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.
Highlights
Journal of Nonlinear Mathematical Physics (2022) 29:349–367 factors leads a barrier for conventional nonlinear techniques
Researchers have carried out the generalization of the concepts of homoclinic orbit, heteroclinic orbit, periodic orbit Melnikov function of non-smooth system with single degree of freedom [11]
The subharmonic Melnikov method of non-smooth double pendulum is significantly improved by using energy-time scale transform and Poincaré map
Summary
There are many factors such as impact, gap and so on. Generally, the impact of manipulator and robot leg with ground can be simplified as the non-smooth double pendulum [1, 2]. It is difficult to deal with the high-dimensional nonlinear due to the non-smoothness It has a wide range of applications in engineering [5, 6], which is a practical and scientific problem to be solved. Researchers have carried out the generalization of the concepts of homoclinic orbit, heteroclinic orbit, periodic orbit Melnikov function of non-smooth system with single degree of freedom [11]. Sun et al [13, 14] extended the subharmonic Melnikov function of smooth system to two degrees of freedom autonomous and non-autonomous nonlinear systems. The periodic solutions of four dimensional smooth system degenerate and non-degenerate resonance were detected by using the generalized subharmonic Melnikov function. The energy-time scale transform is introduced into the decoupled systems and the nonsmooth Melnikov function for high-dimensional non-smooth system is obtained. Numerical simulations are shown to verify the theoretical analysis
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