Stability and nonlinear dynamics of a vibration system with oblique collisions

Two moving rigid bodies can contact with each other as the clearance between them is exceeded. Supposing that the contacting areas are two parallel planes with an oblique angle to the moving direction, periodic responses of the system with oblique-collision are determined by applying Poincaré mapping, impulse-momentum theorem and Poisson's hypothesis in this paper. First, the relations of the body velocities before and after collisions are derived upon an instantaneous collision hypothesis. Then, the existence condition and the stability criterion of the period-n motion with single collision are deduced by using the Poincaré mapping and the Floquet theory. Last, numerical simulations are carried out and bifurcation diagrams, Poincaré maps and phase portraits are adopted to prove the stability analysis and illustrate rich dynamical behaviors of the system over a rather broad range of frequency of the external excitation. Transitions from the single collision period-n motion to chaotic one through three ways, period-doubling bifurcation, Hopf bifurcation and abruptly transition, are revealed.