Eigenvalue density of linear stochastic dynamical systems: A random matrix approach

Abstract Eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics. When uncertainties are considered, the eigenvalue problem becomes a random eigenvalue problem. In this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the Wishart random matrices. An analytical expression involving the Stieltjes transform is derived for the density of the eigenvalues when the dimension of the corresponding random matrix becomes asymptotically large. The mean matrices and the dispersion parameters associated with the mass and stiffness matrices are necessary to obtain the density of the eigenvalues in the frameworks of the proposed approach. The applicability of a simple eigenvalue density function, known as the Marenko–Pastur (MP) density, is investigated. The analytical results are demonstrated by numerical examples involving a plate and the tail boom of a helicopter with uncertain properties. The new results are validated using an experiment on a vibrating plate with randomly attached spring–mass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework.

[1]  Andy J. Keane,et al.  An approximate solution scheme for the algebraic random Eigenvalue problem , 2003 .

[2]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[3]  Cv Clemens Verhoosel,et al.  Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems , 2006 .

[4]  Haym Benaroya,et al.  Finite Element Methods in Probabilistic Structural Analysis: A Selective Review , 1988 .

[5]  Sharif Rahman,et al.  A solution of the random eigenvalue problem by a dimensional decomposition method , 2006 .

[6]  Sondipon Adhikari,et al.  Calculation of derivative of complex modes using classical normal modes , 2000 .

[7]  Rajendra Singh,et al.  Analysis Of Discrete Vibratory Systems With Parameter Uncertainties, Part I: Eigensolution , 1994 .

[8]  Sondipon Adhikari,et al.  Random matrix eigenvalue problems in structural dynamics , 2007 .

[9]  T. J. Page.  Multivariate Statistics: A Vector Space Approach , 1984 .

[10]  A. Rukhin.  Matrix Variate Distributions , 1999, The Multivariate Normal Distribution.

[11]  Sondipon Adhikari,et al.  Wishart Random Matrices in Probabilistic Structural Mechanics , 2008 .

[12]  Nuno M. M. Maia,et al.  Modal analysis and testing , 1999 .

[13]  M. Stephanov,et al.  Random Matrices , 2005, hep-ph/0509286.

[14]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[15]  D. J. Ewins,et al.  Modal Testing: Theory and Practice , 1984 .

[16]  R. Wunderlich,et al.  Random Eigenvalue Problems for Bending Vibrations of Beams , 1999 .

[17]  J. Wishart.  THE GENERALISED PRODUCT MOMENT DISTRIBUTION IN SAMPLES FROM A NORMAL MULTIVARIATE POPULATION , 1928 .

[18]  Athanasios Papoulis,et al.  Probability, Random Variables and Stochastic Processes , 1965 .

[19]  S. Adhikari.  Generalized Wishart distribution for probabilistic structural dynamics , 2010 .

[20]  Nicholas A J Lieven,et al.  Eigenvalue curve veering in stressed structures: An experimental study , 2009 .

[21]  E. Wigner.  On the Distribution of the Roots of Certain Symmetric Matrices , 1958 .

[22]  Alexander Figotin,et al.  Spectra of Random and Almost-Periodic Operators , 1991 .

[23]  V. Serdobolskii.  Multiparametric Statistics , 2007 .

[24]  Geert Degrande,et al.  A non-parametric probabilistic model for ground-borne vibrations in buildings , 2006 .

[25]  O. C. Zienkiewicz,et al.  The finite element method, fourth edition; volume 2: solid and fluid mechanics, dynamics and non-linearity , 1991 .

[26]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[27]  Christian Soize.  Random matrix theory and non-parametric model of random uncertainties in vibration analysis , 2003 .

[28]  Nicholas A J Lieven,et al.  On the Quantification of Eigenvalue Curve Veering: A Veering Index , 2011 .

[29]  Sondipon Adhikari,et al.  Experimental case studies for uncertainty quantification in structural dynamics , 2009 .

[30]  V. Bogachev.  Gaussian Measures on a , 2022 .

[31]  Nuno M. M. Maia,et al.  Theoretical and Experimental Modal Analysis , 1997 .

[32]  F. J. Fahy,et al.  Statistical energy analysis: a critical overview , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[33]  Mircea Grigoriu.  A solution of the random eigenvalue problem by crossing theory , 1992 .

[34]  Haym Benaroya,et al.  Random eigenvalues, algebraic methods and structural dynamic models , 1992 .

[35]  Martin Hála.  Method of Ritz for random eigenvalue problems , 1994, Kybernetika.

[36]  S. Adhiakri.  Rates of Change of Eigenvalues and Eigenvectors in Damped Dynamic System , 1999 .

[37]  Christian Soize.  A nonparametric model of random uncertainties for reduced matrix models in structural dynamics , 2000 .

[38]  Christian Soize,et al.  Nonparametric stochastic modeling of linear systems with prescribed variance of several natural frequencies , 2008 .

[39]  Vi︠a︡cheslav Leonidovich Girko,et al.  Theory of random determinants , 1990 .

[40]  Antonia Maria Tulino,et al.  Random Matrix Theory and Wireless Communications , 2004, Found. Trends Commun. Inf. Theory.

[41]  L. Pastur.  RANDOM MATRICES AS PARADIGM , 2000 .

[42]  F. Mezzadri,et al.  Recent Perspectives in Random Matrix Theory and Number Theory: Frontmatter , 2005 .

[43]  Chris Jones,et al.  Mode count and modal density of structural systems: Relationships with boundary conditions , 2004 .

[44]  Robin S. Langley,et al.  The ensemble statistics of the energy of a random system subjected to harmonic excitation , 2004 .

[45]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[46]  Andy J. Keane,et al.  Statistics of energy flows in spring-coupled one-dimensional subsystems , 1994, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[47]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[48]  V. Girko,et al.  Theory of stochastic canonical equations , 2001 .

[49]  L. Pastur,et al.  Eigenvalue Distribution of Large Random Matrices , 2011 .

[50]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[51]  Achintya Haldar,et al.  Reliability Assessment Using Stochastic Finite Element Analysis , 2000 .

[52]  Michael I. Friswell,et al.  Random Eigenvalue Problems in Structural Dynamics , 2004 .

[53]  L. Pastur,et al.  Introduction to the Theory of Disordered Systems , 1988 .

[54]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of a class of large dimensional random matrices , 1995 .

[55]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[56]  D. Dawe,et al.  Matrix and finite element displacement analysis of structures , 1984 .

[57]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[58]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[59]  Roger Ghanem,et al.  Analysis of Eigenvalues and Modal Interaction of Stochastic Systems , 2005 .

[60]  C Soize,et al.  Maximum entropy approach for modeling random uncertainties in transient elastodynamics. , 2001, The Journal of the Acoustical Society of America.

[61]  Richard L. Weaver,et al.  Power variances and decay curvature in a reverberant system , 2000 .