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

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

王贺元^{1, 2,},
肖胜中^3, ,,
梅鹏飞²,
张熙²

1.
广东科技学院通识教育学院, 广东东莞 523083
2.
沈阳师范大学数学与系统科学学院, 沈阳 110034
3.
广东农工商职业技术学院, 广州 510507

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
计量
- 文章访问数: 455
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- 被引次数: 0
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

Dynamic Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation

WANG Heyuan^{1, 2
,},
XIAO Shengzhong^{3
, ,},
MEI Pengfei²,
ZHANG Xi²

1.
College of General Education, Guangdong University of Science & Technology, Dongguan, Guangdong 523083, P.R.China
2.
School of Mathematics and System Science, Shenyang Normal University, Shenyang 110034, P.R.China
3.
Guangdong AIB College, Guangzhou 510507, P.R.China

摘要

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.
- Kolmogorov系统 /
- 混沌 /
- 混沌水轮 /
- 力学机理
Abstract: To reveal the mechanism of the waterwheel chaotic rotation, the dynamic mechanism and the energy conversion of the waterwheel chaotic rotation were studied with the method of moment analysis. The mathematical model for the Malkus waterwheel rotation was transformed into the Kolmogorov system. Based on the different coupling modes of inertia moments, internal moments, dissipation moments and external moments, the main factors and internal dynamic mechanisms of the Malkus waterwheel chaotic rotation were analyzed and discussed with the method of theoretical analysis and numerical simulation. The conversion among the Hamiltonian energy, the kinetic energy and the potential energy was investigated. The relationship between the energies and the Rayleigh number was discussed. The main factors influencing the chaotic rotation are the external moments and the dissipation moments. The analysis and simulation results show that, the lack-of-moment mode cannot lead to the system chaos, but the full-moment mode can, i.e., the waterwheel chaotic rotation will occur only in the existence of all 4 types of moments and when the dissipation and external forces match well. The Casimir function was introduced to analyze the system dynamics and the energy conversion. The bounds for the chaotic attractor were obtained with the Casimir function. The Casimir function reflects the energy conversion and the distances between the orbits and the equilibria. Numerical simulations depict the relationships among them.
- Kolmogorov system /
- chaos /
- waterwheel system /
- dynamic mechanism

HTML全文

图 1 混沌水轮示意图

Figure 1. The chaotic water wheel diagram

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图 2 分岔和最大Lyapunov指数图

Figure 2. The bifurcation diagram and the Lyapunov exponent spectrum

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图 3 分岔图

Figure 3. The bifurcation diagram

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图 4 系统在惯性力矩下

Figure 4. The system under inertial moments

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图 5 系统在惯性力矩和内力矩下

Figure 5. The system under inertial and internal moments

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图 6 能量函数的时间演化

Figure 6. Time evolutions of the energy function

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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

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图 8 能量的演化

Figure 8. Time evolutions of energy

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图 9 系统在全部力矩下

Figure 9. The system under full moments

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图 10 系统(17)的动力学

Figure 10. The dynamics of system (17)

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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

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图 12 函数与距离D₁，D₂的关系

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

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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

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图 14 D₁与D₂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

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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 r_e=1.058 45, r_g=13.965 6, r_h=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 r_h, 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)

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