Acta Mechanica Solida Sinica

, Volume 22, Issue 2, pp 125–136 | Cite as

Characteristic equations and closed-form solutions for free vibrations of rectangular Mindlin plates

Article

Abstract

The direct separation of variables is used to obtain the closed-form solutions for the free vibrations of rectangular Mindlin plates. Three different characteristic equations are derived by using three different methods. It is found that the deflection can be expressed by means of the four characteristic roots and the two rotations should be expressed by all the six characteristic roots, which is the particularity of Mindlin plate theory. And the closed-form solutions, which satisfy two of the three governing equations and all boundary conditions and are accurate for rectangular plates with moderate thickness, are derived for any combinations of simply supported and clamped edges. The free edges can also be dealt with if the other pair of opposite edges is simply supported. The present results agree well with results published previously by other methods for different aspect ratios and relative thickness.

Key words

Mindlin plate free vibration closed-form solution separation of variables 

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References

  1. [1]
    Leissa, A.W., The free vibration of rectangular plates. Journal of Sound and Vibration, 1973, 31: 257–293.CrossRefGoogle Scholar
  2. [2]
    Liew, K.M., Xiang, Y. and Kitipornchai, S., Research on thick plate vibration: a literature survey. Journal of Sound and Vibration, 1995, 180: 163–176.CrossRefGoogle Scholar
  3. [3]
    Reissner, E., The effect of transverse shear deformation on the bending of elastic plates. Trans. ASME Journal of Applied Mechanics, 1945, 12: A69–A77.MathSciNetMATHGoogle Scholar
  4. [4]
    Mindlin, R.D., Influence of rotatory inertia and shear on flexural motion of isotropic, elastic plates. Trans. ASME Journal of Applied Mechanics, 1951, 18: 31–38.MATHGoogle Scholar
  5. [5]
    Endo, M. and Kimura, N., An alternative formulation of the boundary value problem for the Timoshenko beam and Mindlin plate. Journal of Sound and Vibration, 2007, 301: 355–373.CrossRefGoogle Scholar
  6. [6]
    Shimpi, R.P. and Patel, H.G., Free vibrations of plate using two variable refined plate theory. Journal of Sound and Vibration, 2006, 296: 979–999.CrossRefGoogle Scholar
  7. [7]
    Hashemi, S.H. and Arsanjani, M., Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates. International Journal of Solids and Structures, 2005, 42: 819–853.CrossRefGoogle Scholar
  8. [8]
    Brunelle, E.J., Buckling of transversely isotropic Mindlin plates. AIAA Journal, 1971, 9: 1018–1022.CrossRefGoogle Scholar
  9. [9]
    Gorman, D.J. and Ding, W. Accurate free vibration analysis of point supported Mindlin plates by the superposition method. Journal of Sound and Vibration, 1999, 219: 265–277.CrossRefGoogle Scholar
  10. [10]
    Gorman, D.J. and Ding, W., Accurate free vibration analysis of completely free rectangular Mindlin plates using the superposition method. Journal of Sound and Vibration, 1996, 189: 341–353.CrossRefGoogle Scholar
  11. [11]
    Wang, C.M., Natural frequencies formula for simply supported Mindlin plates. ASME Journal of Vibration and Acoustics, 1994, 116: 536–540.CrossRefGoogle Scholar
  12. [12]
    Xiang, Y. and Wei, G.W., Exact solutions for buckling and vibration of stepped rectangular Mindlin plates. International Journal of Solids and Structures, 2004, 41: 279–294.CrossRefGoogle Scholar
  13. [13]
    Xiang, Y., Vibration of rectangular Mindlin plates resting on non-homogenous elastic foundations. International Journal of Mechanical Sciences, 2003, 45: 1229–1244.CrossRefGoogle Scholar
  14. [14]
    Liew, K.M., Xiang, Y. and Kitipornchai, S., Transverse vibration of thick rectangular plates — I. Comprehensive sets of boundary conditions. Computers and Structures, 1993, 49: 1–29.CrossRefGoogle Scholar
  15. [15]
    Liew, K.M., Hung, K.C. and Lim, M.K., A continuum three-dimensional vibration analysis of thick rectangular plate. International Journal of Solids and Structures, 1993, 24: 3357–3379.CrossRefGoogle Scholar
  16. [16]
    Liew, K.M., Hung, K.C. and Lim, M.K., Vibration of Mindlin plates using boundary characteristic orthogonal polynomials. Journal of Sound and Vibration, 1995, 182: 77–90.CrossRefGoogle Scholar
  17. [17]
    Cheung, Y.K. and Zhou, D., Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions. Computers and Structures, 2000, 78: 757–768.CrossRefGoogle Scholar
  18. [18]
    Shen, H.S., Yang, J., and Zhang, L., Free and forced vibration of Reissner-Mindlin plates with free edges resting on elastic foundations. Journal of Sound and Vibration, 2001, 244(2): 299–320.CrossRefGoogle Scholar
  19. [19]
    Liu, F.L. and Liew, K.M., Vibration analysis of discontinuous Mindlin plates by differential quadrature element method. Transactions of ASME, 1999, 121: 204–208.Google Scholar
  20. [20]
    Malekzadeh, P., Karami, G. and Farid, M., A semi-analytical DQEM for free vibration analysis of thick plates with two opposite edges simply supported. Computer Methods in Applied Mechanics and Engineering, 2004, 193: 4781–4796.CrossRefGoogle Scholar
  21. [21]
    Hou, Y.S., Wei, G.W. and Xiang, Y., DSC-Ritz method for the free vibration analysis of Mindlin plates. International Journal for Numerical Methods in Engineering, 2005, 62: 262–288.CrossRefGoogle Scholar
  22. [22]
    Diaz-Contreras, R.E. and Nomura, S., Green’s function applied to solution of Mindlin plates. Computers & Structures, 1996, 60(1): 41–48.CrossRefGoogle Scholar
  23. [23]
    Sakiyama, T. and Huang, M., Free vibration analysis of rectangular plates with variable thickness. Journal of Sound and Vibration, 1998, 216: 379–397.CrossRefGoogle Scholar
  24. [24]
    Lee, J.M. and Kim, K.C., Vibration analysis of rectangular Isotropic thick plates using Mindlin plate characteristic functions. Journal of Sound and Vibration, 1995, 187(5): 865–877.CrossRefGoogle Scholar
  25. [25]
    Kanrotovich, L.V. and Krylov, V.I., Approximate Methods of Higher Analysis. Groningen, Netherlands: P. Noordhoff Ltd, 1964.Google Scholar
  26. [26]
    Ma, Y.Q. and Ang, K.K., Free vibration of Mindlin plates based on the relative displacement plate element. Finite Elements in Analysis and Design, 2006, 42: 1021–1028.CrossRefGoogle Scholar
  27. [27]
    Srinivas, S., Rao, C.V.J. and Rao, A.K., An exact analysis for vibration of simple-supported homogeneous and laminated thick rectangular plates. Journal of Sound and Vibration, 1970, 12: 187–199.CrossRefGoogle Scholar
  28. [28]
    Wittrick, W.H., Analytical three-dimensional elasticity solutions to some plate problems and some observations on Mindlin’s plate theory. International Journal of Solids and Structures, 1987, 23: 441–464.CrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2009

Authors and Affiliations

  1. 1.The Solid Mechanics Research CenterBeijing University of Aeronautics and AstronauticBeijingChina

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