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The “resultant bifurcation diagram” method and its application to bifurcation behaviors of a symmetric railway bogie system

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Abstract

The concept of symmetric bifurcation for a symmetric wheel-rail system is defined. After that, the time response of the system can be achieved by the numerical integration method, and an unfixed and dynamic Poincaré section and its symmetric section for the symmetric wheel-rail system are established. Then the ‘resultant bifurcation diagram’ method is constructed. The method is used to study the symmetric/asymmetric bifurcation behaviors and chaotic motions of a two-axle railway bogie running on an ideal straight and perfect track, and a variety of characteristics and dynamic processes can be obtained in the results. It is indicated that, for the possible sub-critical Hopf bifurcation in the railway bogie system, the stable stationary solutions and the stable periodic solutions coexist. When the speed is in the speed range of Hopf bifurcation point and saddle-node bifurcation point, the coexistence of multiple solutions can cause the oscillating amplitude change for different kinds of disturbance. Furthermore, it is found that there are symmetric motions for lower speeds, and then the system passes to the asymmetric ones for wide ranges of the speed, and returns again to the symmetric motions with narrow speed ranges. The rule of symmetry breaking in the system is through a blue sky catastrophe in the beginning.

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Acknowledgements

This research was supported by Opening Fund of State Key Laboratory of Traction Power, Southwest Jiaotong University (Grant No. TPL1106), and also supported by National Natural Science Foundation of China (Grant No. 11072204, 11102030, 10902092) and the Fundamental Research Funds for the Central Universities.

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Authors and Affiliations

  1. College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu, 610059, Sichuan, China

    Xue-Jun Gao

  2. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, Sichuan, China

    Ying-Hui Li & Yuan Yue

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  1. Xue-Jun Gao
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  2. Ying-Hui Li
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  3. Yuan Yue
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Correspondence to Xue-Jun Gao.

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Gao, XJ., Li, YH. & Yue, Y. The “resultant bifurcation diagram” method and its application to bifurcation behaviors of a symmetric railway bogie system. Nonlinear Dyn 70, 363–380 (2012). https://doi.org/10.1007/s11071-012-0460-9

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  • DOI: https://doi.org/10.1007/s11071-012-0460-9

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