Analysis of the quasiperiodic response of a generalized van der Pol nonlinear system in the resonance zone

Abstract

The paper addresses the description of the complex behavior of simple nonlinear systems that are excited in the neighborhood of the resonance frequency. Depending on the detuning of the excitation frequency, resonant response can vary from purely stationary to various cases of quasiperiodic or chaotic response. This type of response is characterized by regular or irregular changes of the amplitude, which, in the quasiperiodic case, represents the beating effect. The beating frequency then changes from zero in resonance to a positive value outside the resonance zone. The ratio of the energy content of quasiperiodic and stationary components decreases in the same time. Starting at a certain detuning, the quasiperiodic component fully vanishes and the stationary component absorbs the whole response energy.The motivation of this study originates from the aeroelasticity of large bridges, the tuned mass damper application, and other domains of civil engineering, where beating effects have been observed in the past. Such effects are very dangerous; hence, robust theoretical background for the design of adequate countermeasures should be developed. Nevertheless, investigations of the internal structure of a quasiperiod and its dependence on the difference between excitation frequency and eigenfrequency were conducted on a heuristic basis and an objective theoretical background is still missing.A qualitative analysis of nonlinear systems using combinations of harmonic balance, small-parameter methods, and perturbation techniques is presented in the paper. Parametric evaluations are presented along with a discussion concerning the applicability of the presented approach.