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On the Nature of Random SystemMatrices in Structural DynamicsS. A DHIKARI AND R. S. L ANGLEYCambridge University Engineering DepartmentCambridge, U.K.Nature of Random System Matrices – p.1/20

Outline of the TalkIntroductionSystem randomness: Probabilistic approachParametric and non-parametric modelingMaximum entropy principleGaussian Orthogonal Ensembles (GOE)Random rod exampleConclusionsNature of Random System Matrices – p.2/20

Linear Systems Kyp C M Equations of motion:(1) where M, C and K are respectively the mass, dampingand stiffness matrices, y is the vector of generalizedcoordinates and p is the applied forcing function.Nature of Random System Matrices – p.3/20

System RandomnessKKCK(2) C CM M andM We consider randomness of the system matrices as Here,anddenotes the nominal (deterministic)and random parts ofrespectively.Nature of Random System Matrices – p.4/20

Parametric Modeling The Stochastic Finite Element Method (SFEM)Probability density function q q of randomvectors qhave to be constructed from therandom fields describing the geometry, boundaryconditions and constitutive equations bydiscretization of the fields.Mappings qGq q, where Gdenotes M C or K, have to be explicitlyconstructed. For an analytical approach, this stepoften requires linearization of the functions.For Monte-Carlo-Simulation:Re-assembly of the element matrices is requiredfor each sample.Nature of Random System Matrices – p.5/20

Non-parametric ModelingDirect construction of pdf of M C and K withouthaving to determine the uncertain localparameters of a FE model.Soize (2000) has used the maximum entropyprinciple for non-parametric modeling of systemmatrices in structural dynamics.Philosophy of Jayne’s Maximum Entropy Principle(1957):Make use of all the information that is given andscrupulously avoid making assumptions aboutinformation that is not available.Nature of Random System Matrices – p.6/20

EntropyWhat is entropy? – A measure of uncertainty. , Shannon’s For a continuous random variableMeasure of Entropy (1948):. Constraint: Suppose only the mean is known.Additional constraint:.Nature of Random System Matrices – p.7/20

Maximum Entropy Principle Construct the Lagrangian as where(3)Nature of Random System Matrices – p.8/20

Maximum Entropy Principle (4)Substituting from (3), equation (4) results or From the calculus of variation, forit isrequired thatmust satisfy the Euler-LagrangeequationThat is, exponential distribution.Nature of Random System Matrices – p.9/20

Soize Model (2000) The probability density function of any system matrix(say ) is defined as # & '( " ! % !whereNature of Random System Matrices – p.10/20

Soize Model (2000) The ‘dispersion’ parameter # !and # ifotherwise 0. Hereis the subspace ofconstituted of allpositive definite symmetric real matrices.Nature of Random System Matrices – p.11/20

GOE (Gaussian Orthogonal Ensembles) 1. The ensemble (say H) is invariant under everytransformation HW HW where W is anyorthogonal matrix.are statistically2. The various elementsindependent.3. Standard deviation of diagonals are twice that ofthe off-diagonal terms, H H H H The probability density functionNature of Random System Matrices – p.12/20

GOE in Structural DynamicsThe equations of motion describing free vibration of alinear undamped system in the state-space Ay where Ais the system matrix.Transforming into the modal coordinateswhere uis a diagonal matrix.Nature of Random System Matrices – p.13/20

GOE in Structural Dynamicsconstraints Suppose the system is now subjected toof the formuC Iu where Cconstraint matrix, I is theidentity matrix, u and u are partition of u.If the entries of C are independent, then it can beshown (Langley, 2001) that the random part of the system matrix of the constrained system approaches toGOE.Nature of Random System Matrices – p.14/20

Random Rod Equations of motion:(5) Boundary condition: fixed-fixed (U(0) U(L) 0) are zero mean random fields.Deterministic mode shapes:whereNature of Random System Matrices – p.15/20

Random Rod The random part Consider the mass matrix in the deterministic modalcoordinates:Nature of Random System Matrices – p.16/20

Case 1: is -correlated (white noise): , " " "Results: , ,Nature of Random System Matrices – p.17/20

Case 2: is fully correlated:for Case 2: , ,, " "Results:Nature of Random System Matrices – p.18/20

Conclusions & Future ResearchAlthough mathematically optimal givenknowledge of only the mean values of thematrices, it is not entirely clear how well theresults obtained from Soize model will match thestatistical properties of a physical system.Analytical works show that GOE may be apossible model for the random system matrices inthe modal coordinates for very large and complexsystems.The random rod analysis has shown that thesystem matrices in the modal coordinates is closeto GOE (but not exactly GOE) rather than theSoize model.Nature of Random System Matrices – p.19/20

Conclusions & Future ResearchFuture research will address more complicatedsystems and explore the possibility of using GOE(or close to that, due to non-negative definiteness)as a model of the random system matrices.Such a model would enable us to develop ageneral Monte-Carlo simulation technique to beused in conjunction with FE methods.Nature of Random System Matrices – p.20/20

Non-parametric Modeling Direct construction of pdf of M C and K without having to determine the uncertain local parameters of a FE model. Soize (2000) has used the maximum entropy principle for non-parametric modeling of system matrices in structural dynamics. Philosophy of Jayne's Maximum Entropy Principle (1957):

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