Turn off MathJax
Article Contents
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

Mechanical Mechanism and Energy Evolution of the Waterwheel Chaotic Rotation

doi: 10.21656/1000-0887.420336
  • Received Date: 2021-11-08
  • Rev Recd Date: 2022-01-13
  • Available Online: 2022-11-30
  • In order to reveal the mechanism of water wheel chaotic rotation, the Mechanical Mechanism and energy conversion of water wheel chaotic rotation are studied by the method of moment analysis. The mathematical model of the Malkus water wheel rotation is transformed into Kolmogorov system. Based on the different coupling modes of inertia moment, internal moment, dissipation moment and external moment, the main influencing factors and internal mechanical mechanism of the Malkus water wheel chaotic rotation are analyzed and discussed by using the method of theoretical analysis and numerical simulation. The conversion among Hamiltonian energy, kinetic energy and potential energy is investigated. The relationship between the energies and the Rayleigh number is discussed. The main factors affecting chaotic rotation are internal energy, kinetic energy and Hamiltonian energy. Through analysis and simulation, it is found that the lack of torque mode can not make the system generate chaos, but the full torque mode can make the system produce chaos, at the same time, the system can produce chaos only when the dissipation and external force match, at this time the water wheel is in a chaotic rotating state. The Casimir function is introduced to analyze the system dynamics. The bound of chaotic attractor is obtained by the Casimir function. The Casimir function reflects the energy conversion and the distance between the orbit and the equilibria. These relationships are illustrated by numerical simulations.
  • loading
  • [1]
    Edward N.Lorenz. Deterministic Nonperiodic[J]. Journal of the atmospheric sciences, 1963, 20: 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
    [2]
    R.Teman, Infinite dimensional dynamic system in mechanics and physics[M]. New York, Appl. Math. Sci., Springer-Verlog, 2000,
    [3]
    Pchelintsev, A.N. Numerical and Physical Modeling of the Dynamics of the Lorenz System[J]. Numerical Analysis and Applications, 2014, 7(2): 159-167. doi: 10.1134/S1995423914020098
    [4]
    Knobloch, Edgar. Chaos in the segmented disc dynamo[J]. Physics Letters A, 1981, 82(9): 439-440. doi: 10.1016/0375-9601(81)90274-7
    [5]
    Sparrow, Colin, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors[M] New York, Springer Verlag, 1982.
    [6]
    Miroslav K, Godfrey G. Theory for the exprimental observation of chaos in a rotating water wheel[J]. Physical Review A, 1992, 45(2): 626-637. doi: 10.1103/PhysRevA.45.626
    [7]
    Hilborn R C, Chaos and Nonlinear Dynamics[M], Oxford Univ Press, 1994.
    [8]
    Leslie E M., The Malkus-Lorenz water wheel revisited[J], Am. J. Phys., 75(12): 1114-1122.
    [9]
    Leonov, G.A., Kuznetsov, N.V., Korzhemanova, N.A., Kusakin, D.V. Lyapunov dimension formula for the global attractor of the Lorenz system[J]. Communications in Nonlinear Science, 2016, 41: 84-103. doi: 10.1016/j.cnsns.2016.04.032
    [10]
    Ashish B, VAN G R A. Chaos in a non-autonomous nonlinear system describing asymmetric water wheels[J]. Nonlinear Dyn., 2018, 93: 1977-1988. doi: 10.1007/s11071-018-4301-3
    [11]
    王贺元. Couette-Taylor流的力学机理与能量转换[J]. 数学物理学报, 2020, 41(1): 243-256 doi: 10.3969/j.issn.1003-3998.2020.01.019

    Heyuan Wang. Mechanical Mechanism and Energy Conversion of Couette-Taylor Flow[J]. (in Chinese) Acta Mathematica Scientia, 2020, 41(1): 243-256.(in Chinese) doi: 10.3969/j.issn.1003-3998.2020.01.019
    [12]
    王贺元, 崔进, 旋转流动的低模分析及仿真研究, 应用数学与力学[J], 应用数学和力学, 2017, 38(7): 794-806.

    Heyuan Wang, Jin Cui, Low-Dimensional Analysis and Numerical Simulation of Rotating Flow, Applied Mathematics and Mechanics[J]. 2017, 38(7): 794-806.(in Chinese)
    [13]
    Arnold V. Kolmogorov’S hydrodynamic attractors[J]. Proc R Soc Lond A, 1991, 434(19): 19-22.
    [14]
    Pasini A, Pelino V. A unified view of kolmogorov and lorenz systems[J]. Phys Lett A, 2000, 275: 435-446. doi: 10.1016/S0375-9601(00)00620-4
    [15]
    Xiyin Liang, GuoYuan Qi. Mechanical Analysis and Energy Conversion of Chen Chaotic System[J]. General and applied physics, 2017, 47(4): 288-294.
    [16]
    Xiyin Liang, GuoYuan Qi. Mechanical Analysis of Chen Chaotic System[J]. Chaos, Solitons and Fractals, 2017, 98: 173-177. doi: 10.1016/j.chaos.2017.03.021
    [17]
    Qi G, Liang X. Mechanical analysis of qi four-wing chaotic system[J]. Nonlinear Dyn., 2016, 86(2): 1095-1106. doi: 10.1007/s11071-016-2949-0
    [18]
    V. Pelino, F. Maimone, A. Pasini. Energy cycle for the lorenz attractor[J]. Chaos Soliton Fract., 2014, 64: 67-77. doi: 10.1016/j.chaos.2013.09.005
    [19]
    J. Marsden, T. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems[M]. 2nd edn, Springer, Berlin, 2002.
    [20]
    Strogatz SH, Nonlinear dynamics and chaos[M], Perseus Books, Reading, MA, 1994.
    [21]
    P.J. Morrison. Thoughts on brackets and dissipation: Old and new[J]. Journal of Physics: Conference Series, 2009, 169(1): 012006.
    [22]
    C.R. Doering, J.D. Gibbon. On the shape and dimension of the Lorenz attractor[J]. Dyn. Stab. Syst., 1995, 10(3): 255.
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Figures(14)  / Tables(1)

    Article Metrics

    Article views (71) PDF downloads(5) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return