悬挂点匀速转动时摆锤圆周运动的存在性与稳定性分析

Existence and Stability Analysis on Circular Motion of Pendulums With Uniformly Rotating Pivots

悬挂点水平匀速转动时摆锤的圆周运动及其稳定性问题很少被研究,其中蕴含丰富的动力学现象.首先,考虑摆锤在真空和线性阻尼的介质中运动的情况,建立摆锤的动力学方程.其次,将摆锤水平圆周运动的特解的存在性问题转化为四次多项式的求根问题,通过Descartes(笛卡尔)符号规则及多项式函数的单调性分析,得出了摆的物理参数与特解个数的对应关系,真空中的特解可能是0,1,2个,在介质中时,只能是1个或3个.再次,基于Lyapunov(李雅普诺夫)一次近似理论考察非线性稳定性问题.将运动微分方程在特解附近线性化,特解的稳定性问题通过线性微分方程特征根实部的符号来判断.将涉及的四次特征方程巧妙地转化为二次方程,得出真空中特解线性意义下稳定的条件,以及介质中线性意义下渐近稳定的条件.最后,通过数值仿真,验证和明确了理论分析的结论.

Authough with rich dynamic meanings,the particular circular motion and the stability of pendulums with horizontally uniformly rotating pivots have been seldom studied. Firstly,for the pendulum moving in vacuum and within a medium respectively,the general motion equations under gravity and disturbing force were established. Newton＇s second law in a non-inertial reference frame was used through introduction of a fictitious inertia force. Secondly,the existence of the particular motion was converted into the root finding of a quartic equation. According to Descartes＇ rule of signs and analysis of the monotonicity of quartic polynomials,the relationships between the number of solutions and the physical parameters of the pendulum were given. In vacuum,the number of particular motion solution is 0,1 or2,and within a medium,the number is either 1 or 3. Their judging criteria were also given. Thirdly,Lyapunov＇s first approximation theory was used to investigate the nonlinear stability. The motion equation was linearized around the particular solution,the stability of the particular motion was judged by the signs of the real parts of the eigenvalues related to the linear differential equation. The subsequent quartic characteristic equations were skillfully converted into quadratic equations. Thus,the linearly stability conditions in vacuum and the asymptotic stability conditions within a medium were deduced. Finally,numerical simulations were given to verify and confirm the theoretical conclusions.