Dynamics of an oblique-impact vibrating system of two degrees of freedom

The dynamic analysis of oblique-impact vibration is presented in this paper through an illustrative system, where a spring-pendulum obliquely collides a mass-spring oscillator. The oblique-impact process and the relations between the pre-impact state and the post-impact state of system are analyzed first upon a hypothesis on instantaneous oblique impacts. On the basis of the oblique-impact relations, then, the dynamic equation of such an oblique-impact vibrating system is established. Consequently, the existence and the stability of the periodic motions are analyzed and a series of analytical solutions are derived in some simplified cases. Afterwards, the transitions of the steady state motions of the system with the variation of the excitation, the damping and the contact-friction are numerically studied in general cases. The study shows that the coefficient of restitution for an oblique-impact cannot be regarded as a constant, and that the relations between the pre-impact state and the post-impact state are directly associated with the impact angle and the coefficient of contact-friction between impacting bodies. For the system of concern, the system parameters and initial states have to be carefully selected if any stable periodic motion is expected. Furthermore, very rich dynamic behaviors, such as bifurcations and chaotic motions, are observed in the numerical results.