On the Bifurcation Routes to 1:2 Resonance in Nonlinear Forced Oscillators

Abstract

We investigate the dynamics of nonlinear oscillators. We study the formation of 1:2 resonance in a nonlinear periodically forced oscillator due to period doubling of the primary 1:1 resonance, or born independently. We compute the amplitude–frequency implicit function, the steady-state asymptotic solution, for the effective equation approximating coupled oscillators. Working in the framework of the differential properties of implicit functions, we demonstrate that the birth of 1:2 resonances corresponds to singular isolated points of the implicit functions—amplitude–frequency response functions. We show how to compute such singular isolated points. We can thus compute parameter intervals at which period doubling of the primary 1:1 resonance occurs, or an independent 1:2 resonance emerges. We can therefore avoid the buildup of chaos or emergence of unwanted oscillations, improving the performance of vibration dampers. Based on the cited literature, we infer that the most critical challenges to the vibration-damping approach are the emergence of chaotic dynamics and other redundant vibrations. In this work, we propose tools to predict the appearance of redundant dynamical modes.