The crossing rates,exceedance probabilities,and related statistical properties of the energy frequency response functions of a random built-up system

Abstract

A method is derived for computing a number of key statistical properties of the vibrational energy of each component of a built-up system that has random properties. The energy is considered to be a random function of frequency, and the derived statistical properties include: the mean rate at which the energy crosses a specified level, the probability that the energy will exceed a specified level within a given frequency band, the mean trough-to-peak height, the rate of occurrence of peaks, and the mean quefrency (a measure of the rate of fluctuation of the energy). The analysis is based on combining statistical energy analysis with a non-parametric model of uncertainty based on the Gaussian orthogonal ensemble and avoids the use of Monte Carlo simulations or large computational models. By way of example, the method is applied to a number of coupled plate systems.

Keywords

Probabilistic analysis statistical analysis uncertainty vibration and acoustics vibration theory

Get full access to this article

View all access options for this article.

References

Schroeder

. Frequency-correlation functions of frequency responses in rooms. J Acoust Soc Am 1962; 34: 1819–1823.

Lyon

. Statistical analysis of power injection and response in structures and rooms. J Acoust Soc Am 1969; 45: 545–565.

Keane

Nair

. Computational approaches for aerospace design, Hoboken, NJ: John Wiley and Sons, 2005.

Soize

. A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics. J Sound Vib 2005; 288: 623–652.

Weaver

The unreasonable effectiveness of random matrix theory for the vibrations and acoustics of complex structures. In: Wright

MCM

Weaver

(eds). New directions in linear acoustics and vibration: quantum chaos, random matrix theory, and complexity, Cambridge: Cambridge University Press, 2010, pp. 42–58. .

Mehta

. Random matrices, 2nd ed. San Diego, CA: Academic Press, 1991.

Langley

Brown

AWM

. The ensemble statistics of the energy of a random system subjected to harmonic excitation. J Sound Vib 2004; 275: 823–846.

Weaver

. Spectral statistics in elastodynamics. J Acoust Soc Am 1989; 85: 1005–1013.

Langley

Cotoni

. Response variance prediction in the statistical energy analysis of built-up systems. J Acoust Soc Am 2004; 115: 706–718.

10.

Lyon

DeJong

. Theory and application of statistical energy analysis, Boston, MA: Butterworth-Heinemann, 1990.

11.

Langley

. The level crossing rates and associated statistical properties of a random frequency response function. J Sound Vib 2018; 417: 19–37.

12.

Randall

. Frequency analysis, Naerum: Bruel and Kjaer, 1987.

13.

Fahy

. Statistical energy analysis: a critical overview. Phil Trans R Soc Phys Sci Eng 1994; 346: 431–447.

14.

Finnveden

. A quantitative criterion validating coupling power proportionality in statistical energy analysis. J Sound Vib 2011; 330: 87–109.

15.

Le Bot

Cotoni

. Validity diagrams of statistical energy analysis. J Sound Vib 2010; 329: 221–235.

16.

Langley

The analysis of random built-up engineering systems. In: Wright

MCM

Weaver

(eds). New directions in linear acoustics and vibration: quantum chaos, random matrix theory, and complexity, Cambridge: Cambridge University Press, 2010, pp. 231–250. .

17.

Meirovitch

. Elements of vibration analysis, 2nd ed. New York: McGraw Hill, 1986.

18.

Kessissoglou

Lucas

. Gaussian orthogonal ensemble spacing statistics and the statistical overlap factor applied to dynamic systems. J Sound Vib 2009; 324: 1039–1066.

19.

Langley

Cicirello

Deckers

. The ensemble statistics of the energy of a harmonically excited random system. J Sound Vib 2018; 413: 456–466.

20.

Langley

Legault

Woodhouse

, et al. On the applicability of the lognormal distribution in random dynamical systems. J Sound Vib 2013; 332: 3289–3302.

21.

Langley

Heron

. Elastic wave transmission through plate/beam junctions. J Sound Vib 1990; 143: 241–253.

22.

Shorter

Langley

. Vibro-acoustic analysis of complex systems. J Sound Vib 2005; 288: 669–699.

23.

Brody

Flores

French

, et al. Random-matrix physics: spectrum and strength fluctuations. Rev Mod Phys 1981; 53: 385–479.

24.

Cotoni

Langley

Kidner

MRF

. Numerical and experimental validation of variance prediction in the statistical energy analysis of built-up systems. J Sound Vib 2005; 288: 701–728.

25.

Vanmarcke

. On the distribution of the first-passage time for normal stationary random processes. J Appl Mech 1975; 42: 215–220.

26.

Stratonovich

. Topics in the theory of random noise 1963; vol 1 New York: Gordon and Breach.