Analysis of the Volterra Series Convergence

Abstract

Most of actual structures present some nonlinear characteristics which are due to complex junctions or large displacement effects. The dynamical responses of such systems can be studied by means of different technics: harmonic balance, averaging method, multiple scale approach (NM79) etc… Under some assumptions, these responses can also be expressed as a Volterra series, which can be seen as an extension of the Taylor series applied to functionals (Sch80). This kind of expression provides a direct relation between the input and the output, and allows a physical interpretation of nonlinear phenomena (VCL87). However, the use of such theory is bounded by the convergence domain of the series. A large number of studies have been dedicated to this problem, but in most cases the obtained convergence criteria are difficult to satisfy (KW66) (Bro76) (Bar65). In this paper, we propose to develop an approximated criteria based on the multi-harmonic balance equations (UR66). We use this set of nonlinear equations and the analytical functions properties to establish a criteria depending on the circular frequency of the input. This criteria will be tested on a Duffing oscillator.