The dynamics of two parametrically excited pendula with impacts

We consider a parametrically excited vibro-impact system consisting of two colliding pendula supported on a moving base. Between the impacts the dynamics of the system are described by a nonlinear Mathieu equation, while impacts are represented by an appropriate jump in the state variables, dependent on the coefficient of restitution e. The method of averaging is applied to these equations and a reduced system is developed in which several nontrivial impacting states are identified, including synchronous oscillations, a symmetric impacting state, and a nonsymmetric impacting state. For the symmetric response (including both the synchronous and symmetric impacting states), a two-dimensional map is developed from the averaged equations and for a purely elastic collision (e = 1), a nonlinear Mathieu equation is obtained the impacts play no role in the amplitude of the response. For inelastic collisions (e not equal 1) the dissipation induced by the impacts can be combined with viscous damping into an overall dissipation parameter to determine the equilibrium points of the mapping. Finally, for more general nonsymmetric behavior, the existence of the stable nonsymmetric impacting state can be closely tied to the coexistence of the nontrivial state and the origin as stable states in the symmetric