Post on 20-Jul-2020
2 June 2005
Random Matrix EigenvalueProblems in Probabilistic
Structural MechanicsS Adhikari
Department of Aerospace Engineering, University of Bristol, Bristol, U.K.
URL: http://www.aer.bris.ac.uk/contact/academic/adhikari/home.html
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2 June 2005
Outline of the Presentation
Random eigenvalue problem
Existing methodsExact methodsPerturbation methods
Asymptotic analysis of multidimensionalintegrals
Joint moments and pdf of the naturalfrequencies
Numerical examples & results
Conclusions
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Random Eigenvalue Problem
The random eigenvalue problem of undamped orproportionally damped linear systems:
K(x)φj = ω2jM(x)φj (1)
ωj natural frequencies; φj eigenvectors;M(x) ∈ R
N×N mass matrix and K(x) ∈ RN×N
stiffness matrix.x ∈ R
m is random parameter vector with pdf
px(x) = e−L(x)
−L(x) is the log-likelihood function.
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The Objectives
The aim is to obtain the joint probability densityfunction of the natural frequencies and theeigenvectors
in this work we look at the joint statistics of theeigenvalues
while several papers are available on thedistribution of individual eigenvalues, onlyfirst-order perturbation results are available forthe joint pdf of the eigenvalues
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Exact Joint pdf
Without any loss of generality the originaleigenvalue problem can be expressed by
H(x)ψj = ω2jψj (2)
where
H(x) = M−1/2(x)K(x)M−1/2(x) ∈ R
N×N
and ψj = M1/2φj
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Exact Joint pdf
The joint probability (following Muirhead, 1982)density function of the natural frequencies of anN -dimensional linear positive definite dynamicsystem is given by
p Ω (ω1, ω2, · · · , ωN) =πN2/2
Γ(N/2)
∏
i<j≤N
(ω2
j − ω2i
)
∫
O(N)
pH
(ΨΩ
2Ψ
T)(dΨ) (3)
where H = M−1/2
KM−1/2 & pH(H) is the pdf of H.
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Limitations of the Exact Method
the multidimensional integral over theorthogonal group O(N) is difficult to carry out inpractice and exact closed-form results can bederived only for few special cases
the derivation of an expression of the joint pdf ofthe system matrix pH(H) is non-trivial even ifthe joint pdf of the random system parametersx is known
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Limitations of the Exact Method
even one can overcome the previous twoproblems, the joint pdf of the naturalfrequencies given by Eq. (3) is ‘too muchinformation’ to be useful for practical problemsbecause
it is not easy to ‘visualize’ the joint pdf in thespace of N natural frequencies, andthe derivation of the marginal densityfunctions of the natural frequencies from Eq.(3) is not straightforward, especially when Nis large.
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Eigenvalues of GOE Matrices
Suppose the system matrix H is from a Gaussianorthogonal ensemble (GOE). The pdf of H:
pH(H) = exp(−θ2Trace
(H
2)
+ θ1Trace (H) + θ0
)
The joint pdf of the natural frequencies:
p Ω (ω1, ω2, · · · , ωN) = exp
[−(
N∑
j=1
θ2ω4j − θ1ω
2j − θ0
)]
∏
i<j
∣∣ω2j − ω2
i
∣∣
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Perturbation Method
Taylor series expansion of ωj(x) about the meanx = µ
ωj(x) ≈ ωj(µ) + dTωj
(µ) (x − µ)
+1
2(x − µ)T
Dωj(µ) (x − µ)
Here dωj(µ) ∈ R
m and Dωj(µ) ∈ R
m×m are respec-
tively the gradient vector and the Hessian matrix of
ωj(x) evaluated at x = µ.
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Joint Statistics
Joint statistics of the natural frequencies can beobtained provided it is assumed that the x isGaussian. Assuming x ∼ N(µ,Σ), first fewcumulants can be obtained as
κ(1,0)jk = E [ωj] = ωj +
1
2Trace
(Dωj
Σ),
κ(0,1)jk = E [ωk] = ωk +
1
2Trace (Dωk
Σ) ,
κ(1,1)jk = Cov (ωj, ωk) =
1
2Trace
((Dωj
Σ)(Dωk
Σ))
+ dTωj
Σdωk
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Multidimensional Integrals
We want to evaluate an m-dimensional integral overthe unbounded domain R
m:
J =
∫
Rm
e−f(x) dx
Assume f(x) is smooth and at least twicedifferentiable
The maximum contribution to this integralcomes from the neighborhood where f(x)reaches its global minimum, say θ ∈ R
m
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Multidimensional Integrals
Therefore, at x = θ
∂f(x)
∂xk= 0,∀k or df (θ) = 0
Expand f(x) in a Taylor series about θ:
J =
∫
Rm
e−
f(θ)+ 1
2(x−θ)TDf(θ)(x−θ)+ε(x,θ)
dx
= e−f(θ)∫
Rm
e−1
2(x−θ)TDf(θ)(x−θ)−ε(x,θ) dx
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Multidimensional Integrals
The error ε(x,θ) depends on higher derivatives off(x) at x = θ. If they are small compared to f (θ)their contribution will negligible to the value of theintegral. 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) eval-
uated at x = θ. Under such assumptions ε(x,θ) →0.
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Multidimensional Integrals
Use the coordinate transformation:
ξ = (x − θ)D−1/2f (θ)
The Jacobian: ‖J‖ = ‖Df (θ)‖−1/2
The integral becomes:
J ≈ e−f(θ)∫
Rm‖Df (θ)‖−1/2 e
− 1
2
(ξ
Tξ)
dξ
or J ≈ (2π)m/2e−f(θ) ‖Df (θ)‖−1/2
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Moments of Single Eigenvalues
An arbitrary rth order moment of the naturalfrequencies can be obtained from
µ(r)j = E
[ωr
j (x)]
=
∫
Rm
ωrj (x)px(x) dx
=
∫
Rm
e−(L(x)−r lnωj(x)) dx, r = 1, 2, 3 · · ·
Previous result can be used by choosingf(x) = L(x) − r ln ωj(x)
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Moments of Single Eigenvalues
After some simplifications
µ(r)j ≈ (2π)m/2ωr
j (θ)e−L(θ)
∥∥∥∥DL(θ) +1
rdL(θ)dL(θ)T − r
ωj(θ)Dωj
(θ)
∥∥∥∥−1/2
r = 1, 2, 3, · · ·
θ is obtained from:
dωj(θ)r = ωj(θ)dL(θ)
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Maximum Entropy pdf
Constraints for u ∈ [0,∞]:∫ ∞
0
pωj(u)du = 1
∫ ∞
0
urpωj(u)du = µ
(r)j , r = 1, 2, 3, · · · , n
Maximizing Shannon’s measure of entropyS = −
∫∞0 pωj
(u) ln pωj(u)du, the pdf of ωj is
pωj(u) = e−ρ0+
∑n
i=1ρiu
i = e−ρ0e−∑n
i=1ρiu
i
, u ≥ 0
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Maximum Entropy pdf
Taking first two moments, the resulting pdf is atruncated Gaussian density function
pωj(u) =
1√2πσj Φ (ωj/σj)
exp
−(u − ωj)
2
2σ2j
where σ2j = µ
(2)j − ω2
j
Ensures that the probability of any naturalfrequencies becoming negative is zero
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Joint Moments of TwoEigenvalues
Arbitrary r − s-th order joint moment of two naturalfrequencies
µ(rs)jl = E
[ωr
j (x)ωsl (x)
]
=
∫
Rm
exp − (L(x) − r ln ωj(x) − s ln ωl(x)) dx,
r = 1, 2, 3 · · ·
Choose f(x) = L(x) − r ln ωj(x) − s ln ωl(x)
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Joint Moments of TwoEigenvalues
After some simplifications
µ(rs)jl ≈ (2π)m/2ωr
j (θ)ωsl (θ) exp −L (θ) ‖Df (θ)‖−1/2
where θ is obtained from:
dL(θ) =r
ωj(θ)dωj
(θ) +s
ωl(θ)dωl
(θ)
and Df (θ) = DL(θ) + r
ω2
j(θ)dωj
(θ)dωj(θ)T −
r
ωj(θ)Dωj
(θ) + s
ω2
l (θ)dωl
(θ)dωl(θ)T − s
ωl(θ)Dωl
(θ)
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Joint Moments of MultipleEigenvalues
We want to obtain
µ(r1r2···rn)j1j2···jn
=
∫
Rm
ωr1
j1(x)ωr2
j2(x) · · ·ωrn
jn(x)
px(x) dx
It can be shown that
µ(r1r2···rn)j1j2···jn
≈ (2π)m/2ωr1
j1(θ) ωr2
j2(θ) · · ·ωrn
jn(θ)
exp −L (θ) ‖Df (θ)‖−1/2
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Joint Moments of MultipleEigenvalues
Here θ is obtained from
dL(θ) =r1
ωj1(θ)dωj1
(θ)+r2
ωj1(θ)dωj2
(θ)+· · · rn
ωjn(θ)
dωjn(θ)
and the Hessian matrix is given by
Df (θ) = DL(θ)+
jn,rn∑
j = j1, j2, · · ·r = r1, r2, · · ·
r
ω2j (θ)
dωj(θ)dωj
(θ)T − r
ωj (θ)Dωj
(θ)
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Example System
Undamped three degree-of-freedom randomsystem:
m 1
m 2
m 3 k 4 k 5 k 1 k 3
k 2
k 6
mi = 1.0 kg for i = 1, 2, 3; ki = 1.0 N/m for i =
1, · · · , 5 and k6 = 3.0 N/m
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Example System
mi = mi (1 + ǫmxi), i = 1, 2, 3
ki = ki (1 + ǫkxi+3), i = 1, · · · , 6
Vector of random variables: x = x1, · · · , x9T ∈ R9
x is standard Gaussian, µ = 0 and Σ = I
Strength parameters ǫm = 0.15 and ǫk = 0.20
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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 beconsidered as benchmark.
The percentage error:
Error =(•) − (•)MCS
(•)MCS× 100
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Scatter of the Eigenvalues
1 1.5 2 2.5 3 3.5 4
100
200
300
400
500
600
700
800
900
1000
Sam
ples
Eigenvalues, ωj (rad/s)
ω1
ω2
ω3
Statistical scatter of the natural frequenciesω1 = 1, ω2 = 2, and ω3 = 3
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Error in the Mean Values
1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Eigenvalue number j
Perc
enta
ge e
rror w
rt M
CS
1st−order perturbation
2nd−order perturbation
Asymptotic method
Error in the mean values
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Error in Covariance Matrix
(1,1) (1,2) (1,3) (2,2) (2,3) (3,3)0
2
4
6
8
10
12
14
16
Elements of the covariance matrix
Perc
enta
ge e
rror w
rt M
CS
1st−order perturbation
2nd−order perturbation
Asymptotic method
Error in the elements of the covariance matrix
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Mean and Covariance
Using the asymptotic method, the mean andcovariance matrix of the natural frequencies areobtained as
µΩ = 0.9962, 2.0102, 3.0312T
and ΣΩ =
0.5319 0.5643 0.7228
0.5643 2.5705 0.9821
0.7228 0.9821 8.7292
× 10−2
Individual pdf and joint pdf of the natural frequencies
are computed using these values.
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Individual pdf
0.5 1 1.5 2 2.5 3 3.5 4 4.50
1
2
3
4
5
6
Natural frequencies (rad/s)
pdf o
f the
nat
ural
freq
uenc
ies
Asymptotic methodMonte Carlo Simulation
Individual pdf of the natural frequencies
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Analytical Joint pdf
Joint pdf using asymptotic method
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Joint pdf from MCS
Joint pdf from Monte Carlo Simulation
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Contours of the joint pdf
0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.61.5
2
2.5
3
3.5
4
(ω1,ω
3)
(ω1,ω
2)
(ω2,ω
3)
ωj (rad/s)
ω k (rad
/s)
Asymptotic methodMonte Carlo Simulation
Contours of the joint pdf
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Conclusions
Statistics of the natural frequencies of linearstochastic dynamic systems has beenconsidered
usual assumption of small randomness is notemployed in this study.
a general expression of the joint pdf of thenatural frequencies of linear stochastic systemshas been given
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Conclusions
a closed-form expression is obtained for thegeneral order joint moments of the eigenvalues
it was observed that the natural frequencies arenot jointly Gaussian even they are soindividually
future studies will consider joint statistics of theeigenvalues and eigenvectors and dynamicresponse analysis using eigensolutiondistributions
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References
Muirhead, R. J. (1982), Aspects of Multivariate Statistical Theory, John Wiely
and Sons, New York, USA.
36-1