非线性插值精细积分法在刚柔耦合弹簧摆中的应用

Nonlinear Interpolation Precise Integration Method in Rigid-Flexible Coupling Spring Pendulum

刚柔耦合多体系统变量的特点为既有大范围慢变量,又有小幅度快变量,它们相互耦合,构成时变强非线性的高维动力学方程。由于这一特点往往给系统的数值模拟带来困境,需要对这一特点进行更深入的数值分析。以双时间尺度变量弹簧摆作为研究模型,采用一种三次Lagrange插值精细积分法进行数值计算,该方法是一个显式单步预测一校正的有效算法,能够自起步,且具有精度高、计算量小的特点。将该精细积分法与四阶Runge-Kutta法从能量守恒及计算结果准确度两方面进行比较,结果表明在计算系统快变量的响应时,精细积分法优于四阶Runge-Kutta法。对弹簧摆系统进行动力学行为分析,以大频率比及初始大摆角作为控制参数,研究系统的复杂动力学行为,给出了一定范围内不同动力学性态对应的参数域。

The characteristic of the rigid-flexible coupling multibody system is that there exist both wide range slow variables and slight fast variables. They are coupled to each other, constituting a time-varying, strongly nonlinear and high dimensional kinetic equation. Many difficulties in numerical simulation are brought due to this characteristic. Depth numerical analysis of this characteristic is needed. A two- time-scale variable spring pendulum system was constructed as the research model. A cubic interpolation precise integration method was provided. The method is a single-step prediction-correction algorithm, capable of self-starting, and has the characteristics of high precision and small amount of calculation. Compared with the fourth-order Runge-Kutta method, the precise integration method is better in calcu- lating the response of the fast variable. The dynamical behavior of the spring pendulum system was analyzed. The large frequency ratio and wide initial swing angle were set as control parameters, and the parameter domain corresponding to the different dynamical behavior was given within a certain range.