Random Eigenvalue Problems Revisited

Transcription

1 Random Eigenvalue Problems Revisited S Adhikari Department of Aerospace Engineering, University of Bristol, Bristol, U.K. S.Adhikari@bristol.ac.uk URL: Random eigenvalue problems revisited p.1/38

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 eigenvalue problems revisited p.2/38

3 Motivations Random eigenvalue problems is of fundamental interest in various branches of science and engineering: vibration of complex engineering structures (high frequency vibration) high energy physics (energy levels of atomic nuclei, quantum chaos) stability and control of structures (structural buckling with random imperfections) number theory (zeros of Reimann-Zeta function) Random eigenvalue problems revisited p.3/38

4 Random Eigenvalue Problem The random eigenvalue problem of undamped or proportionally damped linear structural systems: K(x)φ j = ω 2 jm(x)φ j (1) ω j natural frequencies; φ j mode shapes; 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 eigenvalue problems revisited p.4/38

5 The Main Issues 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 eigenvalue problems revisited p.5/38

6 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 eigenvalue problems revisited p.6/38

7 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 eigenvalue problems revisited p.7/38

8 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 ) = [ ( N )] exp θ 2 ωj 4 θ 1 ωj 2 θ 0 j=1 i<j ω 2 j ω 2 i Random eigenvalue problems revisited p.8/38

9 Eigenvalues of Wishart Matrices If H has a Wishart distribution W N (N, λi N ) then the joint pdf of the natural frequencies can be expressed as p Ω (ω 1, ω 2,, ω N ) = exp ( 1 2λ π N2 /2 (2λ) N2 /2 (Γ(N/2)) 2 ) N N 1 N ( ) ω 2 j ωi 2 ωi 2 i=1 i=1 ω i i<j Random eigenvalue problems revisited p.9/38

10 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 eigenvalue problems revisited p.10/38

11 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 eigenvalue problems revisited p.11/38

12 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 eigenvalue problems revisited p.12/38

13 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 eigenvalue problems revisited p.13/38

14 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 eigenvalue problems revisited p.14/38

15 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 eigenvalue problems revisited p.15/38

16 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 eigenvalue problems revisited p.16/38

17 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 eigenvalue problems revisited p.17/38

18 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 eigenvalue problems revisited p.18/38

19 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 eigenvalue problems revisited p.19/38

20 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 eigenvalue problems revisited p.20/38

21 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 eigenvalue problems revisited p.21/38

22 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 eigenvalue problems revisited p.22/38

23 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 eigenvalue problems revisited p.23/38

24 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 eigenvalue problems revisited p.24/38

25 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 eigenvalue problems revisited p.25/38

26 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 eigenvalue problems revisited p.26/38

27 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 eigenvalue problems revisited p.27/38

28 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 eigenvalue problems revisited p.28/38

29 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 eigenvalue problems revisited p.29/38

30 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 eigenvalue problems revisited p.30/38

31 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 eigenvalue problems revisited p.31/38

32 Mean and Covariance Using the asymptotic method, the mean and correlation matrix of the natural frequencies are obtained as µ Ω = {0.9962, , } T and ρ Ω = Individual pdf and joint pdf of the natural frequencies are computed using these values. Random eigenvalue problems revisited p.32/38

33 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 eigenvalue problems revisited p.33/38

34 Analytical Joint pdf Joint pdf using asymptotic method Random eigenvalue problems revisited p.34/38

35 Joint pdf from MCS Joint pdf from Monte Carlo Simulation Random eigenvalue problems revisited p.35/38

36 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 eigenvalue problems revisited p.36/38

37 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 eigenvalue problems revisited p.37/38

38 Conclusions a closed-form expression is obtained for the general order joint moments of the eigenvalues it was observed that the natural frequencies may not be jointly Gaussian even they are individually Gaussian future studies will consider joint statistics of the eigenvalues and eigenvectors and dynamic response analysis using eigensolution distributions Random eigenvalue problems revisited p.38/38

39 References Muirhead, R. J. (1982), Aspects of Multivariate Statistical Theory, John Wiely and Sons, New York, USA. 38-1