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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, 2023, 44(5): 560-572. 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, 2023, 44(5): 560-572. doi: 10.21656/1000-0887.420336
WANG Heyuan, XIAO Shengzhong, MEI Pengfei, ZHANG Xi. Dynamic Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation[J]. Applied Mathematics and Mechanics, 2023, 44(5): 560-572. doi: 10.21656/1000-0887.420336
Citation: WANG Heyuan, XIAO Shengzhong, MEI Pengfei, ZHANG Xi. Dynamic Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation[J]. Applied Mathematics and Mechanics, 2023, 44(5): 560-572. 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: 

国家自然科学基金项目 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)

    通讯作者:

    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

Dynamic 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4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函数分析了水轮系统的动力学行为和能量转换,并估计了混沌吸引子的界. 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.  The chaotic water wheel diagram

    图  2  分岔和最大Lyapunov指数图

    Figure  2.  The bifurcation diagram and the Lyapunov exponent spectrum

    图  3  分岔图

    Figure  3.  The bifurcation diagram

    图  4  系统在惯性力矩下

    Figure  4.  The system under inertial moments

    图  5  系统在惯性力矩和内力矩下

    Figure  5.  The system under inertial and internal moments

    图  6  能量函数的时间演化

    Figure  6.  Time evolutions of the energy function

    图  7  系统在惯性力矩、内力矩和外力矩下

    Figure  7.  The system under inertial, internal and external moments

    图  8  能量的演化

    Figure  8.  Time evolutions of energy

    图  9  系统在全部力矩下

    Figure  9.  The system under full moments

    图  10  系统(17)的动力学

    Figure  10.  The dynamics of system (17)

    图  11  Casimir函数及混沌吸引子的边界

    Figure  11.  Time evolutions of the Casimir energy and the boundary of the chaotic attractor

    图  12  函数与距离D1D2的关系

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

    图  13  能量和Casimir函数相对于r的演化

    Figure  13.  Evolutions of the energy and the Casimir function with r

    图  14  D1D2的距离和与Casimir函数关于r的演化

    Figure  14.  The sum of distances D1 and D2 and the Casimir function evolution with r

    表  1  σ=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 behaviors and energy evolutions of system (1) with corresponding actual rotations of the waterwheel for σ=5 and different r values

    r-value 0<r<1
    r=0, r1=1
    r>1
    re=1.058 45, rg=13.965 6, rh=15.041 2
    equilibrium O stable node saddle node (one direction is unstable, the other two directions are stable)
    equilibrium P± inexistence stable node stable focus stable focus saddle point
    trajectory of system (4) tendency to stable equilibrium O tendency to stable equilibrium P+ or P- spiral line approaching P+ or P- the same as the left, but closer to rh, jumping back and forth between P+ and P-, a transient chaos, ultimately approaching P+ or P- unstable limit cycles (subcritical Hopf bifurcation) leading to chaos
    kinetic energy minimum upgrowth increase sustained increase
    Casimir function minimum gradual increases increase sustained increase
    sum of D1 and D2 inexistence monotone increase increase upgrowth
    state of the water wheel nomotion irregular rotation fig. 1(a)1(b) unstable rotation fig. 1(b)1(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
  • 刊出日期:  2023-05-01

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