Linearization Techniques for Non-Linear Dynamical Systems

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.