Subharmonic resonance of single-degree-of-freedom piecewise-smooth nonlinear oscillator

Abstract

Take the single degree of freedom nonlinear oscillator with clearance under harmonic excitation as an example, the 1/3 subharmonic resonance of piecewise-smooth nonlinear oscillator is investigated. The approximate analytical solution of 1/3 subharmonic resonance of the single-degree-of-freedom piecewise-smooth nonlinear oscillator is presented. By changing the solving process of Krylov-Bogoliubov–Mitropolsky (KBM) asymptotic method for subharmonic resonance of smooth nonlinear system, the classical KBM method is extended to piecewise-smooth nonlinear system. The existence conditions of 1/3 subharmonic resonance steady-state solution are achieved, and the stability of the subharmonic resonance steady-state solution is also analyzed. It is found that the clearance affects the amplitude-frequency response of subharmonic resonance in the form of equivalent negative stiffness. Through a demonstration example, the accuracy of approximate analytical solution is verified by numerical solution, and they have good consistency. Based on the approximate analytical solution, the influences of clearance on the critical frequency and amplitude-frequency response of 1/3 subharmonic resonance are analyzed in detail. The analysis results show that the KBM method is an effective analytical method for solving the subharmonic resonance of piecewise-smooth nonlinear system. And it provides an effective reference for the study of subharmonic resonance of other piecewise-smooth systems.