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黏弹性传动带1:3内共振时的周期和混沌运动

Periodic and chaotic oscillation of a parametrically excited viscoelastic moving belt with 1:3 internal resonance

  • 摘要: 研究了参数激励作用下黏弹性传动带在1:3内共振时的周期解分岔和混沌动力学.同时考虑传动带的线性外阻尼因素和材料内阻尼因素.首先建立了具有线性外阻尼情况下的黏弹性传动带平面运动时的非线性动力学方程,黏弹性材料的本构关系用Kelvin模型描述. 然后考虑黏弹性传动带的横向振动问题,利用多尺度法和Galerkin离散法得到黏弹性传动带系统在1:3内共振时的平均方程.最后利用数值模拟方法研究了黏弹性传动带系统的周期振动和混沌动力学,得到了系统在不同参数下的混沌运动.数值模拟结果说明黏弹性传动带系统存在周期分岔, 概周期运动及混沌运动.

     

    Abstract: In this paper, the bifurcations of periodic solutions andchaotic dynamics for a parametrically excited viscoelastic moving belt with1:3 internal resonance are investigated for the first time. The externaldamping and the internal damping of the material for viscoelastic movingbelt are considered simultaneously. First, the nonlinear equation of planarmotion for viscoelastic moving belt with the external damping isestablished. The Kelvin viscoelastic model is adopted to describe therelation between the stress and strain for viscoelastic material. Then, thetransverse nonlinear oscillations of viscoelastic moving belt areconsidered. The method of multiple scales and the Galerkin approach areapplied directly to the partial differential governing equation ofviscoelastic moving belt to obtain the averaged equations under the case of1:3 internal resonance and primary parametric resonance of the nth mode.Finally, numerical simulation method is used to investigated thebifurcations of periodic solutions and chaotic dynamics for viscoelasticmoving belt. The chaotic motions are found under the cases of differentparameters. The results of numerical simulation demonstrate that there existperiodic, 2-periodic, 3-periodic, 5-periodic and quasiperiodic responses andchaotic motions in viscoelastic moving belt.

     

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