电活性聚合物圆柱壳静态与动态电压下的响应及稳定性

摘要：在电活性聚合物圆柱壳内外表面施加电压，圆柱壳会变薄并且伸长，因此相同的电压会在圆柱壳内产生更大的电场。这个正反馈可能使圆柱壳厚度不断变薄，最终导致其失稳破坏。本文研究了电活性聚合物圆柱壳在静态和周期电压作用下的响应及稳定性问题。采用neo-Hookean材料模型得到描述圆柱壳表面运动的非线性常微分方程。给出了圆柱壳在不同厚度和边界条件下外加电压随圆柱壳变形的变化曲线，结果表明存在一个临界电压，当外加电压大于这一临界值时，圆柱壳将被破坏。同时，也讨论了厚度和边界条件对临界电压的影响。圆柱壳在正弦周期电压作用下，其运动随时间的变化是周期性的或拟周期性的非线性振动。给出了圆柱壳振动固有频率的计算结果，采用打靶法得到圆柱壳振动的周期解，并且用数值法研究了周期解的稳定性。采用数值仿真得到圆柱壳振动振幅随外加动态电压激励频率的变化曲线，结果表明圆柱壳会发生多频共振，共振时圆柱壳振幅发生跳跃，导致圆柱壳失稳破坏。最后给出共振点临近点的振动曲线和相图，并对其振动特性进行讨论。

THE DYNAMICS RESPONSE AND STABILITY OF A DIELECTRIC ELASTOMER CYLINDRICAL SHELL UNDER STATIC AND PERIODIC VOTAGE

When a voltage is applied between the internal and external surface of a dielectric elastomer cylindrical shell,it thins down,and the same voltage will produce an even higher electric field.The positive feedback may make the dielectric elastomer cylindrical shell to thin down drastically,causing an electrical breakdown.We study the dynamics response and stability of the cylindrical shell which is under static and periodic voltage in this article.We get the ordinary differential equation which describes the movement of cylindrical shell using the neo-Hookean material model.The voltage curves as a function of the deformation of cylindrical shell with different thickness and different boundary conditions are given,and the critical voltage is found.The cylindrical shell will be destroyed if the applied voltage is greater than the critical value.In addition,we also discuss the influence of thickness and boundary conditions on the critical voltage.When a periodic voltage is applied,the movement of the shell is cyclical or intends to cyclical nonlinear vibration.We calculate nature frequency of the cylindrical shell,get the periodic solutions using the shooting method,and analyze the stability of the periodic solutions using the numerical method.The vibration amplitude curves as a function of the incentive frequency of periodic voltage are given.The cylindrical shell occur multi-rate resonance,the vibration amplitude jumps at the point of resonance which may cause the cylindrical shell destroyed.We give the time history curves and phase diagrams of the resonance points,and discuss the characteristics of the vibration.