Generalized Wishart Distribution for Probabilistic Structural Dynamics

Comput Mech (2010) 45:495–511 DOI 10.1007/s00466-010-0467-3

ORIGINAL PAPER

Generalized Wishart distribution for probabilistic structural dynamics

Sondipon Adhikari

Received: 9 April 2009 / Accepted: 3 January 2010 / Published online: 23 January 2010 © Springer-Verlag 2010

Abstract An accurate and efficient uncertainty quantifi- Keywords Unified uncertainty quantification · Random cation of the dynamic response of complex structural sys- matrix theory · Wishart distribution · Model validation · tems is crucial for their design and analysis. Among the Parameter identification many approaches proposed, the random matrix approach has received significant attention over the past decade. In this List of symbols paper two new random matrix models, namely (1) gener- f(t) Forcing vector alized scalar Wishart distribution and (2) generalized diag- M, C and K Mass, damping and stiffness matrices, onal Wishart distribution have been proposed. The central respectively aims behind the proposition of the new models are to (1) φ Undamped eigenvectors improve the accuracy of the statistical predictions, (2) sim- j q(t) Response vector plify the analytical formulations and (3) improve computa- δ Dispersion parameter of G tional efficiency. Identification of the parameters of the newly G n Number of degrees of freedom proposed random matrix models has been discussed. Closed- (•)T Matrix transposition form expressions have been derived using rigorous analyti- R Space of real numbers cal approaches. It is considered that the dynamical system is R+ Space n × n real positive definite matrices proportionally damped and the mass and stiffness properties n Rn×m Space n × m real matrices of the system are random. The newly proposed approaches |•| Determinant of a matrix are compared with the existing Wishart random matrix model etr {•} exp {Trace (•)} using numerical case studies. Results from the random matrix • Frobenius norm of a matrix, approaches have been validated using an experiment on a F • = (•)(•)T 1/2 vibrating plate with randomly attached spring-mass oscilla- F Trace ∼ tors. One hundred nominally identical samples have been Distributed as (•) created and separately tested within a laboratory framework. Trace Sum of the diagonal elements of a matrix Relative merits and demerits of different random matrix for- pdf Probably density function mulations are discussed and based on the numerical and experimental studies the recommendation for the best model has been given. A simple step-by-step method for imple- 1 Introduction menting the new computational approach in conjunction with general purpose finite element software has been outlined. Finite element codes implementing physics based models are used extensively for the dynamic analysis of complex systems. Laboratory based controlled tests are often performed B to gain insight into some specific physics of a problem. Such S. Adhikari ( ) tests can indeed lead to new physical laws improving the School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK computational models. Test data can also be used to calibrate e-mail: [email protected] a known model. However, neither of these activities may 123 496 Comput Mech (2010) 45:495–511 be enough to produce a credible numerical tool because of a central Wishart distribution for the system matrices. In a several types of uncertainties which exist in the physics based recent paper [7] it was shown that Wishart matrix with prop- computational framework. Such uncertainties include, but erly selected parameters can be used for systems with both are not limited to (1) parameter uncertainty (e.g, uncertainty parametric uncertainty and nonparametric uncertainty. The in geometric parameters, friction coefficient, strength of the main conclusions arising from this study were: materials involved); (2) model uncertainty (arising from the lack of scientific knowledge about the model which is a priori – Wishart random matrix distribution can be obtained either unknown); (3) experimental error (uncertain and unknown using the maximum entropy approach or using the matrix errors percolate into the model when they are calibrated factorization approach. against experimental results). These uncertainties must be – The maximum entropy approach gives a natural selection assessed and managed for credible computational predic- for the parameters. Through numerical examples it was tions. shown that the parameters obtained using the maximum The predictions from high resolution numerical models entropy approach may yield non-physical results. In that may sometimes exhibit significant differences with the the ‘mean of the inverse’ and the ‘inverse of the mean’ of results from physical experiments due to uncertainty. When the random matrices can be significantly different. substantial statistical information exists, the theory of prob- – The parameters of the pdf of a Wishart random matrix ability and stochastic processes offer a rich mathematical obtained using the maximum entropy approach are not framework to represent such uncertainties. In a probabilis- unique since they depend on what constraints are used in tic setting, the data (parameter) uncertainty associated with the optimization approach. the system parameters, such as the geometric properties and – Considering that the available ‘data’ is the mean and (nor- constitutive relations (i.e. Young’s modulus, mass density, malized) standard deviation of a system matrix, it was Poisson’s ratio, damping coefficients), can be modeled as shown that when the mean of the inverse equals to the random variables or stochastic processes using the so-called inverse of the mean of the system matrices, the predictive parametric approach. These uncertainties can be quantified accuracy is maximum. and propagated, for example, using the stochastic finite element method [8,9,15,17,20,23,25,26,28,34,40–42,49]. Recently, the uncertainty due to modelling error has received The aim of this paper is to further explore the idea of proper attention as this is crucial for model validation [21,22,35]. parameter selection of Wishart matrices. Specifically, we ask The model uncertainty problem poses serious challenges as the question ‘what is the simplest possible Wishart random the parameters contributing to the modelling errors are not matrix model which can be used without compromising the available a priori and therefore precludes the application of predictive accuracy?’. The definition of ‘simplest’ and ‘accu- a parametric approach to address such issues. Model uncer- racy’ will be explained later in the paper. The outline of the tainties do not explicitly depend on the system parameters. paper is as follows. In Sect. 2 dynamic response of linear For example, there can be unquantified errors associated with stochastic systems is discussed. A brief overview of random the equation of motion (linear or non-linear), in the damping matrix models in probabilistic structural dynamics is given model (viscous or non-viscous [3,11]), in the model of struc- in Sect. 3. The identification method of the two new random tural joints. The model uncertainty may be tackled by the so- matrix model is proposed in Sect. 4. In Sect. 5 the accu- called non-parametric method pioneered by Soize [44–46] racy and efficiency the proposed random matrix models are and adopted by others [5,6,31,32]. numerically verified. An experiment conducted on a vibrat- The equation of motion of a damped n-degree-of-freedom ing plate with randomly attached sprung-mass oscillators is linear dynamic system can be expressed as used to validate the proposed approach in Sect. 6. Based on the numerical and experimental studies, a step-by-step ¨( ) + ˙( ) + ( ) = ( ) Mq t Cq t Kq t f t (1) approach is outlined in Sect. 7 to apply the proposed uncer- where f(t) ∈ Rn is the forcing vector, q(t) ∈ Rn is the tainty quantification approach. response vector and M ∈ Rn×n, C ∈ Rn×n and K ∈ Rn×n are the mass, damping and stiffness matrices respectively. In order to completely quantify the uncertainties associated 2 Dynamic response of discrete stochastic systems with system (1) we need to obtain the probability density functions of the random matrices M, C and K.Usingthe Modal analysis, originally due to Lord rayleigh [39], is the parametric approach, such as the stochastic finite element most widely used approach for linear structural dynamics. method, one usually obtains a problem specific covariance Using this approach the dynamic response of a system can structure for the elements of system matrices. The nonpara- be obtained by linear superposition of vibration modes. The metric approach [5,6,44–46] on the other hand results in extension of classical modal analysis to stochastic systems 123 Comput Mech (2010) 45:495–511 497 requires the calculation of natural frequencies and mode- Since is a symmetric and positive definite matrix, it can shapes which are random in nature. The efficiency and be diagonalized by a orthogonal matrix r such that accuracy of a computational method therefore crucially T = 2 dependent on the efficiency and accuracy of the calcula- r r r (9) tion of random natural frequencies and mode-shapes. In this Here the subscript r denotes the random nature of the eigen- section first we transform the coupled equations of motion values and eigenvectors of the random matrix . Recalling in the modal modal domain and then introduce uncertainty T = that r r In, from Eq. (6) we obtain directly in natural frequencies and mode-shapes using ran- −1 dom matrix theory. The parameter estimation and sampling 2 2 ¯ q¯ = s In + sC + f (10) of this random matrix distribution are discussed in details in −1 later sections. = 2 + ζ + 2 T ¯ r s In 2s r r r f (11) Assuming all the initial conditions are zero and taking the Laplace transform of the equation of motion (1)wehave where s2M + sC + K q¯(s) = f¯(s) (2) ζ = diag [ζ1,ζ2,...,ζn] (12) The response in the original coordinate can be obtained as where (•¯) denotes the Laplace transform of the respective − quantities. The aim here is to obtain the statistical properties 1 q¯(s) = q¯ (s)= s2I + 2sζ +2 ( )T f¯(s) of q¯(s) ∈ Cn when the system matrices are random matrices. r n r r r n T ¯( ) The undamped eigenvalue problem is given by xr f s = j x . 2 2 r j (13) φ = ω2 φ , = , ,..., s + 2sζ j ωr + ω K j j M j j 1 2 n (3) j=1 j r j ω2 φ Here where j and j are respectively the eigenvalues and mass- normalized eigenvectors of the system. We define the matri- = diag ω ,ω ,...,ω (14) ces r r1 r2 rn = = , ,..., and Xr r xr1 xr2 xrn (15) = ω ,ω ,...,ω = φ , φ ,...,φ . diag [ 1 2 n] and 1 2 n are respectively the matrices containing random eigenvalues (4) and eigenvectors of the system. The frequency response func- so that tion (FRF) of the system can be obtained by substituting s = iω in Eq. (13). In the next section we discuss the deriva- T = 2 T = Ke and M In (5) tion of the random matrix . where In is an n-dimensional identity matrix. Using these, Eq. (2) can be transformed into the modal coordinates as 3 Overview of random matrix models in probabilistic 2 2 ¯ s In + sC + q¯ = f (6) structural dynamics

where and (•) denotes the quantities in the modal coordi- Recalling that is a symmetric and positive definite nates: matrix, one can use the Wishart random matrix model [5,6,37,44–48]. Consider that a random symmetric and pos- C = T C, q¯ = q¯ and f¯ = T f¯ (7) itive definite matrix G has mean G0 and normalized standard For simplicity let us assume that the system is proportionally deviation or dispersion parameter [2,4,13] damped with deterministic modal damping factors E G − E [G]2 ζ ,ζ ,...,ζ . Modal analysis of such systems [1,29] can δ2 = F (16) 1 2 n G 2 be performed using real normal modes. When we consider E [G] F 2 random systems, the matrix of eigenvalues in Eq. (6) will Suppose that G can be modeled by a Wishart matrix with be a random matrix of dimension n. Suppose this random parameters p and so that G ∼ Wn(p, ). The probability n×n matrix is denoted by ∈ R : density function of G can be expressed as 2 ∼ −1 (8) 1 1 1 1 ( − − ) ( ) = 2 np || 2 p | | 2 p n 1 pG G 2 n p G The distribution of the random matrix will be discussed 2 in details in Sect. 4. The steps in the rest of this section are 1 − etr − 1G (17) however independent of the random matrix distribution. 2 123 498 Comput Mech (2010) 45:495–511 where the multivariate Gamma function is given by where n 1 1 n(n−1) 1 θ = {1 + γG } − (n + 1) (20) n (a) = π 4 a − (k − 1) ; δ2 2 G k=1 1 for (a)> (n − 1) (18) and 2 2 {Trace (G0)} We refer to the books by Muirhead [33], Eaton [14], Griko γG = (21) Trace G 2 [18], Gupta and Nagar [19], Mathai and Provost [27], Tulino 0 and Verdú [50] and Mezzadri and Snaith [30] for the deriva- This parameter selection was proposed by Soize [44,45] tion of the pdf in Eq. (17) and related mathematical details for M, C and K matrices. For proportionally damped on random matrices. systems, this approach requires the simulation of M and Strictly speaking, when the parameter p is a non-integer, K matrices. the probability density function in Eq. (17) is known as the 2. Parameter selection 2: Here one considers that for each matrix variate Gamma distribution, see for example, page 122 system matrices, the mean of the inverse of the ran- in the book by Gupta and Nagar [19]. Soize [44,45] proposed dom matrix equals to the inverse of the deterministic the matrix variate Gamma distribution for structural system matrix and the dispersion parameter is same as the mea- matrices. However, when p is large as in the case of complex sured dispersion parameter. Mathematically this implies systems (recall that p must be more than n + 1), the differ- E G−1 = G −1. Using the theory of inverted Wishart ence between the Gamma distribution and a Wishart distri- 0 distribution and after some simplifications one obtains bution with the closest integer parameter is negligible [7]. As a result we will only consider Wishart distribution since p = n + 1 + θ and = G /θ (22) (1) it is computationally much simpler to simulate, and (2) 0 a very rich body of literature is available. It must be empha- where θ is defined in Eq. (20). This parameter selection sized that the central Wishart distribution used here is just was proposed by Adhikari [7]. one of many possible symmetric random matrix distribution 3. Parameter selection 3: For each system matrices, one available in literature. Other distributions listed by Gupta and considers that the mean of the random matrix and the Nagar [19] such as the noncentral Wishart distribution, matrix mean of the inverse of the random matrix are closest to the variate beta distribution, matrix variate Dirichlet distribution deterministic matrix and its inverse. Mathematically this and generalized quadratic-forms can be used provided their implies G − E [G] and G −1 − E G−1 are parameters can be identified easily. The parameter identifi- 0 F 0 F minimum. This condition results: cation of random matrix distribution is fundamental to their effectiveness as a surrogate model to quantify uncertainty in = + + θ = /α complex dynamical systems. p n 1 and G0 (23) In the previous works [5,6,44–46] the system matrices √ α = θ( + + θ) θ were separately considered as Wishart matrices. Because the where n 1 and is as defined in Eq. (20). matrices , M, C and K have similar mathematical prop- This is known as the optimal Wishart distribution [5]. The erties, for notational convenience we will use the notation rationale behind this approach is that a random system G which stands for any one of the system matrices. The matrix and its inverse should be mathematically treated parameters p and of Wishart matrices can be obtained in a similar manner as both are symmetric and positive- based on what criteria we select. For this parameter estima- definite matrices. 4. Parameter selection 4: This criteria arises from the idea tion problem, the ‘data’ consist of the ‘measured’ mean (G0) that the mean of the eigenvalues of the distribution is and dispersion parameter (δG ) of a system matrix. Adhikari [7] investigated the following parameter selection methods: same as the ‘measured’ eigenvalues of the mean matrix and the dispersion parameter is same as the measured dispersion parameter. Mathematically one can express 1. Parameter selection 1: For each system matrices, one this as considers that the mean of the random matrix is same as the deterministic matrix and the dispersion parameter is −1 −1 E M = M0 , and E [K] = K0 (24) same as the measured dispersion parameter. Mathemat- ically this implies E [G] = G . This condition results 0 Based on extensive numerical studies it was shown [7] that the parameter selection 2 produces least error when the results p = n + 1 + θ and = G0/p (19) were compared with direct Monte Carlo simulation. Even for 123 Comput Mech (2010) 45:495–511 499 proportionally damped systems, following this approach, two of H. The main challenge here is that the dispersion parameter correlated random Wishart matrices (the mass and the stiff- of H needs to be obtained from dispersion parameters of M ness matrices) need to be simulated and their joint eigenvalue and K. We propose a new approach based on the theory of problem need to be solved. Here we consider the possibility inverted Wishart random matrices. of considering the matrix as the only equivalent random It should be note that when M and K are Wishart matri- Wishart matrix. The motivations behind this approach are: ces, theoretically H = M−1/2KM−1/2 cannot be Wishart matrices. The exact distribution of such matrices may be – Because 2 is a diagonal matrix, is an uncorrelated obtained using the Jacobian associated with the non-linear Wishart matrix which is very easy to simulate. transformation of Wishart matrices as discussed by Gupta – Only one matrix needs to be simulated for each sample. and Nagar [19]. On the other hand, theoretically one can – Instead of solving a generalized eigenvalue problem directly derive the Wishart random matrix model for the involving two matrices for each sample, one now needs to symmetric and positive definite matrix H using the maxi- solve only a standard eigenvalue problem involving just mum entropy approach proposed by Soize [44,45]orthe one symmetric and positive definite matrix. matrix factorization approach proposed by Adhikari [7]. In – Because the eigenvalues of the random system are now that case the individual system matrices M and K will not governed by only one random matrix, it is possible to be Wishart matrices. The generalized Wishart approach pro- develop further insights into the distribution of the eigen- posed here therefore do not strictly follow from the original values. Wishart matrix model of the system matrices. The approach – Due to the simplicity of the expression of the frequency presented here should be considered as another approximate response function (13), it may form the essential basis approach derived for mathematical simplicity and computa- for analytical derivation of response moments using the tional efficiency. With this spirit, we fit an equivalent Wishart theory of eigensolutions of Wishart matrices. distribution for the matrix H. From the numerator of the definition of the dispersion For the above points to be realized, one must (1) derive suit- parameter in Eq. (16)wehave ∼ ( , ) able parameters for the random matrix Wn p from − 2 = [ − ][ − ]T δ δ E H E [H] F E Trace H E [H] H E [H] the available data, namely M0, K0, M and K , and (2) numerically verify that this approach results acceptable fidelity = Trace E [H2 − H E [H] − E [H] H − E [H]2] with respect to direct monte carlo simulation and if possible, experimental results. The result of the paper is devoted = Trace E H2 − E [H]2 = Trace E H2 to address these two issues. −Trace E [H]2 2 = −1 −1 − −1 4 Identification of the generalized Wishart random Trace E M KM K Trace E M K matrix model (27)

Because is a symmetric positive-definite matrix, can Let us assume be expressed in terms of its eigenvalues and eigenvectors as M ∼ Wn(p1, 1) and K ∼ Wn(p2, 2) (28) = λT .Here ∈ Rn×n is an orthonormal matrix con- T taining the eigenvectors of , that is, = In and λ is Following Theorems 3.3.15, 3.3.15 and 3.3.17 of Gupta and a real diagonal matrix containing the eigenvalues of .The Nagar [19] we have the following useful results λ2 Frobenius norm of the matrix 2 is given by = E [K] p2 2 (29) 2 = T = λT λT = T + ( ) F Trace Trace E [KBK] p2 2B 2 p2Trace 2B 2 +p2 B (30) = T λλ = λ2 + λ2 +···+λ2 2 2 2 Trace n (25) 1 2 −1 = −1 E M c0 1 (31) Note that the eigenvalues of are same as the eigenvalues − − / − / −1 −1 −1 −1 of M 1K or M 1 2KM 1 2. We deﬁne the matrix H as E M AM = c1 A 1 1 −1 −1 T −1 −1 −1 H = M K (26) +c2 A +Trace A (32) 1 1 1 1 − − Therefore, form the definition of dispersion parameter in and E Trace AM 1 Trace BM 1 Eq. (16) and the expression of the Frobenius norm in Eq. (25), = −1 −1 we can say that the dispersion parameter of is same as that c1Trace A 1 Trace B 1 123 500 Comput Mech (2010) 45:495–511 + −1 −1 Using Eq. (33), the second term in right hand side of Eq. (37) c2 Trace B 1 A 1 can be obtained as − − + Trace B 1AT 1 (33) 1 1 −1 −1 E Trace 2M Trace 2M Here A and B are n × n matrices and the constants c0, c1and 2 2 = −1 + −1 c2 are deﬁned as c1 Trace 1 2 2c2Trace 1 2 (39)

−1 c0 = (p1 − n − 1) , c1 = (p1 − n − 2)c2 Using the notations (34) −1 and c ={(p − n)(p − n − 1)(p − n − 3)} 2 2 2 1 1 1 α = −1 α = −1 a Trace 1 2 and b Trace 1 2 Equations (29)–(33) and the fact that and are symmet- 1 2 (40) ric matrices will be used to derive the dispersion parameter for the H matrix. We use the notation and substituting Eqs. (38) and (39) into Eq. (37)wehave − H = M 1K (35) −1 −1 = ( + 2) {( + )α + α } 0 0 0 Trace E M KM K p2 p2 c1 c2 a c2 b

+p2 {c1αb + 2c2αa} (41) Note that H0 is in general not the mean of the H matrix. Because M and K are statistically independent random matri- For the second term in right-hand side of Eq. (27)wehave ces, we have −1 = −1 = −1 − − − − E M K E M E [K] c0 p2 2 (42) E M 1KM 1K = E M 1 E KM 1K 1 = −1 −1 Therefore, E M p2 2M 2 − 2 − 2 −1 Trace E M 1K = c2 p2Trace 1 +p2Trace 2M 2 0 2 1 2 + 2 −1 = c2 p2α (43) p2 2M 2 0 2 a = ( + 2) −1 −1 From the expressions of the dispersion parameter in Eq. (16) p2 p2 E M 2M 2 we have −1 −1 +p2E M 2Trace 2M 2 Trace E M−1KM−1K − Trace E M−1K (36) δ2 = H 2 Trace E M−1K Taking the trace of the above equation and noting that the p + p 2 ((c +c ) α +c α )+ p (c α +2 c α )−c 2 p 2α = 2 2 1 2 a 2 b 2 1 b 2 a 0 2 a order of the expectation and trace operators can be inter- 2 2 c0 p2 αa changed we have (44) − − Trace E M 1KM 1K Following parameter the Sect. 2 discussed in the previous = ( + 2) −1 −1 section we have p2 p2 Trace E M 2M 2 γ + 1 − − = /θ , = M +p E Trace M 1 Trace M 1 (37) 1 M0 M p1 (45) 2 2 2 δM γ + = /θ , = K 1 Using Eq. (32), the ﬁrst term in right hand side can be obtained and 2 K0 K p2 (46) δK as Using these and following the definition of the parameter γ −1 −1 Trace E M 2M 2 in Eq. (21) one has = −1 −1 + −1 −1 2 −1 2 Trace c1 1 2 1 c2 1 2 1 {Trace (H0)} Trace M0 K0 γH = = − − 2 −1 2 + 1 1 Trace H0 Trace M0 K0 Trace 2 1 1 2 2 − 2 −1 Trace 1 = (c1 + c2)Trace 2 1 2 αb 1 = = (47) 2 α − 2 −1 a + 1 Trace 2 c2 Trace 1 2 (38) 1 123 Comput Mech (2010) 45:495–511 501

Suppose 0 is the diagonal matrix containing the undamped p =[n + 1 + θ]. For complex engineering systems n can natural frequencies corresponding to the baseline system con- be in the order of several thousands or even millions. For sisting the mass and stiffness matrices M0 and K0 respec- this reason, as shown in reference [7], this approximation tively. Then the constant γH can be obtained as introduces negligible error. Now we create an n × p matrix 2 Y with Gaussian random numbers with zero mean and unit 2 2 2 { ( )}2 2 ω Y ∼ N , O, I ⊗ I = /θ Trace H0 Trace 0 j 0 j covariance i.e., n p n p .Note 0 is γH = = = 2 4 ω4 a positive definite diagonal matrix. It can be factorized as Trace H0 Trace 0 j 0 √ √ T j = / θ / θ (48) 0 0 . Using this, we can obtain the matrix Y using the linear transformation Substituting αb = γH αa in Eq. (44)wehave √ = / θ ( + + ) γ + + + + − 2 Y 0Y (55) δ2 = c2 p2c2 c1 H c1 3 c2 p2c1 p2c2 c0 p2 H 2 ∼ c0 p2 Following theorem 2.3.10 in [19] it can be shown that Y , 2/θ ⊗ (49) Nn,p O 0 Ip . One can now obtain the samples of + + θ,2/θ the Wishart random matrices Wn n 1 0 as Using the expression of the constants c0–c2 in Eq. (34), this expression can be further simpliﬁed as = YYT (56) p 2 +(p −2−2n) p +(−n−1) p +n2 +1+2n γ δ2 = 1 2 1 2 H Alternatively, Matlab command wishrnd can be used H (− + )(− + + ) p2 p1 n p1 n 3 to generate the samples of Wishart matrices. One need to 2 2 p1 + (p2 − 2 n) p1 + (1 − n) p2 − 1 + n solve the symmetric eigenvalue problem for every sample + (50) p (−p + n)(−p + n + 3) 2 1 1 2 r = r (57) This equation completely deﬁnes the dispersion parameter r 2 of the generalized random matrix in closed-form. The eigenvalue and eigenvector matrices r and r can then Based on the expression of δH together with the ‘mean be substituted in Eq. (13) to obtain the dynamic response. 2 matrix’ 0, two types of Wishart random matrix models are These two random matrix models together with the original proposed for the generalized system matrix : approach proposed by Soize [44,45] are compared in the next sections. Two numerical and one experimental case studies (a) Scalar Wishart Matrix: In this case it is assumed that are considered. a2 ∼ Wn p, In (51) n 5 Numerical investigations

= 2 δ = δ Considering E [ ] 0 and H the values of A rectangular cantilever steel plate is considered to illustrate the unknown parameters can be obtained as the application of the proposed generalized Wishart random matrices in probabilistic structural dynamics. The determin- 1 + γ = H 2 = 2 / istic properties are assumed to be E¯ = 200 × 109 N/m2, p and a Trace 0 p (52) δ2 3 ¯ H ν¯ = 0.3, ρ¯ = 7,860 kg/m , t = 3.0 mm, L x = 0.998 m, L y = 0.59m. These values correspond to the experimental (b) Diagonal Wishart Matrix with different entries: In this study discussed in the next section. The schematic diagram of case it is assumed that the plate is shown in Fig. 1. The plate is excited by an unit har- monic force and the response is calculated at the point shown ∼ , 2/θ Wn p 0 (53) in the diagram. The plate is divided into 25 elements along the x-axis and 15 elements along the y-axis for the numerical −1 = −2 δ = δ calculations. The resulting system has 1,200 degrees of free- Considering E 0 and H as proposed in [7]wehave dom so that n = 1,200. Response is calculated for the six points shown in the ﬁgure. These points correspond to loca- (1 + γH ) tion of the accelerometers used in the experimental study. p = n + 1 + θ and θ = − (n + 1) (54) δ2 Here we show results corresponding to points 1 and 2 only. H Two different cases of uncertainties are considered. In the The samples of can be generated using the procedure out- ﬁrst case, it is assumed that the material properties are ran- lined in the previous works [5,7]. The constant p needs to domly inhomogeneous. In the second case, we consider that be approximated to the nearest integer of n + 1 + θ so that the plate is ‘perturbed’ by attaching spring-mass oscillators 123 502 Comput Mech (2010) 45:495–511

1 Input with direct stochastic finite element Monte Carlo Simulation results. Recall that these simplified methods require simula- 0.5 tion of only one uncorrelated Wishart random matrix. As a result, we are also interested to understand whether signifi- 5 cant accuracy is lost when comparing these simplified meth- 0 3 6 ods with the original approach involving two fully correlated 1 4 2 Wishart matrices [44,45]. −0.5 Fixed ed From the simulated random mass and stiffness matrices 0.8 ge Outputs δ = . δ = . 0.6 1 we obtain M 0 1133 and K 0 2916. Since 2% constant 0.8 δ = 0.4 0.6 modal damping factor is assumed for all the modes, C 0. 0.2 0.4 The only uncertainty related information used in the random Y direction (width) 0.2 0 0 X direction (length) matrix approach are the values of δM and δK . The informa- Fig. 1 Schematic diagram of a steel cantilever plate. The determinis- tion regarding which element property functions are random tic properties are: E¯ = 200 × 109 N/m2, ν¯ = 0.3, ρ¯ = 7,860 kg/m3, fields, nature of these random fields (correlation structure, ¯ = . = . = . t 3 0 mm, Lx 0 998m, L y 0 59m. A modal damping factor of Gaussian or non-Gaussian) and the amount of randomness 2% is assumed for all modes are not used in the Wishart matrix approach. This is aimed to depict a realistic situation when the detailed information at random locations. The first case corresponds to a paramet- regarding uncertainties in a complex engineering system is ric uncertainty problem while the second case corresponds not available to the analyst. to a non-parametric uncertainty problem. The predicted mean of the amplitude using the direct stochastic finite element simulation and three Wishart matrix approaches are compared in Fig. 2 for the driving-point- 5.1 Plate with randomly inhomogeneous material FRF (node number 481) and a cross FRF (node number properties: parametric uncertainty problem 877). In Fig. 3 percentage errors in the mean of the amplitude of the driving-point and cross-FRF obtained using the It is assumed that the Young’s modulus, Poissons ratio, mass proposed three Wishart matrix approaches are shown. density and thickness are random fields of the form Percentage errors are calculated using the direct Monte Carlo simulation results as the benchmarks. The computa- ¯ E(x) = E (1 + f (x)),ν(x) =¯ν (1 + ν f (x)) (58) E 1 2 tional cost using each of the two single uncorrelated Wishart ¯ ρ(x) =¯ρ 1+ ρ f3(x) and t(x)=t (1+ t f4(x)) (59) random matrix model turns out to be about 30% less compared to the fully correlated case. The computational effi- The two dimensional vector x denotes the spatial coordi- ciency is likely to increase with larger system dimensions nates. The strength parameters are assumed to be E = 0.10, and number of samples used in the Monte Carlo simulation. ν = 0.10, ρ = 0.08 and t = 0.12. The random fields The predicted standard deviation of the amplitude using fi (x), i = 1,...,4 are assumed to be correlated homog- the direct stochastic finite element simulation and three Wis- enous Gaussian random fields. An exponential correlation hart matrix approaches are compared in Fig. 4 for the driving- function with correlation length 0.2 times the lengths in point-FRF and the cross FRF. In Fig. 5 percentage errors in each direction has been considered. The random fields are the standard deviation of the amplitude of the driving-point simulated by expanding them using the Karhunen-Loève and cross-FRF obtained using the proposed three Wishart expansion [17,36] involving uncorrelated standard normal matrix approaches are shown. Percentage errors are variables. A 5000-sample Monte Carlo simulation is per- calculated using the direct Monte Carlo simulation results as formed to obtain the frequency response functions (FRFs) the benchmarks. Error in both mean and standard deviation of the system. The quantities E(x), ρ(x) and t(x) are posi- using any one of the random matrix approaches reduce with tive while −1 ≤ ν(x) ≤ 1/2 for all x. Due to the bounded the increase in frequency. Among the three Wishart matrix nature of these quantities, the Gaussian random field is not approaches discussed here, the diagonal Wishart matrix with an ideal model for these quantities. However, due to small different entries produces most accurate results across the fre- variability considered for these parameters, the probability quency range. As expected, the scalar Wishart matrix model that any of these quantities become non-physical is small. produces least accurate results. Note that this approach only We have explicitly verified that all the realizations of these need the simulation of one random matrix. From these results four random fields are physical in nature in our Monte Carlo we conclude that the diagonal Wishart matrix with different simulation. entries should be used for a system with parametric uncer- We want to identify which of the two Wishart matrix fit- tainty. In the next section we discuss the same system with ting approaches proposed here would produce highest fidelity non-parametric uncertainty. 123 Comput Mech (2010) 45:495–511 503

−60 −60

M and K are fully correlated Wishart M and K are fully correlated Wishart −80 Scalar Wishart −80 Scalar Wishart Diagonal Wishart with different entries Diagonal Wishart with different entries Direct simulation Direct simulation −100 −100 de (dB) de (dB) u u

−120 −120

−140 −140 Mean of amplit Mean of amplit

−160 −160

−180 −180 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 2 Comparison of the mean of the amplitude obtained using the direct stochastic ﬁnite element simulation and three Wishart matrix approaches for the plate with randomly inhomogeneous material properties. a Driving-point-FRF. b Cross FRF

20 20

18 18 M and K are fully correlated Wishart M and K are fully correlated Wishart 16 Scalar Wishart 16 Scalar Wishart Diagonal Wishart with different entries Diagonal Wishart with different entries 14 14

12 12

10 10

8 8 Error in the mean Error in the mean 6 6

4 4

2 2

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 3 Comparison of percentage errors in the mean of the amplitude obtained using the three Wishart matrix approaches for the plate with randomly inhomogeneous material properties. a Driving-point-FRF. b Cross FRF

5.2 Plate with randomly attached spring-mass oscillators: modal damping factor is assumed for all the modes, δC = 0. nonparametric uncertainty problem The only uncertainty related information used in the random matrix approach are the values of δM and δK . The information In this example we consider the same plate but with non- regarding the location and natural frequencies of the attached parametric uncertainty. The baseline model is perturbed by oscillators are not used in the Wishart matrix approach. This attaching ten spring mass oscillators with random natural is aimed to depict a realistic situation when the detailed infor- frequencies at random nodal points in the plate. The natural mation regarding uncertainties in a complex engineering sys- frequencies of the attached oscillators follow a uniform dis- tem is not available to the analyst. tribution between 0.2 and 4.0kHz. The nature of uncertainty The predicted mean of the amplitude using the direct in this case is different from the previous case because here Monte Carlo simulation and three Wishart matrix approaches the sparsity structure of the system matrices change with are compared in Fig. 6 for the driving-point-FRF (node num- different realizations of the system. Again a 5000-sample ber 481) and a cross FRF (node number 877). In Fig. 7, Monte Carlo simulation is performed to obtain the FRFs. percentage errors in the mean of the amplitude of the driv- From the simulated random mass and stiffness matrices we ing-point and cross-FRF obtained using the proposed three obtain δM = 0.1326 and δK = 0.4201. Since 2% constant Wishart matrix approaches are shown. Percentage errors are 123 504 Comput Mech (2010) 45:495–511

−60 −60

−120 −120

−140 −140 Standard deviation (dB) Standard deviation (dB)

−160 −160

−180 −180 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 4 Comparison of the standard deviation of the amplitude obtained using the direct stochastic ﬁnite element simulation and three Wishart matrix approaches for the plate with randomly inhomogeneous material properties. a Driving-point-FRF. b Cross FRF

20 20

12 12

10 10

8 8

6 6 Error in standard deviation Error in standard deviation 4 4

2 2

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 5 Comparison of percentage errors in the standard deviation of the amplitude obtained using the three Wishart matrix approaches for the plate with randomly inhomogeneous material properties. a Driving-point-FRF. b Cross FRF calculated using the direct Monte Carlo simulation results as frequency range. Note that this approach only need the sim- the benchmarks. ulation of one random matrix. From these results we conclude The predicted standard deviation of the amplitude using that the diagonal Wishart matrix with different entries should the direct Monte Carlo simulation and three Wishart matrix be used for a system with nonparametric uncertainty as well. approaches are compared in Fig. 8 for the driving-point and In the next section we discuss the application of the proposed cross-FRF. In Fig. 9 percentage errors in the standard devi- approach in the context of a physical experiment. ation of the amplitude of the driving-point and cross-FRF obtained using the proposed three Wishart matrix approaches are shown. Percentage errors are calculated using the direct 6 Experimental validation Monte Carlo simulation results as the benchmarks. Like the previous example with parametric uncertainty, error using We consider the dynamics of a steel cantilever plate with any one of the random matrix approaches reduce with the homogeneous geometric (i.e. uniform thickness) and consti- increase in frequency. Among the three Wishart matrix app- tutive properties (i.e. uniform Young’s modulus and Poisson’s roaches discussed here, the diagonal Wishart matrix with ratio). This uniform plate deﬁnes (as considered in the numer- different entries produces most accurate results across the ical studies in the previous section) the baseline system. The 123 Comput Mech (2010) 45:495–511 505

−60 −60

−120 −120

−140 −140 Mean of amplit Mean of amplit

−160 −160

−180 −180 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 6 Comparison of the mean of the amplitude obtained using the direct stochastic ﬁnite element simulation and three Wishart matrix approaches for the plate with randomly attached oscillators. a Driving-point-FRF. b Cross FRF

20 20

12 12

10 10

8 8 Error in the mean Error in the mean 6 6

4 4

2 2

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 7 Comparison of percentage errors in the mean of the amplitude obtained using the three Wishart matrix approaches for the plate with randomly attached oscillators. a Driving-point-FRF. b Cross FRF baseline model is perturbed by a set of spring-mass oscilla- Table 1. The plate is clamped along one edge using a clamp- tors with different natural frequencies and attached randomly ing device. The clamping device is attached on the top of a along the plate. Here we investigate the feasibility of adopt- heavy concrete block and the whole assembly is placed on ing the proposed simpliﬁed random matrix model for this a steel table. The plate weights about 12.28kg and special case. care has been taken to ensure its stability and minimizing the vibration transmission. The plate is ‘divided’ into 375 ele- 6.1 Overview of the experiential method ments (25 along the length and 15 along the width). Assuming one corner on the cantilevered edge as the origin, we have The details of this experiment has been described by assigned co-ordinates to all the nodes. Oscillators and accel- Adhikari et al. [10,12]. Here we give a very brief overview. erometers are attached on these nodes. This has been done The test rig has been designed for simplicity and ease of with the view of easy ﬁnite element modeling. The bottom replication and modelling. The overall arrangement of the surface of the plate is marked with node numbers so that the test-rig is shown in Fig. 10. A rectangular steel plate with oscillators can be hung at the nodal locations. This scheme is uniform thickness is used for the experiment. The physical aimed at reducing uncertainly arising from the measurement and geometrical properties of the steel plate are shown in of the locations of the oscillators. A discrete random number 123 506 Comput Mech (2010) 45:495–511

−60 −60

−120 −120

−140 −140 Standard deviation (dB) Standard deviation (dB)

−160 −160

−180 −180 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 8 Comparison of the standard deviation of the amplitude obtained using the direct stochastic ﬁnite element simulation and three Wishart matrix approaches for the plate with randomly attached oscillators. a Driving-point-FRF. b Cross FRF

20 20

12 12

10 10

8 8

6 6 Error in standard deviation Error in standard deviation 4 4

2 2

0 0 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 9 Comparison of percentage errors in the standard deviation of the amplitude obtained using the three Wishart matrix approaches for the plate with randomly attached oscillators. a Driving-point-FRF. b Cross FRF generator is used to generate the X and Y coordinates of the to the spring below over the frequency range considered. A oscillators. In total ten oscillators are used to simulate ran- sample realization of the of the attached oscillators is shown dom unmodelled dynamics. The spring is glue-welded with in Fig. 10b. One hundred such realizations of the oscillators a magnet at the top and a mass at the bottom. The magnet at are created (by hanging the oscillators at random locations) the top of the assembly helps to attach the oscillators at the and tested individually in this experiment. bottom of the plate repeatedly without much difficulty. The A 32 channel LMS™ system and a shaker is employed to stiffness of the ten springs used in the experiments is given perform the modal analysis [16,24,43]. We used the shaker in Table 2. In the same table we have also shown the natural to act as an impulse hammer. The shaker was placed so that it frequency of the individual oscillators. The oscillating mass impacted at the (4,6) node of the plate. The shaker was driven of each of the ten oscillators is 121.4g. Therefore the total by a signal from a Simulink™ and dSpace™ system via a oscillating mass is 1.214Kg, which is 9.8% of the mass of power amplifier. The problem with using the usual manual the plate. The springs are attached to the plate at the pre- hammer is that it is in general difficult to hit the bottom of generated nodal locations using the small magnets located at the plate exactly at the same point with the same amount of the top the assembly. The small magnets (weighting 2g) are force for every sample run. The shaker generates impulses found to be strong enough to hold the 121.4g mass attached at a pulse rate of 20s and a pulse width of 0.01s. Using the 123 Comput Mech (2010) 45:495–511 507

Fig. 10 The test rig for the cantilever plate. The position of the shaker shown in sub-ﬁgure (b). The spring stiffness varies so that the oscillator (used as a impact hammer) and the accelerometers are shown in the frequencies are between 43 and 70Hz sub-ﬁgure (a). A sample of attached oscillators at random locations is

Table 1 Material and geometric properties of the cantilever plate con- Point 5: (18,2), Point 6: (21,10). Note that these are the same sidered for the experiment points used in the simulation study in the previous section. Plate Properties Numerical values Small holes are drilled into the plate and all of the six accelerometers are attached by screwing through the holes. The Length (L) 998mm signal from the force transducer is ampliﬁed using an ampli- Width (b) 530mm ﬁer while the signals from the accelerometers are directly Thickness (th)3.0mminput into the LMS system. For data acquisition and process- 3 Mass density (ρ) 7,800kg/m ing, LMS Test Lab 5.0 is used. In the Impact Scope, we have Young’s modulus (E)2.0 × 105 MPa set the bandwidth to8,192Hz with 8,192 spectral lines (i.e., Total weight 12.38kg 1.00Hz resolution). Five averages are taken for each FRF measurement.

Table 2 Stiffness of the springs and natural frequency of the oscillators used to simulate unmodelled dynamics (the mass of the each oscillator 6.2 Comparison with experiential results is 121.4g)

Oscillator number Spring stiffness Natural frequency The mean of the amplitude from experiment and three (×104 N/m) (Hz) Wishart matrix approaches are compared in Fig. 11 for the 1 1.6800 59.2060 driving-point-FRF (point 1) and a cross FRF (point 2). In 2 0.9100 43.5744 the numerical calculations with Wishart matrices 375 ele- 3 1.7030 59.6099 ments are used and the resulting ﬁnite element model has 4 2.4000 70.7647 1,200 degrees-of-freedom. We have used 2,000 samples in 5 1.5670 57.1801 the Monte Carlo simulations. A modal damping factor of 6 2.2880 69.0938 0.7% is assumed for all of the modes. A constant damping 7 1.7030 59.6099 factor for all the modes is an assumption. Ideally one should 8 2.2880 69.0938 identify modal damping factors for all possible modes and 9 2.1360 66.7592 perhaps take an average over the 100 realizations in the experiment. It is however a widely established practice to treat 10 1.9800 64.2752 damping in a simple manner such as a constant modal damping factor as considered here. This study also veriﬁes if such ad-hoc assumptions on damping can produce any meaningful shaker in this way we have tried to eliminate any uncertain- predictions. ties arising from the input forces. Six accelerometers are used Recall that there are four key parameters needed to imple- as the response sensors. The locations of the six sensors are ment the random matrix approach. They are respectively the selected such that they cover a broad area of the plate. The mean and normalized standard deviation of the mass and stiff- locations of the accelerometers can be seen in Fig. 10.The ness matrices. The mean system is considered to be the canti- nodal locations of the accelerometers are as follows: Point lever plate shown in Fig. 1 discussed in the previous section. 1: (4,6), Point 2: (6,11), Point 3: (11,3), Point 4: (14,14), The mean mass and stiffness matrices are obtained using the 123 508 Comput Mech (2010) 45:495–511

60 60

50 50

40 40

30 30

de (dB) 20 de (dB) 20 u u

10 10

0 0

−10 −10 Mean of amplit Mean of amplit −20 M and K are fully correlated Wishart −20 M and K are fully correlated Wishart Scalar Wishart Scalar Wishart −30 Diagonal Wishart with different entries −30 Diagonal Wishart with different entries Experiment Experiment −40 −40 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 3000 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 11 Comparison of the mean of the amplitude obtained using the experiment and three Wishart matrix approaches for the plate with randomly attached oscillators. a Driving-point-FRF. b Cross FRF

1 1 10 10

M and K are fully correlated Wishart M and K are fully correlated Wishart Scalar Wishart Scalar Wishart Diagonal Wishart with different entries Diagonal Wishart with different entries Experiment Experiment 0 0 10 10

−1 −1 10 10 Relative standard deviation Relative standard deviation

−2 −2 10 10 0 500 1000 1500 2000 2500 3000 3500 4000 0 500 1000 1500 2000 2500 300 0 3500 4000 Frequency (Hz) Frequency (Hz) (a) (b)

Fig. 12 Comparison of relative standard deviation of the amplitude obtained using the experiment and three Wishart matrix approaches for the plate with randomly attached oscillators. a Driving-point-FRF. b Cross FRF standard finite element approach. The mean damping matrix In Fig. 11, observe that the results from the scalar Wishart for the experimental system is not obtained explicitly as con- model is qualitatively different from the other two approa- stant modal damping factors are used. All 100 realizations of ches. In that the mean response curve is very smooth and very the plate and oscillators were individually simulated. Then little ‘fluctuations’ can be seen. The results obtained using the samples of 1,200 × 1,200 mass and stiffness matrices the fully correlated M and K Wishart matrices and the diag- were stored. Normalized standard deviation (the dispersion onal Wishart matrix with different entries are almost simi- parameters) of the mass and stiffness matrices are obtained lar. Qualitatively these simulation results agree well with the as δM = 0.0286 and δK = 0.0017. The only uncertainty experimental results. The discrepancies, especially in the low related information used in the random matrix approach are frequency regions, are perhaps due to incorrect values of the the values of δM and δK . The explicit information regard- damping factors. The relative standard deviation of the ampli- ing the spatial locations of the attached oscillators including tude from experiment and three Wishart matrix approaches their stiffness and mass properties are not used in fitting the are compared in Fig. 12 for the driving-point-FRF and the Wishart matrices. This is aimed to depict a realistic situation cross FRF. We obtained the relative standard deviation by when the detailed information regarding uncertainties in a dividing the standard deviation with the respective mean val- complex engineering system is not available to the analyst. ues. This is done in an attempt to eliminate the effect of 123 Comput Mech (2010) 45:495–511 509 damping on the mean values and consider the variability only. 5. Obtain the dispersion parameter of the generalized As expected, the predictions using the Wishart random matrix Wishart matrix models in the lower frequency ranges are not very accu- p 2 +(p −2−2n) p +(−n − 1) p +n2 +1+2n γ rate. Like the mean predictions, the results using the scalar δ = 1 2 1 2 H H (− + )(− + + ) Wishart random matrix turns out to be most inaccurate. The p2 p1 n p1 n 3 2 2 other two approaches are very similar and in the higher fre- p1 + (p2 − 2 n) p1 + (1 − n) p2 − 1 + n + (64) quencies they agree well with the experimental results. p2 (−p1 + n)(−p1 + n + 3)

6. Calculate the parameters 7 Summary of the computational approach (1 + γ ) θ = H − (n + 1) and p =[n + 1 + θ] (65) δ2 From the numerical and experimental studies conducted so H far it can be concluded that the generalized Wishart matrix + with different diagonal entries is the best random matrix where p is approximated to the nearest integer of n + θ model when both accuracy and computational efficiency are 1 . × considered. Numerical calculations for 5000 samples and 7. Create an n p matrix Y such that with 1,200 × 1,200 matrix size show that is it about 30% √ =ω / θ; = , ,..., ; = , ,..., less computationally expensive compared to fully correlated Yij 0i Yij i 1 2 n j 1 2 p mass and stiffness Wishart matrices. The relative computa- (66) tional efficiency is likely to increase for larger system sizes and increasing samples. This new approach leads to a simple where Yij are independent and identically distributed and general simulation algorithm for probabilistic structural (i.i.d.) Gaussian random numbers with zero mean and dynamics. A step-by-step method for implementing the new unit standard deviation. computational approach in conjunction with general purpose 8. Form the n × n matrix finite element software is given below: ω ω p T 0i 0 j = YY or ij = YikY jk; 1. Form the deterministic mass and stiffness matrices M0 θ = and K using the standard finite element method. Obtain k 1 0 i = 1, 2,...,n; j = 1, 2,...,n (67) n, the dimension of the system matrices and the modal damping factors ζ . j and solve the symmetric eigenvalue problem for every 2. Solve the deterministic undamped eigenvalue problem sample K φ = ω2 M φ , j = 1, 2,...,n (60) 0 0 j 0 j 0 0 j = 2 , ∀ = , ,..., r r r r 1 2 nsamp (68) and create the matrix 9. Obtain the eigenvector matrix = φ , φ ,...,φ 0 01 02 0n (61) Xr = 0r (69) Calculate the ratio and calculate the dynamic response in the frequency ⎛ ⎞ 2 domain as n n γ = ⎝ ω2 ⎠ / ω4 (62) H 0 j 0 j n T ¯( ) xr f s j=1 j=1 q¯ ( ω) = j x r i 2 2 r j (70) −ω + 2iωζ j ωr + ω j=1 j r j 3. Obtain the dispersion parameters δM and δK corresponding to the mass and stiffness matrices. This can be obtai- The above procedure can be implemented very easily. When ned from physical or computer experiments. one implements this approach in conjunction with a general 4. Calculate the scalar constants purpose commercial finite element software, the commercial software needs to be accessed only once to obtain the 1 p = 1 +{Trace (M )}2/Trace M 2 and mean matrices M and K and solve the corresponding deter- 1 δ2 0 0 0 0 M ministic eigenvalue problem. This computational procedure 1 p = 1 +{Trace (K )}2/Trace K 2 (63) proposed here is therefore ‘non-intrusive’ and can be used 2 δ2 0 0 K consistently across the frequency ranges [38]. 123 510 Comput Mech (2010) 45:495–511

8 Conclusions – Numerical as well as experimental results show that the difference between three proposed approaches are more When uncertainties in the system parameters and model- in the low frequency regions and less in the higher fre- ling are considered, the discretized equation of motion of quency regions. linear dynamical systems is characterized by random mass, stiffness and damping matrices. The possibility of using a These results suggest that a generalized diagonal Wishart dis- single Wishart random matrix model for the system is inves- tribution with suggested parameters may be used as a con- tigated in the paper. Closed-form analytical expressions of sistent and unified uncertainty quantification tool. the parameters of the Wishart random matrices have been derived using the theory of matrix variate distributions. The Acknowledgments The author gratefully acknowledges the support proposed Wishart random matrix models are applied to the of UK Engineering and Physical Sciences Research Council (EPSRC) forced vibration problem of a plate with stochastically inho- through the award of an Advanced Research Fellowship and The mogeneous properties and randomly attached oscillators. For Leverhulme Trust for the award of the Philip Leverhulme Prize. both cases it is possible to predict the variation of the dynamic response using the Wishart matrices across a wide range of driving frequency. References The feasibility of adopting a single Wishart random matrix model to quantify uncertainty in structural dynamical sys- 1. Adhikari S (1999) Modal analysis of linear asymmetric non- tems has also been investigated using experimental data. In conservative systems. ASCE J Eng Mech 125(12):1372–1379 2. 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