Analytical period-1 motions to chaos in a two-degree-of-freedom oscillator with a hardening nonlinear spring

Abstract

In this paper, obtained are analytical solutions for period-1 motions to chaos in a two-degree-of-freedom (2-DOF) oscillator with a nonlinear hardening spring. From the finite Fourier series transformation, a dynamical system of coefficients of the finite Fourier series is developed to determine existence, stability and bifurcations of periodic motions in such a 2-DOF nonlinear oscillator. The equilibriums of such a dynamical system of coefficients give the analytical solutions of period-m motions, and the corresponding stability and bifurcations of period-m motions are determined through the eigenvalue analysis of equilibriums. Analytical bifurcation trees of period-1 motions to chaos are presented through frequency–amplitude curves. From the frequency–amplitude curves, the harmonic effects on the periodic motions can be discussed and nonlinear behaviors of periodic motions can be determined. Displacements, velocity, and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator. Through the analytical solutions, the complex dynamics of the 2-DOF nonlinear oscillator is studied. The analytical solutions presented herein for periodic motions in such 2-DOF systems can be used to further discuss the corresponding nonlinear behaviors, and can also be applied to engineering for design.