Higher order spectra and their application to nonlinear mechanical systems

Abstract

This thesis is concerned with the development of useful engineering techniques to detect and analyse non linearities in mechanical systems. The methods developed are based on the concepts of higher order spectra, in particular the bispectrum and trispectrum, and the Volterra series. The study of higher order statistics has been dominated by work on the bispectrum. The bispectrum can be viewed as a decomposition of the third moment (skewness) of a signal over frequency and as such is blind to symmetric non linearities. To study such phenomena one has to go a stage further and resort to the trispectrum, or decomposition of kurtosis over frequency. Techniques are presented here that enable one to estimate and display both auto and cross, bispectra and trispectra.

Initially auto higher order spectra are studied in detail with particular attention being paid to normalisation methods. Two traditional methods based on the bicoherence and skewness function are studied and these are expanded to their fourth order equivalents, the tricoherence and kurtosis functions. Under certain conditions, notably narrow band signals, the above normalisation methods are shown to fail and so a new technique based on pre whitening the signal in the time domain is developed. Examples of these functions are given both for memoryless and dynamic systems. The Volterra series is presented and discussed in some detail. Techniques for calculating a system's Volterra kernals from cross higher order spectra are derived. New methods are presented for the estimation of higher order Volterra kernels which are shown to produce better results than traditional approaches. These are then applied to some simple analytic systems which include the Duffing oscillator.

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