Nonlinear Dynamics

, Volume 79, Issue 3, pp 2225–2235 | Cite as

Bifurcation and chaos analysis of spur gear pair in two-parameter plane

Original Paper


A developed algorithm is designed based on the simple cell mapping method and escape time algorithm to examine every state cell and the dynamic characteristics of the multi-parameter coupling in torsion-vibration gear system. Two different types of bifurcation caused by the intersection of the period-doubling bifurcation curves are researched by analyzing the distribution map and the bifurcation diagram of system’s dynamic characteristic in the parameter plane, \(\omega -F\). The occurrence processes of periodic bubbles and saltatory periodic bifurcation are studied. The stationary solution and its phase trajectory of the fractal of the periodic motion attractor boundary are researched too. The sufficient condition of the fractal structure of the periodic attractor domain is achieved. Homoclinic or heteroclinic trajectory in phase space is found caused by the intersection of the different periodic motion trajectories in non-smooth system.


Gear Bifurcation Fractal Periodic bubbles  Saltatory periodic bifurcation 



This investigation was financially supported by the National Natural Science Foundation of China (Grant No. 51365025, 11462012), by Innovative Research Group Foundation of Gansu Province of China (1308RJIA006) and by Research Fund for the Doctoral Program of Higher Education of Chana (20126204110001).


  1. 1.
    Kahraman, A., Singh, R.: Nonlinear dynamics of a spur gear pair. J. Sound Vib. 142, 49–75 (1990)CrossRefGoogle Scholar
  2. 2.
    Kahraman, A., Singh, R.: Interactions between time-varying mesh stiffness and clearance nonlinearities in a geared system. J. Sound Vib. 146, 135–156 (1991)CrossRefGoogle Scholar
  3. 3.
    Vaishya, M., Singh, R.: Analysis of periodically varying gear mesh systems with coulomb friction using Floquet theory. J. Sound Vib. 243, 525–545 (2001)Google Scholar
  4. 4.
    Theodossiades, S., Natsiavas, S.: Non-linear dynamics of gear-pair systems with periodic stiffness and backlash. J. Sound Vib. 229, 287–310 (2000)CrossRefGoogle Scholar
  5. 5.
    Ma, Q., Kahraman, A.: Subharmonic resonances of a mechanical oscillator with periodically time-varying, piecewise-nonlinear stiffness. J. Sound Vib. 294, 624–636 (2005)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Litak, G., Friswell, M.I.: Vibrations in gear systems. Chaos Solitions Fractals 16, 795–800 (2003)CrossRefMATHGoogle Scholar
  7. 7.
    Litak, G., Friswell, M.I.: Dynamics of a gear system with faults in meshing stiffness. Nonlinear Dyn. 41, 415–421 (2005)CrossRefMATHGoogle Scholar
  8. 8.
    Al-shyyab, A., Kahraman, A.: Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method: period-one motions. J. Sound Vib. 284, 151–172 (2005)Google Scholar
  9. 9.
    Al-shyyab, A., Kahraman, A.: Non-linear dynamic analysis of a multi-mesh gear train using multi-term harmonic balance method: sub-harmonic motions. J. Sound Vib. 279, 417–451 (2005)Google Scholar
  10. 10.
    Giorgio, B., Francesco, P.: Non-smooth dynamics of spur gears with manufacturing errors. J. Sound Vib. 306, 271–283 (2007)Google Scholar
  11. 11.
    Chang-Jian, C.W., Chen, S.M.: Bifurcation and chaos analysis of spur gear pair with and without nonlinear suspension. Nonlinear Anal. Real World Appl. 12, 979–989 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Chang-Jian, C.W.: Strong nonlinearity analysis for gear-bearing system under nonlinear suspension—bifurcation and chaos. Nonlinear Anal. Real World Appl. 11, 1760–1774 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Farshidianfar, A., Saghafi, A.: Global bifurcation and chaos analysis in nonlinear vibration of spur gear systems. Nonlinear Dyn. 75, 783–806 (2014)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Souza, D., Viana, R.L.: Noise-induced basin hopping in a gearbox model. Chaos Solitions Fractals 26, 1523–1531 (2005)CrossRefMATHGoogle Scholar
  15. 15.
    Souza, D.: Sudden changes in chaotic attractors and transient basins in a model for rattling in gearboxes. Chaos Solitions Fractals 21, 763–772 (2004)CrossRefMATHGoogle Scholar
  16. 16.
    Mason, J.F., Homer, M.E., Wilson, R.E.: Mathematical models of gear rattle in Roots blower vacuum pumps. J. Sound Vib. 308, 431–440 (2007)CrossRefGoogle Scholar
  17. 17.
    Mason, J.F., Piiroinen, P.T., Wilson, R.E., et al.: Basins of attraction in nonsmooth models of gear rattle. Int. J. Bifurcat. Chaos. 19, 203–224 (2009)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Mason, J.F., Piiroinen, P.T.: The effect of codimension-two bifurcations on the global dynamics of a gear model. SIAM J. Appl. Dyn. Sys. 8, 1694–1711 (2009)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Liu, H.X., Wang, S.M., Guo, J.S., et al.: Solution domain boundary analysis method and its application in parameter spaces of nonlinear gear system. Chin. J. Mech. Eng. Engl. Ed. 24, 507–513 (2011)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Peng, M.: Rich dynamics of discrete delay ecological models. Chaos Solitions Fractals 24, 1279–1285 (2005)CrossRefMATHGoogle Scholar
  21. 21.
    Yang, X., Peng, M., Hu, J., et al.: Bubbling phenomenon in a discrete economic model for the interaction of demand and supply. Chaos Solitions Fractals 42, 1428–1438 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Peng, M.: Multiple bifurcations and periodic bubbling in a delay population model. Chaos Solitions Fractals 25, 1123–1130 (2005)CrossRefMATHGoogle Scholar
  23. 23.
    Grebogi, C., Ott, E., Yorke, J.A.: Basin boundary metamorphoses: changes in accessible boundary orbits. Physica D 24, 243–262 (1987)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.School of Mechanical EngineeringLanzhou Jiaotong UniversityLanzhouChina

Personalised recommendations