# Two-parameter bifurcation analysis of limit cycles of a simplified railway wheelset model

- 99 Downloads

## Abstract

The effect of the nonlinear terms on bifurcation behaviors of limit cycles of a simplified railway wheelset model is investigated. At first, the stable equilibrium state loses its stability via a Hopf bifurcation. The bifurcation curve is divided into a supercritical branch and a subcritical one by a generalized Hopf point, which plays a key role in determining the occurrence of flange contact and derailment of high-speed railway vehicles, and the occurrence of this critical situation is an important decision-making criteria for design parameters. Secondly, bifurcations of limit cycles are discussed by comparing the bifurcation behavior of cycles for two different nonlinear parameters. Unlike local Hopf bifurcation analysis based on a single bifurcation parameter in most papers, global bifurcation analysis of limit cycles based on two bifurcation parameters is investigated, simultaneously. It is shown that changing nonlinear parameter terms can affect bifurcation types of cycles and division of parameter domains. In particular, near the branch points of cycles, two symmetrical limit cycles are created by a pitchfork bifurcation and then two symmetrical cycles both undergo a period-doubling bifurcation to form two stable period-two cycles. Around the resonant points, period orbits can make several turns, whose number of turns corresponds to the ratio of resonance. Thirdly, near the Neimark–Sacker bifurcation of cycles, a stable torus is created by a supercritical Neimark–Sacker bifurcation, which shows that the orbit of the model exhibits modulated oscillations with two frequencies near the limit cycle. These results demonstrate that nonlinear parameter terms can produce very complex global bifurcation phenomena and make obvious effects on possible hunting motions even though a simple railway wheelset model is concerned.

## Keywords

Wheelset Hunting motion Hopf bifurcation Period-doubling bifurcation Neimark–Sacker bifurcation## Notes

### Acknowledgements

This work is supported by the State Key Laboratory of Rail Traffic Control and Safety (No. RCS2017K002), Beijing Jiaotong University and the Natural Science Foundation of China (NSFC) under Project No. 11171017.

### Compliance with ethical standards

### Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

## References

- 1.Wickens, A.H.: Fundamentals of Rail Vehicle Dynamics: Guidance and Stability. Swets & Zeitlinger Publishers, Lisse (2003)CrossRefGoogle Scholar
- 2.Nath, Y., Jayadev, K.: Influence of yaw stiffness on the nonlinear dynamics of railway wheelset. Commun. Nonlinear Sci. Numer. Simul.
**10**(2), 179–190 (2005)CrossRefMATHGoogle Scholar - 3.Polach, O., Kaiser, I.: Comparison of methods analyzing bifurcation and hunting of complex rail vehicle models. J. Comput. Nonlinear Dyn.
**7**(10), 614–620 (2012)Google Scholar - 4.Jin, X.S., Wu, P.B., Wen, Z.F.: Effects of structure elastic deformations of wheelset and track on creep forces of wheel/rail in rolling contact. Wear
**253**(1–2), 247–256 (2002)CrossRefGoogle Scholar - 5.Koo, J.S., Choi, S.Y.: Theoretical development of a simplified wheelset model to evaluate collision-induced derailments of rolling stock. J. Sound Vib.
**331**(13), 3172–3198 (2012)CrossRefGoogle Scholar - 6.Ahmadian, M., Yang, S.P.: Hopf bifurcation and hunting behavior in a rail wheelset with flange contact. Nonlinear Dyn.
**15**(1), 15–30 (1997)CrossRefMATHGoogle Scholar - 7.Sedighi, H.M., Shirazi, K.H.: Bifurcation analysis in hunting dynamical behavior in a railway bogie: using novel exact equivalent functions for discontinuous nonlinearities. Sci. Iran.
**19**(6), 1493–1501 (2012)CrossRefGoogle Scholar - 8.Dong, H., Zeng, J., Xie, J.H., Jia, L.: Bifurcation instability forms of high speed railway vehicles. Sci. China Technol. Sci.
**56**(7), 1685–1696 (2013)CrossRefGoogle Scholar - 9.Zhang, T.T., Dai, H.Y.: Bifurcation analysis of high-speed railway wheel-set. Nonlinear Dyn.
**83**(3), 1511–1528 (2016)MathSciNetCrossRefMATHGoogle Scholar - 10.Yan, Y., Zeng, J.: Hopf bifurcation analysis of railway bogie. Nonlinear Dyn. (2017). https://doi.org/10.1007/s11071-017-3634-7 Google Scholar
- 11.Yabuno, H., Okamoto, T., Aoshima, N.: Stabilization control for the hunting motion of a railway wheelset. Veh. Syst. Dyn.
**35**, 41–55 (2001)Google Scholar - 12.True, H.: Dynamics of a rolling wheelset. Appl. Mech. Rev.
**46**, 438–444 (1993)CrossRefGoogle Scholar - 13.Wickens, A.H.: The dynamics stability of a simplified four-wheel railway vehicle having profiled wheels. Int. J. Solids Struct.
**1**, 385–406 (1965)CrossRefGoogle Scholar - 14.Xu, G., Troger, H., Steindl, A.: Global analysis of the loss of stability of a special railway body. In: Schiehlen, W. (ed.) Nonlinear Dynamics in Engineering Systems, pp. 345–352. Springer, Berlin (1990)CrossRefGoogle Scholar
- 15.Yabuno, H., Okamoto, T., Aoshima, N.: Effect of lateral linear stiffness on nonlinear characteristics of hunting motion of a railway wheelset. Meccanica
**37**(6), 555–568 (2002)CrossRefMATHGoogle Scholar - 16.Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, New York (2001)CrossRefMATHGoogle Scholar
- 17.Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004)CrossRefMATHGoogle Scholar
- 18.Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw.
**9**(2), 141–164 (2003)MathSciNetCrossRefMATHGoogle Scholar