Dynamic Response of Structures With Frequency Dependent Damping Blanca Pascual & S Adhikari School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/ ∼ adhikaris Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.1/35

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 Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.2/35

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. Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.3/35

Equation of motion The equations of motion of a N -DOF linear system: � t M ¨ u ( t ) + G ( t − τ ) ˙ u ( τ ) d τ + Ku ( t ) = f ( t ) (1) 0 together with the initial conditions u ( t = 0) = u 0 ∈ R N u 0 ∈ R N . and u ( t = 0) = ˙ ˙ (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. Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.4/35

Damping functions - 1 Model Damping function Author and year of publication Number a k s G ( s ) = � n 1 Biot[1] - 1955 k =1 s + b k G ( s ) = E 1 s α − E 0 bs β 2 (0 < α, β < 1) Bagley and Torvik[2] - 1983 1 + bs β s 2 + 2 ξ k ω k s � � sG ( s ) = G ∞ 1 + � 3 k α k Golla and Hughes[3] - 1985 s 2 + 2 ξ k ω k s + ω 2 k and McTavish and Hughes[4] - 1993 ∆ k s G ( s ) = 1 + � n 4 Lesieutre and Mingori[5] - 1990 k =1 s + β k G ( s ) = c 1 − e − st 0 5 Adhikari[6] - 1998 st 0 1 + 2( st 0 /π ) 2 − e − st 0 c 6 G ( s ) = Adhikari[6] - 1998 1 + 2( st 0 /π ) 2 st 0 � � s �� G ( s ) = c e s 2 / 4 µ 1 − erf 7 Adhikari and Woodhouse[7] - 2001 2 √ µ Some damping functions in the Laplace domain. Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.5/35

Damping functions - 2 We use a damping model for which the kernel function matrix: n � µ k e − µ k t C k G ( t ) = (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 Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.6/35

State-space Approach Eq (1) can be transformed to B ˙ z ( t ) = A z ( t ) + r ( t ) (5) where the m ( m = 2 N + nN ) dimensional matrices and vectors are: n � − C 1 /µ 1 − C n /µ n C k M · · · f ( t ) k =1 0 M O O O O 0 C 1 /µ 2 B = , r ( t ) = (6) − C 1 /µ 1 O O O 1 . . ... . . . . O O O 0 C n /µ 2 − C n /µ n O O O n u ( t ) − K O O O O v ( t ) O M O O O O O − C 1 /µ 1 O O y 1 ( t ) A = , z ( t ) = (7) . ... . O O O O . − C n /µ n y n ( t ) O O O O Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.7/35

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. Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.8/35

Dynamic Response - 1 Taking the Laplace transform of equation (1) and considering the initial conditions in (2) we have q = ¯ s 2 M¯ q − s Mq 0 − M˙ q 0 + s G ( s ) ¯ q − G ( s ) q 0 + K¯ f ( s ) q = ¯ or D ( s ) ¯ f ( s ) + M˙ q 0 + [ s M + G ( s )] q 0 . The dynamic stiffness matrix is defined as D ( s ) = s 2 M + s G ( 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) Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.9/35

Dynamic Response - 2 Using the residue-calculus the transfer function matrix can be expressed like a viscously damped system as m z j z T R j � res j H ( s ) = ; R j = s = s j [ H ( s )] = (10) ∂ D ( s j ) s − s j z T z j j =1 j ∂s 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 j M + s j G ( s j ) + K ] z j = 0 , for j = 1 , · · · , m (11) Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.10/35

Dynamic Response - 3 The expression of H ( s ) allows the response to be expressed as modal summation as m j ¯ z T f ( s ) + z T q 0 + s z T j Mq 0 + z T j G ( s ) q 0 ( s ) j M˙ � q ( s ) = γ j ¯ z j s − s j j =1 (12) where 1 γ j = . (13) ∂ D ( s j ) z T z j j ∂s j We aim to derive the eigensolutions in ‘N’-space by solving the non-linear eigenvalue problem. Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.11/35

Non-linear Eigenvalue Problem The eigenvalue problem associated with a linear system with exponential damping model: � n � µ k � s 2 j M + s j C k + K z j = 0 , for j = 1 , · · · , m. s j + µ k k =1 (14) Two types of eigensolutions: 2 N complex conjugate solutions - underdamped/vibrating modes p real solutions [ p = � n k =1 rank ( C k ) ] - overdamped modes Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.12/35

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 ) Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.13/35

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. Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.14/35

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. Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.15/35

Complex-conjugate solutions The eigenvalues of the equivalent viscously damped system: � 1 − ζ 2 s 0 = − ζ n ω n ± i ω n n ≈ − ζ n ω n ± i ω n (15) k u /m u and ζ n = c/ 2 √ k u m u . � ω n = 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 ) . Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.16/35

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 s 0 ( s 0 m u + g ( s 0 )) + k u δ (1) = (17) s 0 (2 m u + g ′ ( s 0 )) + g ( s 0 ) Schaumburg, Illinois, 9 April 2008 Frequency Dependent Damping – p.17/35

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