Journal of Applied Science and Engineering

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Seema Sharma This email address is being protected from spambots. You need JavaScript enabled to view it.1, U. S. Gupta2 and Prag Singhal3

1Department of Mathematics, Gurukul Kangri University, Hardwar, India
2Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India
3Department of Applied Sciences and Humanities, ABES. Engineering College, Ghaziabad, India


 

Received: April 25, 2011
Accepted: August 8, 2011
Publication Date: September 1, 2012

Download Citation: ||https://doi.org/10.6180/jase.2012.15.3.10  


ABSTRACT


Differential Quadrature Method (DQM) has been used to analyse free vibration of non-homogeneous orthotropic rectangular plates of parabolically varying thickness resting on Winkler-type elastic foundation. The behaviour of non-homogeneity has been assumed due to exponential variation in Young’s modulii and density in one direction. Three different combinations of clamped, simply supported and free edge conditions have been considered. Plots are given to show the effect of foundation parameter together with aspect ratio, taper constant, non-homogeneity parameter and density parameter on the natural frequencies for first three modes of vibration. Mode shapes have been computed for different values of plate parameters. A comparison of our results with those available in literature shows good agreement.


Keywords: DQM, Orthotropy, Variable Thickness, Non-Homogeneity, Elastic Foundation


REFERENCES


  1. [1] Leissa, A. W., Vibration of Plates, NASA SP-160, U.S. Government Printing Office, Washington, DC (1969).
  2. [2] Leissa, A. W., “Recent Research in Plate Vibrations, 1973-1976: Classical Theory,” The Shock and Vibration Digest, Vol. 9, pp. 1324 (1977).
  3. [3] Leissa, A. W., “Recent Research in Plate Vibrations, 1973-1976: Complicating Effects,” The Shock and Vibration Digest, Vol. 10, pp. 2135 (1978).
  4. [4] Leissa, A. W., “Plate Vibration Research, 1976-1980: Complicating Effects,” The Shock and Vibration Digest, Vol. 13, pp. 1122 (1981).
  5. [5] Leissa, A. W., “Plate Vibration Research, 1976-1980: Complicating Effects,” The Shock and Vibration Digest, Vol. 13, pp. 1936 (1981).
  6. [6] Leissa, A. W., “Recent Studies in Plate Vibrations: Part I, Classical Theory,” The Shock and Vibration Digest, Vol. 19, pp. 1118 (1987).
  7. [7] Leissa, A. W., “Recent Studies in Plate Vibrations: Part 2: Complicating Effects,” The Shock and Vibration Digest, Vol. 19, pp. 1024 (1987).
  8. [8] Gorman, D. J., “Accurate Free Vibration Analysis of the Completely Free Orthotropic Rectangular Plate by the Method of Superposition,” Jl. of Sound and Vibration, Vol. 165, pp. 409420 (1993).
  9. [9] Lal, R., Gupta, U. S. and Reena, “Quintic Splines in the Study of Transverse Vibrations of Non-Uniform Orthotropic Rectangular Plates,” Jl. of Sound and Vibration, Vol. 207, pp. 113 (1997).
  10. [10] Lal, R., Gupta, U. S. and Goel, C., “Chebyshev Polynomials in the Study of Transverse Vibrations of NonUniform Rectangular Orthotropic Plates,” The Shock and Vibration Digest, Vol. 33, pp. 103112 (2001).
  11. [11] Dhanpati, “Free Transverse Vibrations of Rectangular and Circular Orthotropic Plates,” Ph. D. Thesis, Indian Institute of Technology, Roorkee, India (2007).
  12. [12] Civalek, O., “Fundamental Frequency of Isotropic and Orthotropic Rectangular Plates with Linearly Varying Thickness by Discrete Singular Convolution Method,” Applied Mathematical Modeling, Vol. 33, pp. 3825 3835 (2009).
  13. [13] Lal, R., Kumar, Y. and Gupta, U. S., “Transverse Vibration of Non-Homogeneous Rectangular Plates of Uniform Thickness Using Boundary Characteristic Orthogonal Polynomials,” Int. Jl. of Applied Mathematics and Mechanics, Vol. 6, pp. 93103 (2010).
  14. [14] Fares, M. E. and Zenkour, A. M, “Buckling and Free Vibration of Non-Homogeneous Composite Cross-Ply Laminated Plates with Various Plate Theories,” Composite Structures, Vol. 44, pp. 279287 (1999).
  15. [15] Lal, R. and Dhanpati., “Transverse Vibration of NonHomogeneous Orthotropic Rectangular Plates of Variable Thickness: A Spline Technique,” Jl. of Sound and Vibration, Vol. 306, pp. 203214 (2007).
  16. [16] Lal, R. and Kumar, Y., “Rayleigh-Ritz Method in the Study of Transverse Vibration of Non-Homogeneous Orthotropic Rectangular Plates of Uniform Thickness Resting on Winkler Foundation,” The Fifth Int. Conf. on Vibration Engineering and Technology Machinery, Huazhong University of Science and Technology, Wuhan, China (2009).
  17. [17] Malekzadeh, P. and Shahpari, S. A., “Free Vibration Analysis of Variable Thickness Thin and Moderately Thick Plates with Elastically Restrained Edges by DQM,” Thin-Walled Structures, Vol. 43, pp. 1037 1050 (2005).
  18. [18] Lal, R., Gupta, U. S. and Sharma, S., “Axisymmetric Vibrations of Non-Homogeneous Annular Plate of Quadratically Varying Thickness,” Proc. Int. Conf. on Advances in Applied Mathematics (ICAAM-05) held at Gulbarga, India, pp. 167181, Feb. 2426 (2005).
  19. [19] Civalek, O., “Harmonic Differential Quadrature-Finite Differences Coupled Approaches for Geometrically Nonlinear Static and Dynamic Analysis of Rectangular Plates on Elastic Foundation,” Jl. of Sound and Vibration, Vol. 294, pp. 966980 (2006).
  20. [20] Liu, F.-L., “Rectangular Thick Plates on Winkler Foundation: Differential Quadrature Element Solution,” Int. Jl. of Solids and Structures, Vol. 37, pp. 17431763 (2000).
  21. [21] Biswas, S. K., “Note on the Torsional Vibration of a Finite Circular Cylinder of Non-Homogeneous Material by a Particular Type of Twist on One of the Plane Surface,” Indian Jl. of Physics, Vol. 43, pp. 320323 (1969).
  22. [22] Rao, G. V., Rao, B. P. and Raju, L. S., “Vibrations of Inhomogeneous Thin Plates Using a High-Precision Triangular Element,” Jl. of Sound and Vibration, Vol. 34, pp. 444445 (1974).
  23. [23] Tomar, J. S., Gupta, D. C. and Jain, N. C., “Free Vibrations of an Isotropic Non-Homogeneous Infinite Plate of Parabolically Varying Thickness,” Indian Jl. of Pure and Applied Mathematics, Vol. 15, pp. 211220 (1984).
  24. [24] Gupta, U. S., Lal, R. and Sharma, S., “Vibration Analysis of Non-Homogeneous Circular Plate of NonLinear Thickness Variation by Differential Quadrature Method,” Jl. of Sound and Vibration, Vol. 298, pp. 892906 (2006).
  25. [25] Lekhnitskii, S. G., Anisotropic Plates, New York, (Transl. S. W. Tsai, T. Cheron): Gorden and Breach (1968).
  26. [26] Panc, V., Theories of Elastic Plates, Leydon, The Netherlands: Noordhoff International Publishing (1975).
  27. [27] Jain, R. K. and Soni, S. R., “Free Vibration of Rectangular Plates of Parabolically Varying Thickness,” Indian Jl. of Pure and Applied Mathematics, Vol. 4, pp. 267277 (1973).
  28. [28] Shu, C., Differential Quadrature and Its Application in Engineering, Great Britain: Springer-Verlag (2000).
  29. [29] Biancolini, M. E., Bruti, C. and Reccia, L., “Approximate Solution of Free Vibration of Thin Orthotropic Rectangular Plates,” Jl. of Sound and Vibration, Vol. 288, pp. 321344 (2005).