Response Variability of Viscoelastically Damped Systems S Adhikari - - PowerPoint PPT Presentation

Response Variability of Viscoelastically Damped Systems

S Adhikari & B P Oliver

School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/∼adhikaris

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Outline of the presentation

Overview of viscoelastically damped systems Eigensolutions State-space approach Approximate methods in N-space Dynamic response calculation Parametric sensitivity of eigensolutions Parametric sensitivity of dynamic response Numerical results Conclusions

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Damping models

Viscous damping is the most widely used damping model for complex aerospace dynamic systems. 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 (e.g., viscoelastic) 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.

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Equation of motion

The equations of motion of a N-DOF linear system: M¨ u(t) + t G(t − τ) ˙ u(τ) dτ + Ku(t) = f(t) (1) together with the initial conditions u(t = 0) = u0 ∈ RN and ˙ u(t = 0) = ˙ u0 ∈ RN. (2) u(t): displacement vector, f(t): forcing vector, M, K: mass and stiffness matrices. In the limit when G(t−τ) = C δ(t−τ), where δ(t) is the Dirac-delta function, this reduces to viscous damping.

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Damping functions - 1

Model Damping function Author and year of publication Number 1 G(s) = n

k=1

aks s + bk Biot[1] - 1955 2 G(s) = E1sα − E0bsβ 1 + bsβ (0 < α, β < 1) Bagley and Torvik[2] - 1983 3 sG(s) = G∞

• 1 +

k αk

s2 + 2ξkωks s2 + 2ξkωks + ω2

k

• Golla and Hughes[3] - 1985

and McTavish and Hughes[4] - 1993 4 G(s) = 1 + n

k=1

∆ks s + βk Lesieutre and Mingori[5] - 1990 5 G(s) = c1 − e−st0 st0 Adhikari[6] - 1998 6 G(s) = c st0 1 + 2(st0/π)2 − e−st0 1 + 2(st0/π)2 Adhikari[6] - 1998 7 G(s) = c es2/4µ

• 1 − erf
• s

2õ

• Adhikari and Woodhouse[7] - 2001

Some damping functions in the Laplace domain.

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Damping functions - 2

We use a damping model for which the kernel function matrix: G(t) =

n

• k=1

µke−µktCk (3) The constants µk ∈ R+ are known as the relaxation parameters and n denotes the number relaxation parameters. When µk → ∞, ∀ k this reduces to the viscous damping model: C =

n

• k=1

Ck. (4)

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Non-linear Eigenvalue Problem

The eigenvalue problem associated with a linear system with exponential damping model:

• s2

jM + sj n

• k=1

µk sj + µk Ck + K

• zj = 0,

for j = 1, · · · , m. (5) Two types of eigensolutions: 2N complex conjugate solutions - underdamped/vibrating modes p real solutions [p = n

k=1 rank (Ck)] - overdamped

modes

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State-space Approach - 1

The equation of motion can be transformed to (m = 2N + nN) dimensional system B ˙ z(t) = A z(t) + r(t) (6) B =            

n

• k=1

Ck M −C1/µ1 · · · −Cn/µn M O O O O −C1/µ1 O C1/µ2

1

O O . . . O O ... O −Cn/µn O O O Cn/µ2

n

            , r(t) =                      f(t) . . .                      (7) A =            −K O O O O O M O O O O O −C1/µ1 O O O O O ... O O O O O −Cn/µn            , z(t) =                      u(t) v(t) y1(t) . . . yn(t)                      (8)

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State-space Approach - 2

The eigenvalue problem in the sate-space is given by A zj = λjB zj (9) The ‘size’ of the eigenvalue problem is (2N + nN)-dimensional. although exact in nature, the state-space approach 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

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Approximate eigensolutions

If ωj and xj are the undamped natural frequency and mode shape of the system satisfying Kxj = ω2

jMxj, the eigenvalues of

the viscoelastically damped system obtained using the first-order perturbation method: sj ≈ iωj − G′

jj(iωj)/2,

−iωj − G′

jj(−iωj)/2.

(10) Similarly, the eigenvectors are given by zj ≈ xj −

N

• k=1

k=j

sjG′

kj(sj)xk

ω2

k + s2 j + sjG′ kk(sj).

(11)

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Dynamic Response - 1

Taking the Laplace transform of the equation of motion and considering the initial conditions we have s2M¯ q − sMq0 − M˙ q0 + s G(s)¯ q − G(s)q0 + K¯ q = ¯ f(s)

• r

D(s)¯ q = ¯ f(s) + M˙ q0 + [sM + G(s)] q0. The dynamic stiffness matrix is defined as D(s) = s2M + s G(s) + K ∈ CN×N. (12) The inverse of the dynamics stiffness matrix, known as the transfer function matrix, is given by H(s) = D−1(s) ∈ CN×N. (13)

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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

Rj s − sj ; Rj =

res s=sj [H(s)] =

zjzT

j

zT

j ∂D(sj) ∂sj

zj (14) where m is the number of non-zero eigenvalues (order) of the system, sj and zj are respectively the eigenvalues and eigenvectors of the system, which are solutions of the non-linear eigenvalue problem [s2

jM + sj G(sj) + K]zj = 0,

for j = 1, · · · , m (15)

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Dynamic Response - 3

The expression of H(s) allows the response to be expressed as modal summation as ¯ q(s) =

m

• j=1

γj zT

j ¯

f(s) + zT

j M˙

q0 + szT

j Mq0 + zT j G(s)q0(s)

s − sj zj (16) where the normalization constant γj = 1 zT

j ∂D(sj) ∂sj

zj . (17) We use the approximate eigensolutions in the ‘N’-space.

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Dynamic Response - 4

The response in the time domain can be obtained by taking the inverse transform: q(t) = L−1[¯ q(s)] =

m

• j=

γjaj(t)zj (18) where the time-dependent scalar coefficients (for t > 0) aj(t) = t esj(t−τ) zT

j f(τ) + zT j G(τ)q0

• dτ+esjt

zT

j M˙

q0 + sjzT

j Mq0

• (19)

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Response variability: Direct approach

The dynamic response in the Laplace domain: ¯ q(s) = D−1(s)¯ p(s) (20) where D(s) = s2M + s

n

• k=1

µk s + µk Ck + K (21) ¯ p(s) = ¯ f(s) + M˙ q0 + [sM + G(s)] q0. (22) Suppose the system matrices are functions of some design parameter p. We want to obtain ∂¯ q(s) ∂p .

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Response variability: Direct approach

Differentiating the equation of motion in the Laplace domain ∂¯ q(s) ∂p = ∂D−1(s) ∂p ¯ p(s) + D−1(s)∂¯ p(s) ∂p (23) Using the direct approach, ∂D−1(s) ∂p = D−1(s)∂D(s) ∂p D−1(s) (24) where ∂D(s) ∂p = s2∂M ∂p + s ∂ ∂p n

• k=1

µk s + µk Ck

• + ∂K

∂p (25)

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Response variability: Modal approach

D−1(s) =

m

• j=1

Rj s − sj ; Rj = zjzT

j

θj (26) Using the modal approach, ∂D−1(s) ∂p =

m

• j=1

∂Rj ∂p s − sj − Rj (s − sj)2 ∂sj ∂p (27) ∂Rj ∂p = ∂zj ∂p zT

j + zj

∂zj ∂p

T

/θj (28)

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Eigensolution derivative

It can be shown that (Adhikari: AIAA Journal, 40[10] (2002), pp. 2061-2069) ∂sj ∂p = − 1 θj

• zT

j

∂D(s) ∂p |s=sjzj

• .

(29) ∂zj ∂p = ajjzj −

m

• k=1

k=j

uT

k

∂D(s) ∂p |s=sjzj θk(sj − λk) uk (30) where ajj = − zT

j

∂2[D(s)] ∂s ∂p |s=sjzj 2

• zT

j

∂D(s) ∂s |s=sjzj . (31)

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Example of a 2 DOF system

k m m k2 1 2 g(t) k1 3

The two degrees-of-freedom spring-mass system with non-viscous damping, m = 1 Kg, k1 = 1000 N/m, k3 = 100 N/m, g(t) = c

• µ1e−µ1t + µ2e−µ2t

, c = 4.0 Ns/m, µ1 = 10.0 s−1, µ2 = 2.0 s−1

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Example: system matrices

M =   m m   , K =   k1 + k3 −k3 −k3 k2 + k3   (32) and G(t) = g(t)ˆ I, where ˆ I =   1 −1 −1 1   . (33) The damping function g(t) is assumed to be the GHM model[3, 4] so that g(t) = c

• µ1e−µ1t + µ2e−µ2t

; c, µ1, µ2 ≥ 0, (34)

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System matrix derivative

We consider the derivative of eigenvalues with respect to the relaxation parameter µ1. The derivative of the system matrices: ∂M ∂µ1 = O, ∂G(s) ∂µ1 = ˆ I c s (s + µ1)2 and ∂K ∂µ1 = O. (35) Thus we have ∂G(s) ∂s = −ˆ Ic

• µ1

(s + µ1)2 + µ2 (s + µ2)2

• ∂2[G(s)]

∂s ∂µ1 = −ˆ Ic s − µ1 (s + µ1)3. (36)

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Numerical Results

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02

k3/k1 k2/k1

Real part of derivative of s1

Real part of the derivative of the first eigenvalue with respect to the relaxation parameter µ1.

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Numerical Results

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 −0.06 −0.055 −0.05 −0.045 −0.04 −0.035 −0.03 −0.025 −0.02 −0.015 −0.01

k3/k1 k2/k1

Real part of derivative of s2

Real part of the derivative of the second eigenvalue with respect to the relaxation parameter µ1.

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Numerical Results

0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 0.25 0.3 0.35

k2/k1 Imaginary part of derivative of first eigenvalue

with respect to c with respect to µ1 with respect to µ2

Imaginary part of the derivative of the first eigenvalue with respect to the damping parameters c, µ1 and µ2.

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Numerical Results

0.5 1 1.5 2 2.5 3 0.05 0.1 0.15 0.2 0.25 0.3 0.35

k2/k1 Imaginary part of derivative of second eigenvalue

with respect to c with respect to µ1 with respect to µ2

Imaginary part of the derivative of the second eigenvalue with respect to the damping parameters c, µ1 and µ2.

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Numerical Results

0.5 1 1.5 2 2.5 3 −3 −2.5 −2 −1.5 −1 −0.5 0.5 1 1.5 x 10−3

k2/k1 Derivative of the first eigenvector

dU11/dk2 dU21/dk2 viscously damped dU11/dk2 viscously damped dU21/dk2

Real part of the derivative of the first eigenvector with respect to k2.

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Numerical Results

0.5 1 1.5 2 2.5 3 −2 2 4 6 8 10 12 14 16 x 10−4

k2/k1 Derivative of the second eigenvector

dU12/dk2 dU22/dk2 viscously damped dU12/dk2 viscously damped dU22/dk2

Real part of the derivative of the second eigenvalue with respect to k2.

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Conclusions - 1

Multiple degree-of-freedom linear systems with viscoelastic damping is considered. The transfer function matrix of the system was derived in terms of the eigenvalues and eigenvectors of the second-order system. The eigensolutions are obtained using an approximate perturbation method (although an exact but computationally more expensive state-space method can be used).

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Conclusions - 2

Parametric sensitivity of the dynamic response was derived using two approaches - namely the direct approach and modal approach. The direct approach is easy to implement but computationally expensive as one has to differentiate the dynamic stiffness matrix at every frequency point. The modal approach utilizes derivatives of the complex eigensolutions and generally computationally more efficient.

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Future Directions

The results derived here extend the equivalent results available for viscously damped systems. The expressions can be useful to any problems which require parametric sensitivity information. Such problems include (a) probabilistic analysis, (b) optimal design, (c) model updating and system identification Future work will look into (a) sensitivity of transient dynamic response of viscoelastically damped systems in the time domain (this problem has relevance to vehicle noise reduction), and (b) joint sensitivity analysis of multiple parameters.

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References

[1] Biot, M. A., “Variational principles in irreversible thermodynamics with application to viscoelasticity,” Physical Review, Vol. 97, No. 6, 1955, pp. 1463–1469. [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. 741–748. [3] Golla, D. F. and Hughes, P. C., “Dynamics of vis- coelastic structures - a time domain finite element formulation,” Transactions of ASME, Journal of Ap- plied Mechanics, Vol. 52, December 1985, pp. 897– 906. [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. 103–110. [5] Lesieutre, G. A. and Mingori, D. L., “Finite element modeling of frequency-dependent material proper- ties using augmented thermodynamic fields,” AIAA Journal of Guidance, Control and Dynamics, Vol. 13, 1990, pp. 1040–1050. [6] Adhikari, S., Energy Dissipation in Vibrating Struc- tures, Master’s thesis, Cambridge University Engi- neering Department, Cambridge, UK, May 1998, First Year Report.

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[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. 43–61.

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