COMPLEX MODES IN LINEAR STOCHASTIC SYSTEMS S. Adhikari Department - PDF document

COMPLEX MODES IN LINEAR STOCHASTIC SYSTEMS S. Adhikari Department of Engineering University of Cambridge Trumpington Street Cambridge CB2 1PZ (U.K.) October, 2000

Outline of the Talk • Introduction • Viscously Damped Systems • Complex frequencies and modes • System Randomness • Derivatives of Complex Eigensolutions • Statistics of Complex Eigensolutions • Numerical examples • Summary and Conclusions 1

Viscously Damped Systems M ¨ q ( t ) + C ˙ q ( t ) + Kq ( t ) = 0 . (1) where M , C and K are the mass, damping and stiffness matrices respectively. q ( t ) is the vector of generalized coordinates.

Complex Frequencies and Modes The eigenvalue problem associated with equa- tion (1) can be represented by λ 2 k Mu k + λ k Cu k + Ku k = 0 . The eigenvalues, λ k , are the roots of the char- acteristic polynomial s 2 M + s C + K � � det = 0 . The order of the polynomial is 2 N and the roots appear in complex conjugate pairs. The eigenvalues are arranged as s 1 , s 2 , · · · , s N , s ∗ 1 , s ∗ 2 , · · · , s ∗ N . Each complex mode satisfies the normalization relationship u j = 1 � � u T 2 s j M + C ∀ k = 1 , · · · , 2 N , j γ j

System Randomness Randomness of the system matrices has the following form: M = M + δ M , C = C + δ C , and K = K + δ K . Here, ( • ) and δ ( • ) denotes the nominal (deter- ministic) and random parts of ( • ) respectively. It is assumed that δ M , δ C and δ K are zero- mean random matrices. The random parts are small and also they are such that 1. symmetry of the system matrices is pre- served, 2. the mass matrix M is positive definite, and 3. C and K are non-negative definite.

Statistics of the Eigenvalues If the random perturbations of the system ma- trices are small, s j can be approximated by a first-order Taylor expansion as N N N N ∂s j ∂s j � � � � s j = ¯ s j + δK rs + δC rs ∂K rs ∂C rs r =1 s =1 r =1 s =1 N N ∂s j � � + δM rs ∂M rs r =1 s =1 or in a matrix form as   δ K     s − ¯ s = D s δ C  δ M    where ∂s 1 ∂s 2 ∂s N   · · · ∂ K ∂ K ∂ K   ∂s 1 ∂s 2 ∂s N   ∈ R 3 N 2 × N D T   s = · · ·   ∂ C ∂ C ∂ C     ∂s 1 ∂s 2 ∂s N   · · · ∂ M ∂ M ∂ M

Derivatives of Complex Eigensolutions From Adhikari (1999): [ AIAA Journal , 37(11), pp. 1152–1158] Derivative of the j -th complex eigenvalue ∂s j ∂ M ∂ C ∂α + ∂ K � � s 2 ∂α = − γ j u T ∂α + s j u j . j j ∂α Derivative of the j -th complex eigenvector 2 N ∂ u j a ( α ) � ∂α = jk u k k =1 where ∂ M ∂ C ∂α + ∂ K γ j � � a ( α ) u T s 2 = − ∂α + s j u j j k jk s j − s k ∂α ∀ k = 1 , 2 , · · · , 2 N, � = j = − γ j ∂ M ∂α + ∂ C � � a ( α ) 2 u T and 2 s j u j . j jj ∂α

Derivatives w.r.t. the System Matrices For the eigenvalues: ∂s j � � = − γ j U rj U sj ∂K rs ∂s j ∂s j = s j ∂C rs ∂K rs ∂s j ∂s j = s 2 and . j ∂M rs ∂K rs For the eigenvectors: � � 2 N U rk U sj ∂U lj � = − γ j U lk ∂K rs s j − s k k =1 k � = j ∂U lj ∂U lj = − γ j � � U lj + s j U rj U sj 2 ∂C rs ∂K rs ∂U lj ∂U lj � � U lj + s 2 and = − γ j s j U rj U sj . j ∂M rs ∂K rs

Statistics of the Eigenvalues The covariance matrix of the eigenvalues, Σ s is obtained as s ) ∗ T > Σ s = < ( s − ¯ s ) ( s − ¯ T � �    δ K δ K     D ∗ T s = D s Σ kcm D ∗ T     = D s δ C δ C s .  δ M   δ M      Σ kcm ∈ R 3 N 2 × 3 N 2 , the joint covariance matrix of M , C and K is defined as < δ K δ K T > < δ K δ C T > < δ K δ M T >   < δ C δ K T > < δ C δ C T > < δ C δ M T >   Σ kcm =  .    < δ M δ K T > < δ M δ C T > < δ M δ M T >

Statistics of the Eigenvectors For small random perturbations of the system matrices, u j can be approximated by a first- order Taylor expansion   δ K     u j − ¯ u j = D u j δ C  . δ M    D u j , the matrix containing derivatives of u j with respect to elements of the system matri- ces, is given by   ∂U 1 j ∂U 2 j ∂U Nj · · ·  ∂ K ∂ K ∂ K    ∂U 1 j ∂U 2 j ∂U Nj ∈ R 3 N 2 × N   D T u j = · · ·     ∂ C ∂ C ∂ C   ∂U Nj ∂U 1 j ∂U 2 j    · · ·  ∂ M ∂ M ∂ M The covariance matrix of j -th and k -th eigen- vectors u k ) ∗ T > = D u j Σ kcm D ∗ T � � Σ u j u k = < u j − ¯ u j ( u k − ¯ u k .

Numerical example m m k k k m k m k u u u u u u u u u . . . c u c u Linear array of 8 spring-mass oscillators; nominal system: m u = 1 Kg, k u = 10 N/m and c u = 0 . 1 Nm/s

Statistics of the Eigenvalues (a) (b) 7 0.5 0.45 6 0.4 5 0.35 Standard deviation 0.3 4 Mean 0.25 3 0.2 0.15 2 0.1 1 0.05 0 0 2 4 6 8 2 4 6 8 (a) Absolute value of mean of complex natural frequencies (b) Standard deviation of complex natural frequencies; ‘X-axis’ Mode number; ‘—’ Analytical; ‘-.-.-’ MCS

Statistics of the Eigenvectors Mode: 1 Mode: 2 0.5 0.5 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Mode: 3 Mode: 4 0.5 0.5 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Mode: 5 Mode: 6 0.5 0.5 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Mode: 7 Mode: 8 0.5 0.5 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Real part of mean of the complex modes, ‘X-axis’ DOF; ‘—’ Analytical; ‘-.- -’ MCS

Statistics of the Eigenvectors Mode: 1 Mode: 2 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Mode: 3 Mode: 4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Mode: 5 Mode: 6 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Mode: 7 Mode: 8 0.3 0.3 0.2 0.2 0.1 0.1 0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Standard deviation of the complex modes, ‘X-axis’ DOF; ‘—’ Analytical; ‘-.-.-’ MCS

Summary and Conclusions • An approach has been proposed to obtain the second-order statistics of complex eigen- values and eigenvectors of non-proportionally damped linear stochastic systems. • It is assumed that the randomness is small so that the first-order perturbation method can be applied. • The covariance matrices of the complex eigensolutions are expressed in terms of the covariance matrices of the system proper- ties and derivatives of the eigensolutions with respect to the system parameters. • The proposed method does not require con- version of the equations of motion into the first-order form.