THE NATURE OF RANDOM SYSTEM MATRICES IN STRUCTURAL DYNAMICS S. - PDF document

THE NATURE OF RANDOM SYSTEM MATRICES IN STRUCTURAL DYNAMICS S. Adhikari Department of Engineering University of Cambridge Trumpington Street Cambridge CB2 1PZ (U.K.) May 2001

Outline of the Talk • Introduction • System randomness: Probabilistic approach • Parametric and non-parametric modeling • Maximum entropy principle • Gaussian Orthogonal Ensembles (GOE) • Random rod example • Conclusions 1

Random Systems Equations of motion: M ¨ y ( t ) + C ˙ y ( t ) + Ky ( t ) = p ( t ) (1) where M , D and K are respectively the mass, damp- ing and stiffness matrices, y ( t ) is the vector of gen- eralized coordinates and p ( t ) is the applied forcing function. We consider randomness of the system matrices as M = M + δ M C = C + δ C (2) and K = K + δ K . Here, ( • ) and δ ( • ) denotes the nominal (determin- istic) and random parts of ( • ) respectively.

Probabilistic Approach 1. Parametric modeling: The Stochastic Finite Element Method (SFEM) • Probability density function p q ( q ) of random vectors q ∈ R l have to be constructed from the random fields describing the geometry, boundary conditions and constitutive equa- tions by discretization of the fields. q + q ); R l → R N × N , where • Mappings q → G (¯ G denotes M , C or K , have to be explic- itly constructed. For an analytical approach, this step often requires linearization of the functions. • For Monte-Carlo-Simulation: Re-assembly of the element matrices is re- quired for each sample. 2. Non-parametric modeling: Direct construction of pdf of M , C and K with- out having to determine the uncertain local pa- rameters of a FE model.

Maximum Entropy Principle What is entropy? A measure of uncertainty. For a continuous random variable x ∈ D , Shannon’s Measure of Entropy (1948): � S ( p ( x )) = − D p ( x ) ln p ( x )d x Constraint: � D p ( x )d x = 1 Philosophy of Jayne’s Maximum Entropy Principle (1957): • Speak the truth and nothing but the truth • Make use of all the information that is given and scrupulously avoid making assumptions about information that is not available.

Maximum Entropy Principle Only mean is known: Additional constraint: � D xp ( x )d x = m Construct the Lagrangian as �� � � L = − D p ( x ) ln p ( x )d x − λ 0 D p ( x )d x − 1 �� � − λ 1 D xp ( x )d x − m � = D g ( p ( x ))d x where g ( p ( x )) = − p ( x ) ln p ( x ) − λ 0 p ( x ) − λ 1 xp ( x )+ λ 0 + mλ 1 (3)

Maximum Entropy Principle From the calculus of variation, for δ L = 0 it is re- quired that g ( p ( x )) must satisfy the Euler-Lagrange equation � � ∂g ( p ( x )) ∂g ( p ( x )) − ∂ = 0 (4) ∂p ( x ) ∂x ∂p ( x ) Substituting g ( p ( x )) from (3), equation (4) results − ln p ( x ) − 1 − λ 0 − λ 1 x + λ 1 = 0 p ( x ) = Ae − λ 1 x or That is, exponential distribution. A and λ 1 should be determined from the constraint equations. The analysis can be extended to vector valued random variables and random processes. If mean is unknown then p ( x ) is constant, ie, uni- form distribution. This is also known as the Laplace’s principle of insufficient reason .

Maximum Entropy Principle Mean and standard deviation is known: Additional constraint: � D ( x − m ) 2 p ( x )d x = σ 2 Following previous steps p ( x ) = Ae − λ 1 x − λ 2 x 2 (5) That is, Gaussian distribution.

Soize Model (2000) The probability density function of any system ma- trix (say G ) is defined as N ( R ) ([ G ]) c G (det[ G ]) λ G − 1 p [ G ] ([ G ]) = I M + � � − ( N − 1 + 2 λ G ) × exp Trace( G ) 2 where � N ( N − 1+2 λ G ) / 2 � N − 1 + 2 λ G (2 π ) − N ( N − 1) / 4 2 c G = � � �� ( N − 1 + 2 λ G ) � N l =1 Γ 2 The ‘dispersion’ parameter G ( N − 1) + (Trace[ G ]) 2 � � 1 1 − δ 2 λ G = 2 δ 2 Trace([ G 2 ]) G and � 1 / 2 � E � [ G ] − [ G ] � δ G = � [ G ] � N ( R ) ([ G ]) = 1 if [ G ] ∈ M + N ( R ) otherwise 0. Here I M + M + N ( R ) is the subspace of M N ( R ) constituted of all N × N positive definite symmetric real matrices.

Gaussian Orthogonal Ensembles (GOE) 1. The ensemble (say H ) is invariant under every transformation H → W T HW where W is any orthogonal matrix. 2. The various elements H jk , k ≤ j are statistically independent. 3. Standard deviation of diagonals are twice that of the off-diagonal terms, σ H jj = 2 σ H jk = σ , ∀ j � = k , where σ is some constant. The probability density function � − a Trace( H 2 ) + b Trace( H ) + c � p H ( H ) = exp Probability density function of the eigenvalues of H   N  − 1 x 2 �  � p ( x 1 , x 2 , · · · , x N ) = C N exp | x j − x k | j 2 j =1

GOE in structural dynamics The equations of motion describing free vibration of a linear undamped system in the state-space Ay = 0 where A ∈ R 2 N × 2 N is the system matrix. Trans- forming into the modal coordinates A u = 0 where A ∈ R 2 N × 2 N is a diagonal matrix. Suppose the system is now subjected to n con- straints of the form � � u 1 ( C − I ) = 0 u 2 where C ∈ R n × (2 N − n ) constraint matrix, I is the n × n identity matrix, u 1 and u 1 are partition of u . If the entries of C are independent, then it can be shown (Langley, 2001) that the random part of the system matrix of the constrained system ap- proaches to GOE.

Random Rod Equations of motion: = m ( x ) ∂ 2 U ∂ AE ( x ) ∂U � � (6) ∂t 2 ∂x ∂x Boundary condition: fixed-fixed (U(0)=U(L)=0) m ( x ) = m 0 [1 + ǫ 1 f 1 ( x )] AE ( x ) = AE 0 [1 + ǫ 2 f 2 ( x )] f i ( x ) are zero mean random fields. Deterministic mode shapes: � φ k ( x ) = a sin( kπx/L ) where a = 2 /Lm 0 Consider the mass matrix in the deterministic modal coordinates: � L � L m ′ jk = 0 φ j ( x ) m 0 φ k ( x )d x + ǫ 1 0 φ j ( x ) f 1 ( x ) φ k ( x )d x = m ′ 0 jk + ǫ 1 ∆ m ′ jk The random part � L ∆ m ′ jk = 0 φ j ( x ) f 1 ( x ) φ k ( x )d x < ∆ m ′ jk ∆ m ′ rs > = � L � L 0 φ j ( x 1 ) φ k ( x 1 ) φ r ( x 2 ) φ s ( x 2 ) R f 1 ( x 1 , x 2 )d x 1 d x 2 0

30 25 Mass Matrix 20 15 10 5 0 0 5 10 15 20 25 30

Random Rod Case 1: f 1 ( x ) is δ -correlated (white noise): R f 1 ( x 1 , x 2 ) = Q 1 δ ( x 1 − x 2 ) Results: rr > = 1 • < ∆ m ′ jj ∆ m ′ 4 a 4 Q 1 L , j � = r jj > = 3 • < ∆ m ′ jj ∆ m ′ 8 a 4 Q 1 L kj > = 1 • < ∆ m ′ kj ∆ m ′ 4 a 4 Q 1 L , k � = j • < ∆ m ′ kj ∆ m ′ rs > = 0 • < ∆ m ′ kk ∆ m ′ kr > = 0, k � = r

Random Rod Case 2: f 1 ( x ) is fully correlated: R f 1 ( x 1 , x 2 ) = Q 2 for x 1 , x 2 ∈ [0 , L ] Results: rr > = 1 • < ∆ m ′ jj ∆ m ′ 4 a 4 Q 2 L 2 , j � = r jj > = 3 8 a 4 Q 2 L 2 • < ∆ m ′ jj ∆ m ′ • < ∆ m ′ kj ∆ m ′ kj > = 0, k � = j • < ∆ m ′ kj ∆ m ′ rs > = 0 • < ∆ m ′ kk ∆ m ′ kr > = 0, k � = r

Conclusions and Future Research • Although mathematically optimal given knowl- edge of only the mean values of the matrices, it is not entirely clear how well the results ob- tained from Soize model will match the statis- tical properties of a physical system. • Analytical works show that GOE may be a pos- sible model for the random system matrices in the modal coordinates for very large and com- plex systems. • The random rod analysis has shown that the system matrices in the modal coordinates is close to GOE (but not exactly GOE) rather than the Soize model. • Future research will address more complicated systems and explore the possibility of using GOE (or close to that, due to non-negative definite- ness) as a model of the random system matri- ces. Such a model would enable us to develop a general Monte-Carlo simulation technique to be used in conjunction with FE methods.