CONVERGENCE ANALYSIS OF VOLTERRA SERIES RESPONSE OF NONLINEAR SYSTEMS SUBJECTED TO HARMONIC EXCITATION

Abstract

Volterra series provides a strong platform for non-linear analysis and higher order frequency response functions. However, limited convergence is an inherent difficulty associated with the series and needs to be addressed rigorously, prior to its application to a physical system. The power series representation of the response of non-linear systems, subjected to harmonic excitation is investigated in this study. The problem of convergence is addressed in terms of the convergence of individual frequency harmonics of the non-linear response. Though the procedure is applicable to general polynomial form non-linearity, it is illustrated for a Duffing oscillator subjected to harmonic excitation. A general and structured series expression is obtained for amplitudes of all the response harmonics and convergence is investigated in terms of a non-dimensional non-linear parameter. Critical values of this parameter, representing the upper limit of excitation level for the convergence, are defined for a wide range of excitation frequencies. Zones of convergence and divergence of the response series are presented graphically, for a range of the non-dimensional non-linear parameter and the number of terms included in the approximation of a response harmonic. An algorithm based on ratio test is presented to compute the critical value of the non-dimensional non-linear parameter. Results obtained from the suggested algorithm are found to be in close agreement with the exact values. The method gives better results compared to previous methods and has wider application in terms of excitation frequency. The procedure is also investigated for a two-degree-of-freedom system.