Thermally Induced Vibration of Non-homogeneous Trapezoidal Plate Whose Thickness Varies Linearly in One and Parabolically in Other Direction with Linearly Varying Density
Keywords:
non-homogeneous problem, thickness, density, trapezoidal plate, frequenciesAbstract
Based on classical plate theory, the natural frequencies of thermally induced vibration of non-homogeneous trapezoidal plate with varying thickness linearly in x-direction and parabolically in y-direction has been calculated by Rayleigh–Ritz method. Due to linear variation in density non-homogeneity arises in plate’s material. The frequency equation has been obtained by assuming the two-term deflection function with clamped-simply supported- clamped-simply supported boundary condition. For a symmetric, non-homogeneous trapezoidal plate the effect of non-homogeneity constant, aspect ratios, thermal gradient and taper constants on the frequencies has been studied for first two modes of vibration. All the numerical results which have been obtained presented in tabular and graphical form.
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References
W. Nowac ki, Thermo Elasticity, Pergamon Press, New York, 1962.
S. M. Hassan, M. Makary, “Transverse vibrations of elliptical plate of linearly varying thickness with half of the boundary clamped and the rest simply supportedâ€, International Journal of Mechanical Sciences, vol.45, no.5, pp. 873-890, 2003.
S. Chakraverty, Ragini Jindal, V. K. Agarwal, “Vibration of non-homogeneous orthotropic elliptic and circular plates with variable thicknessâ€, Journal of Vibration and Acoustics, vol. 129, no. 2, pp. 256-259, 2007.
R. Lal, U. S. Gupta, C. Goel, “Chebyshev collocation method in the study of transverse vibrations of non-uniform rectangular orthotropic platesâ€, The Shock and Vibration Digest, vol. 33, no. 2, pp. 103-112, 2001.
P. A. A. Laura, R. H. Gutierrez, R. B. Bhat, “Transverse vibrations of a trapezoidal cantilever plate of variable thicknessâ€, AIAA Journal, vol. 27, no.7, pp. 921-922, 1989.
O.G. McGee, T.S. Butalia, “Natural Vibrations of shear deformable cantilevered skewed trapezoidal and triangular thick platesâ€, Computers and Structures, vol. 45, no.5-6, pp. 1033–1059, 1992.
N. Bhardwaj, A.P. Gupta, “Axisymmetric vibrations of polar orthotropic circular plates of quadratically varying thickness resting on elastic foundationâ€, International Journal of Structural Stability and Dynamics, vol. 5, no.3, pp.387-408, 2005.
Kavita, Satish Kumar, Pragati Sharma, “Study of Thermally Induced Vibration of Non-Homogeneous Trapezoidal Plate with Parabolically Thickness Variation in Both Directionsâ€, Applied Mathematics, vol. 7, no.12, pp. 1283-1296, 2016.
Kavita, Satish Kumar, Pragati Sharma, “Study of Temperature Behaviour on Thermally Induced Vibration of Non-Homogeneous Trapezoidal Plate with Bi-Linearly Varying Thicknessâ€, Journal of Applied Mathematics and Physics, vol. 4, no. 10, pp. 1936-1948, 2016.
K.M. Liew, M.K. Lim, “Transverse Vibration of Trapezoidal Plates of Variable Thickness: Symmetric Trapezoidsâ€, Journal of Sound and Vibration, vol. 165, no. 1, pp. 45-67, 1993.
K. Hosokawa, J. Xie, T. Sakata, “Free vibration analysis of cantilevered laminated trapezoidal platesâ€, Science and Engineering of Composite Materials, vol. 8, no. 1, pp. 1–10, 1999.
H. A. Larrondo, D. R. Avalos, P.A.A. Laura, R. E. Rossi, “Vibration of simply supported rectangular plates with varying thickness and same aspect ratio cutoutsâ€, Journal of Sound and Vibration, vol. 244, no.4, pp. 738-745, 2001.
A. K. Gupta, A. Kumar, Y. K. Gupta, “Vibration of visco-elastic parallelogram plate with parabolic thickness variationâ€, Applied Mathematics, vol. 1, no. 2, pp. 128-136, 2010.
A. Y. T. Leung, C. Xiao, B. Zhu, S. Yuan, “Free vibration of laminated composite plates subjected to in-plane stresses using trapezoidal p-elementâ€, Composite Structures, vol.68, no.2, pp.167–175, 2005.
A. K. Gupta, A. Khanna, S. Kumar, M. Kumar, D. V. Gupta, P. Sharma, “Vibration analysis of visco-elastic rectangular plate with thickness varies linearly in one and parabolically in other directionâ€, Advanced Studies in Theoretical Physics, vol. 4, no. 15, pp. 743–758, 2010.
A. K. Gupta, P. Sharma, “Thermal Effect on vibration on non-homogeneous trapezoidal plate of linearly varying thicknessâ€, International Journal of Applied Mathematics and Mechanics, vol. 7, no. 20, pp. 1-17, 2011.
A. K. Gupta, P. Sharma, “Study the Thermal Gradient Effect on Frequencies of a Trapezoidal Plate of Linearly Varying Thicknessâ€, Applied Mathematics, vol. 1, no. 5, pp. 357-365, 2010.
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