Chaos and bifurcation in nonlinear in-extensional rotating shafts

Document Type : Article


Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Mofatteh Avenue, Tehran, P.O. Box 15719-14911, Iran.


In this paper, bifurcation and chaotic behavior of in-extensional rotating shafts are investigated. The shaft is modeled as a Rayleigh simply supported beam, spinning with constant rotational speed. Using two–mode Galerkin truncation, the partial differential equations of motion are discretized and then with the aid of numerical simulations, the dynamical behavior of the rotating shaft is studied. Using tools from nonlinear dynamics, such as time history, bifurcation diagram, Poincaré map, Lyapunov exponents, and amplitude spectra, a comprehensive analysis is made to characterize the complex behavior of the rotating shaft. Periodic (synchronous), quasi-periodic, chaotic and transient chaotic responses are observed in the neighborhood of the second critical speed. The effect of rotary inertia and damping on the dynamics of the rotating shaft is considered. It is shown that the chaotic response is possible for a shaft with weak nonlinearity without the existence of any internal resonance.


Main Subjects

1.Ishida, Y. Review of research on nonlinear rotordynamics in Japan", Journal of System Design and Dynamics, 7, pp. 151-169 (2013).
2. Ehrich, F. Some observations of chaotic vibration phenomena in high-speed rotor dynamics", ASME Journal of Vibration and Acoustics, 113, pp. 50-57 (1991).
3. Muszynska, A. and Goldman, P. Chaotic responses of unbalanced rotor/bearing/stator systems with looseness or rub", Chaos, Solitons, and Fractals, 5, pp. 1683-1704 (1995). 4. Chu, F.L. and Zhang, Z. Periodic, quasi-periodic and chaotic vibrations of a rub-impact rotor system supported on oil _lm bearings", International Journal of Engineering Science, 35, pp. 963-973 (1997). 5. Chu, F.L. and Zhang, Z. Bifurcation and chaos in a rub-impact Je_cott rotor system", Journal of Sound and Vibration, 210, pp. 1-18 (1998). 6. Dai, X., Zhang, X., and Jin, X. The partial and full rubbing of a ywheel rotor-bearing-stop system", International Journal of Mechanical Science, 43, pp. 505-519 (2001). 7. Sun, Z., Xu, J., and Zhou, T. Analysis on complicated characteristics of a high-speed rotor system with rubimpact", Mechanism and Machine Theory, 37, pp. 659-672 (2002). 8. Luo, Y., Zhang, S., andWen, B. Dynamic characteristics of nonlinear elastics rotor system with rub-impact faults", Turbine Technology, 46, pp. 415-417 (2004). 9. Zhang, W.M. and Meng, G. Stability, bifurcation and chaos of a high-speed rub-impact rotor system", Sensors and Actuators A, 127, pp. 163-78 (2006). 10. Yuan, Z., Chu, F., and Hao, R. Simulation of rotor's axial rub-impact in full degrees of freedom", Mechanism and Machine Theory, 42, pp. 763-775 (2007). 11. Shen, X., Jia, J., and Zhao, M. Nonlinear analysis of a rub-impact rotor-bearing system with initial permanent rotor bow", Archive of Applied Mechanics, 78, pp. 225-240 (2008). 12. Khanlo, H.M., Ghayour, M., and Ziaei-Rad, S. Chaotic vibration analysis of rotating, exible, continuous shaft-disk system with a rub-impact between the disk and the stator", Commun Nonlinear Sci Numer Simulat, 16, pp. 566-582 (2011). 13. Patel, T.H., Zuo, M.J., and Zhao, X. Nonlinear lateral-torsional coupled motion of a rotor contacting a viscoelastically suspended stator", Nonlinear Dynamics, 69, pp. 325-339 (2012). 14. Wang, J., Zhou, J., Dong, D., Yan, B., and Huang, C. Nonlinear dynamic analysis of a rub-impact rotor supported by oil _lm bearings", Archive of Applied Mechanics, 83, pp. 413-430 (2013). 15. Varney, P. and Green, I. Nonlinear phenomena, bifurcations, and routes to chaos in an asymmetrically supported rotor-stator contact system", Journal of Sound and Vibration, 336, pp. 207-226 (2015). S.A.A. Hosseini/Scientia Iranica, Transactions B: Mechanical Engineering 26 (2019) 856{868 867 16. Xiang, L., Hu, A., Hou, L., et al. Nonlinear coupled dynamics of an asymmetric double-disc rotor-bearing system under rub-impact and oil-_lm forces", Applied Mathematical Modelling, 40, pp. 4505-4523 (2016). 17. Yang, Y., Xu, Y., Yang, Y., et al. Dynamics characteristics of a rotor-casing system subjected to axial load and radial rub", International Journal of Non-Linear Mechanics (2017) DOI: 10.1016/j.ijnonlinmec.2017.10.023 18. Jian, C-W.C. and Chen, C-K. Bifurcation and chaos analysis of a exible rotor supported by turbulent long journal bearings", Chaos, Solitons and Fractals, 34, pp. 1160-1179 (2007). 19. Jian, C-W.C. and Chen, C-K. Chaotic response and bifurcation analysis of a exible rotor supported by porous and non-porous bearings with nonlinear suspension", Nonlinear Analysis: Real World Applications, 10, pp. 1114-1138 (2009). 20. Wang, C.-C. Bifurcation and nonlinear analysis of a exible rotor supported by a relative short spherical gas bearing system", Communications in Nonlinear Science and Numerical Simulation, 15, pp. 2659-5571 (2010). 21. Ch_aveza, J.P. and Wiercigrocha, M. Bifurcation analysis of periodic orbits of a non-smooth Je_cott rotor Model", Communications in Nonlinear Science and Numerical Simulation, 18, pp. 2571-2580 (2013). 22. Dakel, M., Baguet, S., and Dufour, R. Nonlinear dynamics of a support-excited exible rotor with hydrodynamic journal bearings", Journal of Sound and Vibration, 333, pp. 2774-2799 (2014). 23. Yan, S., Dowell, E.H., and Lin, B., E_ects of nonlinear damping suspension on nonperiodic motions of a exible rotor in journal bearings", Nonlinear Dynamics, 78, pp. 1435-1450 (2014). 24. Chen, J.H. and Wang, C.C. Chaotic and dynamic analysis of a exible rotor supported by ultra short aero-lubricated bearing system", Journal of Applied Research and Technology, 13, pp. 328-341 (2015). 25. Przybylowicz, P.M., Starczewski, Z., and Korczak- Komorowski, P. Sensitivity of regions of irregular and chaotic vibrations of an asymmetric rotor supported on journal bearings to structural parameters", Acta Mechanica, 227, pp. 3101-3112 (2016). 26. Yamamoto, T. and Ishida, Y., Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications. Wiley-Interscience (2001). 27. Inoue, T. and Ishida, Y. Chaotic vibration and internal resonance phenomena in rotor systems", J. Vib. Acoust., 128, pp. 156-169 (2006). 28. Nagasaka, I., Liu, J., and Ishida Y. Forced vibrations of a very slender continuous rotor with geometrical nonlinearity (harmonic and subharmonic resonances)", J. Vib. Acoust., 132, 021004, 9 pages (2010). 29. Hosseini, S.A.A. and Khadem, S.E. Free vibrations analysis of a rotating shaft with nonlinearities in curvature and inertia", Mech. Mach. Theory, 44, pp. 272-288 (2009). 30. Hosseini, S.A.A. and Khadem, S.E. Combination resonances in a rotating shaft", Mech. Mach. theory, 44, pp. 1535-1547 (2009). 31. Khadem, S.E., Shahgholi, M., and Hosseini, S.A.A. Primary resonances of a nonlinear in-extensional rotating shaft", Mech. Mach. Theory, 45, pp. 1067-1081 (2010). 32. Khadem, S.E., Shahgholi, M., and Hosseini, S.A.A. Two-mode combination resonances of an in-extensional rotating shaft with large amplitude", Nonlinear Dynam., 65, pp. 217-233 (2011). 33. Nayfeh, A.H. and Pai, P.F., Linear and Nonlinear Structural Mechanics, Wiley-Interscience, New York (2004). 34. Argyris, J., Faust, G., and Haase, M., An Exploration of Chaos, North-Holland (1994). 35. Sadoudi, S., Tanougast, C., Azzaz, M.S., et al. Design and FPGA implementation of a wireless hyperchaotic communication system for secure real-time image transmission", EURASIP Journal on Image and Video Processing, 43, pp. 1-18 (2013).