Random Matrix Eigenvalue Problems in Probabilistic Structural Mechanics

Transcription

1 Random Matrix Eigenvalue Problems in Probabilistic Structural Mechanics S Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. URL: Random Matrix Eigenvalue Problems p.1/36

2 Outline of the Presentation Random eigenvalue problem Existing methods Exact methods Perturbation methods Asymptotic analysis of multidimensional integrals Joint moments and pdf of the natural frequencies Numerical examples & results Conclusions Random Matrix Eigenvalue Problems p.2/36

3 Random Eigenvalue Problem The random eigenvalue problem of undamped or proportionally damped linear systems: K(x)φ j = ω 2 jm(x)φ j (1) ω j natural frequencies; φ j eigenvectors; M(x) R N N mass matrix and K(x) R N N stiffness matrix. x R m is random parameter vector with pdf p x (x) = e L(x) L(x) is the log-likelihood function. Random Matrix Eigenvalue Problems p.3/36

4 The Objectives The aim is to obtain the joint probability density function of the natural frequencies and the eigenvectors in this work we look at the joint statistics of the eigenvalues while several papers are available on the distribution of individual eigenvalues, only first-order perturbation results are available for the joint pdf of the eigenvalues Random Matrix Eigenvalue Problems p.4/36

5 Exact Joint pdf Without any loss of generality the original eigenvalue problem can be expressed by where H(x)ψ j = ω 2 jψ j (2) and H(x) = M 1/2 (x)k(x)m 1/2 (x) R N N ψ j = M 1/2 φ j Random Matrix Eigenvalue Problems p.5/36

6 Exact Joint pdf The joint probability (following Muirhead, 1982) density function of the natural frequencies of an N-dimensional linear positive definite dynamic system is given by p Ω (ω 1, ω 2,, ω N ) = πn2 /2 ( ω 2 j ω 2 i ) Γ(N/2) i<j N ( p H ΨΩ 2 Ψ T) (dψ) (3) O(N) where H = M 1/2 KM 1/2 & p H (H) is the pdf of H. Random Matrix Eigenvalue Problems p.6/36

7 Limitations of the Exact Method the multidimensional integral over the orthogonal group O(N) is difficult to carry out in practice and exact closed-form results can be derived only for few special cases the derivation of an expression of the joint pdf of the system matrix p H (H) is non-trivial even if the joint pdf of the random system parameters x is known Random Matrix Eigenvalue Problems p.7/36

8 Limitations of the Exact Method even one can overcome the previous two problems, the joint pdf of the natural frequencies given by Eq. (3) is too much information to be useful for practical problems because it is not easy to visualize the joint pdf in the space of N natural frequencies, and the derivation of the marginal density functions of the natural frequencies from Eq. (3) is not straightforward, especially when N is large. Random Matrix Eigenvalue Problems p.8/36

9 Eigenvalues of GOE Matrices Suppose the system matrix H is from a Gaussian orthogonal ensemble (GOE). The pdf of H: p H (H) = exp ( θ 2 Trace ( H 2) + θ 1 Trace (H) + θ 0 ) The joint pdf of the natural frequencies: p Ω (ω 1, ω 2,, ω N ) = exp [ ( N j=1 θ 2 ω 4 j θ 1 ω 2 j θ 0 )] i<j ω 2 j ω 2 i Random Matrix Eigenvalue Problems p.9/36

10 Perturbation Method Taylor series expansion of ω j (x) about the mean x = µ ω j (x) ω j (µ) + d T ω j (µ) (x µ) (x µ)t D ωj (µ) (x µ) Here d ωj (µ) R m and D ωj (µ) R m m are respectively the gradient vector and the Hessian matrix of ω j (x) evaluated at x = µ. Random Matrix Eigenvalue Problems p.10/36

11 Joint Statistics Joint statistics of the natural frequencies can be obtained provided it is assumed that the x is Gaussian. Assuming x N(µ,Σ), first few cumulants can be obtained as κ (1,0) jk = E [ω j ] = ω j Trace( D ωj Σ ), κ (0,1) jk κ (1,1) jk = E [ω k ] = ω k Trace (D ω k Σ), = Cov (ω j, ω k ) = 1 2 Trace(( D ωj Σ ) (D ωk Σ) ) + d T ω j Σd ωk Random Matrix Eigenvalue Problems p.11/36

12 Multidimensional Integrals We want to evaluate an m-dimensional integral over the unbounded domain R m : J = R e f(x) dx m Assume f(x) is smooth and at least twice differentiable The maximum contribution to this integral comes from the neighborhood where f(x) reaches its global minimum, say θ R m Random Matrix Eigenvalue Problems p.12/36

13 Multidimensional Integrals Therefore, at x = θ f(x) x k = 0, k or d f (θ) = 0 Expand f(x) in a Taylor series about θ: { J = R e f(θ)+ 1 2(x θ) T D } f(θ)(x θ)+ε(x,θ) dx m = e f (θ) 2(x θ) T D f(θ)(x θ) ε(x,θ) dx R m e 1 Random Matrix Eigenvalue Problems p.13/36

14 Multidimensional Integrals The error ε(x, θ) depends on higher derivatives of f(x) at x = θ. If they are small compared to f (θ) their contribution will negligible to the value of the integral. So we assume that f(θ) is large so that 1 f (θ) D(j) (f (θ)) 0 for j > 2 where D (j) (f (θ)) is jth order derivative of f(x) evaluated at x = θ. Under such assumptions ε(x, θ) 0. Random Matrix Eigenvalue Problems p.14/36

15 Multidimensional Integrals Use the coordinate transformation: ξ = (x θ)d 1/2 f (θ) The Jacobian: J = D f (θ) 1/2 The integral becomes: J e f (θ) R D m f (θ) 1/2 e 1 2 ( ξ T or J (2π) m/2 e f (θ) Df (θ) 1/2 ξ ) dξ Random Matrix Eigenvalue Problems p.15/36

16 Moments of Single Eigenvalues An arbitrary rth order moment of the natural frequencies can be obtained from µ (r) j = E [ ωj(x) ] r = R ωr j(x)p x (x)dx m = R e (L(x) r lnωj(x)) dx, r = 1, 2, 3 m Previous result can be used by choosing f(x) = L(x) r ln ω j (x) Random Matrix Eigenvalue Problems p.16/36

17 Moments of Single Eigenvalues After some simplifications µ (r) j (2π) m/2 ωj(θ)e r L (θ) D L(θ) + 1 r d L(θ)d L (θ) T r ω j (θ) D ω j (θ) 1/2 r = 1, 2, 3, θ is obtained from: d ωj (θ)r = ω j (θ)d L (θ) Random Matrix Eigenvalue Problems p.17/36

18 Maximum Entropy pdf Constraints for u [0, ]: 0 0 p ωj (u)du = 1 u r p ωj (u)du = µ (r) j, r = 1, 2, 3,, n Maximizing Shannon s measure of entropy S = 0 p ωj (u) lnp ωj (u)du, the pdf of ω j is p ωj (u) = e {ρ 0 + n i=1 ρ iu i } = e ρ 0 e n i=1 ρ iu i, u 0 Random Matrix Eigenvalue Problems p.18/36

19 Maximum Entropy pdf Taking first two moments, the resulting pdf is a truncated Gaussian density function { } 1 p ωj (u) = 2πσj Φ ( ω j /σ j ) exp (u ω j) 2 2σj 2 where σ 2 j = µ(2) j ω 2 j Ensures that the probability of any natural frequencies becoming negative is zero Random Matrix Eigenvalue Problems p.19/36

20 Joint Moments of Two Eigenvalues Arbitrary r s-th order joint moment of two natural frequencies µ (rs) jl = E [ ωj(x)ω r l s (x) ] = R exp { (L(x) r ln ω j(x) s ln ω l (x))} dx, m r = 1, 2, 3 Choose f(x) = L(x) r ln ω j (x) s ln ω l (x) Random Matrix Eigenvalue Problems p.20/36

21 Joint Moments of Two Eigenvalues After some simplifications µ (rs) jl (2π) m/2 ω r j(θ)ω s l (θ) exp { L (θ)} D f (θ) 1/2 where θ is obtained from: d L (θ) = r ω j (θ) d ω j (θ) + s ω l (θ) d ω l (θ) and D f (θ) = D L (θ) + r ω 2 j(θ) d ω j (θ)d ωj (θ) T r ω j(θ) D ω j (θ) + s ω 2 l(θ) d ω l (θ)d ωl (θ) T s ω l(θ) D ω l (θ) Random Matrix Eigenvalue Problems p.21/36

22 Joint Moments of Multiple Eigenvalues We want to obtain µ (r 1r 2 r n ) { j 1 j 2 j n = ω r 1 R m j 1 (x)ω r 2 j 2 (x) ω r n j n (x) } p x (x)dx It can be shown that µ (r 1r 2 r n ) j 1 j 2 j n (2π) m/2 { ω r 1 j 1 (θ)ω r 2 j 2 (θ) ω r n j n (θ) } exp { L (θ)} D f (θ) 1/2 Random Matrix Eigenvalue Problems p.22/36

23 Joint Moments of Multiple Eigenvalues Here θ is obtained from d L (θ) = r 1 ω j1 (θ) d ω j1 (θ)+ r 2 ω j1 (θ) d ω j2 (θ)+ and the Hessian matrix is given by r n ω jn (θ) d ω j n (θ) D f (θ) = D L (θ)+ j n,r n j = j 1, j 2, r = r 1, r 2, r ω 2 j (θ)d ω j (θ)d ωj (θ) T r ω j (θ) D ω j (θ) Random Matrix Eigenvalue Problems p.23/36

24 Example System Undamped three degree-of-freedom random system: k 6 k 1 m 1 k 4 k 5 m 3 k 3 k 2 m 2 m i = 1.0 kg for i = 1, 2, 3; k i = 1.0 N/m for i = 1,, 5 and k 6 = 3.0 N/m Random Matrix Eigenvalue Problems p.24/36

25 Example System m i = m i (1 + ǫ m x i ), i = 1, 2, 3 k i = k i (1 + ǫ k x i+3 ), i = 1,, 6 Vector of random variables: x = {x 1,, x 9 } T R 9 x is standard Gaussian, µ = 0 and Σ = I Strength parameters ǫ m = 0.15 and ǫ k = 0.20 Random Matrix Eigenvalue Problems p.25/36

26 Computational Methods Following four methods are compared 1. First-order perturbation 2. Second-order perturbation 3. Asymptotic method 4. Monte Carlo Simulation (15K samples) - can be considered as benchmark. The percentage error: Error = ( ) ( ) MCS ( ) MCS 100 Random Matrix Eigenvalue Problems p.26/36

27 Scatter of the Eigenvalues ω 1 ω 2 ω Samples Eigenvalues, ω j (rad/s) Statistical scatter of the natural frequencies ω 1 = 1, ω 2 = 2, and ω 3 = 3 Random Matrix Eigenvalue Problems p.27/36

28 Error in the Mean Values st order perturbation 2nd order perturbation Asymptotic method 0.7 Percentage error wrt MCS Eigenvalue number j Error in the mean values Random Matrix Eigenvalue Problems p.28/36

29 Error in Covariance Matrix st order perturbation 2nd order perturbation Asymptotic method Percentage error wrt MCS (1,1) (1,2) (1,3) (2,2) (2,3) (3,3) Elements of the covariance matrix Error in the elements of the covariance matrix Random Matrix Eigenvalue Problems p.29/36

30 Mean and Covariance Using the asymptotic method, the mean and covariance matrix of the natural frequencies are obtained as µ Ω = {0.9962, , } T and Σ Ω = Individual pdf and joint pdf of the natural frequencies are computed using these values. Random Matrix Eigenvalue Problems p.30/36

31 Individual pdf 6 5 Asymptotic method Monte Carlo Simulation pdf of the natural frequencies Natural frequencies (rad/s) Individual pdf of the natural frequencies Random Matrix Eigenvalue Problems p.31/36

32 Analytical Joint pdf Joint pdf using asymptotic method Random Matrix Eigenvalue Problems p.32/36

33 Joint pdf from MCS Joint pdf from Monte Carlo Simulation Random Matrix Eigenvalue Problems p.33/36

34 Contours of the joint pdf 4 (ω 1,ω 3 ) 3.5 ω k (rad/s) (ω 2,ω 3 ) 2 (ω 1,ω 2 ) Asymptotic method Monte Carlo Simulation ω j (rad/s) Contours of the joint pdf Random Matrix Eigenvalue Problems p.34/36

35 Conclusions Statistics of the natural frequencies of linear stochastic dynamic systems has been considered usual assumption of small randomness is not employed in this study. a general expression of the joint pdf of the natural frequencies of linear stochastic systems has been given Random Matrix Eigenvalue Problems p.35/36

36 Conclusions a closed-form expression is obtained for the general order joint moments of the eigenvalues it was observed that the natural frequencies are not jointly Gaussian even they are so individually future studies will consider joint statistics of the eigenvalues and eigenvectors and dynamic response analysis using eigensolution distributions Random Matrix Eigenvalue Problems p.36/36

37 References Muirhead, R. J. (1982), Aspects of Multivariate Statistical Theory, John Wiely and Sons, New York, USA. 36-1