自由运动双摆的周期性及其解析

Periodic Motion and Analytic Solution for a Free Double Pendulum

描述了一种平面自由运动的双摆,考虑动量矩和能量守恒的条件,根据刘维尔可积定理可知该系统的可积性.利用能量积分,根据惠特克定理对系统的动力学方程进行降阶,并根据旋转数给出系统作周期或准周期运动的判别条件.为了更深入研究系统的运动规律,寻找某些特殊参数以得到系统在相平面内运动方程的解析表达式.通过求解运动方程发现系统在时域的周期解为包含一类椭圆函数的反函数解析解.数值分析及相关的仿真曲线验证了理论分析的正确性.

An analytic method to study the periodic motion of double pendulums is presented. Considering enough first integrals , such as conservation of energy and angular momentum, the system is integrable by Liouville＇ s theorem. The dynamical equations are reduced based on Whittaker＇ s theorem. And rotating number is achieved to decide that the motion of the system is periodic or quasi - periodic. Then with some special parameters selected, the equation of motion in the phase plane is obtained. Because of existence of the elliptic integral in the equation of motion on time domain, the closed form solution is transcendental. The numerical examples and the simulation curves are given to verify the validity of the theoretical conclusion.