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Hamilton体系下介电弹性体圆形薄膜的动力学建模与辛求解
DYNAMIC MODELING AND SYMPLECTIC SOLUTION OF A CIRCULAR MEMBRANE OF DIELECTIC ELASTOMER UNDER HAMILTON SYSTEM

DOI：

 作者 单位 E-mail 李少锋 西北工业大学 lisf1877@163.com 都琳 西北工业大学 lindu@nwpu.edu.cn 邓子辰 西北工业大学

采用辛算法研究了Hamilton体系下介电弹性体圆形薄膜的动力学响应问题。首先，将介电弹性体圆形薄膜的振动问题引入到Hamilton对偶变量体系，借助Legendre变换，给出了系统的广义动量和Hamilton函数，通过对Hamilton函数作用量的变分，得到了Hamilton体系下的正则方程。其次，采用二级四阶辛Runge-Kutta算法对动力学系统进行了数值求解，通过和四级四阶经典Runge-Kutta算法相对比，结果表明二级四阶辛Runge-Kutta算法相比四级四阶Runge-Kutta算法具有保能量以及长时间数值稳定的优势，同时说明了四级四阶经典Runge-Kutta算法对于步长依赖的局限性。最后，对介电弹性体圆形薄膜的自由振动进行了动力学分析，做出了均布载荷与振幅之间的V形曲线，分析了均布载荷与加载电压对系统振幅的影响。

The symplectic algorithm is used to study the dynamic response of the circular membrane of dielectric elastomer under Hamilton system. Firstly, the vibration of circular membrane of dielectric elastomer under electrical and mechanical loading is introduced into the Hamilton dual variable system, and the generalized momentum and Hamilton functions of the system are obtained by means of Legendre transformation. The canonical equation under the Hamilton system is obtained by using the variational priciple to the Hamilton function. Secondly, the two stage and fourth-order symplectic Runge-Kutta algorithm is adopted for the numerical solution, numerical simulation results show that the two stage and fourth-order symplectic Runge-Kutta algorithm has an advantage of preserving energy and long-time numerical stablity by comparing with the four stage and fourth-order classic Runge-Kutta algorithm. In addition, this example also illustrates the limitations of the four stage and fourth-order classical Runge-Kutta algorithm for step dependence. Finally, the dynamic response of the free vibration of circular membrane of dielectric elastomer is analyzed. A v-shaped curve between mechanical loading and the amplitude of the system is plotted.Meanwhile, the influence of mechanical loading and voltage on the dynamic responses of the system is analyzed.
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