带约束多体系统Lyapunov指数的数值计算方法

Numerical method for Lyapunov exponents of multibody systems with constraints

应用第1类Lagrange方程建立的带约束多体系统动力学方程为非线性微分-代数方程组.利用增广法,将其转化成了常微分方程组,并表示成矩阵形式.根据方程的特点,给出了方程的Jacobi矩阵的具体表达式,提高了计算效率.在此基础上,给出了Lyapunov指数的数值计算方法,并通过对具体的树形和非树形多体系统进行Lyapunov指数数值计算,结合相图和Poincare映射对系统的动力学特性进行了分析,表明了该方法的可行性和有效性.

Dynamic equations of muhibody systems with constraints induced by the first kind of Lagrange＇s equations are nonlinear differential-algebraic equations. To be solved numerically, nonlinear differential- algebraic equations were transformed into ordinary differential equations with the augmentation approach. Ordinary differential equations and their Jacobian matrix were given in the matrix form to improve computational efficiency. A numerical method of Lyapunov exponents of nonlinear dynamics of multibody systems was demonstrated. An example was given to analysis dynamical behavior of two muhibody systems with topological tree and non-tree configuration, such as bifurcation and chaos by calculating Lyapunov exponents along with Poincare maps and phase graphs.