# Universal eigenvalue statistics and vibration response prediction

## Abstract

It is well known that, under broad conditions, the local statistics of the eigenvalues of a random matrix tend towards a universal distribution if the matrix is sufficiently random, regardless of the statistics of the matrix entries. This distribution is the same as that associated with a class of random matrix known as the Gaussian Orthogonal Ensemble (GOE). The underlying reason for this behaviour is explored here, and it is concluded that the source of the universal statistics lies in the Vandermonde determinant, which appears in the Jacobian of the transformation between the entries of a matrix and its eigenvalues and eigenvectors. Attention is then turned to the application of this result to natural frequency statistics, and to the prediction of the response statistics of a random built-up system. Recent work is reviewed in which it is shown that, away from low frequencies, the mean and variance of the system response can be predicted in an efficient way without knowledge of the statistics of the underlying system uncertainties.

## Keywords

Acoustical Society Random Matrix Modal Density Random Matrix Theory Joint Probability Density Function## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Ambromowitz, M. & Stegun, I.A. (1964).
*Handbook of Mathematical Functions*. Dover, New York. Google Scholar - 2.Bertelsen, P., Ellegaard, C. & Hugues, E. (2001). Acoustic chaos.
*Physica Scripta***90**, 223–230. Google Scholar - 3.Bohigas, O., Giannoni, M.J. & Schmit, C. (1984). Spectral properties of the Laplacian and random matrix theory.
*J. Physique Lett.***45**, L1015–L1022. CrossRefGoogle Scholar - 4.Cotoni, V., Langley, R.S. & Kidner, M.R.F. (2005). Numerical and experimental validation of variance prediction in the statistical energy analysis of built-up systems.
*Journal of Sound and Vibration***288**, 701–728. CrossRefGoogle Scholar - 5.Dyson, F.J. (1962). Statistical theory of energy levels in complex systems, Parts I, II, and III.
*J. Maths. Phys.***3**, 140–156, 157–165, 166–175. Google Scholar - 6.Forrester, P.J., Snaith, N.C. & Verbaarschot, J.J.M. (2003). Developments in random matrix theory.
*Journal of Physics A: Mathematical and General***36**, R1–R10. zbMATHCrossRefMathSciNetGoogle Scholar - 7.Langley, R.S. & Brown, A.W.M. (2004). The ensemble statistics of the energy of a random system subjected to harmonic excitation.
*Journal of Sound and Vibration***275**, 823–846. CrossRefGoogle Scholar - 8.Langley, R.S. & Cotoni, V. (2004). Response variance prediction in the statistical energy analysis of built-up systems.
*Journal of the Acoustical Society of America***115**, 706–718. CrossRefGoogle Scholar - 9.Langley, R.S. & Cotoni, V. (2005). The ensemble statistics of the vibrational energy density of a random system subjected to single point harmonic excitation.
*Journal of the Acoustical Society of America***118**, 3064–3076. CrossRefGoogle Scholar - 10.Langley, R.S. & Cotoni, V. (2007). Response variance prediction for uncertain vibro-acoustic systems using a hybrid deterministic-statistical method.
*Journal of the Acoustical Society of America***122**, 3445–3463. CrossRefGoogle Scholar - 11.Leff, H.S. (1964). Class of ensembles in the statistical theory of energy-level spectra.
*Journal of Mathematical Physics***5**, 763–768. CrossRefMathSciNetGoogle Scholar - 12.Lobkis, O.I., Weaver, R.L. & Rozhkov, I. (2000). Power variances and decay curvature in a reverberant system.
*Journal of Sound and Vibration***237**, 281–302. CrossRefGoogle Scholar - 13.Lyon, R.H. & DeJong, R.G. (1995).
*Theory and Application of Statistical Energy Analysis*,*Second Edition*. Butterworth-Heinemann, Boston. Google Scholar - 14.Mehta, M.L. (1991).
*Random Matrices, Second Edition*. Academic Press, San Diego. zbMATHGoogle Scholar - 15.Muirhead, R.J. (1982).
*Aspects of multivariate statistical theory*. John Wiley & Sons, Inc. New York. zbMATHCrossRefGoogle Scholar - 16.Shorter, P.J. & Langley, R.S. (2005). Vibro-acoustic analysis of complex systems.
*Journal of Sound and Vibration***288**, 669–700. CrossRefGoogle Scholar - 17.Soize, C. (2005). A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics.
*Journal of Sound and Vibration***288**, 623–652. CrossRefMathSciNetGoogle Scholar - 18.Stratonovich, R.L. (1967).
*Topics in the Theory of Random Noise, Vol. 2*. McGraw-Hill, London. Google Scholar - 19.Weaver, R.L. (1989). On the ensemble variance of reverberation room transmission functions, the efect of spectral rigidity.
*Journal of the Acoustical Society of America***130**, 487–491. Google Scholar - 20.Wigner, E.P. (1955). Characteristic vectors of bordered matrices with infinite dimensions.
*Ann. Math.***62**, 548. CrossRefMathSciNetGoogle Scholar