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Matrix Variate Distributions for Probabilistic Structural Dynamics

Sondipon Adhikari∗

University of Bristol, Bristol, BS8 1TR England, United Kingdom

DOI: 10.2514/1.25512

Matrix variate distributions are proposed to quantify uncertainty in the mass, stiffness, and damping matrices

arising in linear structural dynamics. The proposed approach is based on the so-calledWishart randommatrices. It

is assumed that themean of the systemmatrices are known.Anew optimalWishart distribution is proposed tomodel

the random system matrices. The optimal Wishart distribution is such that the mean of the matrix and its inverse

produce minimum deviations from their respective deterministic values. The method proposed here gives a simple

nonparametric approach for uncertainty quantification and propagation for complex aerospace structural systems.

The new method is illustrated using a numerical example. It is shown that Wishart randommatrices can be used to

model uncertainty across a wide range of excitation frequencies.

Nomenclature

C = space of complex numbers D�!� = dynamic stiffness matrix E����� = an n � n real symmetric matrix E����� = mathematical expectation operator E��� = expectation operator etrf�g = expfTrace���g F = symbol for the inverse of a system matrix, F �

fM�1;C�1;L�1g f�t� = forcing vector G = symbol for a system matrix, G � fM;C;Kg H�!� = frequency response function matrix In = identity matrix of dimension n i = unit imaginary number, i�

������� 1 p

Lf���g = Laplace transform of ��� M, C, and K = mass, damping, and stiffness matrices,

respectively m, � = scalar and matrix parameters of the inverted

Wishart distribution n = number of degrees of freedom On;m = null matrix of dimension n �m p, � = scalar and matrix parameters of the Wishart

distribution p����X� = probability density function of ��� in (matrix)

variable X q�t� = response vector R = space of real numbers R n = space n � n real positive-definite matrices Rn;m = space n �m real matrices Trace��� = sum of the diagonal elements of a matrix Z = an n � n symmetric complex matrix � = consonant for the optimal Wishart distribution �n�a� = multivariate gamma function � = a consonant, �� 2� � = order of the inverse-moment constraint ���� = characteristic function of ���

! = excitation frequency ���T = matrix transposition j � j = determinant of a matrix k � kF = Frobenius norm of a matrix, k � kF �

fTrace�������T �g1=2 � = Kronecker product � = distributed as

I. Introduction

U NCERTAINTIES are unavoidable in the description of real-lifeengineering systems. The quantification of uncertainties plays a crucial role in establishing the credibility of a numerical model. Uncertainties can be broadly divided into two categories. The first type is due to the inherent variability in the system parameters, for example, different cars manufactured from a single production line are not exactly the same. This type of uncertainty is often referred to as aleatoric uncertainty. If enough samples are present, it is possible to characterize the variability using well-established statistical methods and consequently the probability density functions (pdf) of the parameters can be obtained. The second type of uncertainty is mainly due to the lack of knowledge regarding a system, often referred to as epistemic uncertainty. This kind of uncertainty generally arises in the modeling of complex systems, for example, in themodeling of cabin noise in helicopters. Because of its very nature, it is comparatively difficult to quantify and consequently model this type of uncertainty.

Broadly speaking, there are two complimentary approaches to quantify uncertainties in amodel. Thefirst is the parametric approach and the second is the nonparametric approach. In the parametric approach, the uncertainties associated with the system parameters, such as Young’s modulus, mass density, Poisson’s ratio, damping coefficient, and geometric parameters are quantified using statistical methods and propagated, for example, using the stochastic finite element method [1–10]. This type of approach is suitable to quantify aleatoric uncertainties. Epistemic uncertainty, on the other hand, does not explicitly depend on the systems parameters. For example, there can be unquantified errors associated with the equation of motion (linear or nonlinear), in the damping model (viscous or nonviscous), in the model of structural joints, and also in the numerical methods (e.g., discretization of displacement fields, truncation and roundoff errors, tolerances in the optimization and iterative algorithms, step sizes in the time-integration methods). It is evident that the parametric approach is not suitable to quantify this type of uncertainty and a nonparametric approach is needed for this purpose.

In this paper, a general nonparametric uncertainty quantification tool for structural dynamic systems is proposed. Themethod is based

Received 5 June 2006; revision received 6 March 2007; accepted for publication 6 March 2007. Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/07 $10.00 in correspondence with the CCC.

∗Lecturer, Department of Aerospace Engineering, University of Bristol, Queens Building, University Walk, Bristol BS8 1TR, England, United Kingdom; currently Chair of Aerospace Engineering, School of Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom; S.Adhikari@swansea.ac.uk. AIAA SeniorMember.

AIAA JOURNAL Vol. 45, No. 7, July 2007

1748

http://dx.doi.org/10.2514/1.25512

on the random matrix theory and builds upon the existing nonparametric approach proposed by Soize [11]. Uncertainties associatedwith a variable can be characterized using the probabilistic approach or possibilistic approaches based on interval algebra, convex sets, or fuzzy sets. In this paper, the probabilistic approach has been adopted. The equation of motion of a damped n-degree-of- freedom linear structural dynamic system can be expressed as

M �q�t� C _q�t� Kq�t� � f�t� (1)

The importance of considering parametric and/or nonparametric uncertainty also depends on the frequency of excitation. For example, in the high-frequency vibration the wavelengths of the vibration modes become very small. As a result, the vibration response can be very sensitive to the small details of the system. In such situations, a nonparametric uncertaintymodelmay be adequate. Overall, three different approaches are currently available to model stochastic structural dynamic systems across the frequency range:

1) Low-frequency vibration problems use the stochastic finite element method (SFEM) [1–10] which considers parametric uncertainties in details.

2) High-frequency vibration problems use statistical energy analysis [12] (SEA) which does not consider parametric uncertainties in details.

3) Midfrequency vibration problems [13–16] in which both parametric and nonparametric uncertainties need to be considered.

The aim of this paper is to propose a method which will work across the frequency range.Herewewill investigate the possibility of using the random matrix theory as the unified uncertainty modeling tool to be valid for low-, medium-, and high-frequency vibration problems. The probability density functions of the random matrices M,C, andKwill be derived to completely quantify the uncertainties associatedwith system, Eq. (1). In the next section, we briefly outline some aspects of the random matrix theory required for further developments.

II. Background of the Random Matrix Theory

Random matrices were introduced by Wishart [17] in the late 1920s in the context of multivariate statistics. However, random matrix theory (RMT) was not used in other branches until the 1950s whenWigner [18] published his works (leading to the Nobel Prize in physics in 1963) on the eigenvalues of random matrices arising in high-energy physics. Using an asymptotic theory for large dimensional matrices, Wigner was able to bypass the Schrödinger equation and explain the statistics of measured atomic energy levels in terms of the limiting eigenvalues of these random matrices. Since then, research on randommatrices has continued to attract interest in multivariate statistics, physics, number theory, and more recently in mechanical and electrical engineering. We refer the readers to the books by Mezzadri and Snaith [19], Tulino and Verdú [20], Eaton [21], Muirhead [22], and Mehta [23] for history and applications of random matrix theory.

The probability density function of a randommatrix can be defined in a manner similar to that of a random variable or random vector. If A is an n �m real random matrix, the matrix variate probability density function ofA 2 Rn;m, denoted as pA�A�, is a mapping from the space of n �m real matrices to the real line, i.e., pA�A�: Rn;m ! R. Here, we define four types of random matrices which are relevant to this study.

Definition 1. Gaussian random matrix: The random matrix X 2 Rn;p is said to have a matrix variate Gaussian distribution with mean matrixM 2 Rn;p and covariance matrix���, where� 2 R n and � 2 R p provided the pdf of X is given by