Linearization techniques for non-linear dynamical systems

Abstract

This dissertation is concerned with the application of linearization techniques to the study of the response of non-linear dynamical systems subjected to periodic and random excitations. A general method for generating an approximate solution of a multi-degree-of-freedom non-linear dynamical system is presented. This method relies on solving an optimum equivalent linear substitute of the original system. The applicability of the method for determination of the amplitudes and phases of the approximate steady-state solution of a multi-degree-of-freedom non-linear system under harmonic monofrequency excitation is considered. The implementation of the method for several special classes of non-linear functions is discussed in detail. In addition, the manner in which the method may be applied to generate an approximate solution for the covariance matrix of the stationary random response of a multi- degree- of freedom dynamical system subjected to stationary Gaussian excitation is outlined. The potential of the method to treat transient solutions of non-linear systems is indicated in the context of the non-stationary response of a lightly damped and weakly non-linear oscillator subjected to monofrequency harmonic or to a Gaussian white noise disturbance. For both classes of excitation the method produces a first-order differential equation governing the response amplitude. The results pertinent to the harmonically excited oscillator are compared with existing solutions. A non-stationary solution of the Fokker-Planck equation associated with the stochastic differential equation governing the response amplitude of the randomly excited oscillator is accomplished by perturbation techniques; the stationary solution is determined without making any approximation in the Fokker-Planck equation. The new method for transient response is applied to the random response of a Duffing Oscillator and a Hysteretic System. The solution for the Duffing Oscillator is compared with data obtained by a Monte Carlo study.

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