Bifurcation of Periodic Solutions and Numerical Simulation for the Viscoelastic Belt

Abstract

We study the bifurcation of periodic solutions for viscoelastic belt with integral constitutive law in 1: 1 internal resonance. At the beginning, by applying the nonsingular linear transformation, the system is transformed into another system whose unperturbed system is composed of two planar systems: one is a Hamiltonian system and the other has a focus. Furthermore, according to the Melnikov function, we can obtain the sufficient condition for the existence of periodic solutions and make preparations for studying the stability of the periodic solution and the invariant torus. Eventually, we need to give the phase diagrams of the solutions under different parameters to verify the analytical results and obtain which parameters the existence and the stability of the solution are based on. The conclusions not only enrich the behaviors of nonlinear dynamics about viscoelastic belt but also have important theoretical significance and application value on noise weakening and energy loss.