基于球摆模型的离散变分积分子算法研究

在动力学系统长时间的仿真计算中,力学系统固有的结构将影响到计算精度及稳定性.离散变分积分子能够保持力学系统的能量,动量及辛结构的守恒.结合离散变分原理,通过对系统的拉格朗日函数进行离散化以及求变分和积分的过程,可以得到力学系统的离散变分积分子算法.该算法是一种递归算法,给定初始条件便可得到系统的动力学参数的时间历程.使用该原理可以构造具有完整约束的拉格朗日系统的辛-动量积分子方法.与连续算法相比,离散变分积分子算法能够直接在离散拉格朗日函数的基础上得到姿态与角速度的递推公式,而不需要复杂的迭代计算.本文研究是基于第一类拉格朗日函数的离散变分积分子算法.球摆模型是一个具有完整约束的拉格朗日系统.仿真结果表明,系统的能量值在长时间的仿真中得到保持,且计算的精度与步长的数量级呈现二次方的关系,系统角速度和姿态的仿真结果都符合球摆的运动规律.

The discrete variational integrators method of the spherical pendulum

The mechanical system＇s intrinsic structure may influence the long time computation＇s accurate and stability. The discrete variation integrators can conserve the energy momentum and the symplectic structure of the system. Combined with the discrete variation principle, the discrete variation integrator method can be obtained through the process of discretization varaiton and integration. This is a recursive algorithm that the time history of the parameters only need the initial condition. According this theory, a sympletic -momentum integrator can be formulated for the holonomic constraint Lagrange system. This method can get the recursive formula of the attitude and the angle velocity direct form the discrete Lagrange function and don＇t need complicated iterative computa- tion. The discrete variation integrator method explored in this paper is based on the first Lagrange functio spherical pendulum is a Lagrange system with holonomic be conserved in a long time simulation, and the accuracy constraints. The simulation result states that the energy of the computation presents a quadratic relation with the time step. The angle velocity and the attitude also present different character under two different algorithm.