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Research on stochastic stability and stochastic bifurcation of suspended wheelset

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Abstract

We studied the stochastic stability and bifurcation behavior for a suspended wheelset system in the presence of a Gauss white noise stochastic parametric excitation. First, the global stochastic stability was researched by judging the modality of the singular boundary. Then, the diffusion exponent, drift exponent and character value of the two boundaries were calculated. After getting the maximal Lyapunov exponent, the condition of D-bifurcation was obtained. By analyzing the shape and peaks of the stationary probability density function, the condition of stochastic P-bifurcation was also obtained. Finally, the numerical verification and the comparison between deterministic bifurcation and P-bifurcation were performed. The results show that the random excitation shifts the critical velocity to a lower value, and the stochastic system becomes more sensitive and more unstable. The stochastic parametric excitation can destroy the origin subcritical Hopf bifurcation in the deterministic system.

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References

  1. A. H. Wickens, The dynamics of railway vehicles from Stephenson to Carter, Proc. I. Mech. E. Part F, 212 (3) (1998) 209–217.

    Article  Google Scholar 

  2. K. Knothe and F. Bohm, History of stability of railway and road vehicles, Veh. Syst. Dyn., 31 (5) (1999) 283–323.

    Article  Google Scholar 

  3. J. J. Kalker, The computation of three-dimensional rolling contact with dry friction, International J. for Numerical Methods in Engineering, 14 (9) (2005) 1293–1307.

    Article  Google Scholar 

  4. G. Schupp, Bifurcation analysis of railway vehicles, Multibody System Dynamics, 15 (2006) 25–50.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. H. Huang et al., Carbody hunting investigation of a high speed passenger car, J. of Mechanical Science and Technology, 27 (8) (2013) 2283–2292.

    Article  Google Scholar 

  6. A. H. Wickens, Fundamentals of rail vehicle dynamics: Guidance and stability, Swets & Zeitlinger Publishers, Lisse (2003).

    Book  Google Scholar 

  7. C. Knudsen et al., Nonlinear dynamic phenomena in the behavior of a railway wheelset model, Nonlinear Dynamics, 2 (5) (1991) 389–404.

    Article  Google Scholar 

  8. H. True, Railway vehicle chaos and asymmetric hunting, Vehicle System Dynamics, 20 (1) (1992) 625–637.

    Article  Google Scholar 

  9. H. True, Dynamics of a rolling wheelset, Applied Mechanics Reviews, 46 (7) (1993) 438–444.

    Article  Google Scholar 

  10. J. Zeng, Numerical computations of the hunting bifurcation and limit cycles for railway vehicle system, China Railway Society, 18 (3) (1996) 13–19 (in Chinese).

    Google Scholar 

  11. K. W. Lv, Study on the nonlinear dynamic problems for railway vehicle systems, Ph.D. Thesis, Southwest Jiaotong University, China (2004).

    Google Scholar 

  12. K. Zboinski and M. Dusza, Self-exciting vibrations and Hopf’s bifurcation in non-linear stability analysis of rail vehicles in a curved track, European J. of Mechanics -A/Solids, 29 (2) (2010) 190–203.

    Article  Google Scholar 

  13. O. Polach, Comparability of the non-linear and linearized stability assessment during railway vehicle design, Vehicle System Dynamics, 44 (1) (2006) 129–138.

    Article  Google Scholar 

  14. O. Polach, On non-linear methods of bogie stability assessment using computer simulations, Proceedings of the Institution of Mechanical Engineers, Part F: J. of Rail and Rapid Transit, 220 (1) (2006) 13–27.

    Article  Google Scholar 

  15. H. Dong et al., Bifurcationinstability forms of high speed railway vehicles, Science China Technological Sciences, 56 (7) (2013) 1685–1696.

    Article  Google Scholar 

  16. L. Arnold, Random dynamical systems, Springer, New York (1998).

    Book  MATH  Google Scholar 

  17. S. T. Ariaratnam, Bifurcations in nonlinear stochastic systems, New Approaches to Nonlinear Problems in Dynamics, Edited by P. J. Holmes, SIAM, Philadelphia, PA (1980) 470–474.

    Google Scholar 

  18. Y. K. Lin and W. F. Wu, Applications of cumulant closure to random vibration problems, ASME Random Vibrations, AMD, 65 (1) (1983) 113–125.

    Google Scholar 

  19. N. S. Namachchivaya, Hopf bifurcation in the presence of both parametric and external stochastic excitations, J. of Applied Mechanics, 55 (4) (1988) 923–930.

    Article  MATH  Google Scholar 

  20. W. Q. Zhu, Stochastic averaging of the energy envelope of nearly Lyapunov systems, K. Hennig (Ed.), Random Vibrations and Reliability, Proceedings of the IUTAM Symposium, Akademie-Verlag, Berlin (1983) 347–357.

    Google Scholar 

  21. W. Q. Zhu and Y. K. Lin, Stochastic averaging of energy envelope, ASCE J. of Engineering Mechanics, 117 (8) (1991) 1890–1905.

    Article  Google Scholar 

  22. R. Z. Khasminskii, On the averaging principle for stochastic differential Ito equation, Kibernetika(Prague), 4 (1968) 260–279 (in Russian).

    Google Scholar 

  23. L. Arnold, N. S. Namachchivaya and K. R. Schenk-Hoppe, Toward an understanding of stochastic Hopf bifurcation: A case study, International J. of Bifurcation and Chaos, 6 (11) (1996) 1947–1975.

    Article  Google Scholar 

  24. U. V. Wagner, Nonlinear dynamic behaviour of a railway wheelset, Vehicle System Dynamics, 47 (5) (2009) 627–640.

    Article  Google Scholar 

  25. W. Q. Zhu and Z. L. Huang, Stochastic stability of quasinon-integrable Hamiltonian systems, J. of Sound and Vibration, 218 (5) (1998) 769–789.

    Article  MATH  Google Scholar 

  26. W. Q. Zhu and Z. L. Huang, Stochastic Hopf bifurcation of quasi-non-integrable Hamiltonian systems, International J. of Non-Linear Mechanics, 34 (3) (1999) 437–447.

    Article  MathSciNet  MATH  Google Scholar 

  27. W. Feller, Diffusion process in one dimension, Transaction of American Mathematical Society, 77 (1954) 1–31.

    Article  MathSciNet  MATH  Google Scholar 

  28. C. Chiarella et al., The stochastic bifurcation behaviour of speculative financial markets, Physica A: Applications of Physics in Financial Analysis, 387 (15) (2008) 3837–3846.

    Article  MathSciNet  Google Scholar 

  29. G. Ge, Research of stochastic bifurcation and reliability on rectangular thin plate vibration system, Ph. D. Thesis, Tianjin University, China (2009).

    Google Scholar 

  30. X. B. Liu, On the bifurcation behaviours and the variational methods of stochastic mechanical systems, Ph.D. Thesis, Southwest Jiaotong University, China (1995).

    Google Scholar 

  31. K. M. Liew and X. B. Liu, The maximal Lyapunov exponent for a three dimensional stochastic system, Transactions of the ASME, 71 (5) (2004) 677–690.

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Wolf et al., Determining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 16 (3) (1985) 285–317.

    Article  MathSciNet  MATH  Google Scholar 

  33. C. Meunier and A. D. Verga, Noise and bifurcation, J. of Statistical Physics, 50 (1–2) (1988) 345–375.

    Article  MathSciNet  MATH  Google Scholar 

  34. W. V. Wedig, On the optimization of non-negative density solutions of stationary Fokker-Planck equations, PAMM · Proc. Appl. Math. Mech., 4 (2004) 23–26.

    Article  Google Scholar 

  35. H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. of Dynamics and Differential Equations, 10 (2) (1998) 259–274.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Traction Power State Key Laboratory, Southwest Jiaotong University, Chengdu, Sichuan, 0086 610031, China

    Bo Zhang & Jing Zeng

  2. Department of Mechanical Engineering, Emei Campus of Southwest Jiaotong University, Emei, Sichuan, 0086 614202, China

    Weiwei Liu

Authors
  1. Bo Zhang
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  2. Jing Zeng
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  3. Weiwei Liu
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Correspondence to Bo Zhang.

Additional information

Recommended by Associate Editor Ohseop Song

Bo Zhang is a doctoral student in Southwest Jiaotong University, Chengdu, China, in Vehicle Operation Engineering. His research interests are in the area of railway vehicle stochastic dynamics.

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Zhang, B., Zeng, J. & Liu, W. Research on stochastic stability and stochastic bifurcation of suspended wheelset. J Mech Sci Technol 29, 3097–3107 (2015). https://doi.org/10.1007/s12206-015-0708-7

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  • DOI: https://doi.org/10.1007/s12206-015-0708-7

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