Hamilton体系下旋转刚柔耦合楔形梁有限元建模及辛算法

Finite Element Modeling and Simplectic Computation of a Rigid and Flexible Coupled and Tapered Beam Rotating in the Hamilton System

首先利用Hamilton原理对耦合结构进行建模,然后利用有限元方法将空间连续模型离散化,得到有限元模型,然后将模型导入到Hamilton系统中,获得Hamilton正则方程。在建模的基础上,采用半隐式辛Runge-Kutta(SRK)算法对Hamilton系统下的模型进行计算,并与传统Runge-Kutta方法进行了比较。数值仿真结果表明,采用二阶的半隐式SRK算法能够长时间保持系统的能量守恒,是一种保结构算法,而传统的Runge-Kutta方法是一种耗散算法,在求解初期具有高的精度,但是不能保持系统解的长期稳定性。从仿真过程中还可以得到一个非常有意义的结果,二阶的半隐式SRK算法对步长的要求低于传统的四阶Runge-Kutta方法。

We model the coupled structure with the Hamilton principle and discretize the spatial continuous model using the finite element（FE） method, thus obtaining the FE model. Then we introduce the model to the Hamilton system and obtain the Hamilton canonical equation. Furthermore, we calculate the model in the Hamilton system with the half-implicit simplectic Runge-Kutta （SRK） algorithm. Numerical simulation results show that the two-order half-implicit SRK algorithm can maintain the energy conservation of the tapered beam for a long period of time, being a structure-preserving algorithm. More importantly, the two-order half-implicit SRK algorithm has lower requirement for its step length than the traditional four-order Runge-Kutta algorithm.