Nonlinear Dynamics

, Volume 70, Issue 1, pp 363–380 | Cite as

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

  • Xue-Jun Gao
  • Ying-Hui Li
  • Yuan Yue
Original Paper


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.


Railway bogie The ‘resultant bifurcation diagram’ method Symmetry/asymmetry Bifurcation 



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.


  1. 1.
    Knothe, K., Bohm, F.: History of stability of railway and road vehicles. Veh. Syst. Dyn. 31(5), 283–323 (1999) CrossRefGoogle Scholar
  2. 2.
    Wickens, A.H.: Fundamentals of rail vehicle dynamics, guidance and stability. Adv. Eng. 6 (2003) Google Scholar
  3. 3.
    Zeng, J., Wu, P.B.: Stability analysis of high speed railway vehicles. JSME Int. J. Ser. C, Dyn. Control Robot. Des. Manuf. 47(2), 464–470 (2004) Google Scholar
  4. 4.
    True, H.: Dynamics of railway vehicles and rail/wheel contact. In: Dynamics Analysis of Vehicle Systems: Theoretical Foundations and Advanced Applications, Udine, Italy, pp. 75–128 (2007) Google Scholar
  5. 5.
    Cooperrider, N.K.: The hunting behavior of conventional railway trucks. J. Eng. Ind. 94, 752–762 (1972) CrossRefGoogle Scholar
  6. 6.
    Kaas-Petersen, C., True, H.: Periodic, biperiodic and chaotic dynamical behavior of railway vehicles. Veh. Syst. Dyn. 15(6), 208–221 (1986) CrossRefGoogle Scholar
  7. 7.
    True, H.: Railway vehicle chaos and asymmetric hunting. Veh. Syst. Dyn. 20(Supplement), 625–637 (1992) CrossRefGoogle Scholar
  8. 8.
    Jensen, C.N., Golubitsky, M., True, H.: Symmetry generic bifurcations, and mode interaction in nonlinear railway dynamics. Int. J. Bifurc. Chaos 9(7), 1321–1331 (1999) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Zeng, J.: Numerical computations of the hunting bifurcation and limit cycles for railway vehicle system. J. China Railway Soc 15(3), 13–18 (1996) Google Scholar
  10. 10.
    Ahmadian, M., Yang, S.P.: Effect of system nonlinearities on locomotive bogie hunting stability. Veh. Syst. Dyn. 29(6), 365–384 (1998) CrossRefGoogle Scholar
  11. 11.
    Ahmadian, M., Yang, S.P.: Hopf bifurcation and hunting behavior in a rail wheelset with flange contact. Nonlinear Dyn. 15(1), 15–30 (1998) MATHCrossRefGoogle Scholar
  12. 12.
    Yang, S.P., Shen, Y.J.: Bifurcations and Singularities in Systems with Hysteretic Nonlinearity. Science Press, Beijing (2003) Google Scholar
  13. 13.
    Ding, W.C., Xie, J.H., Wang, J.T.: Nonlinear analysis of hunting vibration of truck due to wheel-rail impact. J. Lanzhou Univ. Technol. 30(1), 45–49 (2004) Google Scholar
  14. 14.
    Hoffmann, M.: On the dynamics of European two-axle railway freight wagons. Nonlinear Dyn. 52, 301–311 (2008) MATHCrossRefGoogle Scholar
  15. 15.
    True, H.: On the critical speed of high-speed railway vehicles. In: Noise and Vibration on High-Speed Railways. FEUP, Porto, Portugal, pp. 149–166 (2008) Google Scholar
  16. 16.
    Gao, X.J., Li, Y.H., Gao, Q.: Hunting motion and bifurcation behavior of six-axle locomotive based on continuation method. J. Traffic Transp. Eng. 9(5), 32–36 (2009) Google Scholar
  17. 17.
    Gao, X.J., Li, Y.H., Gao, Q.: Lateral bifurcation behavior of a four-axle railway passenger car. J. Appl. Mech. 77(6), 1–8 (2010) CrossRefGoogle Scholar
  18. 18.
    Wolf, A., Swift, J.B., Swinney, H.L., et al.: Determining Lyapunov exponents from a time series. Physica D 16(3), 285–317 (1985) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Petersen, D.E., Hoffmann, M.: Curving dynamics of railway vehicles. Technical report, IMM, The Technical University of Denmark, Lyngby (2002) Google Scholar
  20. 20.
    Kalker, J.J.: A fast algorithm for the simplified theory of rolling contact. Veh. Syst. Dyn. 11(1), 1–13 (1982) CrossRefGoogle Scholar
  21. 21.
    Kalker, J.J.: On the rolling contact of two elastic bodies in the presence of dry friction. Doctoral Thesis, Delft, the Netherlands (1967) Google Scholar
  22. 22.
    Shen, Z.Y., Hedrick, J.K., Elkins, J.A.: A comparison of alternative creep force models for rail vehicle dynamic analysis. In: Proceedings of 8th IAVSD Symposium on Vehicle System Dynamics, Dynamics of Vehicles on Roads and Tracks, pp. 591–605. MIT Swets and Zeitlinger, Cambridge (1984) Google Scholar
  23. 23.
    Lee, S.Y., Cheng, Y.C.: Hunting stability analysis of high-speed railway vehicle trucks on tangent tracks. J. Sound Vib. 282, 881–898 (2005) CrossRefGoogle Scholar
  24. 24.
    Kaas-Petersen, C.: Chaos in a railway bogie. Acta Mech. 61(1–4), 89–107 (1986) MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yue, Y., Xie, J.H., Xu, H.D.: Symmetry of the Poincare map and its influence on bifurcations in a vibro-impact system. J. Sound Vib. 323(1–2), 292–312 (2009) CrossRefGoogle Scholar
  26. 26.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics, Analytical, Computational, and Experimental Methods. Wiley-VCH, Weinheim (2004) Google Scholar
  27. 27.
    Zhai, W.M.: Vehicle-Track Coupling Dynamics, 2nd edn. China Railway Publishing House, Beijing (2002) Google Scholar
  28. 28.
    Kaas-Petersen, C.: Computation, continuation, and bifurcation of torus solutions for dissipative maps and ordinary differential equations. Physica D 25(1–3), 288–306 (1987) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Brindley, J., Kaas-Petersen, C., Spence, A.: Path-following methods in bifurcation problems. Physica D 34(3), 456–461 (1989) MATHCrossRefGoogle Scholar
  30. 30.
    Doedel, E.J.: Auto-07P: continuation and bifurcation software for ordinary differential equations. California Institute of Technology (2008) Google Scholar
  31. 31.
    True, H., Jensen, J.C.: Parameter study of hunting and chaos in railway vehicles. In: Proceedings of the 13th IAVSD symposium, Chengdu, Sichuan, China, pp. 508–521 (1993) Google Scholar
  32. 32.
    True, H.: On the theory of nonlinear dynamics and its application in vehicle system dynamics. Veh. Syst. Dyn. 31(5), 393–421 (1999) CrossRefGoogle Scholar
  33. 33.
    Hoffmann, M.: Dynamics of European two-axle freight wagons. The Technical University of Denmark, Doctor Thesis (2006) Google Scholar
  34. 34.
    Schupp, G.: Bifurcation analysis of railway vehicles. Multibody Syst. Dyn. 15(1), 25–50 (2006) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Zboinski, K., Dusza, M.: Extended study of railway vehicle lateral stability in a curved track. Veh. Syst. Dyn. 49(5), 789–810 (2011) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.College of Environment and Civil EngineeringChengdu University of TechnologyChengduChina
  2. 2.School of Mechanics and EngineeringSouthwest Jiaotong UniversityChengduChina

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