Subharmonic Bifurcation for a Non-smooth Double Pendulum with Unilateral Impact

Abstract

A unilateral impact double pendulum model with hinge links is constructed to detect subharmonic bifurcation for the high dimensional non-smooth system. The non-smooth and nonlinear coupled factors lead a barrier for high dimensional conventional nonlinear techniques. By introducing reversible transformation and energy time scale transformation, the system is expressed as a smooth decoupling form of energy coordinates. Thus, the concept of subharmonic Melnikov function is extended to high-dimensional nonsmooth systems, and the influence of impact recovery coefficient on the existence of subharmonic periodic orbits of double pendulum is revealed. The efficiency of the theoretical results is verified by phase portraits, time process portraits and Poincaré section.

A compound double pendulum with friction

“…By way of verification of these equations, we note that they match with equations of motion which were assumed by Tian et al [20] for a two degree-of-freedom hinged double rod pendulum which impacts with a rigid wall. We note that with some modifications the method which we have used here could also be used to rigorously derive the equations of motion used in [20]. The method which we have used can also be generalised to any type of torque apart from friction torques.…”

“…In this work, we considered friction torques in the equations of motion, but our analysis could easily be extended to include more general types of nonconservative force. Our method also allows for a rigorous, transparent derivation of similar equations of motion which were stated without justification for a hinged double pendulum in [20]. We have also analysed the stability of solutions at the steady state.…”