Skip to main content
SpringerLink
Log in

Global dynamics of a flexible asymmetrical rotor

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential governing equations of the rotor system. The equations are then transformed into a discretized nonlinear gyroscopic system via Galerkin’s method. The canonical transformation and normal form theory are applied to reduce the system to the near-integrable Hamiltonian standard forms considering zero-to-one internal resonance. The energy-phase method is employed to study the chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. In both the Hamiltonian and the dissipative perturbation, the homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated. In the case of damping dissipative perturbation, the existence of generalized Šilnikov-type multi-pulse orbits which are homoclinic to fixed points on the slow manifold is examined and the parameter region for which the dynamical system may exhibit chaotic motions in the sense of Smale horseshoes is obtained analytically. The global results are finally interpreted in terms of the physical motion of axis orbit. The present study indicates that the existence of multi-pulse homoclinic orbits provides a mechanism for how energy may flow from the high-frequency mode to the low-frequency mode when the rotor system operates near the first-order critical speed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  1. Genta, G.: Dynamics of Rotating Systems. Springer, New York (2004)

    Google Scholar 

  2. Friswell, M.L., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Dynamics of Rotating Machines. Cambridge University Press, New York (2010)

    Book  MATH  Google Scholar 

  3. Yamamoto, T., Ishida, Y.: Linear and Nonlinear Rotordynamics: A Modern Treatment with Applications. Wiley-VCH, Weinheim (2012)

    Google Scholar 

  4. Yamamoto, T.: On the vibrations of a shaft supported by bearings having radial clearances. Trans. Jpn. Soc. Mech. Eng. 21, 186–192 (1955)

    Article  Google Scholar 

  5. Kim, Y.B., Noah, S.T.: Bifurcation analysis for a modified Jeffcott rotor with bearing clearances. Nonlinear Dyn. 1, 221–241 (1990)

    Article  Google Scholar 

  6. Choi, S.K., Noah, S.T.: Mode-locking and chaos in a Jeffcott rotor with bearing clearances. J. Appl. Mech. 61, 131–138 (1994)

    Article  MATH  Google Scholar 

  7. Adiletta, G., Guido, A.R., Rossi, C.: Chaotic motions of a rigid rotor in short journal bearings. Nonlinear Dyn. 10, 251–269 (1996)

    Article  Google Scholar 

  8. Karpenko, E.V., Wiercigroch, M., Cartmell, M.P.: Regular and chaotic dynamics of a discontinuously nonlinear rotor system. Chaos Solitons Fractals 13, 1231–1242 (2002)

    Article  MATH  Google Scholar 

  9. Pavlovskaia, E.E., Karpenko, E.V., Wiercigroch, M.: Non-linear dynamic interactions of a Jeffcott rotor with preloaded snubber ring. J. Sound Vib. 276, 361–379 (2004)

    Article  Google Scholar 

  10. Chávez, J.P., Wiercigroch, M.: Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model. Comm. Nonlinear Sci. Numer. Simul. 18, 2571–2580 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chávez, J.P., Hamaneh, V.V., Wiercigroch, M.: Modelling and experimental verification of an asymmetric Jeffcott rotor with radial clearance. J. Sound Vib. 334, 86–97 (2015)

    Article  Google Scholar 

  12. Chen, C.L., Yau, H.T.: Chaos in the imbalance response of a flexible rotor supported by oil film bearings with non-linear suspension. Nonlinear Dyn. 16, 71–90 (1998)

    Article  MATH  Google Scholar 

  13. Chang-Jian, C.W., Chen, C.K.: Nonlinear dynamic analysis of a flexible rotor supported by micropolar fluid film journal bearings. Int. J. Eng. Sci. 44, 1050–1070 (2006)

    Article  MATH  Google Scholar 

  14. Chang-Jian, C.W., Chen, C.K.: Chaos and bifurcation of a flexible rotor supported by porous squeeze couple stress fluid film journal bearings with non-linear suspension. Chaos Solitons Fractals 35, 358–375 (2008)

    Article  Google Scholar 

  15. Chang-Jian, C.W., Chen, C.K.: Nonlinear analysis of a rub-impact rotor supported by turbulent couple stress fluid film journal bearings under quadratic damping. Nonlinear Dyn. 56, 297–314 (2009)

    Article  MATH  Google Scholar 

  16. Laha, S.K., Kakoty, S.K.: Non-linear dynamics analysis of a flexible rotor supported on porous oil journal bearings. Commun. Nonlinear Sci. Numer. Simul. 15, 1617–1631 (2011)

    Article  Google Scholar 

  17. Shaw, S.W.: Chaotic dynamics of a slender beam rotating about its longitudinal axis. J. Sound Vib. 124, 329–343 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shaw, J., Shaw, S.W.: Non-linear resonance of an unbalanced rotating shaft with internal damping. J. Sound Vib. 147, 435–451 (1991)

    Article  Google Scholar 

  19. Ehrich, F.F.: Some observations of chaotic phenomena in high speed rotordynamics. J. Vib. Acoust. 113, 50–57 (1991)

    Article  Google Scholar 

  20. Ishida, Y., Inoue, T.: Internal resonance phenomena of the Jeffcott rotor with nonlinear spring characteristics. J. Vib. Acoust. 126, 476–484 (2004)

    Article  Google Scholar 

  21. Yabuno, H., Kashimura, T., Inoue, T., Ishida, Y.: Nonlinear normal modes and primary resonance of horizontally supported Jeffcott rotor. Nonlinear Dyn. 66, 377–387 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yu, T.J., Zhou, S., Zhang, W., Yang, X.D.: Multi-pulse chaotic dynamics of an unbalanced Jeffcott rotor with gravity effect. Nonlinear Dyn. 87, 647–664 (2017)

    Article  MATH  Google Scholar 

  23. Zhang, W.M., Meng, G.: Stability, bifurcation and chaos of a high-speed rub-impact rotor system in MEMS. Sens. Actuators A 127, 163–178 (2006)

    Article  Google Scholar 

  24. Zhang, W.M., Meng, G., Chen, D., Zhou, J.B., Chen, J.Y.: Nonlinear dynamics of a rub-impact micro-rotor system with scale-dependent friction model. J. Sound Vib. 309, 756–777 (2008)

    Article  Google Scholar 

  25. Ghommem, M., Nayfeh, A.H., Choura, S., Najar, F., Abdel-Rahman, E.M.: Modeling and performance study of a beam microgyroscope. J. Sound Vib. 329, 4970–4979 (2010)

    Article  Google Scholar 

  26. Nayfeh, A.H., Abdel-Rahman, E.M., Ghommem, M.: A novel differential frequency micro-gyroscope. J. Vib. Control 21, 872–882 (2015)

    Article  MathSciNet  Google Scholar 

  27. Ghayesh, M.H., Farokhi, H., Alici, G.: Size-dependent performance of microgyroscopes. Int. J. Eng. Sci. 100, 99–111 (2016)

    Article  MathSciNet  Google Scholar 

  28. Shaw, S.W., Pierre, C.: Normal modes for non-linear vibratory systems. J. Sound Vib. 164, 85–124 (1993)

    Article  MATH  Google Scholar 

  29. Shaw, S.W., Pierre, C.: Normal modes of vibration for non-linear continuous systems. J. Sound Vib. 169, 319–347 (1994)

    Article  MATH  Google Scholar 

  30. Pesheck, E., Pierre, C., Shaw, S.W.: Modal reduction of a nonlinear rotating beam through nonlinear normal modes. J. Vib. Acoust. 124, 229–236 (2002)

    Article  Google Scholar 

  31. Legrand, M., Jiang, D.Y., Pierre, C., Shaw, S.W.: Nonlinear normal modes of a rotating shaft based on the invariant manifold method. Int. J. Rotating Mach. 10, 319–335 (2004)

    Article  Google Scholar 

  32. Hosseini, S.A.A., Khadem, S.E.: Combination resonances in a rotating shaft. Mech. Mach. Theory 44, 1535–1547 (2009)

    Article  MATH  Google Scholar 

  33. Khadem, S.E., Shahgholi, M., Hosseini, S.A.A.: Primary resonances of a nonlinear in-extensional rotating shaft. Mech. Mach. Theory 45, 1067–1081 (2010)

    Article  MATH  Google Scholar 

  34. Khadem, S.E., Shahgholi, M., Hosseini, S.A.A.: Two-mode combination resonances of an in-extensional rotating shaft with large amplitude. Nonlinear Dyn. 65, 217–233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shahgholi, M., Khadem, S.E.: Primary and parametric resonances of asymmetrical rotating shafts with stretching nonlinearity. Mech. Mach. Theory 51, 131–144 (2012)

    Article  Google Scholar 

  36. Shahgholi, M., Khadem, S.E.: Internal, combinational and sub-harmonic resonances of a nonlinear asymmetrical rotating shaft. Nonlinear Dyn. 79, 173–184 (2015)

    Article  Google Scholar 

  37. Han, Q.K., Chu, F.L.: Dynamic response of cracked rotor-bearing system under time-dependent base movements. J. Sound Vib. 332, 6847–6870 (2013)

    Article  Google Scholar 

  38. Han, Q.K., Chu, F.L.: Dynamic behaviors of a geared rotor system under time-periodic base angular motions. Mech. Mach. Theory 78, 1–14 (2014)

    Article  Google Scholar 

  39. Han, Q.K., Chu, F.L.: Parametric instability of flexible rotor-bearing system under time-periodic base angular motions. Nonlinear Dyn. 39, 4511–4522 (2015)

    MathSciNet  Google Scholar 

  40. Awrejcewicz, J., Krysko, V.A., Papkova, I.V., Krysko, A.V.: Routes to chaos in continuous mechanical systems. Part 1: mathematical models and solution methods. Chaos Solitons Fractals 45, 687–708 (2012)

    Article  MATH  Google Scholar 

  41. Krysko, A.V., Awrejcewicz, J., Papkova, I.V., Krysko, V.A.: Routes to chaos in continuous mechanical systems: part 2. Modelling transitions from regular to chaotic dynamics. Chaos Solitons Fractals 45, 709–720 (2012)

    Article  MATH  Google Scholar 

  42. Awrejcewicz, J., Krysko, A.V., Papkova, I.V., Krysko, V.A.: Routes to chaos in continuous mechanical systems. Part 3: the Lyapunov exponents, hyper, hyper-hyper and spatial–temporal chaos. Chaos Solitons Fractals 45, 721–736 (2012)

    Article  MATH  Google Scholar 

  43. Haller, G., Wiggins, S.: N-pulse homoclinic orbits in perturbations of resonant Hamiltonian systems. Arch. Ration. Mech. Anal. 130, 25–101 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  44. Haller, G., Wiggins, S.: Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schrödinger equation. Physica D 85, 311–347 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  45. Haller, G.: Chaos Near Resonance. Springer, New York (1999)

    Book  MATH  Google Scholar 

  46. Malhotra, N., Namachchivaya, N.S., McDonald, R.J.: Multi-pulse orbits in the motion of flexible spinning discs. J. Nonlinear Sci. 12, 1–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. McDonald, R.J., Namachchivaya, N.S.: Pipes conveying pulsating fluid near a zero to one resonance: global bifurcations. J. Fluids Struct. 21, 665–687 (2005)

    Article  Google Scholar 

  48. Yu, W.Q., Chen, F.Q.: Global bifurcations and chaos in externally excited cyclic system. Commun. Nonlinear Sci. Numer. Simul. 15, 4007–4019 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  49. Zhou, S., Yu, T.J., Yang, X.D., Zhang, W.: Global dynamics of pipes conveying pulsating fluid in the supercritical regime. Int. J. Appl. Mech. 9, 1750029 (2017)

    Article  Google Scholar 

  50. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Localization of hidden Chua’s attractors. Phys. Lett. A. 375, 2230–2233 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Leonov, G.A., Kuznetsov, N.V., Vagaitsev, V.I.: Hidden attractor in smooth Chua systems. Physica D 241, 1482–1486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Sprott, J.C., Wang, X., Chen, G.R.: Coexistence of point, periodic and strange attractors. Int. J. Bifurc. Chaos 23, 1350093 (2013)

    Article  MathSciNet  Google Scholar 

  53. Wei, Z.C., Sprott, J.C., Chen, H.: Elementary quadratic chaotic flows with a single non-hyperbolic equilibrium. Phys. Lett. A 379, 2184–2187 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wei, Z.C., Pham, V.T., Kapitaniak, T., Wang, Z.: Bifurcation analysis and circuit realization for multiple-delayed Wang–Chen system with hidden chaotic attractors. Nonlinear Dyn. 85, 1635–1650 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  55. Nayfeh, A.H., Pai, P.F.: Linear and Nonlinear Structural Mechanics. Wiley-VCH, Weinheim (2004)

    Book  MATH  Google Scholar 

  56. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)

    MATH  Google Scholar 

  57. Yang, X.D., Yang, S., Qian, Y.J., Zhang, W., Melnik, R.V.N.: Modal analysis of the gyroscopic continua: comparison of continuous and discretized models. J. Appl. Mech. 83, 084502 (2016)

    Article  Google Scholar 

  58. Yang, X.D., An, H.Z., Qian, Y.J., Zhang, W., Yao, M.H.: Elliptic motions and control of rotors suspending in active magnetic bearings. J. Comput. Nonlinear Dyn. 11, 054503 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge that the work is supported by National Natural Science Foundation of China (Grant Nos. 11672007, 11290152), Beijing Municipal Natural Science Foundation (Project No. 3172003) and the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

Author information

Authors and Affiliations

  1. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Tian-Jun Yu, Sha Zhou, Xiao-Dong Yang & Wei Zhang

Authors
  1. Tian-Jun Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sha Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-Dong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiao-Dong Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, TJ., Zhou, S., Yang, XD. et al. Global dynamics of a flexible asymmetrical rotor. Nonlinear Dyn 91, 1041–1060 (2018). https://doi.org/10.1007/s11071-017-3927-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3927-x

Keywords

Search

Navigation