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悬挂点匀速转动时摆锤圆周运动的存在性与稳定性分析

李舒航 蒋方华

李舒航, 蒋方华. 悬挂点匀速转动时摆锤圆周运动的存在性与稳定性分析[J]. 应用数学和力学, 2018, 39(2): 183-198. doi: 10.21656/1000-0887.380028
引用本文: 李舒航, 蒋方华. 悬挂点匀速转动时摆锤圆周运动的存在性与稳定性分析[J]. 应用数学和力学, 2018, 39(2): 183-198. doi: 10.21656/1000-0887.380028
LI Shuhang, JIANG Fanghua. Existence and Stability Analysis on Circular Motion of Pendulums With Uniformly Rotating Pivots[J]. Applied Mathematics and Mechanics, 2018, 39(2): 183-198. doi: 10.21656/1000-0887.380028
Citation: LI Shuhang, JIANG Fanghua. Existence and Stability Analysis on Circular Motion of Pendulums With Uniformly Rotating Pivots[J]. Applied Mathematics and Mechanics, 2018, 39(2): 183-198. doi: 10.21656/1000-0887.380028

悬挂点匀速转动时摆锤圆周运动的存在性与稳定性分析

doi: 10.21656/1000-0887.380028
详细信息
    作者简介:

    李舒航(1998—),男(E-mail: lishu.h@foxmail.com);蒋方华(1982—),男,副教授,博士,博士生导师(通讯作者. E-mail: jiangfh@tsinghua.edu.cn).

  • 中图分类号: O317

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

  • 摘要: 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.
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出版历程
  • 收稿日期:  2017-01-25
  • 修回日期:  2017-03-24
  • 刊出日期:  2018-02-15

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