Extremely strong convergence of eigenvalue-density of linear - PowerPoint PPT Presentation

Extremely strong convergence of eigenvalue-density of linear stochastic dynamical systems S Adhikari & L Pastur School of Engineering, Swansea University, Swansea, UK Email: S.Adhikari@swansea.ac.uk URL: http://engweb.swan.ac.uk/ ∼ adhikaris Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.1/33

Outline of the presentation Introduction Uncertainty Propagation (UP) in structural dynamics Wishart random matrices Parameter selection Density of eigenvalues Computational results Experimental results Conclusions Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.2/33

Dynamics of linear systems The equation of motion: M ¨ q ( t ) + C ˙ q ( t ) + Kq ( t ) = f ( t ) (1) Due to the presence of uncertainty M , C and K become random matrices. The main objective of the ‘forward problem’ is to predict the variability in the response vector q . There can be two broad possibilities: quantify uncertainties in the system matrices first and then obtain uncertainties in the response directly quantify uncertainties in the eigenvalues & eigenvectors and then obtain uncertainties in the response Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.3/33

Stochastic dynamic response Taking the Laplace transform of the equation of motion: s 2 M + s C + K q ( s ) = ¯ � � f ( s ) (2) ¯ q ( s ) ∈ C n when the system matrices The aim here is to obtain the statistical properties of ¯ are random matrices. The system eigenvalue problem is given by K φ j = ω 2 j M φ j , j = 1 , 2 , . . . , n (3) where ω 2 j and φ j are respectively the eigenvalues and mass-normalized eigenvectors of the system. We define the matrices Ω = diag [ ω 1 , ω 2 , . . . , ω n ] and Φ = [ φ 1 , φ 2 , . . . , φ n ] . (4) Φ T K e Φ = Ω 2 Φ T MΦ = I n so that and (5) Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.4/33

Stochastic dynamic response Transforming it into the modal coordinates: s 2 I n + s C ′ + Ω 2 � q ′ = ¯ ′ � (6) ¯ f Here ′ = Φ T ¯ C ′ = Φ T CΦ = 2 ζ Ω , ¯ q ′ q = Φ¯ and (7) ¯ f f When we consider random systems, the matrix of eigenvalues Ω 2 will be a random matrix of dimension n . Suppose this random matrix is denoted by Ξ ∈ R n × n : Ω 2 ∼ Ξ (8) Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.5/33

Stochastic dynamic response Since Ξ is a symmetric and positive definite matrix, it can be diagonalized by a orthogonal matrix Ψ r such that Ψ T r ΞΨ r = Ω 2 (9) r Here the subscript r denotes the random nature of the eigenvalues and eigenvectors of the random matrix Ξ . Recalling that Ψ T r Ψ r = I n we obtain q ′ = s 2 I n + s C ′ + Ω 2 � − 1 ¯ ′ � (10) ¯ f � − 1 Ψ T ′ r ¯ s 2 I n + 2 s ζ Ω r + Ω 2 � = Ψ r (11) f r Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.6/33

Stochastic dynamic response The response in the original coordinate can be obtained as � − 1 ( ΦΨ r ) T ¯ q ′ ( s ) = ΦΨ r s 2 I n + 2 s ζ Ω r + Ω 2 � q ( s ) = Φ¯ f ( s ) ¯ r r j ¯ n x T f ( s ) � = x r j . s 2 + 2 sζ j ω r j + ω 2 r j j =1 Here Ω r = diag [ ω r 1 , ω r 2 , . . . , ω r n ] , = ΦΨ r = [ x r 1 , x r 2 , . . . , x r n ] X r are respectively the matrices containing random eigenvalues and eigenvectors of the system. Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.7/33

Wishart random matrix approach Suppose we ‘know’ (e.g, by measurements or stochastic finite element modeling) the mean ( G 0 ) and the (normalized) variance (dispersion parameter) ( δ G ) of the system matrices: � � � G − E [ G ] � 2 E F δ 2 G = . (12) � E [ G ] � 2 F It can be proved that a positive definite symmetric matrix can e expressed by a Wishart matrix G ∼ W n ( p, Σ ) with p = n + 1 + θ Σ = G 0 /θ (13) and where 1 θ = { 1 + γ G } − ( n + 1) (14) δ 2 G and γ G = { Trace ( G 0 ) } 2 (15) Trace � G 02 � Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.8/33

Parameter-selection for structural dynamics Approach 1: M and K are fully correlated Wishart (most complex). For this case M ∼ W n ( p 1 , Σ 1 ) , K ∼ W n ( p 1 , Σ 1 ) with E [ M ] = M 0 and E [ M ] = M 0 . This method requires the simulation of two n × n fully correlated Wishart matrices and the solution of a n × n generalized eigenvalue problem with two fully populated matrices. Here Σ 1 = M 0 /p 1 , p 1 = γ M + 1 (16) δ 2 M Σ 2 = K 0 /p 2 , p 2 = γ K + 1 and (17) δ 2 K γ G = { Trace ( G 0 ) } 2 / Trace 2 � � (18) G 0 Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.9/33

Parameter-selection for structural dynamics Approach 2: Scalar Wishart (most simple) In this case it is assumed that p, a 2 � � Ξ ∼ W n (19) n I n Considering E [ Ξ ] = Ω 2 0 and δ Ξ = δ H the values of the unknown parameters can be obtained as p = 1 + γ H a 2 = Trace Ω 2 � � and /p (20) 0 δ 2 H Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.10/33

Parameter-selection for structural dynamics Approach 3: Diagonal Wishart with different entries (something in p, Ω 2 Ξ − 1 � = Ω − 2 � � � the middle). For this case Ξ ∼ W n 0 /θ with E 0 and δ Ξ = δ H . This requires the simulation of one n × n uncorrelated Wishart matrix and the solution of an n × n standard eigenvalue problem. The parameters can be obtained as θ = (1 + γ H ) p = n + 1 + θ and − ( n + 1) (21) δ 2 H Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.11/33

Parameter-selection for structural dynamics Defining H 0 = M 0 − 1 K 0 , the constant γ H : � 2 �� j ω 2 �� 2 γ H = { Trace ( H 0 ) } 2 Ω 2 � � Trace 0 j 0 = = (22) H 02 � Ω 4 � � � j ω 4 � Trace Trace 0 0 j Obtain the dispersion parameter of the generalized Wishart matrix � p 12 + ( p 2 − 2 − 2 n ) p 1 + ( − n − 1) p 2 + n 2 + 1 + 2 n � γ H δ H = p 2 ( − p 1 + n ) ( − p 1 + n + 3) + p 12 + ( p 2 − 2 n ) p 1 + (1 − n ) p 2 − 1 + n 2 (23) p 2 ( − p 1 + n ) ( − p 1 + n + 3) Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.12/33

A vibrating cantilever plate 1 Input 0.5 5 3 6 0 1 4 2 F i −0.5 x e d e d 0.8 g e Outputs 0.6 1 0.8 0.4 0.6 0.4 0.2 0.2 0 0 Y direction (width) X direction (length) Baseline Model: Thin plate elements with 0.7% modal damping assumed for all the modes. Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.13/33

Uncertainty modeling by random fields The Young’s modulus, Poissons ratio, mass density and thickness are random fields of the form E ( x ) = ¯ E (1 + ǫ E f 1 ( x )) (24) µ ( x ) = ¯ µ (1 + ǫ µ f 2 ( x )) (25) ρ ( x ) = ¯ ρ (1 + ǫ ρ f 3 ( x )) (26) t ( x ) = ¯ and t (1 + ǫ t f 4 ( x )) (27) The strength parameters: ǫ E = 0 . 15 , ǫ µ = 0 . 15 , ǫ ρ = 0 . 10 and ǫ t = 0 . 15 . The random fields f i ( x ) , i = 1 , · · · , 4 are delta-correlated homogenous Gaussian random fields. Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.14/33

Mean of the driving-FRF −60 M and K are fully correlated Wishart −80 Scalar Wishart Diagonal Wishart with different entries Direct simulation Mean of amplitude (dB) −100 −120 −140 −160 −180 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Mean of the amplitude of the response of the driving-FRF of the plate, n = 1200 , σ M = 0 . 078 and σ K = 0 . 205 . Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.15/33

Mean of a cross-FRF −60 M and K are fully correlated Wishart −80 Scalar Wishart Diagonal Wishart with different entries Direct simulation Mean of amplitude (dB) −100 −120 −140 −160 −180 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Mean of the amplitude of the response of the cross-FRF of the plate, n = 1200 , σ M = 0 . 078 and σ K = 0 . 205 . Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.16/33

Standard deviation of the driving-point-FRF −60 M and K are fully correlated Wishart −80 Scalar Wishart Diagonal Wishart with different entries Direct simulation Standard deviation (dB) −100 −120 −140 −160 −180 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Standard deviation of the amplitude of the response of the driving-point-FRF of the plate, n = 1200 , σ M = 0 . 078 and σ K = 0 . 205 . Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.17/33

Standard deviation of a cross-point-FRF −60 M and K are fully correlated Wishart −80 Scalar Wishart Diagonal Wishart with different entries Direct simulation Standard deviation (dB) −100 −120 −140 −160 −180 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Standard deviation of the amplitude of the response of a cross-point-FRF of the plate, n = 1200 , σ M = 0 . 078 and σ K = 0 . 205 . Saint Petersburg, 9 July 2009 Strong convergence of eigenvalue-density – p.18/33