Skip to main content
SpringerLink
Log in

Effect of aspect ratio on large amplitude free vibrations of simply supported and clamped rectangular Mindlin plates using coupled displacement field method

  • Published:
Journal of Mechanical Science and Technology Aims and scope Submit manuscript

Abstract

We propose a novel method, known as Coupled displacement field (CDF) method, an alternative to study large amplitude free vibration behavior of moderately thick rectangular plates. An admissible trial function was assumed for one of the variables, say, the total rotations (in both X, Y directions). The function for lateral displacement field is derived in terms of the total rotations with the help of coupling equations, where the two independent variables become dependent on one another. This method makes use of the energy formulation, where it contains only half the number of undetermined coefficients when compared with conventional Rayleigh-Ritz method. The vibration problem is simplified significantly due to the reduction in number of undetermined coefficients. The frequency-amplitude relationship for the moderately thick rectangular plates with various aspect ratios for all edges simply supported and clamped boundary conditions was obtained. Closed form expressions for linear and nonlinear fundamental frequency parameters were derived.

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

Similar content being viewed by others

References

  1. D. Zhou, Free vibration of multi-span Timoshenko beams using static Timoshenko beam functions, J. of Sound and Vibration, 241 (2001) 725–734.

    Article  Google Scholar 

  2. G. V. Rao, K. MeeraSaheb and G. Rangajanardha, Concept of coupled displacement field for large amplitude free vibrations of shear flexible beams, American Society of Mechanical Engineering, 128 (2006) 251–255.

    Google Scholar 

  3. T. Wah, Large amplitude flexural vibration of rectangular plates, International J. of Mechanical Sciences, 5 (1963) 425–438.

    Article  Google Scholar 

  4. H. N. Chu and G. Herrman, Influence of large amplitudes on free flexural vibrations of rectangular elastic plates, J. of Applied Mechanics, 23 (1956) 532–540.

    MathSciNet  MATH  Google Scholar 

  5. C. Mei, Finite element displacement method for large amplitude free flexural vibrations of beams and plates, Computers & Structures, 3 (1973) 163–174.

    Article  Google Scholar 

  6. C. M. Wang, C. Y. Wang and J. N. Reddy, Exact solutions for buckling of structural members, CRC Press(2005).

    Google Scholar 

  7. W. Leissa, The free vibration of rectangular plates, J. of Sound and Vibration, 31 (3) (1973) 257–293.

    Article  MATH  Google Scholar 

  8. M. Batista, Analytical solution for free vibrations of simply supported transversally inextensible homogeneous rectangular plate, arXiv:1007.2539(2010) [physics.gen-ph].

    Google Scholar 

  9. S. H. Hashemi and M. Arsanjani, Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates, International J. of Solids and Structures, 42 (2005) 819–853.

    Article  MATH  Google Scholar 

  10. K. M. Liew and T. M. Teo, Three-dimensional vibration analysis of rectangular plates based on differential quadrature method, J. of Sound and Vibration, 220 (4) (1999) 577–599.

    Article  Google Scholar 

  11. K. K. Raju, G. V. Rao and I. S. Raju, Effect of geometric nonlinearity on large amplitude free flexural vibrations of moderately thick rectangular plates, Computers and Structures, 9 (1978) 441–444.

    Article  MATH  Google Scholar 

  12. K. K. Raju and E. Hinton, Natural frequencies and modes of rhombic Mindlin plates, Earthquake Engineering Structural Dynamics, 8 (1980) 55–62.

    Article  Google Scholar 

  13. B. S. Sarma, Nonlinear free vibrations of beams, plates, and nonlinear panel flutter, Ph.D. Thesis, Department of Aerospace Engineering, I.I.T., Madras(1987).

    Google Scholar 

  14. K. K. Raju and E. Hinton, Nonlinear vibrations of thick plates using Mindlin plate elements, International J. of Numerical Merhods in Engineering, 15 (1980) 241–257.

    MATH  Google Scholar 

  15. C. Mei and K. Decha-Umphai, A finite element method for nonlinear forced vibrations of rectangular plates, AIAA J., 23 (1985) 1104-l 110.

    Article  MATH  Google Scholar 

  16. Y. Shi and C. Mei, A finite element time domain modal formulation for large amplitude free vibrations of beams and plates, J. of Sound and Vibration, 193 (2) (1996) 453–464.

    Article  MATH  Google Scholar 

  17. S. A. Eftekhari and A. A. Jafari, A simple and accurate Ritz formulation for free vibration of thick rectangular and skew plates with general boundary conditions, Acta Mechanica, 224 (2013) 193–209.

    Article  MathSciNet  MATH  Google Scholar 

  18. X. Liu and J. R. Banerjee, Free vibration analysis for plates with arbitrary boundary conditions using a novel spectraldynamic stiffness method, Computers & Structures, 164 (2016) 108–126.

    Article  Google Scholar 

  19. K. K Pradhan and S. Chakraverty, Transverse vibration of isotropic thick rectangular plates based on new inverse trigonometric shear deformation theories, International J. of Mechanical Sciences, 94–95 (2015) 211–231.

    Article  Google Scholar 

  20. R. P. Shimpi, H. G. Patel and H. Arya, New first-order shear deformation plate theories, J. of Applied Mechanics, 74 (2007) 523–533.

    Article  MATH  Google Scholar 

  21. J. S. Shabnam, A. Reza and K. F. Rahmat, Free vibration analysis of variable thickness thin plates by two-dimensional differential transform method, Acta. Mech., 224/8 (2013) 1643–1658.

  22. H. T. Thai and D.-H. Choi, Analytical solutions of refined plate theory for bending, buckling and vibration analyses of thick plates, Applied Mathematical Modelling, 37 (18–19) (2013) 8310–8323.

    Article  MathSciNet  Google Scholar 

  23. A. Mahi, E. Abbas, A. Tounsi and A. Benkhedda, A new simple shear deformation theory for free vibration analysis of isotropic and FG plates under different boundary conditions, Multidiscipline Modeling in Materials and Structures, 11 (3) (2015) 437–470.

    Article  Google Scholar 

  24. A. Mahi, E. Abbas, A. Tounsi and A. Benkhedda, A new hyperbolic shear deformation theory for bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates, Applied Mathematical Modelling, 39 (2015) 2489–2508.

    Article  MathSciNet  Google Scholar 

  25. A. S. Sayyada and Y. M. Ghugal, Bending and free vibration analysis of thick isotropic plates by using exponential shear deformation theory, Applied and Computational Mechanics, 6 (2012) 65–82.

    Google Scholar 

  26. A. M. A. Neves, A. J. M. Ferreira, E. Carrera, M. Cinefra, C. M. C. Roque, R. M. N. Jorge and C. M. M. Soares, Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higherorder shear deformation theory and a meshless technique, Composites: Part B, 44 (2013) 657–674.

    Article  Google Scholar 

  27. Y. T. Leung and S. G. Mao, A symplectic galerkin method for non-linear vibration of beams and plate, J. of Sound and Vibration, 183 (3) (1995) 475–491.

    Article  MATH  Google Scholar 

  28. S. L. Lau, Incremental harmonic balance method for nonlinear structural vibrations, Ph. D. Thesis, University of Hong Kong(1982).

  29. R. D. Mindlin, Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates, J. of Applied Mechanics, 18 (1951) 31–38.

    MATH  Google Scholar 

  30. K. M. Liew, Y. Xiang and S. Kitipornchai, Transverse vibration of thick rectangular plates-1. Comprehensive sets of boundary conditions, Computers & Structures, 49 (1)(1993) 1–29.

    Article  Google Scholar 

  31. D. J. Gorman, Free vibration analysis of rectangular plates, Elsevier North Holland: New York(1982).

  32. D. J. Dawe and O. L. Roufaeil, Rayleigh-Ritz vibration analysis of Mindlin plates, J. of Sound and Vibration, 69 (1980) 345–359.

    Article  MATH  Google Scholar 

  33. K. C. Hung, A treatise on three-dimensional vibration of a class of elastic solids, Ph. D. Thesis, Nanyang Technological University, Singapore(1996).

  34. K. Saeedi, A. Leo, R. B. Bhat and I. Stiharu, Vibration of circular plate with multiple eccentric circular perforations by the Rayleigh-Ritz method, J. of Mechanical Science and Technology, 26 (5) (2012) 1439–1448.

    Article  Google Scholar 

  35. E. Bahmyari and A. Rahbar-Ranji, Free vibration analysis of orthotropic plates with variable thickness resting on nonuniform elastic foundation by element free Galerkin method, J. of Mechanical Science and Technology, 26 (9) (2012) 1–11, Doi 10.1007/s12206-011-0913-y.

    Article  Google Scholar 

  36. L. Azrar and R. G. White, A semi -Analytical approach to the nonlinear dynamic response problem of s-s and c-c beams at large vibration amplitudes. Part1: General theory and application to the single mode approach to free and forced vibration analysis, J. of sound and Vibration, 224 (1999) 183 -207.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University College of Engineering, JNTUK, Kakinada, 533003, India

    K. KrishnaBhaskar & K. MeeraSaheb

Authors
  1. K. KrishnaBhaskar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. MeeraSaheb
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. KrishnaBhaskar.

Additional information

Recommended by Associate Editor Eung-Soo Shin

K. Krishna Bhaskar is a Ph.D. Research scholar working in the field of nonlinear vibrations. He is an Assistant Professor of Mechanical Engineering, University College of Engineering, JNT University Kakinada, Andhra Pradesh, India.

K. Meera Saheb received Ph.D. in Nonlinear vibrations from JNT University, Hyderabad, India in 2010. Now he is a Professor of Mechanical Engineering and Head of PE & PCE Department, University College of Engineering, JNT University Kakinada, Andhra Pradesh, India.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

KrishnaBhaskar, K., MeeraSaheb, K. Effect of aspect ratio on large amplitude free vibrations of simply supported and clamped rectangular Mindlin plates using coupled displacement field method. J Mech Sci Technol 31, 2093–2103 (2017). https://doi.org/10.1007/s12206-017-0406-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12206-017-0406-8

Keywords

Search

Navigation