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轮对系统的Hopf分岔研究

武世江 张继业 隋皓 殷中慧 胥奇

武世江, 张继业, 隋皓, 殷中慧, 胥奇. 轮对系统的Hopf分岔研究. 力学学报, 2021, 53(9): 2569-2581 doi: 10.6052/0459-1879-21-321
引用本文: 武世江, 张继业, 隋皓, 殷中慧, 胥奇. 轮对系统的Hopf分岔研究. 力学学报, 2021, 53(9): 2569-2581 doi: 10.6052/0459-1879-21-321
Wu Shijiang, Zhang Jiye, Sui Hao, Yin Zhonghui, Xu Qi. Hopf bifurcation study of wheelset system. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(9): 2569-2581 doi: 10.6052/0459-1879-21-321
Citation: Wu Shijiang, Zhang Jiye, Sui Hao, Yin Zhonghui, Xu Qi. Hopf bifurcation study of wheelset system. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(9): 2569-2581 doi: 10.6052/0459-1879-21-321

轮对系统的Hopf分岔研究

doi: 10.6052/0459-1879-21-321
基金项目: 国家重点研发计划课题 (2020YFA0710902)和国家自然科学基金资助项目 (12172308)
详细信息
    作者简介:

    张继业, 教授, 主要研究方向: 高速列车动力学与控制. E-mail: 185335104@qq.com

  • 中图分类号: U270.1+1

HOPF BIFURCATION STUDY OF WHEELSET SYSTEM

  • 摘要: 针对轮对系统中的非线性动力学问题, 本文基于Hopf分岔代数判据得到考虑陀螺效应的轮对系统Hopf分岔点解析表达式, 即轮对系统蛇形失稳的线性临界速度解析表达式. 基于分岔理论得到轮对系统的第一、第二Lyapunov系数表达式, 并结合打靶法分别得到不同纵向刚度下, 考虑陀螺效应与不考虑陀螺效应的轮对系统分岔图. 通过对比有无陀螺效应的轮对系统分岔图发现, 在同一纵向刚度下, 考虑陀螺效应的轮对系统线性临界速度和非线性临界速度均大于不考虑陀螺效应的轮对系统, 即陀螺效应可以提高轮对系统的运动稳定性. 基于Bautin分岔理论, 以纵向刚度和纵向速度作为参数, 分别得到考虑陀螺效应和不考虑陀螺效应的轮对系统, 从亚临界Hopf分岔到超临界Hopf分岔, 再从超临界Hopf分岔到亚临界Hopf分岔的迁移机理拓扑图. 通过对比有、无陀螺效应的轮对系统Bautin分岔拓扑图发现, 陀螺效应将改变轮对系统的退化Hopf分岔点, 但对于轮对系统Bautin分岔拓扑图的影响不大.

     

  • 图  1  轮对模型图

    Figure  1.  Wheelset model diagram

    图  2  轮对系统线性临界速度与横向转动惯量的关系

    Figure  2.  The relationship between the linear critical speed of the wheelset system and the lateral moment of inertia

    图  3  轮对系统的线性临界速度与横向自旋蠕滑系数关系

    Figure  3.  The relationship between the linear critical speed of the wheelset system and the lateral spin-slip coefficient

    图  4  不同纵向刚度下不考虑陀螺效应的轮对系统分岔图

    Figure  4.  The bifurcation diagram of wheelset system without considering the gyroscopic action under different longitudinal stiffness

    图  5  不同纵向刚度下考虑陀螺效应的轮对系统分岔图

    Figure  5.  The bifurcation diagram of wheelset system considering the gyroscopic action under different longitudinal stiffness

    图  6  轮对系统退化Hopf分岔图

    Figure  6.  Degenerate Hopf bifurcation diagram of the wheelset system

    图  7  Bautin分岔图($ \xi = - 1 $)

    Figure  7.  Bautin bifurcation diagram ($ \xi = - 1 $)

    图  8  轮对系统考虑陀螺效应时的Bautin分岔拓扑图

    Figure  8.  Bautin bifurcation topology diagram of wheelset system when considering gyroscopic action

    图  9  图8中区域①对应的相图

    Figure  9.  Phase diagram corresponding to area ① in Fig 8

    图  12  图8T区域对应的相图

    Figure  12.  Phase diagram corresponding to the T region in Fig 8

    图  10  图8中区域②对应的相图

    Figure  10.  Phase diagram corresponding to area ② in Fig 8

    图  11  图8中区域③对应的相图

    Figure  11.  Phase diagram corresponding to area ③ in Fig 8

    图  13  轮对系统不考虑陀螺效应时的Bautin分岔拓扑图

    Figure  13.  Bautin bifurcation topology diagram of wheelset system without considering gyroscopic action

    图  14  图9中区域①对应的相图

    Figure  14.  Phase diagram corresponding to area ① in Fig 9

    图  17  图9T区域对应的相图

    Figure  17.  Phase diagram corresponding to the T region in Fig 9

    图  15  图9中区域②对应的相图

    Figure  15.  Phase diagram corresponding to area ② in Fig 9

    图  16  图9中区域③对应的相图

    Figure  16.  Phase diagram corresponding to area ③ in Fig 9

    A1  轮对参数表

    A1.   Wheelset parameter list

    ParameterValue
    m/kg 1612
    $ {I}_{z}/(\text{kg} \cdot {\text{m}}^{2}) $881
    Iy$ /(\text{kg} \cdot {\text{m}}^{2}) $100
    $ {\lambda } $0.13
    $ {{r}_{0}}/{\text{m}} $0.46
    $L/{\rm{m}}$1
    $b/{\rm{m}}$0.7175
    $ {{f}_{{\text{11}}}}/{\text{MN}} $13.7
    $ {{f}_{{\text{22}}}}/{\text{MN}} $11.1
    $ {{f}_{{\text{23}}}}/{\text{MN}} $0.13
    $ {{f}_{{\text{33}}}}/{\text{MN}} $0
    ky$/(\text{MN} \cdot {\text{m} }^{{-1} })$4
    $ {W/}{\text{kN}} $112.2
    $ {\delta _{\text{1}}} $−1.6 × 1011
    $ {\delta _{\text{2}}} $1.6 × 1015
    下载: 导出CSV

    A2  图4中各速度值

    A2.   Values of each speed in Fig. 4

    v/(m·s−1)Value
    $ {v_{1,1}} $ 135.3351
    $ {v_{1,2}} $ 127.4978
    $ {v_{2,1}} $ 150.7902
    $ {v_{2,2}} $ 147.0965
    $ {v_{3,1}} $ 159.8320
    $ {v_{3,2}} $ 158.8470
    $ {v_{3,3}} $ 158.8609
    $ {v_{3,4}} $ 158.8112
    $ {v_{4,1}} $ 170.1243
    $ {v_{4,2}} $ 172.6321
    $ {v_{4,3}} $ 170.5156
    $ {v_{5,1}} $ 182.0883
    $ {v_{5,2}} $ 189.3157
    $ {v_{5,3}} $ 181.1072
    $ {v_{6,1}} $ 196.3457
    $ {v_{6,2}} $ 210.3911
    $ {v_{6,3}} $ 190.7989
    下载: 导出CSV

    A3  图5中各速度值

    A3.   Values of each speed in Fig. 5

    v/(m·s−1)Value
    $ {v_{7,1}} $140.5364
    $ {v_{7,2}} $133.0938
    $ {v_{8,1}} $157.5975
    $ {v_{8,2}} $155.0780
    $ {v_{9,1}} $167.7721
    $ {v_{9,2}} $167.6546
    $ {v_{9,3}} $168.6460
    $ {v_{9,4}} $167.8276
    $ {v_{10,1}} $179.5391
    $ {v_{10,2}} $184.9333
    $ {v_{10,3}} $179.4192
    $ {v_{11,1}} $193.4721
    $ {v_{11,2}} $205.2436
    $ {v_{11,3}} $189.8764
    $ {v_{12,1}} $210.4280
    $ {v_{12,2}} $231.7092
    $ {v_{12,3}} $199.4705
    下载: 导出CSV

    A4  图6中各速度值

    A4.   Values of each speed in Fig. 6

    v/(m·s−1)Value
    $ {v_{13,2}} $169.9825
    $ {v_{13,3}} $168.5057
    $ {v_{14,2}} $172.7866
    $ {v_{14,3}} $171.1294
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-07-01
  • 录用日期:  2021-08-12
  • 网络出版日期:  2021-08-13
  • 刊出日期:  2021-09-18

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