector of A(c) . Then,

∂A ψ T φ ∂λ ∂ = c . (4.41) ∂c ψ Tφ

Proof: From the definition of Rayleigh’s quotient (4.32) for the non-symmetric matrix,

86 ψ T Aφ λ = . (4.42) ψ Tφ

By differentiating (4.42), we obtain

 ∂ T ∂ ∂   ∂ T ∂  ψ T A T φ T T ψ T φ  Aφ + ψ φ + ψ A ψ φ − ()ψ Aφ  φ + ψ  ∂λ ∂c ∂c ∂c ∂c ∂c =     2 . (4.43) ∂c ()ψ Tφ

Substitution of (4.39) and (4.40) in (4.43) gives,

 ∂ψ T ∂A ∂ψ −T   ∂ψ T ∂ψ −T   λφ + ψ T φ + λψ T  − λ φ + ψ T ψ Tφ ∂λ  ∂c ∂c ∂c   ∂c ∂c  =   2 , (4.46) ∂c ()ψ Tφ that leads to the desired relationship (4.41).

4.2.2 Development of an Algorithm for Non-uniform Continuous System

Denoting the systems described in (4.13) and (4.14) as following,

= () = m Fi det A i , for i 1,2, , n (4.47) where,

= ()m ω A i A x1 , x2 , , xn , i . (4.48)

Denote the functional,

(k ) = ()(k ) (k ) (k ) T F F1 F2 Fn , (4.49) where, k = 0 represents the initial estimate and,

(k ) = ()(k ) (k ) (k ) T x x1 x2 xn . (4.50)

Taylor series expansion in the neighborhood of x (k ) is given by,

2 F (k ) ()()()x (k ) + ε (k ) = F (k ) x(k ) + J (k )ε (k ) + O ε (k ) , (4.51) where, the Jacobian matrix J ∈ R n×n is defined as,

87 ∂F (k ) ∂F (k ) ∂F (k )   1 1 1  ∂ (k ) ∂ (k ) ∂ (k )  x1 x2 xn  ∂ (k ) ∂ (k ) ∂ (k )  F2 F2 F2 (k ) J =  ∂ (k ) ∂ (k ) ∂ (k )  , (4.52)  x1 x2 xn    ∂F (k ) ∂F (k ) ∂F (k )  n n n  ∂ (k ) ∂ (k ) ∂ (k )   x1 x2 xn  and the neighborhood values of x (k ) can be defined as,

(k ) = ()ε (k ) ε (k ) ε (k ) T ε 1 2 n . (4.53)

By neglecting the higher order terms in (4.51) and by setting F (k ) (x (k ) + ε (k ) ) = o the following system of equations is obtained,

−1 ε (k ) = −()J (k ) F (k ) . (4.54)

From the fundamental definition of determinant (4.28), the functional can also be written as,

()= λ i ()λ i ()λ i ()λ i () Fi x 1 x 2 x 3 x m x (4.55)

λ i = where, l , for l 1,2,3, , m , are the eigenvalues of the system,

()− λ i = A i I φ o . (4.56)

Partial derivative of equation (4.55) gives,

∂F ∂()λ i ∂()λ i ∂()λ i i = 1 ()λ i ()λ i + 2 ()λ i ()λ i ++ m ()λ i ()λ i , (4.57) ∂ ∂ 2 m ∂ 1 m ∂ 1 m−1 x j x j x j x j which can be written as,

∂F m ∂()λ i m i = ∑ r ∏()λ i , (4.58) ∂ ∂ s x j r=1 x j s=1 s≠r

∂()λ i where, the values of l can be obtained from (4.41) as, ∂ x j

88 T ∂()λ i ()ψ i ()B()ϖ , x ()φi l = l i j l . (4.59) ∂x ()()i T i j ψ l φl

()ϖ ∈ m×m In (4.59) the elements, of the matrix B i , x j R , are defined as,

∂()a()ϖ b()ϖ ,x = i , (4.60) i j ∂ x j

()i th where, b and a are the elements of the matrices A and B respectively, φ l is the l

th ( i ) th right eigenvector corresponding to l eigenvalue of (4.56), and ψ l is the l left eigenvector corresponding to l th eigenvalue of (4.56) is evaluated using (4.26) as,

− ()i T = ()i T ψ l φ l . (4.61)

∂F (k ) By using (4.59)-(4.61) the Jacobian matrix is obtained by evaluating i and ∂ (k ) x j by substituting this value in the matrix equation (4.52). The linear system (4.54) then can be solved iteratively until the convergence criterion ( ε < δ , where δ is a pre-selected small positive tolerance) is satisfied.

Similarly, each element of the Jacobian matrix is evaluated by using the eigenvalue-eigenvector relationship developed in section 4.2.1. In the next section the above procedure is summarized in an algorithmic form that can be used to identify physical parameters of vibrating rods and beams based upon their eigenvalues.

4.2.3 Algorithm 2

Newton’s eigenvalue iterative method for an inverse transcendental eigenvalue problem is as follows.

89 INPUT

ϖ ω ω l ω µ µ l µ (a) The set of spectrum, n , consisting 1 , 2 , , r and 1 , 2 , , s ,

where r + s = n is the number of unknown to be identified.

(b) An initial estimate of x (0) .

(d) A convergence tolerance δ (a small positive number)

ITERATION

(i) ()(0) ϖ (a) Based upon the initial estimate, evaluate the matrices A x , n and

ij (0) B for each variable x j .

(b) Evaluate functional vector F (0) by using (4.49).

−1 (d) Solve the linear system (4.54) asε (0) = −()J (0) F (0) .

(e) Compute new estimate x(1) = x(0) + ε (0) .

(f) Iterate step (a) to (e) until p iteration such that the value of ε ( p) is

sufficiently small.

OUTPUT

At convergence ε < δ , the solution vector x( p) is obtained.

4.3 Reconstruction of Continuous Vibrating System from its Spectral Data

Based upon the algorithm developed in the previous section, physical parameters of the piecewise constant rod is identified based upon their eigenvalues.

90 4.3.1 Example: Reconstruction of the Piecewise Continuous Rod

Considering the example of piecewise continuous rod which is discussed in section 4.1 and 4.2 and the matrices (4.9)-(4.11) can be defined as,

 ω ω ω   sin 1 − sin 1 − cos 1   2c1 2c2 2c2  ω ω ω ω ω ω ()ϖ = ω =  1 1 − 1 1 1 1  A1 c1 ,c2 ,r1 , 1 1  cos r1 cos r1 sin  , (4. 62) c1 2c1 c2 2c2 c2 2c2  ω ω ω ω   0 1 cos 1 − 1 sin 1     c2 c2 c2 c2 

 ω ω ω   sin 2 − sin 2 − cos 2   2c1 2c2 2c2  ω ω ω ω ω ω ()ϖ = ω =  2 2 − 2 2 2 2  A 2 c1 ,c2 ,r1 , 2 2  cos r1 cos r1 sin  , (4.63) c1 2c1 c2 2c2 c2 2c2  ω ω ω ω   0 2 cos 2 − 2 sin 2     c2 c2 c2 c2  and,

 µ µ µ   sin 1 − sin 1 − cos 1   2c1 2c2 2c2  µ µ µ µ µ µ ()ϖ = µ =  1 1 − 1 1 1 1  A 3 c1 ,c2 , r1 , 3 1  cos r1 cos r1 sin  . (4.64) c1 2c1 c2 2c2 c2 2c2  µ µ   0 sin 1 cos 1     c2 c2 

The available data are

    ω = −1  3  ω = π − −1  3  µ = π 1 2cos  , 2 2 2cos  , and 1 , (4.65)  3   3  and they are the eigenvalues obtained form the system (4.3) and (4.6), by substituting,

= = = c1 1 c2 1 r1 0.5. (4.66)

91 The motivation here is to identify physical parameters of (4.66) with the data available from (4.65) in order to reconstruct the system (4.4). In algorithm 2 the values

of c1 ,c2 and r1 are to be taken appropriately as the initial guesses of the initial vector,

(0) = ()T x c1 c2 r1 . (4.67)

The matrices from (4.60) are obtained as,

 ()ω  cos 1 2c1  2 0 0  2c1  ω sin()ω 2c cos()ω 2c  B()ω ,c = 1 1 1 − 1 1 0 0 , (4.68) 1 1  2c 3 c 2   1 1   0 0 0  

 ()ω  cos 2 2c1  2 0 0  2c1  ω sin()ω 2c cos()ω 2c  B()ω ,c = 2 2 1 − 2 1 0 0 , (4.69) 2 1  2c 3 c 2   1 1   0 0 0  

 cos()µ 2c  − 1 1  2 0 0  2c1  µ sin()µ 2c cos()µ 2c  B()µ ,c = 1 1 1 − 1 1 0 0 , (4.70) 1 1  2c 3 c 2   1 1   0 0 0  

  ω   ω    cos 1  sin 1     2c   2c   0 2 − 2  2c 2 2c 2   2 2    ω   ω     ω   ω      1  ω  1   ω  1   1   cos  1 sin  1 cos  sin     2c   2c     2c   2c   ()ω =  2 − 2 − 2 + 2  , ( 4.71) B 1 ,c2 0 r1  2 3  r1  3 2    c2 c2   c2 c2             ω  ω  ω  ω   ω  1   1  ω  1   1   1 sin  cos  1 cos  sin    c   c   c   c  0 2 − 2 2 + 2   3 2 3 2   c2 c2 c2 c2 

92   ω   ω    cos 2  sin 2     2c   2c   0 2 − 2  2c 2 2c 2   2 2    ω   ω     ω   ω      2  ω  2   ω  2   2   cos  2 sin  2 cos  sin     2c   2c     2c   2c   ()ω =  2 − 2 − 2 + 2  , (4.72) B 2 ,c2 0 r1 2 3  r1  3 2    c2 c2   c2 c2             ω  ω  ω  ω   ω  2   2  ω  2   2   2 sin  cos  2 cos  sin    c   c   c   c  0 2 − 2 2 + 2   3 2 3 2   c2 c2 c2 c2 

  µ   µ    cos 1  sin 1     2c   2c   0 2 − 2  2c 2 2c 2   2 2     µ   µ     µ   µ     1  µ  1    µ  1   1    cos  1 sin  1 cos  sin     2c   2c     2c   2c   B()µ ,c = 0 r  2 − 2  − r  2 + 2  ,(4.73) 1 2  1 2 3 1 3 2   c2 c2   c2 c2              µ   µ    cos 1  sin 1     c2   c2   0 −  2 2   c2 c2 

0 0 0   cos(ω 2c ) sin(ω 2c ) ()ω =  − 1 2 1 2  B 1 ,r1 0 , (4.74)  c2 c2  0 0 0 

0 0 0   cos(ω 2c ) sin(ω 2c ) ()ω =  − 2 2 2 2  B 2 ,r1 0 , (4.75)  c2 c2  0 0 0  and,

0 0 0   cos(µ 2c ) sin(µ 2c ) ()µ =  − 1 2 1 2  B 1 ,r1 0 . (4.76)  c2 c2  0 0 0 

93 λ1 λ 2 l λ 2 λ 3 ()ϖ The eigenvalues 1 , 1 , , 3 , 3 of A c1 ,c2 ,r1 , 1 are evaluated from (4.62)-(4.64) and the functional,

(0) = ()T F F1 F2 F3 , (4.77) is obtained where,

 F ()x (0) = λ1 ()x(0) λ1 ()x (0) λ1 ()x(0)  1 1 2 3 ()(0) = λ 2 ()(0) λ 2 ()(0) λ 2 ()(0) F2 x 1 x 2 x 3 x . (4.78)  ()(0) = λ 3 ()(0) λ 3 ()(0) λ 3 ()(0) F3 x 1 x 2 x 3 x

The Jacobian matrix is then computed for the first iteration as,

∂F ∂x ∂F ∂x ∂F ∂x   1 1 1 2 1 3  (0) = ∂ ∂ ∂ ∂ ∂ ∂ J  F2 x1 F2 x2 F2 x3  , (4.79) ∂ ∂ ∂ ∂ ∂ ∂   F3 x1 F3 x2 F3 x3  where each element of the matrix (4.79) can be calculated numerically by (4.57), (4.58)

∂F and (4.59). For example, 1 of the Jacobian matrix can be computed as, ∂ x1

∂F ∂()λ1 ∂()λ1 ∂(λ1 ) 1 = 1 ()λ1 ()λ1 + 2 ()λ1 ()λ1 + 3 ()λ1 ()λ1 . (4.80) ∂ ∂ 2 3 ∂ 1 3 ∂ 1 2 x1 x1 x1 x1

∂()λ1 ∂()λ1 ∂(λ1 ) The values of 1 , 2 and 3 can be obtained from (4.59) as, ∂ ∂ ∂ x1 x1 x1

T  ∂()λ1 ()ψ1 ()B()ω ,c ()φ1  1 = 1 1 1 1 ∂ ()()1 1  x1 ψ1 φ1  T ∂()λ1 ()ψ1 ()B()ω ,c ()φ1  2 = 2 1 1 2 , (4.81) ∂ ()()1 1  x1 ψ 2 φ 2 T ∂()λ1 ()ψ1 ()B()ω ,c ()φ1  3 = 3 1 1 3 ∂ ()()1 1  x1 ψ 3 φ3

94 ψ i where the values of k can be calculated from (4.61). Similarly, all the elements of the matrix (4.79) can be evaluated numerically. Now applying algorithm 2 using (4.62)-

(4.81) the solution vector is obtained as,

= ()()T = T x c1 c2 r1 1 1 0.5 . (4.82)

Thus the piecewise continuous rod has been reconstructed accurately. It is observed that along with the true solution, other possible solutions are obtained for different starting point. The significance of other solutions is discussed in the next section in detail.

4.4 Significance of the Multiple Solutions and Uniqueness of the Results

In this section the uniqueness of the results are discussed. The physical significance of multiple solutions obtained using algorithm 2 is discussed here. These solutions are analyzed in order to differentiate the true solution from all the possible solutions.

4.4.1 Multiplicity of the Solutions

Analyzing the problem of reconstructing physical parameters for a piecewise continuous rod yields multiple solutions for different initial guess. Solution sets are tabulated below in table 4.1 with their initial guess. These multiple solutions are expected since the given spectrum is not complete spectrum of the system. Unlike discrete models, which lend themselves to a unique solution, continuous systems lead to multiple solutions.

It is evident from table 4.1, that different initial guesses lead to different solution sets which are the singular solutions of (4.9)-(4.11). It is interesting to note that some solutions (with negative entries) do not have any physical significance.

95 Table 4. 1: Solution sets of estimated parameters with different initial guess

Number of c(0) c c(0) c r (0) r Iterations 1 1 2 2 1 1 7 0.7833 1 0.6808 1 0.4611 0.5 6 0.9601 1 0.7266 1 0.4 0.5 7 0.8903 1 0.7349 1 0.6873 0.5 7 0.6649 1 0.8704 1 0.0099 0.5 6 0.4784 0.5 0.5548 0.5 0.1210 0.1250 11 0.2731 0.5 0.2548 0.1667 0.8656 0.1917 9 0.2379 0.3333 0.6458 0.2 0.9669 0.1354 8 0.2622 0.25 0.1165 0.1250 0.0693 0.1328 7 0.1121 0.1111 0.1121 0.1111 0.2916 0.8468 7 0.6101 -0.2 0.7015 -1.0 0.0922 -0.2273

The aim here is to study the physical significance of these solutions and understand the possible relationship between the solution sets. In order to further analyze these solutions, two different rods with different physical parameters are considered.

These two rods are obtained from solution set from table 4.1, and they are shown in figure 4.2.

======c1 1,c2 1,r1 1 2 c1 1 2,c2 1 2,r1 1 8

x =1 2 x =1 2 1 L =1 1 L =1

(a) (b)

======c1 1,c2 1,r1 1 2 c1 1 2,c2 1 2,r1 1 8

= x1 1 2 L = 1

(c) (d) Figure 4. 2: Piecewise continuous axially vibrating rod with different physical parameters and different boundary configuration

96 The frequency equation for the non-trivial solution based on (4.3) for the rod shown in figure 4.2(a) is obtained as,

3 ω  1 − cos 2   + = 0 . (4.83) 2  2  2

The closed form solution of frequency equation (4.83) can be obtained as,

   − 3 ω = π − + 1   = i 2 (i 1) 2cos  , where i 1,3,5,   3   . (4.84)   ω = π − −1  3  =  j 2 j 2cos  , where i 2,4,6,   3 

Table 4. 2: Spectrum of rods shown in figure 4.2

Fixed-free configuration Fixed-fixed configuration ω µ (Lower natural frequencies= i ) (Lower natural frequencies= i ) 1 1 1 1 1 1 1 1 c = 1, c = 1, r = c = , c = , r = c = 1, c = 1, r = c = , c = , r = 1 2 1 2 1 2 2 2 1 8 1 2 1 2 1 2 2 2 1 8 1.91063323624902 1.23095941734077 4.14159265358979 1.57079632679490 4.37255207093057 1.91063323624902 6.28318530717959 4.14159265358979 8.19381854342861 4.37255207093057 9.42477796076938 4.71238898038469 10.65573737811015 5.05222588983881 12.56637061435917 6.28318530717959 14.47700385060819 7.51414472452036 15.70796326794897 7.85398163397448 16.93892268528974 8.19381854342861 18.84955592153876 9.42477796076938 20.76018915778778 10.65573737811015 21.99114857512855 10.99557428756428 24.22210799246933 11.33541119701840 25.13274122871835 12.56637061435917 27.04337446496736 14.79733003169995 28.27433388230814 14.13716694115407 29.50529329964891 14.47700385060819 31.41592653589793 15.70796326794897

Similarly, the frequency equation for rod shown in figure 4.2(b) can be written as,

9 1 − cos 2 ()ω + = 0 , (4.85) 4 4 and their closed form solution yields,

97  −  1  ω = π (i −1) + cos 1  , where i = 1,3,5,  i  3  (4.86) −  1  ω = πj − cos 1  , where i = 2,4,6,  j  3 respectively.

Two rods for fixed-fixed boundary configurations as shown in figure 4.2(c)-(d) also lend themselves to these frequency equations,

 µ   µ  sin cos  = 0 , (4.87)  2   2 

sin()µ cos ()µ = 0 , (4.88) and yields the following closed form solutions,

µ = π = i i , where i 1,2,3,4,5 , (4.89) and,

π µ = i , where i = 1,2,3,4,5, (4.90) i 2 respectively.

Numerical values of the non-dimensional natural frequencies from equations

(4.84), (4.86), (4.89) and (4.90) are tabulated in table 4.2. Note that the frequencies obtained from (4.84) and (4.89) are the input sets of spectrum towards the identification of piecewise rod. Frequencies obtained from (4.86) and (4.90) are the spectrum coming from another solution set obtained while starting from different initial guess.

Some more rods with different physical parameters exhibit the same behavior as shown in table 4.2. The spectrum sets obtained for different rods with fixed-free and fixed-fixed configuration are tabulated in table 4.3.

98 Table 4. 3: Spectrum set available from dual rods that contain the common eigenvalues

c = 1 1 1 1 1 c = c = c = = 1 2 1 1 c2 1 6 8 1 1 1 1 c = = = r = 2 c2 c2 1 2 2 6 4 = 1 69 = 17 r1 r = r 8 1 40 1 32 λ µ λ µ λ µ λ µ i i i i i i i i 1.9106 3.1416 1.2310 1.5708 0.4363 0.7177 0.3398 0.4456 4.3726 6.2832 1.9106 3.1416 1.2309 1.5708 0.7854 1.1252 8.1938 9.4248 4.3726 4.7124 1.9106 2.4239 1.2310 1.5708 10.6557 12.5664 5.0522 6.2832 2.7053 3.1416 1.5708 2.0164 14.4770 15.7080 7.5141 7.8540 3.5778 3.8593 1.9106 2.6960 16.9389 18.8496 8.1938 9.4248 4.3725 4.7124 2.3562 3.1416 20.7602 21.9911 10.6557 10.9956 5.0522 5.5655 2.8018 3.5872 23.2221 25.1327 11.3354 12.5664 5.8469 6.2832 3.1416 4.2668

It has been observed clearly from table 4.2 that two different rods with different configurations do not have distinct spectrum but they have some common natural frequencies. Thus, input spectrum may also be the spectrum for another rod that is not being investigated. There exist multiple rods as tabulated in table 4.3 that may contain the same spectrum set which is used as input for the inverse problem. It explains the possibility of another solution that will satisfy the input spectrum while solving any continuous system. Therefore, it is very difficult to obtain a unique solution of the inverse problems associated with any continuous system if the initial guess is taken arbitrary. However, the true solution may be obtained when initial guess is very close to the true solution or they can be obtained after applying some modifications to the algorithm 2. In order to obtain the true solution using the proposed algorithm some modifications are further investigated and discussed in the following section.

99 4.4.2 Modification towards Obtaining the True Solution: Least Square Method

In theory the continuous system contains infinite number of spectrum whereas only the finite number of the spectrum can be obtained experimentally and they can be used to solve the associated inverse transcendental eigenvalue problem. The transcendental models associated with continuous systems are of smaller model order.

However, it is possible to have more spectral data set from the experiments than the number of physical parameters to be determined. In that case the inverse problem can be solved in a least square sense where the system is considered to be over determined.

Let the number of data set m be greater than the number of the physical

= parameter to be estimated n p 3 for piecewise rod. For this case Jacobian matrix can be written as,

∂ ∂ ∂ ∂ ∂ ∂  F1 x1 F1 x2 F1 x3 ∂ ∂ ∂ ∂ ∂ ∂   F2 x1 F2 x2 F2 x3  ∂F ∂x ∂F ∂x ∂F ∂x  J =  3 1 3 2 3 3  , (4.91) ∂ ∂ ∂ ∂ ∂ ∂  F4 x1 F4 x2 F4 x3      ∂ ∂ ∂ ∂ ∂ ∂  Fm x1 Fm x2 Fm x3  and in the algorithm 2, the linear system

L(x (k ) ) = ((J (k ) )ε (k ) + F (k ) ), (4.92) can be solved in the least square sense.

Thus for the least squares problem usually there is no x that satisfies

L(x) = 0 , (4.93) instead, such solutions are searched, for which

2 2 L(x (k ) ) = ()()J (k ) ε (k ) + F (k ) (4.94)

100 is minimum.

Such least squares scheme can be carried out to algorithm 2 during the iteration process. Note that the inversion of the non-square Jacobian matrices for the over- determined system does not exists. Thus the linear least square problem can be solved by the method of normal equations or evaluating its equivalent pseudo inverse (Golub and

Van Loan, 1996).

In order to ensure the least square solution following modification is proposed in algorithm 2.

ITERATION

()(0) ϖ (a) Based upon the initial estimate, evaluate the matrices A i x , n and

ϖ (0) B( i , x j ) for each variable x j .

(b) Evaluate functional vector F (0) by using (4.49).

(d) Solve the linear least squares system (J (0) )ε (0) + F (0) = o .

(e) Compute new estimate x(1) = x(0) + ε (0) .

(f) Iterate step (a) to (e) until p iteration such that the value of

2 ()()J ( p) ε ( p) + F ( p) is sufficiently small which can be achieved by setting

the tolerance ε p − ε p−1 < δ .

However it is not guaranteed that the unique solution will be achieved by this modification but the number of solution set will be reduced when the initial guess is taken arbitrary.

101 4.4.3 Modification towards Obtaining the True Solution: Verification of Solution Set

The following conjecture is based on many observations with the developed algorithm.

Conjecture 1: “There is a unique rod that satisfies the given data by its lowest spectrum”,

Hence, in order to make sure that the obtained solution is the true solution, the recovered parameters are substituted back in the transcendental eigenvalue model of the associated structure and using algorithm 1 its spectrum can be reevaluated. If the substitution of obtained solution set yields input spectrum as the lowest spectrum of the transcendental eigenvalue problem, then they are indeed the true solution. This conjecture can be added to algorithm 2 to ensure the uniqueness of solution.

λ λ λ µ µ µ Suppose that for a given spectrum set 1 , 2 , , r and 1 , 2 , , s , the output of algorithm 2 yields a solution vector x( p) , where r is the number of eigenvalues corresponding to resonant frequencies and s is the number of eigenvalues corresponding to anti-resonant frequencies. In order to achieve true solution following modification can be added to the algorithm 2 at the end of the iterative process. In this modification

Algorithm 2 is combined with the algorithm 1. The identified physical parameters from algorithm 2 are used as the input for algorithm 2.

OUTPUT

(a) At convergence the solution vector x( p) is obtained.

(b) Substituting x( p) as input parameters and using algorithm 1 eigenvalues

associated with resonant and anti-frequencies of the structure are

λˆ λˆ m λˆ µ µ µ evaluated as 1 , 2 , , r and ˆ1 , ˆ 2 , , ˆ s .

102 λ λ m λ ≈ λˆ λˆ m λˆ µ µ m µ ≈ µ µ m µ (c) If 1 , 2 , , r 1 , 2 , , r and 1 , 2 , , s ˆ1 , ˆ 2 , , ˆ s then the

true solution is obtained.

This safeguard is implemented effectively while solving the inverse problem as illustrated in the next section.

4.5 Numerical Example and Conclusions

Based upon algorithm 2 and using the modification proposed in the section 4.4.3, the reconstruction of the shape of a non-uniform exponential rod, from its spectral data, is presented here.

4.5.1 Example: Reconstruction of the Shape of the Exponential Rod

Consider the free vibration of a unit length exponential rod that can be expressed by following differential equation,

∂  ∂u  ∂ 2u  Ee x  = ρe x , (4.95) ∂x  ∂x  ∂t 2 with these boundary conditions,

u(0) = 0, u′(1) = 0 . (4.96)

Suppose that the modulus of elasticity and density are known as non-dimensional parameters α and β respectively. For the unit values α and β the corresponding eigenvalue problem can be written as,

 ′ ()x ′ + λ x = < <  e v e v 0, 0 x 1. (4.97) v(0) = 0, v′(1) = 0

The solution of eigenvalue problem (4.97) can be obtained as, (see Ram 1994-a),

     − x     − x    =  2   ω 2 − 1  +  2   ω 2 − 1  v(x) Ae sin i x Be cos i x , (4.98)  4   4 

103 ω = λ where, i are the natural frequencies of the fixed-free exponential rod.

Substituting the boundary conditions of (4.97), the following frequency equation is obtained,

     ω 2 − 1  −  ω 2 − 1  = tan i  2 i  0 (4.99)  4   4 

The natural frequencies are obtained analytically by solving the frequency equation

(4.99).

Similarly, for the fixed-fixed configuration of the exponential rod the eigenvalue problem can be written as,

 ′ ()x ′ + µ x = < <  e v e v 0, 0 x 1. (4.100) v(0) = 0, v′(1) = 0

Substituting the boundary conditions of (4.100) in the solution form (4.98), the natural frequencies of the fixed-fixed rod can be obtained by solving the frequency equation,

   ω 2 − 1  = sin i  0 . (4.101)  4 

ω = µ where, i are the natural frequencies of the fixed-fixed exponential rod.

λ Suppose that the eigenvalues of fixed-free rod i and fixed-fixed exponential rod

µ i are given. Then the estimation of variation in the area of exponential rod is shown in figure 4.3.

= λ µ Assuming that for i 1,2, ,8 , eigenvalues i and i are available then for the piecewise model of exponential rod of model order n = 3 , n = 4 , n = 6 and n = 8 are

λ µ developed. Using the data set i and i for different model order, corresponding inverse transcendental eigenvalue problems are solved using algorithm 2. Results of area

104 variation are then plotted with the actual cross sectional area variation of the rod in figure

4.3.

Theoretical area Estimated area

0 0.10.2 0.30.4 0.50.6 0.70.8 0.9 1 0 0.1 0.20.3 0.40.5 0.60.7 0.80.9 1

Model order n=3 Model order n=4

0 0.1 0.2 0.30.4 0.50.6 0.70.8 0.9 1 0 0.1 0.2 0.30.40.5 0.6 0.70.8 0.9 1 Model order n=6 Model order n=8

Figure 4. 3: Estimation of the variation in area of the exponential rod from its spectral data using different model order

It is evident from figure 4.3 that a low dimensional model is capable of

reconstructing the approximate shape of the rod. Such reconstructions are in agreement

with the exact shape of the rod. If sufficient data set are available then increasing the

model order a desired accuracy may be achieved for reconstruction.

105 Numerical evidence shown in 4.3.1 and 4.5.1 demonstrates that continuous system can be approximated by the associated transcendental eigenvalue problems and the solutions of such inverse transcendental eigenvalue problems are capable of reconstruction of the physical parameters from its spectral data. In this chapter the results related to an axially vibrating non-uniform rods are presented. They can be readily extended towards the solution of transversely vibrating beams.

In the following chapter, an inverse problem associated with beam is addressed.

Some laboratory experiments are conducted using non-uniform beams in order to demonstrate the effectiveness of presented method and algorithm. The physical significance of the input data set along with the availability of such data sets is also discussed in the following chapter.

106 5. IDENTIFICATION OF THE BEAM FROM SPECTRAL DATA: EXPERIMENT

In the previous chapters a new theoretical model based upon transcendental eigenvalue problem for non-uniform continuous systems has been developed. Numerical algorithms have also been developed to solve the associated direct and inverse problems.

In this chapter, an experiment is carried out to identify the physical properties of a piecewise beam from measured spectral data. A frequency response function of the beam is measured using modal analysis technique. An inverse problem is developed for piecewise beam and based upon algorithm 2 physical properties, of this beam, are identified. The physical significance of the spectral data obtained from these experiments and the results, of the reconstruction procedure, are presented in this chapter.

5.1 Physical Significance of Spectral Data

Theoretically, two sets of data are necessary for the identification of the physical parameters of axially vibrating rods and three sets of data are required for transversely vibrating beams (Barcilon, 1982, 1986 and Gladwell, 1986-a). It is shown here that these data set or eigenvalues can be evaluated appropriately from experiments by extracting the frequency response functions.

5.1.1 Spectral Data: Resonant and Anti-resonant Frequencies

Using modal analysis technique, a frequency response function can be measured consisting of the resonant and the anti-resonant frequencies of vibrating system. The aim here is to understand the physical significance of resonant and anti-resonant frequencies associated with vibrating structures.

107 Let us consider a harmonically excited conservative n-degree-of-freedom mass- spring system, as shown in figure 5.1. The governing equation of motion can be expressed as,

+ = ω Mx Kx e a sin t , (5.1) where, M ∈ ()n × n and K ∈ ()n × n , are symmetric positive definite mass and stiffness matrices respectively and,

= ()m T ∈ ()× ea 0 0 1 n 1 , (5.2) is the vector representing location ‘a’ of harmonic excitation from an actuator. The particular solution of (5.1) is of the following form,

()ω = ω x ,t u a sin t . (5.3)

"s" "a" kn kn−1 k2 k1 sin ωt

mn m3 m2 m1

Figure 5.1: Harmonically excited n-degree-of-freedom mass spring system

Substituting (5.3) in (5.1) gives,

()−ω 2 = K M u a e a . (5.4)

Suppose the response is measured from the sensor at location ‘s’, then applying

Crammer’s rule (Kreyszig, 1999), the response of the vibrating system can be obtained as,

det()Kˆ −ω 2Mˆ u (ω) = , (5.5) as det()K −ω 2M

108 where, Kˆ and Mˆ is obtained by deleting ath row and sth column of K and M

ω th ω λ λ m λ respectively and uas ( ) is s element of u a ( ) . The eigenvalues 1 , 2 , , n represent resonant frequencies of discrete system and they are obtained from the matrix pencil

()− λ µ µ m µ K M . Also, the eigenvalues 1 , 2 , , n−1 represent the anti-resonant frequencies K K ( − λ ) ω of the system. They are obtained from the matrix pencil K M . The function uas ( )

≠ ω is known as the sine response function at ‘s’ due to excitation at ‘a’. If a s then uas ( )

= ω is known as the non-collocated frequency response function. If a s then uaa ( ) is known as the collocated frequency response function as shown in figure (5.2).

100 λ λ λ λ 1 2 3 4 10-1

10-2 10-3 µ µ µ 10-4 1 2 3

-5 10 0 5 10 15 20 25

Figure 5. 2: Collocated frequency response function for the mass-spring system of model order n=4

It is clear from figure 5.2 that a single frequency response function may contain the information about two sets of spectral data corresponding to two different boundary configurations. Note that the anti-resonant frequencies can be obtained distinctly when collocated measurements are taken.

109 5.1.2 Frequency Response Function of an Axially Vibrating Rod

The proposed research mainly deals with a parameter estimation problem associated with distributed parameter systems e.g. axially vibrating rod and transversely vibrating beam, based upon spectral data obtained from a frequency response function.

Thus, it is necessary to understand the physical meaning of the spectral data obtained from the continuous systems.

Consider a unit length axially vibrating fixed-free rod, which is subjected to a harmonic excitation force at the free end as shown in figure 5.3 (a). The eigenvalue problem associated with this rod can be written as,

sin ωt x x x

L =1 L = 1 (a) (b)

Figure 5. 3: (a) Harmonically excited axially vibrating fixed-free uniform rod (b) Axially vibrating uniform rod with fixed-fixed configuration

 ω 2 v′′ + λv = 0, λ =  c 2 , (5.6)  v(0) = 0, v′(1) = 1 where, c is the wave propagation speed in the rod.

General solution of the eigenvalue problem (5.6) is written as,

v(x) = Asin λ x + B cos λ x . (5.7)

Substituting the boundary condition of (5.6) in the solution (5.7) gives,

1 A = , B = 0 . (5.8) λ cos λ

110 Thus, frequency response function of the axially vibrating rod can be obtained as,

sin λ x v(ω) = . (5.9) λ cos λ

105 λ = π 1 2 λ = π λ = 5π 2 2 3 2 3

100 µ = π µ = π 1 2 2 -5 10 0 1 2 3 4 5 6 7 89 λ

Figure 5. 4: Collocated frequency response function of the uniform rod

The collocated frequency response function of the rod can be obtained from (5.9) by substituting for x = 1, as shown in the figure (5.4). The anti-resonant frequencies from frequency response function are the natural frequencies of same rod with fixed-fixed boundary configuration as shown as figure 5.3(b). Natural frequencies of the fixed-fixed configuration of the uniform rod can be obtained analytically by solving the following eigenvalue problem,

 ω 2 v′′ + µv = 0, µ =  c 2 . (5.10)  v(0) = 0, v(1) = 0

λ λ Thus by obtaining a single frequency response function, two spectral data sets 1 , 2 ,...

µ µ and 1 , 2 ,... for a longitudinally vibrating rod can be obtained.

5.1.3 Frequency Response Function of the Transversely Vibrating Beam

Similarly for a transversely vibrating structure, consider a uniform beam with fixed-free configuration as shown in figure 5.5(a) that is subjected to a harmonic

111 transverse excitation at its free end. The eigenvalue problem associated with this system can be written as,

β 4 EI v'''' − v = 0 where, β 4 = ω 2 , c 4 = , 0 < x < L , (5.11) c 4 ρA which is subject to following boundary conditions,

v(0) = 0 v′(0) = 0  . (5.12) v′′(1) = 0 v′′′(1) = 1

sin ωt

x x x L =1 L =1

(a) (b)

Figure 5. 5: (a) Harmonically excited transversely vibrating fixed-free uniform beam (b) Transversely vibrating uniform beam with fixed-pinned configuration

The general solution of (5.11) is expressed as,

β β β β v(x) = Asin x + B cos x + C sinh x + D cosh x , (5.13) c c c c and applying the boundary conditions from (5.12), the solution of (5.11) can be written as,

 0 1 0 1   β β   0 0   c c   βx   βx   βx   βx   A 0  − β 2 sin  − β 2 cos  β 2 sinh  β 2 cosh       c   c   c   c  B  = 0 . (5.14) C 0  2 2 2 2      c c c c  D 1  βx   βx   βx   βx  − β 3 cos  β 3 sin  β 3 cosh  β 3 sinh     c   c   c   c      c 3 c 3 c 3 c 3 

112 Considering the uniform beam with c =1, the system of equations (5.14) is solved and the following coefficients are calculated.

 β β + + 2β = − 2cos( )e 1 e  A 0.5 3 β 2β  β (2e + cos(β )e + cos β )  β β − + 2β = 2sin( )e 1 e  B 0.5 β β  β 3 (2e + cos(β )e 2 + cos β )  (5.15) 2cos(β )e β +1+ e 2β  c = 0.5  β 3 (2e β + cos(β )e 2β + cos β )  β β 2sin(β )e −1+ e 2 D = −0.5  β 3 (2e β + cos(β )e 2β + cos β )

These coefficients are substituted back in (5.13) to obtain the collocated frequency response function at x = 1.

β β 1 2 β β 3 4 β β 5 6

µ 1 µ 2 µ µ µ 5 6 µ 4 3

0 2 4 6 8 10 12 14 16 18 20 β

Figure 5. 6: Collocated frequency response function of the uniform beam

This frequency equation is plotted for different values of β and the resonant and anti- resonant frequencies obtained are shown in figure 5.6. The resonant frequencies corresponding to the eigenvalues of fixed-free uniform beam and the anti-resonance frequencies corresponding to the eigenvalues of fixed-pinned beam are shown in figure

5.5(b). The eigenvalues of fixed-pinned configuration can be obtain analytically by solving the following eigenvalue problem,

113 µ 4 EI v '''' − v = 0 where, µ 4 = ω~ 2 , c 4 = , 0 < x < L , (5.16) c 4 ρA with the following boundary conditions,

v(0) = 0 v′(0) = 0  . (5.17) v(1) = 0 v"(1) = 0

A single transfer function obtained from collocated modal analysis experiment performed on the beam produces two sets of data corresponding to the resonant frequencies of the fixed-free and the fixed-pinned configurations of the beam. These examples demonstrate the physical significance of the spectral data set that are used in following section for determining the solution of the inverse problems associated with the transversely vibrating beam.

Reconstruction of the physical parameters of beams using discrete models requires three sets of spectral data corresponding to three different boundary configurations (Golub and Van Loan 1996). Moreover, these data sets are not necessarily physically realizable, as the boundary conditions such as fixed-sliding configuration are difficult to obtain in practice. The possible means of extracting the spectral data, corresponding to fixed-sliding boundary conditions, are shown later in the chapter. The reconstruction method proposed here utilizes all the three set of spectral data and reconstruct the physical parameters of the beam.

5.1.4 Obtaining Fixed-sliding Frequencies of the Transversely Vibrating Beam

In order to estimate the physical properties of the beam, spectral data set corresponding to the fixed-sling boundary condition is required (Barcilon 1987).

However, these data sets are difficult to obtain in practice and need advanced instrumentation techniques. One way to obtain the resonant frequencies corresponding to

114 the fixed-sliding condition is to apply moment at the free end, measure corresponding rotations at the free end and evaluate the transfer function. Such method requires advanced instrumentation such as Laser vibration meter or high-speed cameras. Another way to acquire data requires shakers at the free end and those frequencies resulting no slope at the free end can be measured during the experiments ensuring fixed-sliding boundary conditions. Such experiments also require sophisticated instrumentation as they require measurement of slopes.

(a)

(b)

Figure 5.7: (a) Clamped-free piecewise beam – original beam (b) Clamped-clamped piecewise beam symmetric from the mid-point – modified beam

Due to the availability of limited instrumentation techniques available to us these data set can be obtained by the method of per-symmetry. The analysis of such approach is presented here. In this chapter, our aim is to solve the inverse problem associated with the piecewise beam. Assume a piecewise beam, as shown in the figure 5.7(a), and the spectral data corresponding to fixed-free, fixed-pinned and fixed-sliding frequencies are needed from the experiments. The resonant frequencies of the fixed-free beam can be obtained by performing modal analysis test by appropriately clamping one end of the beam to a very heavy structure. By measuring the collocated frequency response function, the zeros of the fixed-free configuration can also be obtained which are the

115 resonant frequencies of the fixed-pinned beam. Thus, we can obtain these two sets of data from a single experiment as shown analytically in the previous section.

MODE 1 1

0.5 0 -0.5

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

MODE 2 1 0.5

0 -0.5 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MODE 3 1 0.5 0

-0.5 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MODE 4 1

0.5 0

-0.5 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.8: Mode shapes of the modified beam containing the mode shapes of original beam for clamped-pinned and clamped sliding configuration

116 Now, consider another piecewise beam as shown in figure 5.7(b) such that half of the beam is of the shape of the original beam and other half of the beam is a mirror image

(Per-symmetric) of the original piecewise beam. This beam is clamped at the both ends.

The mode shape of such beam is evaluated and shown in the figure 5.8. It is clear from the figure 5.8 that all the mode shapes are symmetric from the mid-point. First mode shape of the fixed-fixed beam assures no slope at the mid point and it is corresponding to the first mode of the original beam with fixed-sliding configuration. Second mode shape ensures no displacement at the mid-point and it is the second mode of the original beam with fixed-pinned configuration.

Thus, the odd mode shape corresponding to the modified beam are the mode shapes of the piecewise beam with fixed-sliding boundary configuration and the even mode shapes corresponding to the modified beam are the mode shapes of the piecewise beam with fixed-pinned boundary conditions. Therefore by performing the modal analysis experiment on the modified beam with fixed-fixed boundary condition as shown in figure 5.7(b), the resonant frequencies of the original beam as shown in the figure

5.7(a) can be evaluated corresponding to the two boundary configurations i.e. fixed- pinned and fixed-sliding.

It is clear from the above analysis that all the frequencies or spectral data set can be obtained from the modal analysis experiments. In order to acquire three set of spectral data two modal analysis experiments were carried out, one with the original piecewise beam and other with the modified piecewise beam.

In this section, the analysis of spectral data is introduced. Basic definitions with necessary details are shown. The physical significance of poles and zeroes relating to the appropriate boundary conditions are shown for discrete and continuous systems. An

117 approach has been suggested to extract the desired spectral data corresponding to all three boundary conditions. In subsequent sections, the modal analysis experimental setup, formulation of the inverse problem of the piecewise beam, experimental data sets and the results of parameter estimation based upon experimental frequencies are shown.

5.2 Modal Analysis Experiment for Frequency Response Measurement

Modal analysis is a widely used technique for determining the natural characteristics or modal parameters, i.e. natural frequencies, mode shapes and damping factors, of any time-invariant structure. This technique is effectively used here to establish the relationship between the theoretical and the experimental results. The process of determining modal parameters using experiments involves experimental data acquisition, signal processing, analysis of raw input and output data in terms of time or frequency response functions. Shakers or impact hammers are commonly used for modal analysis experiments. In the present experimental method, only modal impact hammers and accelerometers are used. Experimental setup, data acquisition and data analysis techniques are briefly introduced. The main motivation here is to obtain frequency response functions of a non-uniform beam so that the all the three spectral data sets corresponding to the three different boundary conditions can be measured from the experiments. The theory of modal analysis and experimental modal analysis methods are well documented and detailed information regarding this technique can be found in

Ewins (2000), Braun (1986) and Goldman (1999).

5.2.1 Experimental Setup

The following components are used for the experimental modal analysis.

118 (a) Test structure: A piecewise steel beam is used where one end of beam is

clamped to a very heavy structure to ensure fixed-free boundary

configuration.

(b) Impact hammer: In order to obtain transient excitation, an impact hammer

from PCB Piezotronics (Model 291M78-086C05) is used. A force sensor

is mounted on striking end of hammerhead.

accelerometer from Piezoelectric (Model 353B33). Frequency range of the

accelerometer is 5 Hz - 4000 Hz and its sensitivity is 100mV/g.

Computer with DSA Software

I/O device (NI4551) Clamps

Impact hammer

Accelerometer

Piecewise beam Heavy Structure Figure 5.9: Experimental setup

(d) Data acquisition card: The raw data from impact hammer and

accelerometer are collected in the computer using data acquisition card

119 NI-4551 from National Instrument. It consists of one input and one output

channel and data can be collected simultaneously using two channels. The

hammer and accelerometer are connected to the data acquisition card

using BNC cables. A dynamic signal analyzer (DSA) software is used to

modify the setting of input and output response and to interpret and collect

raw data.

The experimental setup is shown in the figure (5.9) and the parameters, for the experiment, are summarized in the table (5.1).

Table 5.1: Experimental parameters

Sampling rate (∆t) 1.5625×10−4 sec Sampling frequency 6400 Hz. Resolution (∆f ) 3.90625×10−1 Hz. Number of data point 1024

Increasing the range of the sampling frequency may yield information about higher natural frequencies, but it tends to deteriorate the resolution due to the fixed number of data points in the sample period. Therefore, it is important to note that the sampling frequency must be low in order to achieve high resolution of frequencies lower than the Nyquist frequency. Our main concern here is to obtain the best resolution of data, as the proposed model is highly sensitive to perturbation. This phenomenon is explained later in this chapter and reason for choosing low sampling frequency is justified.

Some precautions are also taken to achieve successful modal analysis for the beam. It is important that the accelerometer be mounted appropriately on the beam structure. Inappropriate mounting of accelerometer will cause a relative motion between

120 accelerometer and surface during modal analysis test. Therefore, a stud mounting or a good quality of glue is recommended for accelerometer mounting. The sensitivity of hammer used for impact purpose is also an important issue. Selection of very sensitive settings will trigger hammer without any impact or for any movement of the person performing experiment. A low sensitivity setting requires a large impact force on the structure, which is not recommended, as it may damage the structure and may generate inconsistent impact. Hence, it is recommended to have 0.3 volt setting for triggering the hammer. Most important issue was to make sure that data acquisition is being performed correctly. An anti-alias filter in the hardware ensures that no aliasing is present when sampling the input and output response. The frequency response functions are evaluated using the DSA software. However, the time domain and frequency domain analysis are performed using the acquired data obtained from the modal analysis tests to ensure an accurate data acquisition process. It is ensured that all wiring connections are rigid so that small movement of wires would not lead to triggering.

5.3 Modeling of an Inverse Problem for the Piecewise Beam

In order to solve the inverse problem of piecewise beam by using algorithm 2, a non-dimensional model for the piecewise beam is developed here. The governing equation of motion for transversely vibrating Bernoulli-Euler’s beam is

∂ 2  ∂ 2 w(x,t)  ∂ 2 w(x,t)  E(x)I(x)  + ρ(x)A(x) , 0 < x < L . (5.18) ∂x 2  ∂x 2  ∂t 2

Substituting the harmonic motion of the form,

w(x,t) = v(x)sinωt , (5.19) in (5.18), leads to,

121 ∂ 2  ∂ 2v(x)   E(x)I(x)  − ρ(x)A(x)ω 2v(x) = 0, 0 < x < L . (5.20) ∂x 2  ∂x 2 

Above stated equation can be written in the dimensionless form as,

∂ 2  ∂ 2   v(s)  − ρ ρ ω 2 = < <  E0 E(s)I(x0 )I(s)  0 (s)A(x0 )A(s) v(s) 0, 0 s 1, (5.21) ∂s 2  ∂s 2  where,

 I(x)  x = Ls, v(s) = v(x), I(s) =  I  0 , (5.22) A(x) E(x) ρ(x) A(s) = , E(s) = , ρ(s) =  ρ  A0 E0 0

ρ and the values of A(x0 ), I(x0 ), E(x0 ) and (x0 ) are assumed to be constant.

The dimensionless form of eigenvalue problem (5.21) thus can be written as,

″ ()E(s)I(s)v′′(s) − λρ(s)A(s)v(s) = 0, 0 < s < 1, (5.23) where,

ρ A(x )ω 2 L4 λ = 0 0 . (5.24) E0 I(x0 )

Now, consider a piecewise Bernoulli-Euler’s beam as shown in figure 5.10 (a), such that the physical properties of the beam are distributed along the length with modulus of elasticity,

E 0 < x < a E(x) =  1 , (5.25) < < E2 a x L density,

ρ 0 < x < a ρ(x) =  1 , (5.26) ρ < <  2 a x L cross sectional area

122  A 0 < x < a A(x) =  1 , (5.27) < < A2 a x L and moment of inertia

I 0 < x < a I(x) =  1 . (5.28) < < I 2 a x L

ρ ρ E1, 1, I1, A1 E2 , 2 , I2 , A2

(a)

ρ ρ E1, 1, I1, A1 E2 , 2 , I2 , A2

L (b)

ρ ρ E1, 1, I1, A1 E2 , 2 , I2 , A2

(c)

Figure 5.10: Piecewise constant beam (a) fixed-free boundary configuration (b) fixed- sliding boundary configuration (c) fixed-pinned boundary configuration

The following eigenvalue problem can be derived for the piecewise beam as,

 λρ A ρ A ω 2 L4 ρ A  v '''' − 1 1 v = 0 v '''' − 0 0 1 1 v = 0  1 E I 1 1 E I E I 1  1 1 0 0 1 1 , (5.29) λρ A ρ A ω 2 L4 ρ A v'''' − 2 2 v = 0 v '''' − 0 0 2 2 v = 0  2 2 2 2  E2 I 2 E0 I 0 E2 I 2

123 where, the physical parameters are assumed to be constant for each element of the beam, such as, for a = a / L , modulus of elasticity,

E 0 < s < a E(s) =  1 , (5.30) < < E2 a s 1 density,

ρ 0 < s < a ρ(s) =  1 , (5.31) ρ < <  2 a s 1 cross sectional area

 A 0 < s < a A(s) =  1 , (5.32) < < A2 a s 1 and moment of inertia

I 0 < x < a I(s) =  1 . (5.33) < < I 2 a x 1

The eigenvalue problem for the beam can be written as,

 β 4 ω 2 L4 ρ A E I v'''' − v = 0, where, β 4 = 0 0 ,c 4 = 1 1 , 0 < s < a  1 c 4 1 E I 1 ρ A  1 0 0 1 1 , (5.34) β 4 ω 2 L4 ρ A E I v'''' − v = 0, where, β 4 = 0 0 ,c 4 = 2 2 , a < s < 1  2 4 1 2 ρ  c2 E0 I 0 2 A2 with the following boundary conditions and matching conditions,

v (0) = 0 v (a) = v (a) v′′(a) = q v′′(a) v′′(1) = 0  1 1 2 2 1 2 2 , (5.35) ′ = ′ = ′ ′′′ = ′′′ ′′′ = v1 (0) 0 v1 (a) v2 (a) v1 (a) q1v2 (a) v2 (1) 0 where,

E I = 2 2 q1 . (5.36) E1I1

The solution of (5.34) can be written as,

124  β β β β  v (s) = z sin s + z cos s + z sinh s + z cosh s, 0 < s < a  1 1 c 2 c 3 c 4 c  1 1 1 1 β β β β . (5.37) v (s) = z sin s + z cos s + z sinh s + z cosh s, a < s < 1  2 5 6 7 8  c2 c2 c2 c2

Applying the boundary and matching conditions from (5.35) in the solution (5.36) leads to the formulation of transcendental eigenvalue form,

A(β )z = o , (5.38) where,

 0 1 0 1 0 0 0 0     1 c1 0 1 c1 0 0 0 0 0   βa βa βa βa βa βa βa βa   s c sh ch − s − c − sh − ch   c1 c1 c1 c1 c2 c2 c2 c2   βa βa βa βa βa βa βa βa  c − s ch sh − c s − ch − sh  c c c c c c c c   1 1 1 1 2 2 2 2   c1 c1 c1 c1 c2 c2 c2 c2   β β β β β β β β  − a − a a a a a − a − a  s c sh ch q1s q1c q1 sh q1ch   c1 c1 c1 c1 c2 c2 c2 c2  A =  c 2 c 2 c 2 c 2 c 2 c 2 c 2 c 2   1 1 1 1 2 2 2 2  βa βa βa βa βa βa βa βa − c s ch sh q c − q s − q ch − q sh   1 1 1 1  c1 c1 c1 c1 c2 c2 c2 c2  3 3 3 3 3 3 3 3   c1 c1 c1 c1 c2 c2 c2 c1   β β β β   − s − c sh ch  c c c c  2 2 2 2  0 0 0 0 2 2 2 2  c2 c2 c2 c2   β β β β   − c s ch sh  , (5.39)  c c c c  0 0 0 0 2 2 2 2  3 3 3 3   c2 c2 c2 c2  and where, the following notations have been used in (5.39)

s → sin, c → cos, sh → sinh, ch → cosh, (5.40) with

= ()l T z z1 z2 z8 . (5.41)

Now, for the fixed-sliding configuration of beam, as shown in figure 5.10(b), the eigenvalue problem can be written as,

125  σ 4 ω 2 L4 ρ A E I v'''' − v = 0, where, σ 4 = 0 0 ,c 4 = 1 1 , 0 < s < a  1 c 4 1 E I 1 ρ A  1 0 0 1 1 , (5.42) σ 4 ω 2 L4 ρ A E I v'''' − v = 0, where, σ 4 = 0 0 ,c 4 = 2 2 , a < s < 1  2 4 1 2 ρ  c2 E0 I 0 2 A2 with the following boundary and matching conditions,

v (0) = 0 v (a) = v (a) v′′(a) = q v′′(a) v′ (1) = 0  1 1 2 2 1 2 2 . (5.43) ′ = ′ = ′ ′′′ = ′′′ ′′′ = v1 (0) 0 v1 (a) v2 (a) v1 (a) q1v2 (a) v2 (1) 0

Applying these boundary and matching conditions to the general solution form,

 σ σ σ σ v (s) = z sin s + z cos s + z sinh s + z cosh s, 0 < s < a  1 1 2 3 4 c1 c1 c1 c1  σ σ σ σ , (5.44) v (s) = z sin s + z cos s + z sinh s + z cosh s, a < s < 1  2 5 6 7 8  c2 c2 c2 c2 the following transcendental eigenvalue problem,

B(σ )z = o , (5.45) where,

 0 1 0 1 0 0 0 0     1 c1 0 1 c1 0 0 0 0 0   σa σa σa σa σa σa σa σa  s c sh ch − s − c − sh − ch  c c c c c c c c   1 1 1 1 2 2 2 2  σa σa σa σa σa σa σa σa  c − s ch sh − c s − ch − sh   c c c c c c c c   1 1 1 1 2 2 2 2   c1 c1 c1 c1 c2 c2 c2 c2   σa σa σa σa σa σa σa σa  − s − c sh ch q s q c − q sh − q ch  c c c c 1 c 1 c 1 c 1 c   1 1 1 1 2 2 2 2  = 2 2 2 2 2 2 2 2 B  c1 c1 c1 c1 c2 c2 c2 c2   σ σ σ σ σ σ σ σ  − a a a a a − a − a − a  c s ch sh q1c q1s q1ch q1sh   c1 c1 c1 c1 c2 c2 c2 c2   3 3 3 3 3 3 3 3  c1 c1 c1 c1 c2 c2 c2 c1  σ σ σ σ   c s ch sh   c c c c  0 0 0 0 2 − 2 2 2  c c c c   2 2 2 2  σ σ σ σ  − c s ch sh  . (5.46)  c2 c2 c2 c2   0 0 0 0 3 3 3 3   c2 c2 c2 c2 

126 Similarly, for a fixed-pinned configuration of the beam, as shown in figure 5.10(c), the eigenvalue problem can be written as,

 µ 4 ω 2 L4 ρ A E I v'''' − v = 0, where, µ 4 = 0 0 ,c 4 = 1 1 , 0 < s < a  1 c 4 1 E I 1 ρ A  1 0 0 1 1 , (5.47) µ 4 ω 2 L4 ρ A E I v'''' − v = 0, where, µ 4 = 0 0 ,c 4 = 2 2 , a < s < 1  2 4 1 2 ρ  c2 E0 I 0 2 A2 with the following boundary and matching conditions,

v (0) = 0 v (a) = v (a) v′′(a) = q v′′(a) v (1) = 0  1 1 2 2 1 2 2 . (5.48) ′ = ′ = ′ ′′′ = ′′′ ′′ = v1 (0) 0 v1 (a) v2 (a) v1 (a) q1v2 (a) v2 (1) 0

Applying these boundary and matching conditions to the general solution form,

 µ µ µ µ v (s) = z sin s + z cos s + z sinh s + z cosh s, 0 < s < a  1 1 2 3 4 c1 c1 c1 c1  µ µ µ µ , (5.49) v (s) = z sin s + z cos s + z sinh s + z cosh s, a < s < 1  2 5 6 7 8  c2 c2 c2 c2 the following transcendental eigenvalue problem,

C(µ )z = o , (5.50) where,

 0 1 0 1 0 0 0 0     1 c1 0 1 c1 0 0 0 0 0   µa µa µa µa µa µa µa µa  s c sh ch − s − c − sh − ch  c c c c c c c c   1 1 1 1 2 2 2 2  µa µa µa µa µa µa µa µa  c − s ch sh − c s − ch − sh   c c c c c c c c   1 1 1 1 2 2 2 2   c1 c1 c1 c1 c2 c2 c2 c2   µa µa µa µa µa µa µa µa  − s − c sh ch q s q c − q sh − q ch  c c c c 1 c 1 c 1 c 1 c  =  1 1 1 1 2 2 2 2  C 2 2 2 2 2 2 2 2  c1 c1 c1 c1 c2 c2 c2 c2   µ µ µ µ µ µ µ µ  − a a a a a − a − a − a  c s ch sh q1c q1s q1ch q1sh   c1 c1 c1 c1 c2 c2 c2 c2   3 3 3 3 3 3 3 3  c1 c1 c1 c1 c2 c2 c2 c1  µ µ µ µ   0 0 0 0 s c sh ch   c2 c2 c2 c2   µ µ µ µ   − s − c sh ch  c c c c  0 0 0 0 2 2 2 2  . (5.51)  2 2 2 2   c2 c2 c2 c2 

127 After evaluating the transcendental matrices corresponding to three different boundary conditions, algorithm 2 is successfully applied to solve the inverse problem of piecewise constant beam.

5.4 Identification of the Physical Properties of the Piecewise Beam

Determination of the physical parameters of the experimental piecewise beam, as shown in the figure 5.11(a), is developed here.

(a) (b)

0.1016 (4”) 0.0508 (2”)

0.3048 (12”) 0.0127 (0.5”) 6096 (24”) (c)

0.0508 (2”) 0.1016 (4”)

0.3048 (12”) 0.6096 (24”) 0.0127 (0.5”)

1.2198 (48”)

(d)

Figure 5.11: Experimental beams (a) original beam with fixed-free configuration (b) modified beam with fixed-fixed configuration (c) dimensions of the original piecewise beam (d) dimensions of the modified piecewise beam

Two beams are selected for experimental purposes in order to obtain the three sets of spectral data. The physical parameter estimation of the original piecewise beam as

128 shown in figure 5.11(a) is considered here. The modified beam as shown in figure 5.11(b) is used for obtaining the fixed-sliding and fixed-pinned frequencies of the original beam.

The original beam is clamped at one end, other end is free, and the modified beam is clamped at the both ends. The dimensions of both beams are shown in figure 5.11(c) and

5.11(d). The dimensions, shown in this figure, are in meters and those in the brackets are in inches.

In order to estimate the four physical parameters, c1 , c2 , q1 and a of the beam four natural frequencies are required. First two natural frequencies are corresponds to the fixed-free boundary condition of the original beam. Other two natural frequencies are the natural frequencies obtained from the modified beam with fixed-fixed boundary conditions and they are the fundamental natural frequencies of the original beam corresponding to the fixed-sliding and fixed-pinned boundary conditions.

Table 5.2: Physical parameters of the piecewise beam

Physical parameters Physical Values Physical Values (actual) parameters (actual) parameters (non- (units) (non-dimensional) dimensional) = = Pa × 12 = = 1 E1 E2 E 2 10 E1 E2 E ρ = ρ = ρ 3 ρ = ρ = ρ 1 2 kg/ m 7872 1 2 1 m2 × −3 2 A1 1.29032 10 A1 A 6.4516×10−4 1 2 A2 m4 × −8 2 I1 1.7343 10 I1 I 8.6715×10−9 1 2 I 2 a m a 0.5 L L 1 ρ c1 = E1I1 1 A1 341.4846 c1 1 ρ 341.4846 1 c2 = E2 I 2 2 A2 c2

E2 I 2 E1I1 0.5 q1 0.5

129 Now, let us define,

= ()β F1 det A(c1 ,c2, q,a, 1 F = det()A(c ,c q,a, β 2 1 2, 2 , (5.52) = ()σ F3 det B(c1 ,c2,q,a, 1 = ()µ F4 det C(c1 ,c2, q,a, 1 where the necessary matrices are obtained from (5.39), (5.46) and (5.51).

(a) (b)

Figure 5.12: Modal analysis results for piecewise beam using ANSYS tetrahedral elements for fixed-free and fixed-pinned configuration

The values of non-dimensional constants in (5.24) are assumed to be,

= × 12 ρ = = × −8 = E0 2 10 , 0 7872, I 0 1.73 10 , A0 0.0013 , (5.53)

130 for the eigenvalue problem (5.34), (5.42) and (5.47). Actual values and the corresponding non-dimensional values, of the physical parameters of the piecewise beam, are tabulated in table 5.2.

Response of the hammer

Response of the accelerometer

Frequency response function

Hammer

Accelerometer

Test Structure

Figure 5.13: Response of the accelerometer, hammer and frequency response function of the beam from modal analysis experiment

These four natural frequencies are also obtained from finite element analysis and analytical analysis in order to verify the experimental results. Modal analysis simulation is performed in ANSYS for both the beams in order to obtain the natural frequencies from finite element method. The results from ANSYS are shown in figure 5.12.

131 Analytical natural frequencies are obtained by solving the associated transcendental eigenvalue problem (5.34), (5.42) and (5.47) as described in chapter 3 by using the developed algorithm 1.

(38.580 Hz.) (183.05 Hz.) β β 1 2

105

100

0 50 100 150 200 250 300

hz. (a)

(54.60 Hz.) (129.92 Hz.) (248.50 Hz.) σ µ σ 1 1 2

5 10

0 10

0 50 100 150 200 250 300

hz. (b)

Figure 5.14: The frequency response function (a) original beam fixed-free boundary condition (b) modified beam fixed-fixed boundary condition

Modal analysis experiments are carried out for both the beams as described in the previous section. The response from the accelerometer, hammer and their corresponding frequency response function can be extracted from modal analysis experiments.

132 Examples, of hammer, accelerometer and the frequency response function, are shown in the figure (5.13). Two experiments were carried out, one with the original beam and other with the modified beam and two sets of frequency response functions are extracted.

Ten sets of modal analysis tests are carried out for both the beams and the frequency response function of the original beam and the modified beam are plotted in the figure 5.14.

The mean values and variances of the resonant frequencies from the experiments corresponding to the three boundary conditions are compared with the analytical natural frequencies of the piecewise beam and those obtained by the finite element method using

ANSYS. The natural frequencies are compared and tabulated in the table 5.2.

Table 5.2: Comparison of natural frequencies of piecewise beam (in Hz.)

Trans ANSYS Experiments Fixed free 37.5205 37.783 38.58 ± 0.12 beam 176.8519 176.356 183.05 ± 0.43 493.2921 495.167 ------Fixed sliding 52.8827 53.522 54.60 ± 0.10 beam 237.0875 236.195 ------Fixed pinned 125.0400 125.732 129.92 ± 0.04 beam 399.2687 399.44 ------

It is clear from table 5.2 that the obtained natural frequencies from experiments are closer to the analytical natural frequencies. The error between the experimental and the analytical natural frequencies may be further reduced by using a modal analysis kit with high-resolution data acquisition capability and ensuring perfect boundary conditions.

These natural frequencies can be written as non-dimensional eigenvalues from (5.34),

(5.42) and (5.47) and they are compared with the analytical eigenvalues in table 5.3.

133 Table 5.3: Comparison of non-dimensional eigenvalues

Analytical Experimental Fixed-free Fixed-sliding Fixed-pinned Fixed-free Fixed- Fixed- sliding pinned 2.1773 2.5849 3.9748 β = σ = µ = 1 2.2078 1 2.6267 1 4.0516 4.7271 5.4733 7.1027 β = 4.8098 ------7.8949 8.6725 10.1646 2 ------

Now, the inverse problem associated with piecewise constant beam can be stated as follows:

β β “Given non-dimensional eigenvalues 1 , 2 corresponding to the fixed-free boundary

σ µ conditions, 1 corresponding to the fixed-sliding boundary condition and 1 corresponding to the fixed-pinned boundary condition from the modal analysis

experiments then evaluate the physical parameters of the piecewise beam c1 ,c2 , q1 and a ”.

System of equations (5.52) is solved by algorithm 2 and using analytical eigenvalues from table (5.3). The estimated parameters are identified accurately as shown in table 5.4. It can be concluded from table 5.4 that if the true eigenvalues are used for parameter estimation of such piecewise beam then the true identification can be obtained.

Table 5.4: Identification of the physical properties of piecewise constant beam from true eigenvalues (0) = (0) = (0) = (0) = Initial Guess: c1 0.9 , c2 0.9 , q1 0.4 , a 0.4 Natural Input spectrum Estimated Exact values Error Frequencies (Eigenvalues) Parameters Theoretical β = = 37.5205 1 =2.17734086976699 c1 1 c1 1 ------β = = 52.8827 2 =4.72712385081856 c2 1 c2 1 ------σ = = 125.0400 1 =2.58493082960977 q1 0.5 q1 0.5 ------µ = = 176.8519 1 =3.97481205674621 x1 0.5 x1 0.5 ------

134 Now, the system of equations (5.52) is resolved by using experimental eigenvalues. In order to demonstrate the results, five sets of experimental eigenvalues, with their mean value, are used as input spectrum, and the parameters are identified using algorithm 2. The results of parameter estimation are tabulated in the table 5.5.

Table 5.5: Identification of the physical properties of piecewise constant beam from experimental eigenvalues

(0) = (0) = (0) = (0) = Initial Guess: c1 0.9 , c2 0.9 , q1 0.4 , a 0.4 Data Natural Input spectrum Estimated True Error set Frequencies ()β σ µ β T values values 1 1 1 2 ()T ()T c1 c2 q1 a c1 c2 q1 a

1. 38.28 2.1993 1.0341 1 3.41 % 54.30 2.6193 0.9973 1 0.27 % 129.69 4.0480 0.4737 0.5 5.26 % 182.03 4.7958 0.4711 0.5 5.77 % 2. 38.21 2.1973 1.0331 1 3.31 % 54.29 2.6191 1.0018 1 0.18 % 130.07 4.0540 0.4861 0.5 2.77 % 182.81 4.8061 0.4741 0.5 5.19 % 3. 38.67 2.2104 1.0159 1 1.58 % 54.68 2.6285 1.0196 1 1.95 % 129.68 4.0479 0.5097 0.5 1.94 % 183.21 4.8113 0.4971 0.5 0.57 % 4. 38.67 2.2104 1.0243 1 2.42 % 54.68 2.6285 1.0134 1 1.34 % 130.07 4.0540 0.4977 0.5 0.45 % 183.59 4.8163 0.4918 0.5 1.65 % 5. 39.06 2.2216 1.0202 1 2.01 % 55.07 2.6378 1.0177 1 1.76 % 130.07 4.0541 0.4942 0.5 1.16 % 183.59 4.8163 0.4948 0.5 1.04 % Mean 38.97 2.2078 1.0241 1 2.42 % 54.99 2.6267 1.0107 1 1.07 % 130.31 4.0516 0.4942 0.5 1.15 % 183.44 4.8092 0.4866 0.5 2.68 %

It is clear from table 5.5 that the physical parameters are identified with fair accuracy. The small error between true and identified parameters suggests that the

135 effective length of the experimental beam may be slightly different from the original beam because of clamping.

It is also clear from table 5.5 that the inverse algorithm is not very sensitive to small perturbation because all the data set converge to different solution that are the perturbations of the actual solution.

Based upon the experimental results following conclusions are drawn for solving inverse transcendental eigenvalue problem associated with vibrating beams:

• If the experimental natural frequencies are true natural frequencies of the

structure, the parameter estimation can be achieved accurately.

• In order to achieve accurate eigenvalues boundary conditions i.e. fixed end,

must be imposed properly.

• Either two separate experiments or sophisticated instrumentation techniques

are needed in order to obtain the three set of spectrum for parameter

identification.

• Small perturbation in the experimental frequencies may not change the

identification process significantly.

• With appropriate spectral data sets a non-uniform beam can be identified with

fair accuracy by approximating the beam with piecewise constant physical

properties.

• Similar results can be obtained towards the identification of non-uniform rods

from experiments. Due to the limitation of instrumentation capabilities, these

experiments were not carried out, however theoretically such identification is

shown in chapter 4.

136 • In conclusion, a non-uniform continuous system such as rods and beams can

be modeled accurately by approximating it using another continuous system

with piecewise constant properties and associated inverse problems can be

solved towards the physical parameter estimation from experimental data.

137 6. OTHER ENGINEERING APPLICATIONS

6.1 Introduction

It was shown in the previous chapters that the mathematical models developed using piecewise constant approximation of continuous system, can be used for solving direct and inverse problems related to vibrating non-uniform beams and rods. In this chapter, other possible future contributions, using this mathematical model, are presented.

They involve vibration control, structural health monitoring (damage detection), and dynamics of composite structures. It is illustrated here that this new mathematical model, along with the algorithms developed in this dissertation can be used to solve various problems in engineering applications.

Problems related to vibration control of continuous system and damage detection techniques in beams are discussed here. A few fundamental problems are solved and their results are compared with the existing results. Some derivations and examples relevant to related research are also presented in the appendices. The idea here is to show that the new mathematical model is effective in solving other challenging engineering problems.

Also, the algorithms developed for solving direct and inverse vibration problems are versatile and they can be used efficiently for solving other practical problems.

6.2 Application in Vibration Control

In most of structures and machinery, vibration is often undesirable because it may cause damage to the machine parts, increase the noise and result in malfunctioning.

Suppressing or controlling, such undesirable vibrations, is an important engineering problem and different methods are studied and developed for this purpose. Vibration

138 control can be achieved by active means (use of actuators) and passive means (use of passive elements such as springs or damping materials). One popular method of designing control for reducing the steady state vibrations of a harmonically excited system is by using the Frahm dynamic absorber (Den Hartog 1947, Inman 1994). The vibration absorption is done by appropriately designing a mass-spring system that counters the steady state force applied on a particular degree of freedom.

m mn−1 mn 1 sin ϖt

k1 kn−1 kn

x1 xn-1 xn

(a) Original system

m m m 1 n−1 n sin ϖt

k1 kn−1 kn

= α T u(t) en−1x

(b) Controlled system

Figure 6.1: Discrete mass-spring system excited by harmonic excitation (a) original system to be controlled (b) controlled system with the modification in structure

Absorption of the steady state motion of a particular degree of freedom in discrete mass spring or mass spring damper system is closely related to zero assignment problems

(Mottershead, 1998, Mottershead and Lallement, 1999, and Ram 1998-b). However, in order to assign zeros to the system, the knowledge of complete state is required.

Recently, a method of dynamic absorption by means of passive and active control is developed by Singh and Ram (2000). It is shown that under certain conditions only one

139 zero can be assigned by a single sensor and a single actuator. This method allows attaching the absorber to any other degree of freedom for which the motion is to be absorbed. The theory of dynamic absorber for un-damped and damped systems is developed in this paper. Criterion for passive or active control implementation and stability of the system is also developed.

Consider a multi-degree-of-freedom mass-spring system, as shown in figure 6.1, that is governed by

DD + = ω + Mx Kx en sin t en−1u(t ), (6.1)

where ei is the i-th unit vector of appropriate dimension and

= α T = α u(t) en−1x xn−1 . (6.2)

The control force u(t) may be realized, whenever α < 0 , by a spring of constant

−α connected between (n −1) − th degree of freedom and the ground, or alternatively by active feedback control. The problem of dynamic absorption of the steady state motion of a particular degree of freedom can be stated as,

“If the values of M and K are known then determine the control gain α such that

= = xn (t) vn 0 ”,

where xn and vn are the n-th elements of the vectors x(t) and v , respectively.

A method for achieving such absorption in conservative vibratory systems is derived in appendix B. An example is also presented demonstrating such absorption. This method can also be used for achieving the dynamic absorption in continuous structures i.e. beams and rods by discretizing the continuous system by appropriate mass spring or mass spring rod model.

140 Recently, a new method is developed towards the elimination of steady state vibration at a desired location in beams by means of nodal control (Singh, A., 2002-a). A theoretical approach and experimental verification is shown to demonstrate that the steady state motion at the point of excitation of an uniform beam can be absorbed by means of a control force determined from displacement information at the point of application. Closed form solution for the control gain and a criterion for implementing the control by active and passive means are also developed.

However, this method requires a closed form solution of the continuous system, which may not be available for beams with arbitrary physical parameter distribution. In order to overcome this difficulty, approximation based upon new mathematical model

(see, chapter 3) can be used to demonstrate the successful extension of nodal control method for any non-uniform structure.

Consider an exponential beam clamped at one end and harmonically excited at the other end as shown in figure 6.2(a). Assume the distribution of area to be A(,x) = e x moment of inertial to be I(x) = e x / 12 and constant E and ρ . Thus the governing differential equation can be written as,

E ∂ 2  ∂ 2 w  ∂ 2 w e x  + ρe x = 0, 0 < x < L , t > 0 . (6.3) 12 ∂x2  ∂x2  ∂t 2 with these boundary conditions,

 ∂w(0,t) w(0,t) = 0, = 0,  ∂  x . (6.4) ∂ 2 w(L,t) E ∂  ∂w(L,t)   = 0, e x  = cosωt  ∂x 2 12 ∂x  ∂x 

The problem for consideration here can be described as follows:

141 “Determine the control gain α at the location a on the beam to eliminate the motion at the free end of harmonically excited exponential beam”.

Let the exponential beam be approximated by another beam with piecewise constant physical properties, as shown in figure 6.2(b). The objective here is to apply a control force at the distance a from fixed-support to achieve no steady state motion at the point of excitation. For the case, a = 0.5 control gain α is evaluated for different excitation frequency by using the approximated piecewise constant beam model with the method shown by Singh A. (2002-a).

w(x,t)

a αw(a,t)

L ω cos t (a)

a αw(a,t) L cosωt (b)

Figure 6.2: Exponential beam with harmonic excitation at the free end (a) controlled beam (b) controlled beam with piecewise constant approximation

142 Exponential beam is discretized by finite difference method by equivalent mass- spring-rod model as shown in chapter 2 and associated mass and spring matrices M and

K are obtained. These matrices are then used to calculate the control gain by solving 6.1 and 6.2 (Singh and Ram, 2000). The control gain calculated by piecewise constant approximation and finite difference approximations are compared in figure 6.3 with the results obtained by Singh A. and Ram (2002). The system of equations and associated matrices involved in such calculations are shown in appendix B.

By analytical method (Singh, A. and Ram, 2002)

By developed approximating method of model order n=6 By mass-spring-rod approximation of model order n=20,

(Singh and Ram, 2000) 2000

1500

1000 α

500

-500

-1000

-1500

-2000 51015202530 ω Figure 6.3: Comparison of the control gain

It is clear from the figure 6.3 that for fairly small model order n=6, control gain can be achieved with a fair accuracy by using piecewise constant approximation. Control

143 gain obtained by finite difference approximation of high dimension n=20 may provide relatively better approximation for low frequencies but it deteriorates for higher frequencies. Whereas, the piecewise constant approximated model provides uniform accuracy for all the excitation frequencies.

It is demonstrated here, that the piecewise constant approximation of continuous system can be successfully used in developing practical solutions to certain structural control problems. This approximated model can also be successfully used for future research problem associated with vibration control and pole assignment in any non- uniform rods and beams by circumventing the need for associated closed form solutions.

6.3 Application in Structural Health Monitoring

Monitoring of the integrity of the structures during routine maintenance for any possible damage is known as structural health monitoring. There are various testing methods for detection of damage in structures. Most commonly used non-destructive testing methods are vibration based damage detection techniques. In such testing methods detection of damage in structures are performed through measurement of their modal parameters i.e. natural frequencies and mode shapes. These parameters can be easily obtained by performing modal analysis test and these tests are cheaply available.

Significant research has been done in the past to detect damage by using different non- destructive vibration based detection techniques. The results are summarized in the technical report by Doebling et. al. (1996) and a review paper of Salawu (1997).

One of the benchmark problems associated with structural health monitoring is to detect the location and quantification of crack in transversely vibrating beam by using natural frequencies. The basic idea is that modal parameters (notably frequencies, mode shapes, and modal damping) are functions of the physical properties of the structure

144 (mass, damping, and stiffness). Therefore, changes in the physical properties will cause changes in the modal properties (Doebling et. al., 1996). Most of the recent research work is limited to the modeling and detection of crack in the uniform beams (see, Liang et. al.,

1991, Yokoyama and Chen, 1998, Just Agosto, 1997) by performing the parametric analysis. Existence of a crack in the beam is commonly represented by two beams connected with a rotational spring at the location of crack (Rizos et. al., 1990 and Liang,

1991). Stiffness of the rotational spring is proportional to the crack severity and is equivalent to the jump in the slope due to additional rotation caused by reduction in the moment of inertia.

a E, A, ρ, I

(a)

θ K

a L

(b)

Figure 6.4: (a) Damaged beam, (b) Damaged beam model where a rotational spring represents damage in the beam

The idea here is to demonstrate that such modeling leads to transcendental eigenvalue problems and they can be solved by using algorithm 1, developed in the previous chapters. Direct problem related to damage detection, where the natural

145 frequencies of the cracked beam are evaluated by knowing the location and magnitude of the crack, is shown here. The associated inverse problem of determining the magnitude and location of the crack based upon natural frequencies is solved using algorithm 2 and circumventing parametric analysis, are also presented here. Moreover piecewise constant approximation model of the beam may be used to solve direct and inverse problem associated with damage detection in non-uniform beams.

Consider a uniform cantilever beam shown in figure 6.4(a), which is cracked at the location a from the fixed end. Let the depth of the crack be d , length, width and thickness of the beam are L,b and t respectively.

Damage in the beam is modeled by dividing the beam into two parts and placing a rotational spring at the location of crack. The stiffness of rotational beam represents the change in moment of inertia, and it can be calculated by following formula (see, Rizos et. al. 1990),

θ EI K = , (6.5) 0.5346tJ where, E is the modulus of elasticity, I is the moment of inertia and J is a local compliance function that can be evaluated by the following equation,

1.864α 2 − 3.95α 3 +16.375α 4 − 37.226α 5 + 76.81α 6 −126.9α 7 J =  ,(6.6)  +1721α 8 −143.97α 9 + 66.56α 10

d with, α = . t

The equation of motion of the cracked beam shown in figure 6.4(b) can be written as the transcendental eigenvalue problem of the following form,

A(β , K θ ,a) = o . (6.7)

146 Table 6.1: Lower two natural frequencies of the cracked and un-cracked beam for different locations and magnitudes of crack β β 1 2 Un-cracked beam K θ = 0 1.87510406871196 4.69409113297420 a = 0.1 K θ = 0.1 1.80483652904047 4.66852922335703 K θ = 0.2 1.74202113599894 4.64626043566315 θ 1.68562441933304 4.62672085593214 K = 0.3 1.54668671740515 4.58040924809461 K θ = 0.6 a = 0.3 K θ = 0.1 1.84111327437560 4.68430881621360 K θ = 0.2 1.80888775279854 4.67504202695181 θ 1.77831152411678 4.66625633285685 K = 0.3 1.69541278029246 4.64247699142554 K θ = 0.6 a = 0.5 K θ = 0.1 1.86364903601104 4.64438872157657 K θ = 0.2 1.85234014785731 4.59706598356534 θ 1.84117865479808 4.55200497084703 K = 0.3 1.80858587938744 4.42925208264385 K θ = 0.6 a = 0.8 K θ = 0.1 1.87469588454521 4.68499545288847 K θ = 0.2 1.87428723168271 4.67583650402145 θ 1.87387811082445 4.66661624212004 K = 0.3 1.87264794738111 4.63860809445018 K θ = 0.6

Details of (6.7) are shown in the appendix C. For simplicity, by assuming

E = I = 1, the solution of the transcendental eigenvalue problem (6.7) is evaluated for different locations and magnitudes of the crack. The obtained eigenvalues,

ω 2 ρA β = 4 , (6.8) i EIL4

ω for the cracked beam is tabulated in table 6.1, where, i are the natural frequencies of the beam.

The fundamental eigenvalues are also plotted in figure 6.5 for different values of a and K θ . Such plots are used for solving the associated inverse problem for determining the location and magnitude of the crack (Just Agosto, 1997). These parameters can also be identified by plotting the parametric curves of K θ as a function of

147 ω assumed damage location a for a given set of natural frequencies i using finite element based techniques (Liang et. al., 1991).

1.90 θ = β 1.85 K 0 1

θ 1.80 K = 0.1

1.75 θ K = 0.2

1.70 θ K = 0.3 1.65

1.60

1.55 K θ = 0.6 1.50

θ 1.45 K = 0.9

1.40 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a

Figure 6.5: Fundamental eigenvalues of the cracked beam

The above-mentioned inverse problem can be resolved by solving associated inverse transcendental eigenvalue problem using algorithm 2. The associated inverse problem can be stated as follows,

β β “For the given values of first two eigenvalues 1 and 2 from the experiments, identify the crack location a and magnitude K θ ”.

β β Assuming the spectra 1 and 2 are available from the experiments then the associated functional as shown in chapter 4 can be written as,

 F = det()A()β ,a, K θ = 0  1 1 . (6.9) = ()()β θ = F2 det A 2 ,a, K 0

148 By solving the system of equation (6.9) using algorithm 2 for different set of spectrum as an input (highlighted in table 6.1), the location and the magnitude of crack are evaluated. By choosing different initial random guess for a particular set of input spectrum, the parameters are estimated and they are tabulated in table 6.2.

Table 6.2: Identification of crack location and its magnitude based upon their spectrum

Input spectrum Initial Guess Initial Guess Estimated Estimated (0) (0) θ a ()K θ a (K ) 1.80483652904047 0.1509 0.6979 0.1 0.1 4.66852922335703 1.80888775279854 0.3784 0.8600 0.3 0.2 4.67504202695181 1.84117865479808 0.6068 0.4860 0.5 0.3 4.55200497084703 1.87264794738111 0.4966 0.8998 0.8 0.6 4.63860809445018

It is evident from table 6.2 that the location of crack and its magnitude can be identified accurately by solving an inverse transcendental eigenvalue problem. Moreover, for the non-uniform beams (see, figure 6.6-a), closed form solution in the form of equation (6.7) may not be available. In that case, the non-uniform beam may be approximated as piecewise constant model (see, figure 6.6-b) and the corresponding transcendental eigenvalue problem can be developed and solved.

It is clear from the above section that identification of crack or damage can be successfully achieved by solving associated inverse transcendental eigenvalue problems.

Such method eliminates the evaluation of parametric curves and directly identifies the crack parameters using algorithm 2. Also, this piecewise constant model can be used for future research study for the damage models of non-uniform beams and rods. In order to develop an effective and robust methodology for damage detection techniques in non-

149 uniform rods and beams, based upon their spectral data, the developed mathematical model and algorithms can be used for future research.

E(x), A(x), ρ(x), I(x)

(a)

hk hk +1 h h2 θ h 1 K n

A Ak Ak+1 2 A1 An

a L (b)

Figure 6.6: (a) Non-uniform beam with a crack (b) Approximated model to solve associated direct and inverse crack problems

6.4 Possible Future Applications

The existing analytical models of composite laminates (figure 6.7-a) and sandwich beam (figure 6.7-b) lend themselves to the form of transcendental eigenvalue problem (equation 3.1). Such formulations are developed in Vinson, (1986) and Whitney

(1987).

Similarly, the approximation of laminated beams based upon coupled bending- tensional beam theory leads to a transcendental eigenvalue problem (equation 28,

Banerjee, 2001). In the case of sandwich structures, the transcendental model can be developed based on the assumption that the laminates are decoupled from the core but contact forces between them are maintained. By applying the continuity of the

150 displacement and stresses between the layers of sandwich beam a transcendental eigenvalue problem of the form (3.1) can be developed (Frostig, 1990, Tavallaey 2001, and Nilsson, 2002).

z z h a y l β ht i y hc h x x b l (a) (b)

Figure 6.7: (a) Laminate structure (b) sandwich structure with flexible core

Thus, the new mathematical model and inverse transcendental eigenvalue algorithm can be extended towards the identification of the material parameters for the composite sandwich structures and laminates. Currently, the problem of estimating the material constants for composite structures is limited to finite element methods (Soares et. al., 1993 and Rikard, 2001). Such methods require large number of experimental frequencies as they are of high model order. In engineering practice, higher frequencies are very difficult to obtain experimentally because most of the dynamical behavior of system which is governed by high-energy modes corresponding to lower frequencies. In order to address these difficulties, the presented research work on transcendental eigenvalue problem can be extended for composite structures. Solution of inverse transcendental eigenvalue problems associated with laminates and sandwich beams may be obtained.

In addition, problems of the evaluation of approximated critical buckling load for non-uniform column may also be solved. Possible future research can also be extended in

151 addressing inverse buckling problems (Elishakoff, 1991) and solving inverse problems

(Barcilon 1990), related to the multi-dimensional vibrating structures such as plates and frames.

152 7. CONCLUSIONS

Vibration characteristics, i.e. natural frequencies and mode shapes are widely used for regulating and controlling the response of structures. These characteristics can be obtained analytically by using appropriate mathematical models. Most of the structures, in real life, are continuous systems with certain distribution of their physical parameters. In the presence of non-homogeneous distribution of geometry, rigidity and density, these structures may not lend themselves to the closed form solution. In practice, finite dimensional mathematical models that are based upon finite-element and finite- difference techniques are used to approximate continuous system. The dynamic behavior of such systems cannot be accurately estimated because the spectrum of a continuous system is different from that of the associated finite dimensional model. It is therefore necessary to develop mathematical models that approximate the dynamic behavior of non-uniform continuous systems accurately.

In engineering design, certain desirable dynamic behaviors of the structure (such as beams, rods) are needed. In order to obtain these characteristics, a suitable mathematical modeling is also required. The true characteristics of a structure can also be obtained by dynamic tests e.g. modal analysis technique. Spectral data from the dynamic tests contain the information of both the eigenvalues and eigenvectors. During design process, the prototype can be tested to verify the desired behavior. Based upon the requirement of the cost involved or for the purpose of optimization, design may be changed. It can be achieved either by redesigning the structure or by making appropriate changes in the existing design which may involve change in their physical parameters

153 e.g. rigidity, density. Thus, it is required to identify the appropriate physical parameters based upon desired dynamic characteristics i.e. eigenvalues and eigenvectors. Such identification problems are known as inverse eigenvalue problems. These problems are of great importance as they can reveal important information about the physical parameters that can be further used for accurate mathematical modeling. Solutions of the direct and inverse problems associated with the continuous systems are developed and illustrated with few examples in this presented dissertation.

In chapter 2, mathematical models for the continuous systems e.g. axially vibrating rods and beams that are based upon finite difference and finite element techniques were studied. Finite dimensional matrices corresponding to approximated models are developed to evaluate eigenvalues and eigenvectors of distributed parameter systems. It is shown with examples that such approximation techniques cannot capture the true spectral characteristic of the continuous system. Since these methods use matrix approximation techniques leading to algebraic eigenvalue problem, they approximate the solution of continuous system by using trial functions in the form of polynomials. On the other hand, the solutions of continuous systems are transcendental equations and no polynomial function can approximate the strong variation of transcendental equations accurately. Approximated models of order n provide about n/3 eigenvalues that are accurate but at the same time they give almost two-third of the inaccurate eigenvalues.

Thus, there exists a need for such mathematical model that approximates any continuous system accurately.

Inverse eigenvalue problems related to vibrating structures were also studied in chapter 2. In the past, inverse problems related to discrete mass-spring systems can be solved uniquely. For illustration purpose, a continuous system is approximated by

154 equivalent mass-spring system and associated inverse problem is solved. It is shown that reconstruction of physical parameter for the continuous system is not possible with such an approximation. Difficulties associated in obtaining the solution for non-uniform rods and beams are demonstrated. Reasons for such inaccurate estimation were identified and studied. It is concluded that an accurate mathematical model is required to solve the associated direct and inverse eigenvalue problems.

Theoretically, to solve inverse eigenvalue problems associated with the continuous system, the complete knowledge of all the eigenvalues is required (Gladwell,

1986). In real life engineering applications, only partial information of eigenvalues is available. Thus, it is often necessary to develop a reconstruction procedure that involves system of smaller dimension. Recently, Chu (2001) has addressed following open research problem in the field of inverse eigenvalue problem related to vibrating structures.

“In structure design, often we are only interested in a few low order natural frequencies. Indeed, for large structures, it is impractical to calculate all of the frequencies and modes. How should one solve the problem if only a few low order frequencies are given?”

Considering the difficulties associated with existing research in the field of inverse problems associated with continuous system, a new low order analytical model is developed in chapter 3. In this approximation technique, a non-uniform continuous system is approximated by another continuous system with piecewise constant physical parameter distribution. Due to the availability of the closed form solution for an individual element, approximated matrix eigenvalue problem can be developed, by applying the appropriate continuity conditions between the elements. Such eigenvalue

155 problem is termed as transcendental eigenvalue problem where the elements of the matrices involved are transcendental functions. For non-uniform axially vibrating rods and transversely vibrating beams, transcendental eigenvalue problems were developed for arbitrary model order n. Obtaining the solution for such eigenvalue problems generally involves the evaluation of determinants of the matrices. Evaluations of such determinant should be circumvented, as they require symbolic computations. In order to solve transcendental eigenvalue problems, a Newton’s eigenvalue iteration method is developed. New mathematical models along with the algorithm were successfully used for evaluating the spectrum of non-uniform rod and beam structures. It has been shown that compared to finite element and finite difference methods, these mathematical models can contain the information of all the eigenvalues of the system. Moreover, all the eigenvalues obtained are of uniform accuracy. These mathematical models are further used to solve buckling problems associated with the composite loads and quadratic eigenvalue problems. It has been illustrated successfully that the new mathematical model approximate the true system more accurately and this model could be used for solving associated inverse problems for rods and beams.

In chapter 4, inverse problem associated with the non-uniform rod is developed.

The algorithm for solving direct problem is extended for solving the system of equations with multiple parameters. Based upon Newton’s eigenvalue iteration method, an algorithm for solving inverse transcendental eigenvalue problem is developed.

Mathematical expressions are derived to calculate the elements of Jacobian matrix in the algorithm. Physical parameters of piecewise rod have been identified based upon input spectrum. Due to the dual behavior of continuous systems, multiple solutions are obtained with certain initial guesses that were not close to the true solution. Physical

156 significance, of such solutions, is explored and a conjecture has been established. Based upon this conjecture only those solutions that will provide the lowest spectrum corresponding to the structure are accepted. The algorithm has also been modified to solve inverse problem in the least-square sense that leads to the true solution. In order to demonstrate the effectiveness of developed algorithm, non-uniform distribution of the cross sectional area for an exponential rod is identified based upon the eigenvalues. It is illustrated that a low model order (n=8) system can estimate the cross section variation with high accuracy.

The inverse problem for transversely vibrating beam is developed in chapter 5.

Identification of the physical parameters of a piecewise beam based upon spectral data has been developed. Physical significance of spectral data in solving inverse problems for vibrating structures has been illustrated with examples. In order to validate the developed model and algorithm, some laboratory experiments are carried out. . Two experiments were carried out in order to obtain the required three spectral data of the beam corresponding to three different boundary conditions. Spectral data for the piecewise beam is obtained from the modal analysis tests. Experimental results are observed to be accurate in comparison with the analytical results obtained from the transcendental eigenvalue problem. The experimental frequencies were taken as the input spectrum to solve the corresponding inverse problems. It was shown that the estimated parameters were identified accurately. If the input spectral data is the true spectral data for the system then the reconstruction procedure is observed to be accurate. However, experimental spectrum leads to a different solution that are in the neighborhood of the accurate solution. Modern instrumentation techniques, such as Scanning Laser Doppler

Velocimeter (SLDV) and high-speed camera, may avoid the experiments on two different

157 beams and all spectral data may be obtained by single experiments. Such advanced instrumentation along with the developed algorithm is recommended for the solution of such inverse problems.

In chapter 6, developed mathematical model and algorithms are used to solve other engineering problems such as vibration control and damage detection. It has been shown that this model can be used for solving vibration control problems associated with the non-uniform vibrating beam. Algorithm for solving inverse transcendental eigenvalue problems can be extended to estimate the location and magnitude of the crack in the beam based upon their measured frequencies.

The research work can be summarized as follow:

(a) A mathematical model is developed for non-uniform continuous systems

e.g. rods and beams.

(b) An effective algorithm based on Newton’s eigenvalue iteration method is

developed. It uses an algebraic eigenvalue solver to find the eigenvalues of

the transcendental eigenvalue problem iteratively.

mathematical model is solved to evaluate spectra of the non-uniform rods,

which may not be accurately evaluated by other numerical methods.

(d) This mathematical model is extended towards the solution of the inverse

problem of vibrating rods and beams.

(e) Physical parameters of the rods and beams are identified by true spectrum

of the system.

(f) Based upon the observation of the results, a conjecture has been

established by which a unique solution is identified.

158 (g) A set of modal tests are performed to validate the developed model and

algorithm

(h) Possible future contribution of the developed mathematical model and

algorithms has been demonstrated by solving problems for vibration

control and damage detection.

The conditions, under which an inverse transcendental eigenvalue problem has a solution, are as follow:

a) In order to estimate the physical parameters of the non-uniform rods, two

sets of spectral data corresponding to the fixed-free and fixed-fixed

boundary conditions are required. One non-spectral data i.e. mass and

static deflection are also required.

b) In order to estimate the physical parameters of the non-uniform beams,

three sets of spectral data corresponding to the fixed-free, fixed-sliding

and fixed-pinned boundary conditions are required. Two non-spectral data

i.e. mass and static deflection are also required.

c) The numerical scheme developed to solve transcendental eigenvalue

problem or inverse transcendental eigenvalue problems is based upon

Newton’s method, thus an appropriate initial guess close to the solution set

is required.

d) True spectrum as an input identifies the physical parameters accurately.

Spectrum set from experiments should be close to true eigenvalues.

Two fundamental issues associated with the inverse eigenvalue problems (Chu,

2001), (i) Solvability: obtaining a necessary or sufficient condition under which an inverse eigenvalue problem has a solution and (ii) Computability: development of a

159 numerical procedure by which, knowing a priori that the given spectral data are feasible, a matrix can be constructed numerically, are addressed in this research work.

In future research work, challenging issues such as robustness of developed algorithms, per-conditioning of the involved matrices, should be addressed in order to improve the developed numerical techniques. In order to obtain the precise experimental data for realistic structures latest modal testing techniques using Scanning Laser Doppler

Velocimeter (SLDV) may be used (Stanbridge and Ewins, 1998). Newton’s method based on higher order approximation of Taylor’s series or non-gradient optimization techniques may be used for developing robust numerical algorithm. The important problems of estimating physical parameters of two-dimensional and three-dimensional system are still open research problems. The developed approximation method may be extended for solving such problems.

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167 APPENDIX

Appendix A: Natural Frequencies of the Non-uniform Rods

1) A(x) = 2 − x , L = 1

Natural Exact element Transcendental eigenvalue Exact frequencies method problem (Eisenberger, 1991) (Eisenberger, 1991) n = 4 n = 8 n = 50 ω 1 1.79402 1.7803 1.7906 1.79401 ω 2 4.80863 4.7554 4.7912 4.80206 ω 3 7.91675 7.8109 7.8899 7.90896 ω 4 11.05686 10.7861 11.0063 11.03509 ω 5 14.21460 14.3466 14.1264 14.16799

2) A(x) = 3 − 4x + 2x 2 , L = 1

Natural Exact element Transcendental eigenvalue Exact frequencies method problem (Eisenberger, 1991) (Eisenberger, 1991) n = 4 n = 8 n = 50 ω 1 1.9708 1.9414 1.9634 1.97090 ω 2 4.8223 4.7642 4.8073 4.82076 ω 3 7.9259 7.8021 7.8958 7.91820 ω 4 11.1283 10.6250 11.0080 11.04144 ω 5 14.2193 14.5078 14.1247 14.17284

168 3) A(x) = (1+ x) 2 , L = 1

Natural Transcendental eigenvalue Rayleigh-Ritz frequencies problem approach n = 4 n = 8 (Abrate, 1995)

ω 1 1.2269 1.2268 1.16556 ω 2 4.6716 4.6421 4.60421 ω 3 7.8947 7.8235 7.78988 ω 4 11.3394 10.9865 10.94994 ω 5 13.7933 14.1462 14.101725

169 Appendix B: Dynamic Absorption in Conservative Vibratory System

Following derivations and an example are the excerpts of Singh and Ram (2000) demonstrating the method and an example of dynamic absorption in mass-spring system.

Given: M and K

α = = Find: such that xn (t) vn 0 , (B-1)

where xn and vn are the n-th elements of the vectors x(t) and v , respectively. We partition M and K in the form

 Mˆ mˆ   Kˆ kˆ  M =  n  and K =  n  , (B-2) T ˆ T mˆ n mnn  k n knn  where Mˆ and Kˆ are the ()()n −1 × n −1 leading sub-matrices of M and K , respectively, and obtain the following result.

Theorem 1: If

( ˆ −α T −ω 2 ˆ )= det K en−1en−1 M 0 , (B-3) then

= v.n 0 (B-4)

Proof: Substituting,

= α T = α u(t) en−1x xn−1 . (B-5) and

x(t) = v sinωt , (B-6) in

DD + = ω + Mx Kx en sin t en−1u(t) (B-7) gives

170 −ω 2 = +α (K M)v en en−1vn−1 . (B-8-a) or

−α T −ω 2 = (K en−1en−1 M)v en . (B-8-b)

The Cramer's Rule gives

det(Kˆ −αe eT − ω 2Mˆ ) v = n−1 n−1 , (B-9) n ()−α T − ω 2 det K en−1en−1 M

= and hence vn 0 whenever (B-6) is satisfied.

Let us now consider the problem of determining α such that (B-6) holds.

Evaluating the determinant of

ˆ −α T −ω 2 ˆ (K en−1en−1 M ) (B-10) by the (n −1) − th row gives

( ˆ −α T −ω 2 ˆ )= ( ˆ −ω 2 ˆ )−α ()~ − ω 2 ~ det K en−1en−1 M det K M det K M , (B-11)

~ ~ where K and M are the ()()n − 2 × n − 2 leading sub-matrices of K and M , respectively. Hence, for (B-6) to hold we choose

det()Kˆ −ω 2Mˆ α = ~ ~ . (B-12) det()K −ω 2M

If α < 0 then the control can be realized by passive element, i.e. a spring of constant −α . In any case the desired force can be applied using active control with a

α sensor measuring xn−1 (t ) and an actuator applying the force xn−1en−1 in real time.

Consider the mass-spring system shown in Figure B-1(a). Its differential equation of motion is

+ = Mx Kx en sin3t

171 where

1 0 0  4 −1 −1      M = 0 1 0 and K = −1 2 −1  . 0 0 1 −1 −1 2 

1 sin 3t 1 11 2 11

x1 x2 x3

(a) Original system

1 2 11sin 3t 1 1 1

6.8

(b) Modified system Figure B1. Three degree-of-freedom mass-spring system (a) Original system. (b) Modified system

Suppose we wish to eliminate the motion x3 by adding a spring between the middle mass and the ground. Then by (12)

  −     4 1 1 0  det   − 9   −1 2  0 1 α = = −6.8 . det()[]4 − 9 []1

The required spring constant is thus

k = 6.8.

The equation of motion for the modified system, shown in Figure B-1(b), is thus

172  DxD   4 −1 −1  x   0   1   1    DD + − −  =  x2   1 8.8 1  x2   0  ,  DD  − −      x3   1 1 2  x3  sin3t  which has a particular solution of the form (B-3)

 x (t)  v   1   1  =  x2 (t) v2 sin3t .      x3 (t) v3 

It thus follows that

 x (t)  v   1   1  = −  x2 (t) 9v2 sin3t ,      x3 (t) v3  and hence

 − 9   4 −1 −1  v  0      1     − + − −  =  9   1 8.8 1  v2  0 ,   −  − −       9  1 1 2 v3  1 or equivalently

− 5 −1 −1  v  0  1    − − −  =  1 0.2 1 v2  0 − − −      1 1 7 v3  1

The solution of this system is

= = − = v1 0.25 v2 1.25 v3 0.

and as expected, the motion of x3 has been absorbed by the structural modification.

173 Appendix C: Parameters for Crack Problem

The expansion of the matrix A(β, K θ ,a ) of equation 6.7 is given below.

A A  A =  11 12  , (C-1) A 21 A 22  where,

 1 0 1 0    0 1 0 1 A =   , (C-2) 11 cosh λ sinh λ cos λ sin λ    sinh λ cosh λ sin λ − cos λ

 0 0 0 0    0 0 0 0 A =   , (C-3) 12 − cosh λ − sinh λ − cosλ − sin λ   − sinh λ − cosh λ − sin λ cosλ 

 cosh λ sinh λ − cosλ − sin λ    0 0 0 0 A =   21  0 0 0 0    cosh λ + K sinh λ sinh λ + K cosh λ − cosλ − K sin λ − sin λ + K cosλ

(C-4)

 − cosh λ − sinh λ cosλ sin λ    cosh λ sinh λ − cosλ − sin λ A =   , (C-5) 22  sinh λ cosh λ sin λ − cosλ    − K sinh λ − K cosh λ K sin λ − K cosλ

βa  K θ L  λ = , and K =   . (C-6) L  EIβ 

174 VITA

Kumar Vikram Singh was born and brought up in Ranchi, Jharkhand, India. He completed his schooling from St. John’s High School, Ranchi and intermediate education from St. Xavier’s College, Ranchi. He joined Birla Institute of Technology, Mesra, in

1993 and earned a bachelor of engineering degree in mechanical engineering in May

1997. After his graduation, he worked as Graduate Trainee Engineer at HINDALCO

Industries Limited till July 1998. He decided to pursue higher education in United States and enrolled at Louisiana Tech University, Ruston, in August 1998. He transferred to

Louisiana State University, Baton Rouge, in the spring 1999 and enrolled in the integrated doctoral program in the Mechanical Engineering Department. He has been awarded Outstanding Research Assistant Award for the Year 2002 in the Mechanical

Engineering Department. He realized his ambition by earning a Doctorate of Philosophy degree in Mechanical engineering in the spring 2003. His research interests are vibration, control, design and inverse problems. Kumar is intended to pursue his career in the research environment.

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