Periodic Motions and Bifurcations of a Spring-Driven Joint System with Periodic Excitation

Spring-driven joint system is a nonlinear dynamic system that is applied in flexible robot arms. Similar to the forced pendulum system, it has state variable in the sinusoidal function, but are more complicated with driven spring length coupling with the joint angle. With traditional analytical approaches, it is difficult to analyze such a nonlinear system since both the system state and the excitation are correlated with the nonlinear sinusoidal term. In this paper, a spring-driven joint system with periodic excitation will be discussed. The implicit discrete maps approach will be applied to solve the periodic motions for such a joint system, and the stability condition will be discussed. The analytical expressions for periodic motions for such a spring-driven system can be recovered with a series of Fourier functions. The bifurcation diagram for such a system will be given to show the complexity of the motions when the frequency of the excitation varies. From analytical bifurcation for period-1 and period-2 motions, the jump phenomenon and the continuity of periodic motion can be analytically explained.