A mathematical model for predicting the vibrational response of a non-homogeneous visco-elastic rectangular plate was developed to assist design engineers and researchers. In the presented model, thermally induced vibrations of a four-sided clamped rectangular plate of non-uniform thickness is discussed. Non-homogeneity in the material is characterized exponentially in Poisson’s ratio while temperature variation is considered bi-parabolic. A boundary value fourth order partial differential equation of motion is formulated for the parabolic tapered rectangular plate. Visco-elastic properties of the material are of Kelvin type, and deflection is considered small and linear. This paper focuses on the effects of structural parameters, i.e. thermal gradient, taper constant, aspect ratio, and non-homogeneity constant on the vibrational behaviour of rectangular plates. The Rayleigh–Ritz method is used to obtain results for the time period and deflection for the first two modes of vibration. Comparison of the results of the present paper with others available in the literature is visualized with the help of graphs.
1. Abu, A. I., Turhan, D., and Mengi, D. Two dimensional transient wave propagation in visco-elastic layered media. J. Sound Vib., 2001, 244(5), 837–858.
2. Avalos, D. R. and Laura, P. A. Transverse vibrations of a simply supported plate of generalized anisotropy with an oblique cut-out. J. Sound Vib., 2002, 258(2), 773–776.
3. Chakraverty, S. Vibrations of Plate. Vol. 10. Taylor and Francis, 2009.
4. Hasheminejad, S. M., Ghaheri, A., and Vaezian, S. Exact solution for free in-plane vibration analysis of an eccentric elliptical plate. Acta Mech., 2013, 224(8), 1609–1624.
5. Hosseini-Hashemi, S., Derakhshani, M., and Fadaee, M. An accurate mathematical study on the free vibration of stepped thickness circular/annular Mindlin functionally graded plates. Appl. Math. Model., 2013, 37(6), 4147–4164.
6. Gupta, A. K. and Khanna, A. Vibration of visco-elastic rectangular plate with linearly thickness variation in both directions. J. Sound Vib., 2007, 301(3–5), 450–457.
7. Khanna, A. and Arora, P. Theoretical analysis on thermally induced vibration of tapered parallelogram plate with mixed boundary conditions. J. Vibroeng., 2014, 16(3), 1276–1283.
8. Khanna, A. and Kaur, N. Effect of non-homogeneity on free vibration of visco-elastic rectangular plate with varying structural parameters. J. Vibroeng., 2013, 15(4), 2146–2155.
9. Khanna, A. and Kaur, N. Vibration of non-homogeneous plate subject to thermal gradient. J. Low Freq. Noise V. A., 2014, 33(1), 13–26.
10. Khanna, A. and Sharma, K. A. Natural vibration of visco-elastic plate of varying thickness with thermal effect. Journal of Applied Science and Engineering, 2013, 16(2), 135–140.
11. Lal, R. and Kumar, Y. Characteristic orthogonal polynomials in the study of transverse vibrations of nonhomogeneous rectangular orthotropic plates of bilinearly varying thickness. Meccanica, 2012, 47(1), 175–193.
12. Lal, R., Kumar, Y., and Gupta, U. S. Transverse vibrations of non-homogeneous rectangular plates of uniform thickness using boundary characteristic orthogonal polynomials. Int. J. Appl. Math. Mech., 2010, 6(14), 93–109.
13. Leissa, A. W. Vibration of Plate. NASA SP-160, U. S. Govt. Printing Office, 1969.
14. Patel, D. S., Pathan, S. S., and Bhoraniya, I. H. Infuence of stiffeners on the natural frequencies of rectangular plate with simply supported edges. International Journal of Engineering Research & Technology, 2012, 1(3), 1–6.
15. Sharma, S., Gupta, U. S., and Lal, R. Effect of pasternak foundation on axisymmetric vibration of polar orthotropic annular plates of varying thickness. J. Vib. Acoust., 2010, 132(4), 1–13.
16. Tariverdilo, S., Shahmardani, M., Mirzapour, J., and Shabani, R. Asymmetric free vibration of circular plate in contact with incompressible fluid. Appl. Math. Model., 2013, 37(1–2), 228–239.