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An approximate method for analysing non - linear systems subject to random excitation

A solution technique based on the representation of the response of the non-linear system by a polynomial of the response of the linearized system is presented. The relation between the original non-linear system and the linearized system is introduced by considering the so-called extended moment equations and their closed set is to be solved to determine unknowns. | Vietnam Journal of Mechanics, NOST of Vietnam Vol. 22, 2000, No 1 (1 - 10) AN APPROXIMATE METHOD FOR ANALYSING NON-LINEAR SYSTEMS SUBJECT TO RANDOM EXCITATION NGUYEN DONG ANH - NINH QUANG HAI Insti'tute of Mechanics, NCST of Vietnam ABSTRACT. A solution technique based on the representation of the response of the non-linear system by a polynomial of the response of the linearized system is presented. The relation between the original non-linear system and the linearized system is introduced by considering the so-called extended moment equations and their closed set is to be solved to determine unknowns. For the Vanderpol oscillator subject to white noise excitation, the technique gives good approximation to the response moments as well as the probability density function. 1. Introduction Since all real engineering systems are, more or less , non-linear and for those systems the exact solutions are known only for a number of special cases, it is necessary to develop approximate techniques to determine the response of nonlinear systems under excitations. Several books examine approximate techniques for solving deterministic and / or random vibration problems, for instance, see [1, 2] and see [3, 4], respectively. This paper presents a solution technique based on the representation of the response of a non-linear system by a polynomial of the response of the linearized system. As the result of considering an original and the corresponding linear systems together, the extended moment equations are developed and their closed set is to be solved to determine unknowns. In [18] the application of the method to the Duffing system is presented. Herein, we continue our investigation on the Vanderpol oscillator subject to white noise excitation. It turns out that, the technique also gives good approximation to the response moments as well as the probability density function. 2. Extended moment equations and polynomial form for the system response Consider the equation of