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Natural Frequencies of Rectangular Plate With- and Without-Rotary Inertia

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Abstract

A nine-node isoparametric plate element, in conjunction with first-order shear deformation theory, was used for free vibration analysis of rectangular plates. Both thick and thin plate problems were solved for various aspect ratios and boundary conditions. In this work, the primary focus is on the effect of rotary inertia on the natural frequencies of rectangular plates. It is found that rotary inertia significantly affects thick plates, while it can be ignored for thin plates. The numerical convergence is very rapid and based on a comparison with data from the literature; it is proposed that the present formulation can yield highly accurate results. Finally, some numerical solutions are provided here, which may serve as benchmarks for future research on similar problems.

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Abbreviations

[B]:

Strain displacement matrix

[D]:

Rigidity matrix

[K]:

Global stiffness matrix

[N]:

Shape function

[N0]:

Null matrix

[M]:

Consistent mass matrix

|J|:

Jacobian matrix

[Nr]:

Interpolation function of the rth point

[K0]:

Overall stiffness matrix

[M0]:

Overall Mass matrix

w:

Transverse displacement

θxθy :

Total rotations in bending

E:

Modulus of elasticity

G:

Modulus of rigidity

ν:

Poisson’s ratio

h:

Thickness of plate

a, b:

Plate dimensions

D:

Flexural rigidity

ω:

Natural frequency

ϕ x ϕ y :

Average shear rotation

θ x θ y :

Total rotation in bending

{σ}:

Stress vector

{ε}:

Strain vector

M x , M y :

Bending moments in x and y direction

M xy :

Twisting moment

QxQy :

Transverse shear forces

\( \xi , \eta \) :

Natural coordinates

ρ :

Density

References

  1. A.W. Leissa, Recent research in plate vibrations, 1973–1976: classical theory. Shock Vib. Inf. Cent. Shock Vib. Dig. 9(10), 13–24 (1978)

    Article  Google Scholar 

  2. A.W. Leissa, Recent research in plate vibrations, 1973-1976: complicating effects. Shock Vib. Inf. Cent. Shock Vib. Dig. 10(12), 21–35 (1978)

    Article  Google Scholar 

  3. A.W. Leissa, Plate vibration research, 1976–1980: classical theory. Shock Vib. Inf. Cent. Shock Vib. Dig. 13(9), 11–22 (1980)

    Article  Google Scholar 

  4. A.W. Leissa, Plate vibration research, 1976–1980: complicating effects. Shock Vib. Inf. Cent. Shock Vib. Dig. 13(10), 19–36 (1980)

    Google Scholar 

  5. A.W. Leissa, Plate vibration research, 1981–1985, part I: classical theory. Shock Vib. Inf. Cent. Shock Vib. Dig. 19(2), 11–18 (1987)

    Article  Google Scholar 

  6. A.W. Leissa, Plate vibration research, 1981–1985, part II: complicating effects. Shock Vib. Inf. Cent. Shock Vib. Dig. 19(3), 10–24 (1987)

    Article  Google Scholar 

  7. K.M. Liew, Y. Xiang, S. Kitipornchai, Research on thick plate vibration: a literature survey. J. Sound Vib. 180(1), 163–176 (1995)

    Article  Google Scholar 

  8. G. Yamada, T. Irie, Plate vibration research in Japan. Appl. Mech. Rev. 40(7), 879–892 (1987)

    Article  Google Scholar 

  9. O. G. McGee (2013) Flexural vibrations of clamped-free rhombic plates with corner stress singularities Part 1 review of research. J. Vib. Control 1077546312456864

  10. S. Haldar, A.H. Sheikh, Bending analysis of composite folded plates by finite element method. Finite Elem. Anal. Des. 47(4), 477–485 (2011)

    Article  Google Scholar 

  11. P. Dey, A.H. Sheikh, D. Sengupta, A new element for the analysis of composite plates. Finite Elem. Anal. Des. 82, 62–71 (2014)

    Article  MathSciNet  Google Scholar 

  12. T. Rock, E. Hinton, Free vibration and transient response of thick and thin plates using the finite element method. Earthq. Eng. Struct. Dyn. 3(1), 51–63 (1974)

    Article  Google Scholar 

  13. S. Natarajan et al. Natural frequencies of cracked isotropic and specially orthotropic plates using the extended finite element method (2011)

  14. S. Natarajan et al., Natural frequencies of cracked functionally graded material plates by the extended finite element method. Compos. Struct. 93(11), 3082–3092 (2011)

    Article  Google Scholar 

  15. D.J. Gorman, Free vibration analysis of completely free rectangular plates by the superposition–Galerkin method. J. Sound Vib. 237(5), 901–914 (2000)

    Article  Google Scholar 

  16. D.J. Gorman, W. Ding, The superposition-Galerkin method for free vibration analysis of rectangular plates. J. Sound Vib. 194(2), 187–198 (1996)

    Article  Google Scholar 

  17. J.T. Chen et al., A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function. Eng. Anal. Boundary Elem. 28(5), 535–545 (2004)

    Article  MathSciNet  Google Scholar 

  18. G.R. Liu, X.L. Chen, A mesh-free method for static and free vibration analyses of thin plates of complicated shape. J. Sound Vib. 241(5), 839–855 (2001)

    Article  Google Scholar 

  19. P. Malekzadeh, S.A. Shahpari, Free vibration analysis of variable thickness thin and moderately thick plates with elastically restrained edges by DQM. Thin Wall. Struct. 43(7), 1037–1050 (2005)

    Article  Google Scholar 

  20. C.W. Bert, W. Xinwei, A.G. Striz, Differential quadrature for static and free vibration analyses of anisotropic plates. Int. J. Solids Struct. 30(13), 1737–1744 (1993)

    Article  Google Scholar 

  21. Ö. Civalek, Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method. Int. J. Mech. Sci. 49(6), 752–765 (2007)

    Article  Google Scholar 

  22. G.W. Wei, Y.B. Zhao, Y. Xiang, The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution. Int. J. Mech. Sci. 43(8), 1731–1746 (2001)

    Article  Google Scholar 

  23. R.B. Bhat, Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. J. Sound Vib. 102(4), 493–499 (1985)

    Article  Google Scholar 

  24. E. Carrera, F.A. Fazzolari, L. Demasi, Vibration analysis of anisotropic simply supported plates by using variable kinematic and Rayleigh-Ritz method. J. Vib. Acoust. 133(6), 061017 (2011)

    Article  Google Scholar 

  25. C.V. Srinivasa, J.S. Yalaburgi, W.P. Prema Kumar, Experimental and finite element studies on free vibration of skew plates. Int. J. Adv. Struct. Eng. (IJASE) 6(1), 1–11 (2014)

    Google Scholar 

  26. Srinivasa et al., Experimental and finite element studies on free vibration of cylindrical skew panels. Int. J. Adv. Struct. Eng. 6, 1 (2014)

    Google Scholar 

  27. R.W. Clough, J.L. Tocher, Finite element stiffness matrices for analysis of plates in bending, Proceedings of conference on matrix methods in structural analysis. 1 (1965)

  28. J.-L. Batoz, K.-J. Bathe, L.-W. Ho, A study of three-node triangular plate bending elements. Int. J. Numer. Meth. Eng. 15(12), 1771–1812 (1980)

    Article  Google Scholar 

  29. M.M. Hrabok, T.M. Hrudey, A review and catalogue of plate bending finite elements. Comput. Struct. 19(3), 479–495 (1984)

    Article  Google Scholar 

  30. S. Sahoo, Free vibration of laminated composite hypar shell roofs with cutouts. Adv. Acoust. Vib. (2011)

  31. S. Sahoo, Laminated composite stiffened shallow spherical panels with cutouts under free vibration–A finite element approach. Eng. Sci. Technol. Int. J. 17(4), 247–259 (2014)

    Article  Google Scholar 

  32. S. Sahoo, Laminated composite stiffened cylindrical shell panels with cutouts under free vibration. Int. J. Manuf. Mater. Mech. Eng. (IJMMME) 5(3), 37–63 (2015)

    Google Scholar 

  33. K.-N. Koo, Effects of shear deformation and rotary inertia on the natural frequencies of axially loaded beams. J. Mech. Sci. Technol. 28(3), 849–857 (2014)

    Article  Google Scholar 

  34. A. Majumdar, M.C. Manna, S. Haldar, Bending of skewed cylindrical shell panels. Int. J. Comput. Appl. 1(8), 89–93 (2010)

    Google Scholar 

  35. M.K. Pandit, S. Haldar, M. Mukhopadhyay, Free vibration analysis of laminated composite rectangular plate using finite element method. J. Reinf. Plast. Compos. 26(1), 69–80 (2007)

    Article  Google Scholar 

  36. K. Kalita, S. Haldar, Parametric study on thick plate vibration using FSDT. Mech. Mech. Eng. 19(2), 81–90 (2015)

    Google Scholar 

  37. K. Kalita, S. Haldar, Free vibration analysis of rectangular plates with central cutout. Cogent Eng. 3(1), 1163781 (2016)

    Article  Google Scholar 

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

    MATH  Google Scholar 

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Authors and Affiliations

  1. SVKM’S Narsee Monjee, Institute of Management Studies, Shirpur, 425405, Maharashtra, India

    Kanak Kalita

  2. Indian Institute of Engineering Science and Technology, Shibpur, 711103, West Bengal, India

    Salil Haldar

Authors
  1. Kanak Kalita
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  2. Salil Haldar
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Correspondence to Kanak Kalita.

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Kalita, K., Haldar, S. Natural Frequencies of Rectangular Plate With- and Without-Rotary Inertia. J. Inst. Eng. India Ser. C 99, 539–555 (2018). https://doi.org/10.1007/s40032-016-0327-9

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  • DOI: https://doi.org/10.1007/s40032-016-0327-9

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