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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu


Geometrically Nonlinear Forced Transverse Vibrations of C-S-C-S Rectangular Plate: Numerical and Experimental Investigations

Journal of Applied Nonlinear Dynamics 10(4) (2021) 739--757 | DOI:10.5890/JAND.2021.12.012

A. Majid , E.Abdeddine, Kh.Zarbane, Z.Beidouri

Laboratoire M'ecanique, Productique et G'enie Industriel (LMPGI) Ecole Sup'erieure de Technologie (ESTC), Hassan II University of Casablanca, BP 8012 Oasis, Casablanca, Maroc

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Abstract

The present paper describes a theoretical and an experimental investigation of Clamped-Simply-Clamped-Simply supported isotropic rectangular plates subject to a large amplitudes forced transverse vibration. The theoretical model is based on Lagrange equations and the harmonic balance method which is used to obtain the non-linear algebraic equation of the problem. To examine our results, a test rig is designed and made. Furthermore, an electrodynamic exciter is used to provide the transverse harmonic vibrations, whereas an accelerometer measures the displacements. The approximate analytical solutions are obtained, and the experimental measurements are discussed and compared which are highlight the nonlinear hardening resonance.

References

  1. [1]  Leissa, A.W. (1969), Vibrations of plates, NASA SP-160, U.S. Government Printing Office, Washington, D.C., Washington.
  2. [2]  Leissa, A.W. (1973), The free vibration of rectangular plates, Journal of Sound and Vibration, 31(3), 257-293.
  3. [3]  Liguore, S.L., Pitt, D.M., Thomas, M.J., and Gurtowski, N. (2011), Air {Vehicle} {Integration} and {Technology} {Research} ({AVIATR}), {Delivery} {Order} 0013: {Nonlinear}, {Low}-{Order}/{Reduced}-{Order} {Modeling} {Applications} and {Demonstration}, page 269.
  4. [4]  Bert, C.W. and Malik, M. (1994), Frequency equations and modes of free vibrations of rectangular plates with various edge conditions, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 208(5), 307-319.
  5. [5]  Eisenberger, M. and Deutsch, A. (2019), Solution of thin rectangular plate vibrations for all combinations of boundary conditions, Journal of Sound and Vibration, 452, 1-12.
  6. [6]  Amabili, M. (2004) Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments, Computers Structures, 82(31-32), 2587-2605.
  7. [7]  Amabili, M. (2006), Theory and experiments for large-amplitude vibrations of rectangular plates with geometric imperfections, Journal of Sound and Vibration, 291(3-5), 539-565.
  8. [8]  Amabili, M. (2008), {Nonlinear vibrations and stability of shells and plates}, Cambridge University Press, Cambridge.
  9. [9]  Amabili, M. and Farhadi, S. (2009), Shear deformable versus classical theories for nonlinear vibrations of rectangular isotropic and laminated composite plates, Journal of Sound and Vibration, 320(3), 649-667.
  10. [10]  Amabili, M. (2018), Nonlinear damping in nonlinear vibrations of rectangular plates: Derivation from viscoelasticity and experimental validation, Journal of the Mechanics and Physics of Solids, 118, 275-292.
  11. [11]  Beidouri, Z., Benamar, R., and El~Kadiri, M. (2006), Geometrically non-linear transverse vibrations of C-S-S-S and C-S-C-S rectangular plates, International Journal of Non-Linear Mechanics, 41(1), 57-77.
  12. [12]  Benamar, R., Bennouna, M.M.K., and White, R.G. (1991), The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures part {I}: Simply supported and clamped-clamped beams, Journal of Sound and Vibration, 149(2), 179-195.
  13. [13]  Benamar, R., Bennouna, M.M.K., and White, R.G. (1992), The {effects} {of} {large} {vibration} {amplitudes} {on} {the} {mode} {shapes} {and} {natural} {frequencies} {of} {thin} {elastic} {structures}, {part} {II}: {fully} {clamped} {rectangular} {isotropic} {plates}, Journal of Sound and Vibration, 175(3), 377-395.
  14. [14]  Benamar, R., Bennouna, M.M.K., and White, R.G. (1993), The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, part III: fully clamped rectangular isotropic plates measurements of the mode shape amplitude dependence and the spatial distribution of harmonic distortion, Journal of Sound and Vibration, 175(3), 377-395.
  15. [15]  El Bikri, K., Benamar, R., and Bennouna, M. (2003), Geometrically non-linear free vibrations of clamped simply supported rectangular plates. {Part} {I}: the effects of large vibration amplitudes on the fundamental mode shape, Computers $\&$ Structures, 81(20), 2029-2043.
  16. [16]  El Kadiri, M., Benamar, R., and White, R.G. (1999), The non-linear free vibration of fully clamped rectangular plates: second non-linear mode for various plate aspect ratios, Journal of Sound and Vibration, 228(2), 333-358.
  17. [17]  El~Kadiri, M. and Benamar, R. (2002), Improvement of the semi-analytical method, based on {Hamilton}s principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. {Part} {III}: steady state periodic forced response of rectangular plates, Journal of Sound and Vibration, 264(1), 1-35.
  18. [18]  Herrmann, G. and Chu, H.N, (1956), Influence of large amplitude on free flexural vibrations of rectangular elastic plates, Journal of Applied Mechanics, pages 532-540.
  19. [19]  Ribeiro, P. and Petyt, M. (2000), Non-linear free vibration of isotropic plates with internal resonance, International Journal of Non-Linear Mechanics, 35(2), 263-278.
  20. [20]  Amabili, M. and Augenti, C. (2005), Nonlinear Vibrations of Rectangular Plates With Different Boundary Conditions: Theory and Experiments, 2005 ASME International Mechanical Engineering Congress and Exposition, Orlando, Florida USA.
  21. [21]  Rahmouni, A., Beidouri, Z., and Benamar, R. (2013), A discrete model for geometrically nonlinear transverse free constrained vibrations of beams with various end conditions, Journal of Sound and Vibration, 332(20), 5115-5134.
  22. [22]  Merrimi, E., El Bikri, K., and Benamar, R. (2012), Non-linear forced vibration analysis of rectangular plates including the coupling between transverse and in-plane displacements, MATEC Web of Conferences, 10014.
  23. [23]  Eddanguir, A., Beidouri, Z., and Benamar, R. (2012), Geometrically nonlinear transverse steady-state periodic forced vibration of multi-degree-of-freedom discrete systems with a distributed nonlinearity, Ain Shams Engineering Journal, 3(3), 191-207.
  24. [24]  Awodola, T.O. and Omolofe, B. (2018), Flexural motion of elastically supported rectangular plates under concentrated moving masses and resting on bi-parametric elastic foundation, Journal of Vibrational Engineering and Technologies, 6(3), 165-177.
  25. [25]  Chladni, E.F.F. (1830), Die akustik. Breitkopf \& H\"{a}rtel.
  26. [26]  Azrar, L., Benamar, R., and White, R.G. (1998), Semi-analytical approach to the non-linear dynamic response problem of {S} {S} and {C} {C} beams at large vibration amplitudes part {I}: general theory and application to the single mode approach to free and forced vibration analysis, Journal of Sound and Vibration, 224(2), 183-207.
  27. [27]  Rougui, M., Moussaoui, F., and Benamar, R.(2007), Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach, International Journal of Non-Linear Mechanics, 1102-1115.
  28. [28]  Harras, B., Benamar, R., and White, R.G. (2001), Geometrically non-linear free vibration of fully clamped symmetrically laminated rectangular composite plates, Journal of Sound and Vibration, 579-619.
  29. [29]  Ribeiro, P. (2001), Periodic vibration of plates with large displacements, In 19th AIAA Applied Aerodynamics Conference.
  30. [30]  Amabili, M. (2004), \newblock Nonlinear vibrations of rectangular plates with different boundary conditions: Theory and experiments, Computers and Structures, 82(31-32), 2587-2605.
  31. [31]  Alijani, F. and Amabili, M.(2013), Theory and experiments for nonlinear vibrations of imperfect rectangular plates with free edges, Journal of Sound and Vibration, 332(14), 3564-3588.
  32. [32]  Beidouri, Z. (2006), Contribution a une th\eorie danalyse modale non-lin\eaire : application aux structures continues et aux syst\emes discrets a non-lin\earites localis\ees. Sciences appliqu\ees, Universite Mohammed V, Agdal Ecole Mohamadia Des Ing\enieurs Rabat.
  33. [33]  Hosseini-Hashemi, S., Fadaee, M., and Taher, H.R.D. (2011), Exact solutions for free flexural vibration of {L\evy} type rectangular thick plates via third-order shear deformation plate theory, Applied Mathematical Modelling, 708-727. %