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 ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.1/30

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 ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.2/30

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. ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.3/30

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. ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.4/30

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 st 0 1 + 2( st 0 /π ) 2 � � s �� G ( s ) = c e s 2 / 4 µ 7 1 − erf Adhikari and Woodhouse[7] - 2001 2 √ µ Some damping functions in the Laplace domain. ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.5/30

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 ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.6/30

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 (5) Two types of eigensolutions: 2 N complex conjugate solutions - underdamped/vibrating modes p real solutions [ p = � n k =1 rank ( C k ) ] - overdamped modes ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.7/30

State-space Approach - 1 The equation of motion can be transformed to ( m = 2 N + nN ) dimensional system B ˙ z ( t ) = A z ( t ) + r ( t ) (6) n     � − C 1 /µ 1 − C n /µ n C k M · · · f ( t )     k =1          0    M O O O O             0   B = C 1 /µ 2 , r ( t ) = (7) − 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                 − C 1 /µ 1 y 1 ( t ) O O O O A = , z ( t ) = (8)     . ...     .       . O O O O                 − C n /µ n y n ( t ) O O O O   ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.8/30

State-space Approach - 2 The eigenvalue problem in the sate-space is given by A z j = λ j B z j (9) The ‘size’ of the eigenvalue problem is ( 2 N + 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 ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.9/30

Approximate eigensolutions If ω j and x j are the undamped natural frequency and mode shape of the system satisfying Kx j = ω 2 j Mx j , the eigenvalues of the viscoelastically damped system obtained using the first-order perturbation method: s j ≈ i ω j − G ′ − i ω j − G ′ jj (i ω j ) / 2 , jj ( − i ω j ) / 2 . (10) Similarly, the eigenvectors are given by N s j G ′ kj ( s j ) x k � z j ≈ x j − kk ( s j ) . (11) ω 2 k + s 2 j + s j G ′ k =1 k � = j ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.10/30

Dynamic Response - 1 Taking the Laplace transform of the equation of motion and considering the initial conditions 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 . (12) The inverse of the dynamics stiffness matrix, known as the transfer function matrix, is given by H ( s ) = D − 1 ( s ) ∈ C N × N . (13) ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.11/30

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 )] = (14) ∂ 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 (15) ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.12/30

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 (16) ¯ z j s − s j j =1 where the normalization constant 1 γ j = . (17) ∂ D ( s j ) z T z j j ∂s j We use the approximate eigensolutions in the ‘N’-space. ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.13/30

Dynamic Response - 4 The response in the time domain can be obtained by taking the inverse transform: m � q ( t ) = L − 1 [ ¯ q ( s )] = γ j a j ( t ) z j (18) j = where the time-dependent scalar coefficients (for t > 0 ) � t e s j ( t − τ ) � z T j f ( τ ) + z T d τ + e s j t � z T q 0 + s j z T � � a j ( t ) = j G ( τ ) q 0 j M˙ j Mq 0 0 (19) ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.14/30

Response variability: Direct approach The dynamic response in the Laplace domain: q ( s ) = D − 1 ( s ) ¯ p ( s ) (20) ¯ where n µ k � D ( s ) = s 2 M + s C k + K (21) s + µ k k =1 p ( s ) = ¯ f ( s ) + M˙ q 0 + [ s M + G ( s )] q 0 . (22) ¯ Suppose the system matrices are functions of some design parameter p . We want to obtain ∂ ¯ q ( s ) ∂p . ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.15/30

Response variability: Direct approach Differentiating the equation of motion in the Laplace domain = ∂ D − 1 ( s ) ∂ ¯ q ( s ) p ( s ) + D − 1 ( s ) ∂ ¯ p ( s ) (23) ¯ ∂p ∂p ∂p Using the direct approach, ∂ D − 1 ( s ) = D − 1 ( s ) ∂ D ( s ) D − 1 ( s ) (24) ∂p ∂p where � n � ∂ D ( s ) = s 2 ∂ M ∂p + s ∂ µ k + ∂ K � (25) C k ∂p ∂p s + µ k ∂p k =1 ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.16/30

Response variability: Modal approach m ; R j = z j z T R j j � D − 1 ( s ) = (26) s − s j θ j j =1 Using the modal approach, ∂ R j m ∂ D − 1 ( s ) ∂s j R j ∂p � = − (27) ( s − s j ) 2 ∂p s − s j ∂p j =1 T � ∂ R j � ∂ z j ∂ z j ∂p z T ∂p = j + z j /θ j (28) ∂p ICNPAA, Genoa, 25 June 2008 Viscoelastically damped systems – p.17/30