水轮混沌旋转的力学机理与能量演化研究

Dynamic Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation

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.

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 mathe⁃matical 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 Hamilto⁃nian 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 mo⁃ments 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 oc⁃cur 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.