Hamilton体系下介电弹性体圆形薄膜的动力学建模与辛求解
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摘要
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。
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.
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.
引文
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[2] Wissler M,Mazza E.Modeling of a pre -strained circular actuator made of dielectric elastomers[J].Sensors and Actuators A:Physical,2005,120(1):184-192.
[3] Wissler M,Mazza E.Modeling and simulation of dielectric elastomer actuators[J].Smart Materials and Structures,2005,14(6):1396-1402.
[4] Lee B Y,Kim J,Kim H,et al.Low-cost flexible pre -ssure sensor based on dielectric elastomer film with micro -pores[J].Sensors and Actuators A:Physical,2016,240:103-109.
[5] McKay T,O’Brien B,Calius E,et al.An integrated,self-priming dielectric elastomer generator[J].Applied Physics Letters,2010,97(6):062911.
[6] Jung K,Koo J C,Nam J D,et al.Artificial annelid robot driven by soft actuators[J].Bioinspiration & Biomimetics,2007,2(2):S42-S49.
[7] Zhu J,Cai S Q,Suo Z G.Nonlinear oscillation of a dielectric elastomer balloon[J].Polymer International,2010,59(3):378-383.
[8] Lu T Q,Huang J S,Jordi C,et al.Dielectric elastomer actuators under equal-biaxial forces,uniaxial forces,and uniaxial constraint of stiff fibers[J].Soft Matter,2012,8(22):6167-6173.
[9] Dai H L,Wang L.Nonlinear oscillations of a dielectric elastomer membrane subjected to in-plane stret-ching[J].Nonlinear Dynamics,2015,82(4):1709-1719.
[10] Wang F F,Lu T Q,Wang T J.Nonlinear vibration of dielectric elastomer incorporating strain stiffening[J].International Journal of Solids and Structures,2016,87:70-80.
[11] Ren J S.Dynamical response of hyper-elastic cylindrical shells under periodic load [J].Applied Mathematics and Mechanics (English Edition),2008,29(10):1319-1327.
[12] Tang D F,Lim C W,Hong L,et al.Analytical asymptotic approximations for large amplitude nonlinear free vibration of a dielectric elastomer balloon[J].Nonlinear Dynamics,2017,88(3):2255-2264.
[13] Wu B S,Sun W P,Lim C W.An analytical approximate technique for a class of strongly non-linear osci -llators [J].International Journal of Non -Linear Mechanics,2006,41(6-7):766-774.
[14] 冯康,秦孟兆.哈密尔顿系统的辛几何算法[M].杭州:浙江科学技术出版社,2003.(FENG Kang,QIN Meng -zhao.Symplectic Geometric Algorithms for Hamiltonian Systems[M].Hangzhou:Zhejiang Science and Technology Publishing House,2003.(in Chinese))
[15] 姚伟岸,钟万勰.辛弹性力学[M].北京:高等教育出版社,2002.(YAO Wei-an,ZHONG Wan-xie.Symplectic Elasticity[M].Beijing:High Education Press,2002.(in Chinese))
[16] 钟万勰.应用力学对偶体系[M].北京:科学出版社,2002.(ZHONG Wan-xie.Applied Mechanics Dual System [M].Beijing:Science Press,2002.(in Chinese))
[17] 魏乙,邓子辰,李庆军,等.空间太阳能电站的轨道、姿态和结构振动的耦合动力学建模及辛求解[J].动力学与控制学报,2016,14(6):513-519.(WEI Yi,DENG Zi-chen,LI Qing-jun,et al.Coupling dynamic modeling among orbital motion,attitude motion and structural vibration and symplectic solution of SPS [J].Journal of Dynamics and Control,2016,14(6):513-519.(in Chinese))
[18] 邓子辰,李庆军.精细指数积分法在卫星编队飞行动力学中的应用[J].北京大学学报(自然科学版),2016,52(4):669-675.(DENG Zi-chen,LI Qing-jun.Precise exponential integrator and its application in dynamics of spacecraft formation flying [J].Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(4):669-675.(in Chinese))
[19] Iserles A,Sanz-Serna J M ,Calvo M P.Numerical Hamiltonian problem[J].Mathematics of Computation,1995,64(211):1346.