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Spectral analysis of vibration in weakly non-linear systems

This paper deals with the above mentioned problem. The main idea of this work is the use of an analytical simulation of random excitation given by its spectral density function and afterward application of the well known procedure of the asymptotic method to obtain an asymptotic expression of the response spectral density function. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 22, 2000, No 3 (181 - 192) SPECTRAL ANALYSIS OF VIBRATION IN WEAKLY NON-LINEAR SYSTEMS NGUYEN TIEN KHIEM Institute of Mechanics, NCST of Vietnam The weakly nonlinear systems subjected to deter:m:inistic excitations have been fully and deeply studied by use of the well developed asymptotic methods [14]. The systems excited by a random load have been investigated mostly using the Fokker-Plank-Kolmogorov equation technique combined with the asymptotic methods [5-8]. However, the last approach in most successful cases allows to obtain only a stationary single point probability density function, that contains no information about the correlation nor' consequently, the spectral structure of the response. The linearisation technique [9, 10] in general permits the spectral density of the response to be determined, but the spectral function obtained by this method because of the linearisation eliminates the effect of the nonlinearity. Thus, spectral structure of response of weakly nonlinear systems to random excitation, to the author's knowledge, has not been studied enough. This paper deals with the above mentioned problem. The main idea of this work is the use of an analytical simulation of random excitation given by its spectral density function and afterward application of the well known procedure of the asymptotic method to obtain an asymptotic expression of the response spectral density functi.on. The obtained spectral relationship covers the linear system case and especially emphasizes the nonlinear effect on the spectral density of response. The theory will be illustrated by an example and at the end of this paper there will be a discussion about the obtained results. 1. Simulation of stationary random process with given spectral density Let's consider a stationary random process X( t) with zero the spectral density function Sx(w). This means that +oo (X{t)) = 0; (X(t)X(t + r)) = Rx(r) .