Piecewise-Linear Approximation and Global Perturbation of Homoclinic Motions for a Two-Degree-of-Freedom Coupled Mechanical System

In this paper, a novel two-degree-of-freedom mechanical system under sinusoidal excitation is established by coupling the archetypal Smooth and Discontinuous (SD) oscillator with a harmonic oscillator to analytically study the global perturbation dynamics of homoclinic motions. First, a piecewise-linear approximation of the irrational restoring force induced by the SD oscillator is carried out to reconstruct a four-dimensional piecewise-smooth nonautonomous system with a saddle-center geometrical structure. Second, the Melnikov method for piecewise-defined homoclinic orbits under periodic perturbation in the four-dimensional phase space is generalized to obtain the parameter threshold of chaos. Third, the effectiveness of this global perturbation technique is verified by numerical simulations, through which the regulation of the coupled system’s dynamic response from both damping and periodic external excitation is revealed, along with the observation of rich nonlinear dynamic phenomena induced by their combined effects. Finally, a novel strategy is proposed to control the chaotic motion of the SD oscillator through a linear spring-coupled harmonic oscillator. The dynamic response analysis demonstrates that this coupled system enables passive vibration control of the SD oscillator, establishing a foundation for optimizing vibration absorber designs.