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.
Highlights
The research about the nonlinear dynamics is one of the frontier application problems of the international dynamics area
In 1991, Perdios et al [2] investigated the families of threedimensional periodic orbits which branch off the families of planar periodic around the triangular equilibrium point
In 2008, Liu et al [16] investigated the problem of the transverse nonlinear nonplanar oscillations of an axially moving viscoelastic belt with the integral constitutive law in the case of 1 : 1 internal resonance
Summary
The research about the nonlinear dynamics is one of the frontier application problems of the international dynamics area. In 2009, Liu and Han [5] investigated the bifurcation of periodic solutions of a 4-dimensional system depending on a small parameter and gave a new method to use the Melnikov function. In 1998, Zhang and Zu [13, 14] studied the nonlinear vibration and stability of a parametrically excited viscoelastic belt by using the multiscale method. In 2004, Zhang et al [15] investigated the bifurcation of periodic solutions and chaotic dynamics for a parametrically excited viscoelastic moving belt with 1 : 3 internal resonance and obtained that there exist. In 2008, Liu et al [16] investigated the problem of the transverse nonlinear nonplanar oscillations of an axially moving viscoelastic belt with the integral constitutive law in the case of 1 : 1 internal resonance.
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