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Nonlinear Dynamics

, Volume 93, Issue 2, pp 671–687 | Cite as

Study on primary resonance and bifurcation of a conductive circular plate rotating in air-magnetic fields

  • Y. D. Hu
  • W. Q. Li
Original Paper
  • 92 Downloads

Abstract

Based on the Kirchoff plate theory and Hamiltonian principle, the magneto-aeroelastic nonlinear governing equation for the forced vibration of the rotating conductive circular plate is derived. According to principles of electromagnetic field combined with a simplified aerodynamic model, the expressions of electromagnetic force and aerodynamic load of rotating circular plate are presented. The transverse nonlinear forced vibration differential equation of simply edge rotating circular plate is achieved by Galerkin method, where Bessel functions are utilized to a mode shape. Amplitude–frequency response equation of the system is obtained by using averaging. By numerical calculation, the amplitude–frequency curves of circular plate are plotted, and the influences of different parameters on amplitude–frequency characteristics of systems are analyzed, respectively. The dynamic behaviors of the system are investigated by means of bifurcation diagrams, maximum Lyapunov exponents and system responses under different controlling parameters.

Keywords

Conductive circular plate Rotating motion Magneto-aeroelasticity Primary resonance Bifurcation and chaos 

Notes

Acknowledgements

This project was supported by the National Natural Science Foundation of China (11472239) and Hebei Provincial Natural Science Foundation of China (No. A2015203023).

References

  1. 1.
    Chona, S., Jiang, Z.W., Shyu, Y.J.: Stability analysis of a 2” floppy disk drive system and the optimum design of the disk stabilizer. J. Vib. Acoust. 114(1), 283–286 (1992)CrossRefGoogle Scholar
  2. 2.
    Hoska, H.C., Randall, S.: Self-excited vibrations of a flexible disk rotating on an air film above a surface. Acta Mech. 3, 115–127 (1992)Google Scholar
  3. 3.
    Renshaw, A.A., Mote, C.D.: Absence of one nodal diameter critical speed modes in an axisymmetric rotating disk. J. Appl. Mech. 59, 687–688 (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Renshaw, A.A.: Critical speeds for floppy disks. J. Appl. Mech. 65(1), 116–120 (1998)CrossRefGoogle Scholar
  5. 5.
    Renshaw, A.A., D’Angelo, C., Mote, C.D.: Aerodynamically excited vibration of a rotating disk. J. Sound Vib. 177(5), 577–590 (1994)CrossRefzbMATHGoogle Scholar
  6. 6.
    Kim, H.R., Renshaw, A.A.: Aeroelastic flutter of circular rotating disks: a simple predictive model. J. Sound Vib 256(2), 227–248 (2002)CrossRefGoogle Scholar
  7. 7.
    Yasuda, K., Torii, T., Shimizu, T.: Self-excited oscillations of a circular disk rotating in air. JSME Int. J. 35(3), 347–352 (1992)Google Scholar
  8. 8.
    Hansen, M.H., Raman, A., Mote, C.D.: Estimation of nonconservative aero-dynamic pressure leading to flutter of spinning disks. J. Fluids Struct. 15(1), 39–57 (2005)CrossRefGoogle Scholar
  9. 9.
    Kim, B.C., Raman, A., Mote, C.D.: Prediction of aeroelastic flutter in a hard disk drive. J. Sound Vib. 238(2), 309–325 (2000)CrossRefGoogle Scholar
  10. 10.
    Wang, X.Z., Huang, X.Y.: A simple modeling and experiment on dynamic stability of a disk rotating in air. J. Struct. Stab. Dyn. 8(1), 41–60 (2008)CrossRefGoogle Scholar
  11. 11.
    Huang, X.Y., Wang, X., Yap, F.F.: Feedback control of rotating disk flutter in an enclosure. J. Fluids Struct. 19(4), 917–932 (2004)CrossRefGoogle Scholar
  12. 12.
    Touzé, C., Thomas, L.O., Chaigne, A.: Asymmetric non-linear forced vibrations of free-edge circular plates. Part 1: theory. J. Sound Vib. 258(4), 649–676 (2002)CrossRefGoogle Scholar
  13. 13.
    Touzé, C., Thomas, L.O., Chaigne, A.: Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: experiments. J. Sound Vib. 265, 1075–1101 (2003)CrossRefGoogle Scholar
  14. 14.
    Camier, C., Touzé, C., Thomas, O.: Non-linear vibrations of imperfect free-edge circular plates and shells. Euro. J. Mech. A-Solids 28, 500–515 (2009)CrossRefzbMATHGoogle Scholar
  15. 15.
    Touzé, C., Thomas, O., Amabili, M.: Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates. Int. J. Nonlinear Mech. 46, 234–246 (2011)CrossRefGoogle Scholar
  16. 16.
    Rad, A.B., Shariyat, M.: Three-dimensional magneto-elastic analysis of asymmetric variable thickness porous FGM circular plates with non-uniform tractions and Kerr elastic foundations. Compos. Struct. 125, 558–574 (2015)CrossRefGoogle Scholar
  17. 17.
    Arefi, M., Allam, M.N.M.: Nonlinear responses of an arbitrary FGP circular plate resting on the Winkler-Pasternak foundation. Smart Struct. Syst. 16(1), 81–100 (2015)CrossRefGoogle Scholar
  18. 18.
    Dai, T., Dai, H.L.: Investigation of mechanical behavior for a rotating FGM circular disk with a variable angular speed. J. Mech. Sci. Technol. 29(6), 3779–3787 (2015)CrossRefGoogle Scholar
  19. 19.
    Hu, Y.D., Wang, T.: Nonlinear resonance of a rotating circular plate under static loads in magnetic field. Chin. J. Mech. Eng. 28(3), 1277–1284 (2015)CrossRefGoogle Scholar
  20. 20.
    Hu, Y.D., Wang, T.: Nonlinear free vibration of a rotating circular plate under the static load in magnetic field. Nonlinear. Dyn. 85(3), 1825–1835 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Li, X.Y., Ding, H.J., Chen, W.Q.: Three-dimensional analytical solution for functionally graded magneto- electro-elastic circular plates subjected to uniform load. Compos. Struct. 83, 381–90 (2008)CrossRefGoogle Scholar
  22. 22.
    Dai, H.L., Dai, T., Yang, L.: Free vibration of a FGPM circular plate placed in a uniform magnetic field. Meccanic 48, 2339–2347 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hu, Y.D., Zhang, Z.Q.: Bifurcation and chaos of thin circular functionally graded plate in thermal environment. Chaos Solitons Fractals 44(6), 739–750 (2011)zbMATHGoogle Scholar
  24. 24.
    Hu, Y.D., Zhang, Z.Q.: The bifurcation analysis on the circular functionally graded plate with combination resonances. Nonlinear Dyn. 67, 1779–1790 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yeh, Y.L., Chen, C.K.: Chaotic and bifurcation for a simply supported thermo-mechanical coupling circular plate with variable thickness. Chaos Solitons Fractals 22, 1013–1030 (2004)CrossRefzbMATHGoogle Scholar
  26. 26.
    Korenev, B.G.: Bessel Functions and Their Applications. Taylor & Francis Inc, London and New York (2002)zbMATHGoogle Scholar
  27. 27.
    Nayfeh, A.H., Mok, D.T.: Nonlinear Oscillations. Wiley, New York (1995)CrossRefGoogle Scholar
  28. 28.
    ASME.: 2015 ASME boiler and pressure vessel code. Section II, Materials. American Society of Mechanical Engineers, New York (2015)Google Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Civil Engineering and MechanicsYanshan UniversityQinhuangdaoChina
  2. 2.Hebei Provincial Key Laboratory of Mechanical Reliability for Heavy Equipments and Large Structures of Hebei ProvinceYanshan UniversityQinhuangdaoChina

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