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Acta Mechanica Sinica

, Volume 35, Issue 1, pp 242–263 | Cite as

Instability inspection of parametric vibrating rectangular Mindlin plates lying on Winkler foundations under periodic loading of moving masses

  • E. Torkan
  • M. PirmoradianEmail author
  • M. Hashemian
Research Paper
  • 35 Downloads

Abstract

Parametric resonance is one of the most important issues in the study of dynamical behavior of structures. In this paper, dynamic instability of a moderately thick rectangular plate on an elastic foundation is investigated in the case of parametric and external resonances due to periodic passage of moving masses. The governing coupled partial differential equations (PDEs) of the system, with consideration of the first-order shear deformation theory (FSDT) or Mindlin plate theory, are presented and they are reduced to a set of ordinary differential equations (ODEs) with time-dependent coefficients using the Galerkin procedure. All inertial components of the moving masses are adopted in the dynamical formulation. Instability survey is carried out for three different loading trajectories considerably interested in many practical applications of the issue, i.e. rectilinear, diagonal and orbiting trajectories. In order to analyze the resonance conditions, the incremental harmonic balance (IHB) method is introduced to calculate instability boundaries, as well as external resonance curves in parameters plane. A comprehensive study is done to assess effects of thickness ratio and foundation stiffness on the resonance conditions. It is found that an increase in the plate’s thickness ratio leads to a reduction in values of critical parameters. Moreover, it is observed that increasing the foundation stiffness moves the instability regions and resonance curves to higher frequencies of the moving masses and also leads to further stability of the parametrically excited system at lower frequencies. Time response simulations done via Runge–Kutta method confirmed the results predicted by IHB method.

Keywords

Mass-plate interaction Mindlin plate Parametric resonance External resonance Incremental harmonic balance method 

References

  1. 1.
    Nikkhoo, A., Rofooei, F.R., Shadnam, M.R.: Dynamic behavior and modal control of beams under moving mass. J. Sound Vib. 306, 712–724 (2007)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Yang, Y., Ding, H., Chen, L.Q.: Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation. Acta Mech. Sin. 29, 718–727 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Rao, G.V.: Linear dynamics of an elastic beam under moving loads. J. Vib. Acoust. 122, 281–289 (2000)CrossRefGoogle Scholar
  4. 4.
    Pirmoradian, M., Keshmiri, M., Karimpour, H.: Instability and resonance analysis of a beam subjected to moving mass loading via incremental harmonic balance method. J. Vibroengineering 16, 2779–2789 (2014)Google Scholar
  5. 5.
    Pirmoradian, M., Keshmiri, M., Karimpour, H.: On the parametric excitation of a Timoshenko beam due to intermittent passage of moving masses: instability and resonance analysis. Acta Mech. 226, 1241–1253 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Shadnam, M.R., Mofid, M., Akin, J.E.: On the dynamic response of rectangular plate, with moving mass. Thin-Walled Struct. 39, 797–806 (2001)CrossRefGoogle Scholar
  7. 7.
    Gbadeyan, J.A., Dada, M.S.: Dynamic response of a Mindlin elastic rectangular plate under a distributed moving mass. Int. J. Mech. Sci. 48, 323–340 (2006)CrossRefzbMATHGoogle Scholar
  8. 8.
    Wu, J.J.: Vibration analyses of an inclined flat plate subjected to moving loads. J. Sound Vib. 299, 373–387 (2007)CrossRefGoogle Scholar
  9. 9.
    Kiani, K., Nikkhoo, A., Mehri, B.: Assessing dynamic response of multispan viscoelastic thin beams under a moving mass via generalized moving least square method. Acta Mech. Sin. 26, 721–733 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Nikkhoo, A., Rofooei, F.R.: Parametric study of the dynamic response of thin rectangular plates traversed by a moving mass. Acta Mech. 223, 15–27 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Rofooei, F.R., Enshaeian, A., Nikkhoo, A.: Dynamic response of geometrically nonlinear, elastic rectangular plates under a moving mass loading by inclusion of all inertial components. J. Sound Vib. 394, 497–514 (2017)CrossRefGoogle Scholar
  12. 12.
    Amiri, J.V., Nikkhoo, A., Davoodi, M.R., et al.: Vibration analysis of a Mindlin elastic plate under a moving mass excitation by eigenfunction expansion method. Thin-Walled Struct. 62, 53–64 (2013)CrossRefGoogle Scholar
  13. 13.
    Nikkhoo, A., Hassanabadi, M.E., Azam, S.E., et al.: Vibration of a thin rectangular plate subjected to series of moving inertial loads. Mech. Res. Commun. 55, 105–113 (2014)CrossRefGoogle Scholar
  14. 14.
    Esen, I.: A new finite element for transverse vibration of rectangular thin plates under a moving mass. Finite Elem. Anal. Des. 66, 26–35 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Esen, I.: A new FEM procedure for transverse and longitudinal vibration analysis of thin rectangular plates subjected to a variable velocity moving load along an arbitrary trajectory. Lat. Am. J. Solids Struct. 12, 808–830 (2015)CrossRefGoogle Scholar
  16. 16.
    Ghazvini, T., Nikkhoo, A., Allahyari, H., et al.: Dynamic response analysis of a thin rectangular plate of varying thickness to a traveling inertial load. J. Braz. Soc. Mech. Sci. Eng. 38, 403–411 (2016)CrossRefGoogle Scholar
  17. 17.
    Frýba, L.: Vibration of Solids and Structures Under Moving Loads. Thomas Telford House, London (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ouyang, H.: Moving-load dynamic problems: a tutorial (with a brief overview). Mech. Syst. Signal Process. 25, 2039–2060 (2011)CrossRefGoogle Scholar
  19. 19.
    Fang, F., Xia, G., Wang, J.: Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations. Acta Mech. Sin. 34, 561–577 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, Y.Q., Zu, J.W.: Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerosp. Sci. Technol. 69, 550–562 (2017)CrossRefGoogle Scholar
  21. 21.
    Qian, Y.J., Yang, X.D., Wu, H., et al.: Gyroscopic modes decoupling method in parametric instability analysis of gyroscopic systems. Acta Mech. Sin. (2018).  https://doi.org/10.1007/s10409-018-0762-3 MathSciNetGoogle Scholar
  22. 22.
    Wang, Y.Q., Zu, J.W.: Instability of viscoelastic plates with longitudinally variable speed and immersed in ideal liquid. Int. J. Appl. Mech. 9, 1750005 (2017)CrossRefGoogle Scholar
  23. 23.
    Jazar, G.N.: Stability chart of parametric vibrating systems using energy-rate method. Int. J. Non-Linear Mech. 39, 1319–1331 (2004)CrossRefzbMATHGoogle Scholar
  24. 24.
    Karimpour, H., Pirmoradian, M., Keshmiri, M.: Instance of hidden instability traps in intermittent transition of moving masses along a flexible beam. Acta Mech. 227, 1213–1224 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Torkan, E., Pirmoradian, M., Hashemian, M.: Occurrence of parametric resonance in vibrations of rectangular plates resting on elastic foundation under passage of continuous series of moving masses. Modares Mech. Eng. 17, 225–236 (2017)Google Scholar
  26. 26.
    Pirmoradian, M., Karimpour, H.: Parametric resonance and jump analysis of a beam subjected to periodic mass transition. Nonlinear Dyn. 89, 2141–2154 (2017)CrossRefGoogle Scholar
  27. 27.
    Pirmoradian, M., Torkan, E., Karimpour, H.: Parametric resonance analysis of rectangular plates subjected to moving inertial loads via IHB method. Int. J. Mech. Sci. 142, 191–215 (2018)CrossRefGoogle Scholar
  28. 28.
    Torkan, E., Pirmoradian, M., Hashemian, M.: On the parametric and external resonances of rectangular plates on an elastic foundation traversed by sequential masses. Arch. Appl. Mech. 88, 1411–1428 (2018)CrossRefGoogle Scholar
  29. 29.
    Leissa, A.W.: Vibration of Plates. US Government Printing Office, Washington (1969)Google Scholar
  30. 30.
    Wang, Y.Q., Zu, J.W.: Nonlinear dynamic thermoelastic response of rectangular FGM plates with longitudinal velocity. Compos. Part B Eng. 117, 74–88 (2017)CrossRefGoogle Scholar
  31. 31.
    Wang, Y.Q., Zu, J.W.: Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid. Compos. Struct. 164, 130–144 (2017)CrossRefGoogle Scholar
  32. 32.
    Reddy, J.N.: Theory and Analysis of Elastic Plates and Shells. CRC Press, Boca Raton (2006)Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Young Researchers and Elite Club, Khomeinishahr BranchIslamic Azad UniversityKhomeinishahrIran
  2. 2.Department of Mechanical Engineering, Khomeinishahr BranchIslamic Azad UniversityKhomeinishahrIran

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