Analytical approximation of the primary resonance response of a periodically excited piecewise non-linear–linear oscillator

An analytical approximate solution is constructed for the primary resonance response of a periodically excited non-linear oscillator, which is characterized by a combination of a weakly non-linear and a linear differential equation. Without eliminating the secular terms, a valid asymptotic expansion solution for the weakly non-linear equation is analytically determined for the case of primary resonances. Then, a symmetric periodic solution for the overall system is obtained by imposing continuity and matching conditions. The stability characteristic of the symmetric periodic solution is investigated by examining the asymptotic behaviour of perturbations to the steady state solution. The validity of the developed analysis is highlighted by comparing the first order approximate solutions with the results of numerical integration of the original equations.