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Bifurcation of Periodic Solutions and Numerical Simulation for the Viscoelastic Belt

机译：粘弹性带周期解的分叉与数值模拟

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摘要

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.
机译：我们研究具有整体本构律的粘弹性带在1：1内部共振中的周期解的分歧。首先，通过应用非奇异线性变换，将系统转换为另一个系统，该系统的无扰动系统由两个平面系统组成：一个是哈密顿系统，另一个具有焦点。此外，根据梅尔尼科夫函数，我们可以获得周期解存在的充分条件，并为研究周期解和不变环的稳定性做好了准备。最终，我们需要给出不同参数下溶液的相图，以验证分析结果并获得溶液的存在性和稳定性所基于的参数。结论不仅丰富了粘弹性带的非线性动力学行为，而且对降低噪声和降低能量损失具有重要的理论意义和应用价值。

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