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水轮混沌旋转的力学机理与能量演化研究

王贺元 肖胜中 梅鹏飞 张熙

王贺元,肖胜中,梅鹏飞,张熙. 水轮混沌旋转的力学机理与能量演化研究 [J]. 应用数学和力学,2022,43(X):1-13 doi: 10.21656/1000-0887.420336
引用本文: 王贺元,肖胜中,梅鹏飞,张熙. 水轮混沌旋转的力学机理与能量演化研究 [J]. 应用数学和力学,2022,43(X):1-13 doi: 10.21656/1000-0887.420336
WANG Heyuan, XIAO Shengzhong, MEI Pengfei, ZHANG Xi. Mechanical Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420336
Citation: WANG Heyuan, XIAO Shengzhong, MEI Pengfei, ZHANG Xi. Mechanical Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420336

水轮混沌旋转的力学机理与能量演化研究

doi: 10.21656/1000-0887.420336
基金项目: 国家自然科学基金(11572146);辽宁省科技计划重点研发项目(2019JH8/10100086)
详细信息
    作者简介:

    王贺元(1963—),男,教授,博士(E-mail:wangheyuan6400@sina.com

    肖胜中(1965—),男,教授,博士(通讯作者. E-mail:xszong@sohu.com

  • 中图分类号: O175.1; O192; O193; O302

Mechanical Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation

  • 摘要: 为了揭示水轮混沌旋转的生成机制,采用力矩分析方法研究了水轮混沌旋转的力学机理与能量转换问题。把Malkus水轮的数学模型转换为Kolmogorov系统,基于惯性力矩、内力矩、耗散力矩和外力矩的不同耦合模式,利用理论分析和数值仿真相结合的方法,分析探讨了Malkus水轮混沌旋转的主要影响因素和内在的力学机理。研究了水轮系统Hamilton能量。动能和势能之间的相互转换,讨论了能量与Rayleigh数之间的关系。影响水轮系统混沌生成的主要因素是外力矩和耗散力矩。通过分析和仿真得知:力矩缺失模式并不能使系统生成混沌,全力矩模式才能使系统产生混沌,即混沌发生时四种力矩缺一不可,与此同时只有耗散和外力相匹配时系统才能产生混沌,此时水轮发生混沌旋转。引进Casimir函数分析水轮系统的动力学行为和能量转换,并估计混沌吸引子的界。Casimir函数反映了能量转换和轨道与平衡点间的距离,数值结果仿真刻画了它们之间的关系。
  • 图  1  混沌水轮示意图

    Figure  1.  Chaotic water wheel diagram

    图  2  分岔和最大Lyapunov指数图:(a) 分岔图;(b) 最大Lyapunov指数

    Figure  2.  bifurcation diagram and Lyapunov exponent spectrum: (a) bifurcation diagram; (b) Lyapunov exponent spectrum

    图  3  分岔图:(a) $r=r_1 $处的音叉分岔;(b) $P^{\pm} $$r=r_{\rm{h}} $处的亚临界Hopf分岔

    Figure  3.  bifurcation diagram: (a) Pitchfork bifurcation at $r=r_1 $; (b) Subcritical bifurcation when $r=r_{\rm{h}} $ at $P^{\pm} $

    图  4  系统在惯性力矩下:(a) 周期轨道的3维视图;(b) 状态变量z的轨迹

    Figure  4.  System is under inertial torques: (a) 3-D view of periodic orbit; (b) The trajectory of state z

    图  5  系统在惯性力矩和内力矩下:(a) 周期轨道的3维视图;(b) 状态变量z的轨迹

    Figure  5.  System is under inertial and internal torques: (a) 3-D view of periodic orbit; (b) The trajectory of state z

    图  6  能量函数的时间演化

    Figure  6.  Time evolution of the energy function

    图  7  系统在惯性力矩,内力矩和外力矩下:(a) 螺旋状轨道的3维视图;(b) 状态变量z的轨迹

    Figure  7.  The system is under inertial, internal and external torques: (a) 3-D view of spiral-like orbit; (b) The trajectory of state z

    图  8  能量的演化。

    Figure  8.  Time evolution of Energy.

    图  9  系统在全部力矩下。

    Figure  9.  The system is under full torques.

    图  10  系统(2.5)的动力学:(a) 拟周期轨道的3维视图;b) 动能和势能

    Figure  10.  The dynamics of system (17): (a) 3-D view of quasi periodic attractor; (b) Kinetic Energy and Potential Energy

    图  11  Casimir函数及混沌吸引子的边界:(a) Casimir能量时间演化;(b) 混沌吸引子的边界

    Figure  11.  Time evolution of Casimir energy and the boundary of chaotic attractor: (a) Time evolution of Casimir energy; (b) The boundary of chaotic attractor

    图  12  函数与距离 D1,D2的关系。

    Figure  12.  The relationship between Casimir function and the D1 and D2.

    图  13  能量和Casimir函数相对于$ r$的演化:(a) 能量;b) Casimir函数

    Figure  13.  Evolution of energy and Casimir function with $ r$ increase: (a) energy; (b) Casimir function

    图  14  $D_1$$D_2$的距离和与 Casimir函数关于$r$的演化:(a) $D_1 $$D_2 $的距离和;(b) Casimir函数

    Figure  14.  The sum of distances $D_1$ and $D_2$ and Casimir function with $r$ increase: (a) The sum of distances $D_1 $ and $D_2 $; (b) Casimir function

    表  1  $ \sigma =5,r$取不同值时水轮系统(1)的动力学行为与能量演化及其相应的旋转状态

    Table  1.   Dynamics behavior and energy evolution of system (1) and corresponding actual rotation of the waterwheel for $\sigma=5$ and different $r$ values

    $r$-value$0<r<1$
    r = 0, $r_1=1$
    $r>1$
    $r_{\rm{e}}=1.058\;45,\quad r_{\rm{g}}=13.965\;6,\quad r_{\rm{h}}=$15.041 2
    equilibria $O$stable node Saddle node
    (one direction is unstable,
    the other two directions are stability)
    equilibria $P^{\pm}$inexistencestable nodestable focusstable focussaddle point
    trajectory of system (4)tends to stable equilibria $O$tends to stable equilibria $P^{+}$ or $P^{-}$ spiral line tends to $P^{+}$ or $P^{-}$same as left, but closer to $r_{\rm{h}}$, jumping back and forth between $P^{+}$ and $P^{-}$, a transient chaos, ultimately tends to $P^{+}$ or $P^{-}$unstable limit cycles (subcritical Hopf bifurcation) lead to chaos
    the kinetic energyminim grow biggerincreasesmaintain increaseing
    the Casimir functionminimumgradual increasesincreaseskeep increasing the trend
    the sum of $D_1$ and $D_2$inexistencemonotone increasesincreasesgrow bigger
    the state of the water wheelmotionlessirregular rotation fig. 1(a)、(b)unstable rotation fig. 1(b)、(c)chaotic rotation fig. 1(c)
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-11-08
  • 修回日期:  2022-01-13
  • 网络出版日期:  2022-11-30

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