圆筒连续型水轮混沌旋转的力学机制与能量演化

Mechanism and Energy Evolution of the Chaotic Rotation of a Continuous Cylindrical Waterwheel

混沌水轮是展示混沌现象的典型力学装置,国内外众多学者对其进行了深入的研究.本文探讨了圆筒连续型水轮混沌旋转的力学机理与能量转换问题.把圆筒连续型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方法,分析探讨了圆筒连续型水轮混沌旋转的主要影响因素和内在的力学机制.研究了水轮系统哈密顿能、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.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,数值仿真结果展现出它们之间的关系.

The chaotic water wheel is a typical mechanical device to show chaotic phenomena.There have been a number of investigations which concern the chaotic rotation.The dynamic mechanism and energy conversion of the continuous cylindrical waterwheel are investigated in this paper.The cylinder waterwheel system is transformed into a Kolmogorov system.Based on the different coupling modes of the four types of torques,i.e.,the inertial torque,the internal torque,the dissipation torque and the external torque,and combining both theoretical and numerical approaches,the major influential factors and the intrinsic mechanisms of the chaotic rotation of the continuous cylindrical waterwheel are studied.The conversion among Hamiltonian energy,kinetic energy and potential energy is investigated.The relationship between the energies and the Rayleigh number is discussed.Further,the Casimir function is introduced to analyze the dynamic behavior and energy transition of the water wheel system,with the boundary of the chaotic attractor evaluated.The Casimir function can reflect the energy conservation and the orbit-balance distance,whose relationship is also simulated by the numerical simulations.