Exact Global Representation of All Symmetric Periodic Solutions for Zero Stiffness Impact Oscillator (Derivation by Cesaro's Summation Method)

Abstract

A new analytical method to derive exact global form of all symmetric periodic solutions appearing in one degree of freedom forced zero stiffness impact oscillator is proposed. The author has been proposed a method to obtaining general solutions globally by linearize piecewise linear system to convert it's nonlinearity to virtual external force, and applying the principle of superposition for equivalent exact linearized system. According to this method, the function form of periodic solutions are represented by superposition of stationary periodic solution for linear system and finite number of periodic functions which are constructed by infinite number of responses caused by converted nonlinearity. However, since this infinite summation becomes a divergent series for zero-stiffness case, the value of this series is unknown. In order to overcome this difficulty, we formerly proposed a calculation method of these divergent series using Cesaro's summation, a kind of technique to extract physically reasonable finite value of divergent series. In this paper, we extend this formerly derived result to determine global form of all symmetric periodic solutions. The value of divergent series is calculated as (C.1). Cesaro's summation for viscous damped case, and (C.2) summation for undamped case, respectively. The validation of proposed method is confirmed by numerical example.