Technical Briefs
Journal of Applied Mechanics Technical Briefs
Modal Analysis of Nonviscously where r is the spatial position vector and t is time, specified in some domain D, is governed by a linear partial differential Damped Beams equation ͑ ͒ ͑ ͒ L ͑ ͒ L ͑ ͒ ͑ ͒ D ͓ ͔ r u¨ r,t + 1u˙ r,t + 2u r,t = p r,t ; r , t 0,T 1 S. Adhikari ͑1͒ e-mail: [email protected] with homogeneous linear boundary conditions of the form M ͑ ͒ ⌫ M ͑ ͒ ⌫ ͑ ͒ M. I. Friswell 1u r,t =0; r 1 and 2u˙ r,t =0; r 2 2 Sir George White Professor of Aerospace Engineering ⌫ ⌫ specified on some boundary surfaces 1 and 2. In the above equation ͑r͒ is the mass distribution of the system: p͑r,t͒ is the Department of Aerospace Engineering, L distributed time-varying forcing function; and 2 is the spatial M M Queens Building, self-adjoint stiffness operator and 1 and 2 are linear opera- University of Bristol, tors acting on the boundary. For external ͑or foundation͒ damping L Queens Walk, the operator 1 can be written in the form Bristol BS8 1TR, UK t L u˙͑r,t͒ = ͵ ͵ C ͑r,,t − ͒u˙͑,͒d d ͑3͒ 1 1 D −ϱ Y. Lei where C ͑r,,t͒ is the kernel function. The velocities u˙͑,͒ at Associate Professor 1 different time instants and spatial locations are coupled through College of Aerospace and Material Engineering, this kernel function. Lei et al. ͓5͔ considered both external and National University of Defense Technology, internal damping, and although internal damping is not considered Changsha 410073, PRC further in this paper, the proposed method may be extended to this case. Kernel functions that serve similar purposes have been de- scribed by different names in different subjects ͑for example, re- tardation functions, heredity functions, after-effect functions, re- Linear dynamics of Euler–Bernoulli beams with nonviscous non- laxation functions͒, and different models have been used to local damping is considered. It is assumed that the damping force describe them. Equation ͑1͒ together with Eq. ͑3͒ represents a at a given point in the beam depends on the past history of ve- continuous dynamic system with general linear damping. It may locities at different points via convolution integrals over exponen- L ͑ ͒ be noted that if 1 =0 in Eq. 1 , i.e., an undamped system, or if tially decaying kernel functions. Conventional viscous and vis- the system satisfies the criteria given by Caughey and O’Kelly ͓6͔, coelastic damping models can be obtained as special cases of this then the system will possess classical normal modes. However, general damping model. The equation of motion of the beam with L due to the general nature of the operator 1 as described by Eq. such a general damping model results in a linear partial integro- ͑3͒, there is no definite reason why the system should have clas- differential equation. Exact closed-form equations of the natural sical normal modes. Thus the mode shapes and natural frequen- frequencies and mode shapes of the beam are derived. Numerical cies of such systems in general will be complex in nature. In this examples are provided to illustrate the new results. ͑ ͒ ͓ ͔ context we wish to note that the system expressed by Eq. 1 and DOI: 10.1115/1.2712315 the damping operator defined in Eq. ͑3͒ represents a partial integro-differential equation with the boundary conditions given in Eq. ͑2͒. In this technical brief we are interested in the natural 1 Introduction frequencies and mode shapes of the system. Exact closed-form expressions of such quantities for the general case are difficult to Viscous damping is the most common damping model used for obtain. We make the following general assumptions: linear dynamic systems. However, within the scope of linear theory, more general nonviscous models have been used in the 1. The mass and stiffness distributions are homogeneous, that recent past ͓1–5͔. Nonviscous damping models in general have is, they do not vary with the position vector r; and more parameters and therefore are more likely to have a better 2. The damping kernel function is separable in space and time match with experimental measurements. A linear damped continu- so that ͑ ͒ ous dynamic system in which the displacement variable u r,t , ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒͑͒ C1 r, ,t − = C r c r − g t − 4
1 Corresponding author. Depending on the nature of the functions c͑•͒ and g͑•͒, several Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received February 8, 2006; final manu- special cases, starting from the simple viscous model to the more script received August 15, 2006. Review conducted by Oliver M. O’Reilly. general nonviscous model, may arise ͓5͔. For example, if both c͑•͒
1026 / Vol. 74, SEPTEMBER 2007 Copyright © 2007 by ASME Transactions of the ASME
Downloaded 02 Dec 2007 to 137.222.10.58. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm continuously differentiable up to fourth order with respect to x. Here we assume that w͑x,t͒ is continuously differentiable up to fifth order. In what follows, Eqs. ͑7͒ and ͑8͒ are solved separately and the solutions are combined to obtain the eigensolutions. We begin with the solution of Eq. ͑7͒. Fig. 1 Euler–Bernoulli beam with nonviscous damping patch
3 Solution for the Section With Nonlocal Viscoelastic and g͑•͒ are delta functions then the result is the “locally reacting” Damping viscous damping model. If only c͑•͒ is a delta function, then the resulting model becomes a locally reacting viscoelastic damping Assuming zero initial conditions, the Laplace transform of the which is also known as the time hysteresis model. For the case displacement ͑with no external force͒ satisfies ͑ ͒ when only g • is a delta function, the resulting model becomes ␣ x2 nonlocal viscous damping or the spatial hysteresis model. Here EIWIV͑x,s͒ + s2AW͑x,s͒ + sG͑s͒͵ exp͑− ␣͉x − ͉͒W͑,s͒d 2 the general case, that is, when none of these two functions are the x1 delta functions, is considered. Based on the transfer matrix ͑ ͒͑͒ method ͓7͔, we propose a new method for modal analysis of a =0, x x1,x2 9 Euler–Bernoulli beam with general linear damping given by Eq. Here s is the complex Laplace parameter; and W͑x,s͒ is the ͑4͒. Laplace transform of w͑x,t͒. The roman superscripts, for example ͑•͒IV, denote the order of derivative with respect to the spatial 2 Governing Equation of Motion variable x. It is useful to separate the contribution arising from the The Euler–Bernoulli beam considered in this study is shown in term ͉x−͉ in Eq. ͑9͒ as Fig. 1. The left and right coordinates of the beam are denoted by ␣ x x and x , respectively. The beam has a nonlocal viscoelastic IV͑ ͒ 2 ͑ ͒ ͑ ͒͵ ͓ ␣͑ ͔͒ ͑ ͒ L R EIW x,s + s AW x,s + sG s exp − x − W ,s d 2 damping patch between x1 and x2. It is assumed that the elastic x1 properties of the beam are uniformly distributed with bending x2 rigidity EI and mass density A. ␣ + sG͑s͒͵ exp͓␣͑x − ͔͒W͑,s͒d =0 ͑10͒ In order to formulate and solve the equation of motion, it is 2 x necessary to use some kind of plausible functional form of the kernel functions in space and time. In this study we choose the The function W͑x,s͒ is continuously differentiable up to fifth or- following functional forms ͓5͔ der with respect to x because w͑x,t͒ is assumed to be continuously differentiable up to fifth order. Differentiating Eq. ͑10͒ with re- g͑t͒ = gϱ exp͑− t͒͑5a͒ spect to the spatial variable x one obtains so that ␣2 x V͑ ͒ 2 I͑ ͒ ͑ ͒͵ ͓ ␣͑ ͔͒ ͑ ͒ gϱ EIW x,s + s AW x,s − sG s exp − x − W ,s d G͑s͒ = , gϱ, ജ 0 ͑5b͒ 2 s + x1 x and ␣2 2 + sG͑s͒͵ exp͓␣͑x − ͔͒W͑,s͒d =0 ͑11͒ ␣ 2 x c͑x − ͒ = exp͑− ␣͉x − ͉͒, C͑x͒ =1 ͑6͒ 2 Differentiating again gives These models imply that the correlations in both space and time ␣3 x decay exponentially. If ␣→ ϱ ,→ϱ one obtains the standard EIWVI͑x,s͒ + s2AWII͑x,s͒ + sG͑s͒͵ exp͓− ␣͑x − ͔͒W͑,s͒d ␣→ϱ 2 viscous model; if and is finite one obtains the time hys- x1 teresis model; and if ␣ is finite but →ϱ one obtains the spatial ␣3 x2 ␣2 hysteresis model. + sG͑s͒͵ exp͓␣͑x − ͔͒W͑,s͒d −2 sG͑s͒W͑x,s͒ =0 The equation of motion of the part with the damping patch can 2 x 2 be expressed by ͑ ͒ x t 12 ␣ 2w͑x,t͒ 2ץ 4w͑x,t͒ץ EI + A + ͵ ͵ exp͑− ␣͉x − ͉͒ͯ Using Eqs. ͑10͒ and ͑12͒ we have 2 ץ 4 ץ x t ϱ 2 x1 − EIWVI͑x,s͒ + s2AWII͑x,s͒ − ␣2͓EIWIV͑x,s͒ + s2AW͑x,s͔͒ w͑,t͒ 2ץ ϫ ͓ ͑ ͔͒ ͯ − ␣ sG͑s͒W͑x,s͒ =0 ͑13͒ gϱ exp − t − d d tץ t= Equation ͑13͒ is a sixth-order ordinary differential equation which ͓ ͔͑͒ = 0 when x x1,x2 7 can be solved by transforming into the first-order form. Recall that the solution of a sixth-order ordinary differential equation requires The equation of motion of the part outside the damping patch can six boundary conditions corresponding to W͑x,s͒,...,WVI͑x,s͒. be expressed by However, the compatibility with the other beam solutions only w͑x,t͒ gives four boundary conditions. The other boundary conditionsץ 2w͑x,t͒ץ 4w͑x,t͒ץ EI + A + C t are implicit in the integro-differential equation. For example, atץ t2 0ץ x4ץ x=x , Eq. ͑10͒ becomes ͑ ͒ ഫ ͑ ͒͑͒ 1 = 0 when x xL,x1 x2,xR 8 ␣ x2 IV 2 Appropriate boundary conditions must be satisfied at x=xL and at EIW ͑x ,s͒ + s AW͑x ,s͒ + sG͑s͒͵ exp͓␣͑x − ͔͒W͑,s͒d 1 1 2 x=xR. In addition to this we need to satisfy relevant continuity x1 conditions at the internal points x1 and x2. No forcing is assumed ͑ ͒ because the central interest in this study is to obtain the eigenso- =0 14 lutions. The function w͑x,t͒ in Eqs. ͑7͒ and ͑8͒ is smooth and and Eq. ͑11͒ becomes
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Downloaded 02 Dec 2007 to 137.222.10.58. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm ␣2 x2 ␣s2A EIWV͑x ,s͒ + s2AWI͑x ,s͒ + sG͑s͒͵ exp͓␣͑x − ͔͒W͑,s͒d 1 1 2 s2A x1 0 =0 ͑15͒ b ͑s͒ = ͑26b͒ 2 0 Notice the integrals are the same in Eqs. ͑14͒ and ͑15͒ and hence Ά · ␣ combining these equations gives EI EI V͑ ͒ ␣ IV͑ ͒ 2 I͑ ͒ ␣ 2 ͑ ͒ EIW x1,s − EIW x1,s + s AW x1,s − s AW x1,s =0 From Eq. ͑24͒ and Eq. ͑25b͒ ͑16͒ ͑ ͒T ͑ ͒ ͑ ͒T⌿͑ ͒ ͑ ͒ ͑ ͒ b2 s 2 s = b2 s s 1 s =0 27 A similar procedure for x=x2 gives ͑ a͒ ͑ ͒ V͑ ͒ ␣ IV͑ ͒ 2 I͑ ͒ ␣ 2 ͑ ͒ Combining Eqs. 25 and 27 we have EIW x2,s + EIW x2,s + s AW x2,s + s AW x2,s =0 ͑ ͒ ͑ ͒ ͑ ͒ ͑17͒ E s 1 s = 0 28 To find the eigenvalues using the transfer matrix approach, we where need to relate the boundary conditions at the ends of the beam b ͑s͒T E͑s͒ = ͫ 1 ͬ segments. For the beam with the damping layer, we define the ͑ ͒T⌿͑ ͒ state vector ͑x,s͒ and partition it as b2 s s We partition E͑s͒ as u͑x,s͒ ͑x,s͒ = ͫ ͬ C6 ͑18͒ ͑ ͒ ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ v͑x,s͒ E s = E1 s E2 s 29 ͑ ͒ ϫ ͑ ͒ ϫ ͑ ͒ where where E1 s is 2 4 and E2 s is 2 2. From Eq. 28 we have I II III T 4 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ u͑x,s͒ = ͓W͑x,s͒,W ͑x,s͒,W ͑x,s͒,W ͑x,s͔͒ C ͑19͒ E1 s u x1,s + E2 s v x1,s = 0 30 and or ͑ ͒ ͑ ͒−1 ͑ ͒ ͑ ͒͑͒ v͑x,s͒ = ͓WIV͑x,s͒,WV͑x,s͔͒T C2 ͑20͒ v x1,s =−E2 s E1 s u x1,s 31 ͑ ͒ Using the state vector in Eq. ͑18͒, Eq. ͑13͒ can be cast in matrix Using Eq. 24 we finally have form as ͑ ͒ ͑ ͒ ͑ ͒͑͒ u x2,s = T s u x1,s 32 d where ͑x,s͒ = ⌽͑s͒͑x,s͒, x ͑x ,x ͒͑21͒ dx 1 2 I ͑ ͒ ͓ ͔⌿͑ ͒ͫ 4ϫ4 ͬ ͑ ͒ T s = I ϫ 0 ϫ s 33 where 4 4 4 2 ͑ ͒−1 ͑ ͒ − E2 s E1 s 0 10000 0 01000 0 00100 4 Eigensolutions of the Complete Beam ⌽͑s͒ = 0 00010͑22͒ Applying the Laplace transform to the equation of motion for 0 00001 the viscously damped segments Eq. ͑8͒, and considering the state ΄ ΅ ͑ ͒ vector u x,s we can obtain ␣2sG͑s͒ + ␣2s2A s2A 0 − 0 ␣2 0 d EI EI ͑ ͒ ⌽¯ ͑ ͒ ͑ ͒ ͑ ͒ ഫ ͑ ͒͑͒ u x,s = s u x,s , x xL,x1 x2,xR 34 The solution of Eq. ͑21͒ can be expressed as dx ͑ ͒ ͓⌽͑ ͒͑ ͔͒ ͑ ͒ ͑ ͒͑͒ where x,s = exp s x − x1 1 s , x x1,x2 23 ͑ ͒ ͑ ͒ 0 100 where 1 s = x1 ,s . In particular ͑ ͒ ͓⌽͑ ͒͑ ͔͒ ͑ ͒ ⌿͑ ͒ ͑ ͒͑͒ 0 010 2 s = exp s x2 − x1 1 s = s 1 s 24 ⌽¯ ͑s͒ = 0 001 ͑35͒ ͑s͒ ͑x s͒ ⌿͑s͒ ͓⌽͑s͒͑x x ͔͒ where 2 = 2 , and =exp 2 − 1 . We also have ΄ 2 ΅ s A + sC from Eqs. ͑16͒ and ͑17͒ that − 0 0 00 EI b ͑s͒T ͑s͒ =0 ͑25a͒ 1 1 The transfer function matrices can be obtained in the usual man- and ner from Eq. ͑34͒ as discussed before. The boundary conditions of ͑ ͒T ͑ ͒ ͑ ͒ the beam can be expressed as b2 s 2 s =0 25b ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ where M s u xL,s + N s u xR,s = 0 36 4ϫ4 4ϫ4 ͑ − ␣s2A where M C and N C are the boundary matrices see, for example, Ref. ͓7͔͒. Using the transfer matrices corresponding to s2A the three parts, u͑xR ,s͒ can be expressed in terms of u͑xL ,s͒ as 0 ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒͑͒ b ͑s͒ = ͑26a͒ u xR,s = TR s T s TL s u xL,s 37 1 0 ͑ ͒ ͑ ͒ Ά ␣ · where T s is defined in Eq. 33 and − EI ͑ ͒ ͓⌽¯ ͑ ͒͑ ͔͒ ͑ ͒ EI TR s = exp s xR − x2 38a and and
1028 / Vol. 74, SEPTEMBER 2007 Transactions of the ASME
Downloaded 02 Dec 2007 to 137.222.10.58. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm Table 1 The first five eigenvalues of the damped pinned-pinned beam
ϫ −3 j 10 j Proportional viscous =100, ␣=10 =100, ␣=0.1 =1, ␣=0.01
1 −0.1161±0.0725i −0.0604±0.1369i −0.0214±0.0702i −0.0214±0.0698i 2 −0.1161±0.2901i −0.0622±0.2949i −0.0585±0.2853i −0.0585±0.2853i 3 −0.1161±0.6528i −0.0696±0.6490i −0.0703±0.6453i −0.0703±0.6453i 4 −0.1161±1.1606i −0.0574±1.1575i −0.0576±1.1550i −0.0576±1.1550i 5 −0.1161±1.8134i −0.0505±1.8124i −0.0505±1.8113i −0.0505±1.8113i
T ͑s͒ = exp͓⌽¯ ͑s͒͑x − x ͔͒ ͑38b͒ ered as special cases of the general formulation derived in the L 1 L paper. The method was applied to a uniform beam with a nonlocal Substituting Eq. ͑37͒ into Eq. ͑36͒, one concludes that the eigen- viscoelastic damping patch. Future work will discuss computa- values are the roots of characteristic equation tional issues and forced vibration problems. ͓ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͔͒ ͑ ͒ det M s + N s TR s T s TL s =0 39 Assuming j are the eigenvalues, the corresponding mode shapes Acknowledgment can be determined by Sondipon Adhikari gratefully acknowledges the support of the ͑ ͒ ͑ ͒ j x = u x, j Engineering and Physical Sciences Research Council through the ͓⌽¯ ͑ ͒͑ ͔͒ ͑ ͒ ഛ ഛ award of an Advanced Research Fellowship. Michael Friswell exp j x − xL u0 j , xL x x1 gratefully acknowledges the support of the Royal Society through ͑ ͒ ͑ ͒ ͑ ͒ ഛ ഛ = ΆTm x, j TL j u0 j , x1 x x2 ͓⌽¯ ͑ ͒͑ ͔͒ ͑ ͒ ͑ ͒ ͑ ͒ ഛ ഛ exp j x − x2 T j TL j u0 j , x2 x xR ͑40͒ Here I ͑ ͒ ͓ ͔ ͓⌽͑ ͒͑ ͔͒ͫ 4ϫ4 ͬ T x, = I ϫ 0 ϫ exp x − x m j 4 4 4 2 j 1 ͑ ͒−1 ͑ ͒ − E2 j E1 j ͑41͒ ͑ ͒∀ ͑ ͒ and u0 j j is a vector in the null space of the matrix in Eq. 39 evaluated at s=j. 5 Numerical Example A damped pinned–pinned beam similar to that shown in Fig. 1 is used to illustrate the proposed method. The numerical values used are as follows: xR =0 m, xL =1 m, x1 =0.25 m, and x2 2 3 =0.75 m, E=70 GN/m , =2700 kg/m , C0 =gϱ =15.667 Ns/m, and the cross section is 5ϫ5mm2. The first five eigenvalues for different values of the relaxation parameter, including the case when the whole beam is uniformly viscously damped with param- eter C0, is shown in Table 1. Fig. 2 The real parts of the first four modes The real parts of the first four modes are shown in Fig. 2 for the four sets of parameter values given in Table 1. The corresponding imaginary parts of the first four modes are shown in Fig. 3. Note that, unlike the eigenvalues, and ␣ do significantly af- fect the eigenvectors.
6 Conclusions The increasing use of advanced composite materials and active control mechanisms demand sophisticated treatment of damping forces within a distributed parameter system. This technical brief proposes a new method to obtain the natural frequencies and mode shapes of Euler–Bernoulli beams with general linear damp- ing models. It was assumed that the damping force at a given point in the beam depends on the past history of velocities at different points via convolution integrals over exponentially de- caying kernel functions. Due to the general nature of the damping forces, the equation of motion becomes an integro-differential equation which couples the deflections at different time instants and spatial locations. The conventional transfer matrix method was extended to such integro-differential equations. Commonly used viscous and viscoelastically damped systems can be consid- Fig. 3 The imaginary parts of the first four modes
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Downloaded 02 Dec 2007 to 137.222.10.58. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm a Royal Society-Wolfson Research Merit Award. Y. Lei gratefully Eng. Mech., 128͑3͒, pp. 328–339. acknowledges the support of China Scholarship Council through a ͓4͔ Wagner, N., and Adhikari, S., 2003, “Symmetric State-Space Formulation for ͑ ͒ Scholarship Fund Award. a Class of Non-viscously Damped Systems,” AIAA J., 41 5 , pp. 951–956. ͓5͔ Lei, Y., Friswell, M. I., and Adhikari, S., 2006, “A Galerkin Method for Dis- tributed Systems with Non-Local Damping,” Int. J. Solids Struct., 43͑11–12͒, References pp. 3381–3400. ͓ ͔ ͓1͔ Banks, H. T., and Inman, D. J., 1991, “On Damping Mechanisms in Beams,” 6 Caughey, T. K., and O’Kelly, M. E. J., 1965, “Classical Normal Modes in ASME J. Appl. Mech., 58, pp. 716–723. Damped Linear Dynamic Systems,” ASME J. Appl. Mech., 32, pp. 583–588. ͓2͔ Woodhouse, J., 1998, “Linear Damping Models for Structural Vibration,” J. ͓7͔ Yang, B., and Tan, C. A., 1992, “Transfer Functions of One-Dimensional Sound Vib., 215͑3͒, pp. 547–569. Distributed Parameter Systems,” Trans. ASME, J. Appl. Mech., 59͑4͒, pp. ͓3͔ Adhikari, S., 2002, “Dynamics of Non-Viscously Damped Linear Systems,” J. 1009–1014.
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