Analysis of transition regions in the parametrically forced system of bi-linear oscillators: Resonant excitation in the neighborhood of similar modes

Abstract

Present paper is devoted to the analysis of resonant response of symmetric, two-oscillator bi-linear models subject to a low-amplitude parametric forcing. The central goal of the present study is the analytical description of 1:1 and sub-harmonic m:1 (m > 2) resonance regimes and corresponding transition regions of two-oscillator, parametric bi-linear system in the vicinity of similar nonlinear normal mode through the system reduction to a single, parametric bi-linear oscillator. Asymptotic analysis of transition regions is applied further to the reduced order model and the transition regions corresponding to resonant parametric excitation in the vicinity of a similar, nonlinear normal mode are recovered. Asymptotically approximated transition curves derived from the analysis of parametrically forced effective oscillators are in agreement with the results of numerical simulations of the original system.