Hamilton体系下介电弹性体圆形薄膜的动力学建模与辛求解

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。首先，将该问题引入Hamilton对偶变量体系，借助Legendre变换，给出系统的广义动量和Hamilton函数，通过对Hamilton函数作用量的变分，得到Hamilton体系下的正则方程。其次，对于得到的正则方程给出了辛Runge-Kutta的计算格式。最后，采用二级四阶辛Runge-Kutta算法对动力学系统进行了数值求解，和四级四阶经典Runge-Kutta算法进行对比，结果表明，二级四阶辛Runge-Kutta算法具有保能量以及长时间数值稳定的优势，同时说明四级四阶经典Runge-Kutta算法对于步长依赖的局限性。

Dynamic modeling and symplectic solution of a circular membrane of dielectric elastomer under Hamilton system

The symplectic algorithm is used to study the dynamic response of the circular membrane of dielectric elastomer in a Hamiltonian system.Firstly,this model is introduced into the dual Hamiltonian variable system,and the generalized momentum and Hamilton functions of the system are obtained by means of Legendre transformation.The canonical equation is obtained by using the variational principle to the Hamilton function.Secondly,for the obtained canonical equations,the calculation scheme of the symplectic Runge-Kutta algorithm is given.Finally,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 stability by comparing with the four-stage and fourth-order classic Runge-Kutta algorithm.In addition,this example also illustrates the limitations of step dependence of the four-stage and fourth-order classical Runge-Kutta algorithm.