Dynamic Response of Structures With Frequency Dependent Damping

Transcription

1 Dynamic Response of Structures With Frequency Dependent Damping Blanca Pascual & S Adhikari School of Engineering, Swansea University, Swansea, UK S.Adhikari@swansea.ac.uk URL: adhikaris Frequency Dependent Damping p.1/35

2 Outline of the presentation Damping models in structural dynamics Review of current approaches Dynamic response of frequency dependent damped systems Non-linear eigenvalue problem SDOF systems: real & complex solutions MDOF systems: real & complex solutions Conclusions & discussions Frequency Dependent Damping p.2/35

3 Damping models In general a physically realistic model of damping may not be a viscous damping model. Damping models in which the dissipative forces depend on any quantity other than the instantaneous generalized velocities are non-viscous damping models. Possibly the most general way to model damping within the linear range is to use non-viscous damping models which depend on the past history of motion via convolution integrals over kernel functions. Frequency Dependent Damping p.3/35

4 Equation of motion The equations of motion of a N-DOF linear system: Mü(t) + t 0 G(t τ) u(τ) dτ + Ku(t) = f(t) (1) together with the initial conditions u(t = 0) = u 0 R N and u(t = 0) = u 0 R N. (2) u(t): displacement vector, f(t): forcing vector, M, K: mass and stiffness matrices. In the limit when G(t τ) = Cδ(t τ), where δ(t) is the Diracdelta function, this reduces to viscous damping. Frequency Dependent Damping p.4/35

5 Damping functions - 1 Model Damping function Author and year of publication Number 1 G(s) = n k=1 a k s s + b k Biot[1] G(s) = E 1s α E 0 bs β 1 + bs β (0 < α, β < 1) Bagley and Torvik[2] sg(s) = G [1 + ] k α s 2 + 2ξ k ω k s k s 2 + 2ξ k ω k s + ωk 2 Golla and Hughes[3] and McTavish and Hughes[4] G(s) = 1 + n k s k=1 Lesieutre and Mingori[5] s + β k 5 G(s) = c 1 e st 0 Adhikari[6] st 0 6 G(s) = c 1 + 2(st 0 /π) 2 e st0 st (st 0 /π) 2 Adhikari[6] [ ( )] 7 G(s) = c e s2 /4µ s 1 erf 2 Adhikari and Woodhouse[7] µ Some damping functions in the Laplace domain. Frequency Dependent Damping p.5/35

6 Damping functions - 2 We use a damping model for which the kernel function matrix: n G(t) = µ k e µ kt C k (3) k=1 The constants µ k R + are known as the relaxation parameters and n denotes the number relaxation parameters. When µ k, k this reduces to the viscous damping model: n C = C k. (4) k=1 Frequency Dependent Damping p.6/35

7 State-space Approach Eq (1) can be transformed to B ż(t) = A z(t) + r(t) (5) where the m (m = 2N + nn) dimensional matrices and vectors are: B = n C k M C 1 /µ 1 C n /µ n k=1 M O O O O C 1 /µ 1 O C 1 /µ 2 1 O O, r(t) =.... O O O C n /µ n O O O C n /µ 2 n f(t) (6) A = K O O O O u(t) O M O O O v(t) O O C 1 /µ 1 O O, z(t) = y 1 (t)... O O O O. O O O O C n /µ n y n (t) (7) Frequency Dependent Damping p.7/35

8 Main issues The main reasons for not using a frequency dependent non-viscous damping model include, but not limited to: although exact in nature, the state-space approach usually needed for this type of damped systems is computationally very intensive for real-life systems; the physical insights offered by methods in the original space (eg, the modal analysis) is lost in a state-space based approach the experimental identification of the parameters of a frequency dependent damping model is difficult. Frequency Dependent Damping p.8/35

9 Dynamic Response - 1 Taking the Laplace transform of equation (1) and considering the initial conditions in (2) we have s 2 M q smq 0 M q 0 + sg(s) q G(s)q 0 + K q = f(s) or D(s) q = f(s) + M q 0 + [sm + G(s)]q 0. The dynamic stiffness matrix is defined as D(s) = s 2 M + sg(s) + K C N N. (8) The inverse of the dynamics stiffness matrix, known as the transfer function matrix, is given by H(s) = D 1 (s) C N N. (9) Frequency Dependent Damping p.9/35

10 Dynamic Response - 2 Using the residue-calculus the transfer function matrix can be expressed like a viscously damped system as H(s) = m j=1 R j s s j ; R j = res s=s j [H(s)] = z T j z j z T j (10) D(s j ) s j z j where m is the number of non-zero eigenvalues (order) of the system, s j and z j are respectively the eigenvalues and eigenvectors of the system, which are solutions of the non-linear eigenvalue problem [s 2 jm + s j G(s j ) + K]z j = 0, for j = 1,,m (11) Frequency Dependent Damping p.10/35

11 Dynamic Response - 3 The expression of H(s) allows the response to be expressed as modal summation as q(s) = where m j=1 z T f(s) j + z T j M q 0 + sz T j Mq 0 + z T j G(s)q 0 (s) γ j z j s s j γ j = z T j 1 (12) D(s j ) s j z j. (13) We aim to derive the eigensolutions in N -space by solving the non-linear eigenvalue problem. Frequency Dependent Damping p.11/35

12 Non-linear Eigenvalue Problem The eigenvalue problem associated with a linear system with exponential damping model: [ ] n s 2 µ k jm + s j C k + K z j = 0, s j + µ k k=1 Two types of eigensolutions: 2N complex conjugate solutions - underdamped/vibrating modes for j = 1,,m. p real solutions [p = n k=1 rank (C k)] - overdamped modes (14) Frequency Dependent Damping p.12/35

13 Non-linear Eigenvalue Problem The following four cases are considered: single-degree-of-freedom system with single exponential kernel (N = 1,n = 1) single-degree-of-freedom system with multiple exponential kernels (N = 1,n > 1) multiple-degree-of-freedom system with single exponential kernel (N > 1,n = 1) multiple-degree-of-freedom system with multiple exponential kernels (N > 1,n > 1) Frequency Dependent Damping p.13/35

14 SDOF systems Computational cost and other relevant issues identified before do not strictly affect the eigenvalue problem of a single-degree-of-freedom system (SDOF) with exponential damping. The main reason for considering a SDOF system is that in many cases the underlying approximation method can be extended to MDOF systems in a relatively straight-forward manner. Frequency Dependent Damping p.14/35

15 Complex-conjugate solutions The main motivation of the approximations is that the approximate solution can be constructed from the solution of equivalent viscously damped system. The solution of equivalent viscously damped system can in turn be expressed in terms of the undamped eigensolutions. Combining these together, one can therefore obtain the eigensolutions of frequency-dependent systems by simple post-processing of undamped solutions only. Frequency Dependent Damping p.15/35

16 Complex-conjugate solutions The eigenvalues of the equivalent viscously damped system: s 0 = ζ n ω n ± iω n 1 ζ 2 n ζ n ω n ± iω n (15) ω n = k u /m u and ζ n = c/2 k u m u. Viscous damped system is a special case when the function g(s) is replaced by g(s ). For that case this solution would have been the exact solution of the characteristic equation. The difference between the viscous solution and the true solution is essentially arising due to the varying nature of the function g(s). Frequency Dependent Damping p.16/35

17 Complex-conjugate solutions The central idea here is that the actual solution can be obtained by expanding the solution in a Taylor series around s 0. We assume s = s 0 + δ, (δ is small). Substituting this into the characteristic equation we have (s 0 + δ) 2 m u + (s 0 + δ)g(s 0 + δ) + k u = 0. (16) First-order approximation δ (1) = s 0(s 0 m u + g(s 0 )) + k u s 0 (2m u + g (s 0 )) + g(s 0 ) (17) Frequency Dependent Damping p.17/35

18 Complex-conjugate solutions Second-order approximation: δ (2) = B B 2 4AC 2A (18) where A = (m u + g (s 0 ) s 0 + g (s 0 )) 2! (19) B = (2m u s 0 + s 0 g (s 0 ) + g(s 0 )) (20) and C = (s 2 0m u + s 0 g(s 0 ) + k). (21) g (s 0 ) and g (s 0 ) are respectively the first and second order derivative of g(s) evaluated at s = s 0. Frequency Dependent Damping p.18/35

19 Real solutions When N = 1,n = 1 the eigenvalue equation: s 2 m u + sg(s) + k u = 0 where g(s) = µ c. (22) s + µ While the complex-conjugate solution can be expected to be close to the solution of the equivalent viscously damped system, no such analogy can be made for the real solution as the equivalent viscously damped system doesn t have one. Rewrite the characteristic equation as (s 2 m u + k u )(µ + s) + scµ = 0. (23) Frequency Dependent Damping p.19/35

20 Real solutions Consider that the damping is small so that scµ 0. Since (s 2 m u + k u ) 0 as we are considering the real solution only, the first guess is obtained as µ + s + 0 = 0 or s 0 = µ. We take the first approximation of the real root as s = s 0 + = µ + Substituting into the characteristic equation and neglecting all the higher-order terms: µ 2 c µ 2 m u + k u + µc (24) Frequency Dependent Damping p.20/35

21 Real solutions: General case The characteristic equation: s 2 m u + s n k=1 µ k s + µ k c k + k u = 0. (25) This a polynomial in s of order (n + 2) - it has (n + 2) roots [2 complex conjugate and n real]. Multiplying the characteristic equation by the product n j (s + µ j): (s 2 m u + k u ) n (s + µ j ) + s j=1 n n (µ k c k (µ j + s)) = 0. (26) k=1 j=1 j k Frequency Dependent Damping p.21/35

22 Real solutions: General case Use the approximation s k = µ k + k, k = 1, 2, n. Substituting into the characteristic equation and retaining only the first-order terms in k, after some simplifications: k where θ k = c k µ 2 k n (µ j µ k ) j=1 j k (µ 2 k m + k) n µ k c k nj=1 j k r=1 n k nm=1 m j m k (µ r µ k ) + µ k θ k for k = 1, 2,, n. (µ m µ k ) n r=1 r k µ r c r nj=1 j n j k (µ j µ k ) + n j=1 j k c k (µ j µ k ) Frequency Dependent Damping p.22/35

23 Numerical Results SDOF system with eight exponential kernels: m = 1 kg, k = 2 N/m, µ k for k = 1, 2.., 8 are , , , , , , , Frequency Dependent Damping p.23/35

24 Numerical Results µ Exact solution (statespace) Proposed solution approximate Percentage error Complex solution Conjugate ± i ±1.4718i ±0i Exact and approximate eigenvalues of the SDOF system. Frequency Dependent Damping p.24/35

25 Frequency response function 10 1 exact solution approximation 10 2 log(abs(h)) ω (rad/s) Frequency response function of the SDOF system obtained using exact and approximate eigenvalues. Frequency Dependent Damping p.25/35

26 Complex solutions: MDOF Complex modes can be expanded as a complex linear combination of undamped modes z j = N l=1 α(j) l x l Substituting in to the eigenvalue equation (11): N l=1 s 2 jα (j) l Mx l + s j α (j) l G(s j )x l + α (j) l Kx l = 0. (27) Premultiplying by x T k and using the mass-orthogonality property of the undamped eigenvectors: s 2 jα (j) k +s j N l=1 α (j) l G kl(s j )+ωkα 2 (j) k = 0, G kl(s j ) = x T kg(s j )x l (28) Frequency Dependent Damping p.26/35

27 Complex solutions: MDOF Considering the j-th set of equation and neglecting the second-order terms involving α (j) k and G kl (s j), k l: s 2 j + s j G jj(s j ) + ω 2 j 0 (29) Similar to the SDOF case (replace m u by 1, k u by ω 2 j and g(s) by G jj(s)). To obtain the eigenvectors rewrite Eq. (28) for j k as s 2 j α(j) k + s j G kj (s j) + α (j) k G kk (s j) + N l k j l G kl (s j) +ωk 2 α(j) k = 0, α (j) k = 1,, N; j. (30) Frequency Dependent Damping p.27/35

28 Complex solutions: MDOF Retaining only the product terms α (j) l G kl : z j x j N k=1 k j s j G kj (s j)x k ω 2 k + s2 j + s jg kk (s j). (31) A second-order expressions is given in the paper. The results derived here are based on small non-proportional damping. Frequency Dependent Damping p.28/35

29 Real solutions: MDOF For systems with single exponential kernel assume s l = µ + l (32) Substituting in the approximate characteristic Eq (29) l µ 2 C ll ; l = 1, 2,,N (33) µ 2 + ωl 2 + µc ll Frequency Dependent Damping p.29/35

30 Real solutions: MDOF Assuming all coefficient matrices are of full rank, for systems with n kernels there are in general nn number of purely real eigenvalues. The approximate eigenvalues can be obtained as s lk = µ k + lk (34) where lk C kll µ 2 k (µ 2 k + ω2 l ) n θ lk = µ k C kll nj=1 j k r=1 n k nm=1 n (µ j µ k ) j=1 j k m j m k and C kll = x T l C kx l. (µ r µ k ) + µ k θ lk k = 1, 2,, n; l = 1, 2,, N (µ m µ k ) n r=1 r k µ r c r nj=1 j n j k (µ j µ k )+ n j=1 C kll (µ j µ k ) j k Frequency Dependent Damping p.30/35

31 Numerical example We consider a three degree-of-freedom system: M = m u m u 0, K = 2k u k u 0 k u 2k u k u (35) 0 0 m u 0 k u 2k u G(s) = C 6 k=1 µ k s + µ k, where C = (36) m u = 1 kg, k u = 1 N/m and µ k are , , , , , Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping p.31/35

32 Numerical results Exact solution (state-space) Proposed approximate solution Real solutions Percentage error Frequency Dependent Damping p.32/35

33 Numerical results Exact solution (state-space) Proposed approximate solution Real solutions Complex Conjugate solutions Percentage error ± i ± i ± i ± i ± i ± i ± i ± i ± i Frequency Dependent Damping p.33/35

34 Conclusions - 1 Multiple degree-of-freedom linear systems with frequency depended damping kernels is considered. The transfer function matrix of the system was derived in terms of the eigenvalues and eigenvectors of the second-order system. The response can be expressed as a sum of two parts, one that arises in usual viscously damped systems and the other that occurs due to non-viscous damping. The calculation of the eigensolutions of frequency-depended damped systems requires the solution of a non-linear eigenvalue problem. Frequency Dependent Damping p.34/35

35 Conclusions - 2 Approximate expressions are derived for the complex and real eigenvalues of the SDOF system with single and multiple exponential kernels. These results are extended to MDOF systems. These approximations allow one to obtain the dynamic response of general frequency-depended damped systems by simple post-processing of undamped eigensolutions. The accuracy of the proposed approximations were verified using numerical examples. The complex conjugate eigensolutions turn out to be more accurate compared to the real eigensolutions. Frequency Dependent Damping p.35/35

36 References [1] Biot, M. A., Variational principles in irreversible thermodynamics with application to viscoelasticity, Physical Review, Vol. 97, No. 6, 1955, pp [2] Bagley, R. L. and Torvik, P. J., Fractional calculus a different approach to the analysis of viscoelastically damped structures, AIAA Journal, Vol. 21, No. 5, May 1983, pp [3] Golla, D. F. and Hughes, P. C., Dynamics of viscoelastic structures - a time domain finite element formulation, Transactions of ASME, Journal of Applied Mechanics, Vol. 52, December 1985, pp [4] McTavish, D. J. and Hughes, P. C., Modeling of linear viscoelastic space structures, Transactions of ASME, Journal of Vibration and Acoustics, Vol. 115, January 1993, pp [5] Lesieutre, G. A. and Mingori, D. L., Finite element modeling of frequency-dependent material properties using augmented thermodynamic fields, AIAA Journal of Guidance, Control and Dynamics, Vol. 13, 1990, pp [6] Adhikari, S., Energy Dissipation in Vibrating Structures, Master s thesis, Cambridge University Engineering Department, Cambridge, UK, May 1998, First Year Report. 35-1

37 [7] Adhikari, S. and Woodhouse, J., Identification of damping: part 1, viscous damping, Journal of Sound and Vibration, Vol. 243, No. 1, May 2001, pp