Effect of Thermal Gradient on Vibration of Non-Homogeneous Square Plate with Exponentially Varying Thickness

Journal of Solid Mechanics

مقاله 8، دوره 7، شماره 4، اسفند 2015، صفحه 477-484   

نوع مقاله: Research Paper

نویسندگان

A Khanna^{* 1}؛ R Deep²؛ D Kumar²

¹Department of Mathematics, DAV College Sadhaura, Yamuna Nagar, Haryana

²Department of Mathematics, Maharishi Markandeshwar University- Mullana

چکیده

Vibrations of plate and plate type structures made up of composite materials have a significant role in various industrial mechanical structures, aerospace industries and other engineering applications. The main aim of the present paper is to study the two dimensional thermal effect on the vibration of non-homogeneous square plate of variable thickness having clamped boundary. It is assumed that temperature varies bi-parabolic i.e. parabolic in x-direction & parabolic in y-direction and thickness is considered to vary exponentially in x direction. Also, density is taken as the function of “x” due to non-homogeneity present in the plate’s material.  Rayleigh Ritz technique is used to calculate the natural frequency for both the modes of vibration for the various values of taper parameter, non-homogeneity constant and thermal gradient. All the calculations are carried out for an alloy of Aluminum, Duralumin, by using mathematica.

کلیدواژه‌ها

Vibration؛ frequency؛ Thermal gradient؛ Taper constant؛ Non-homogeneity constant

مراجع

[1] Gupta A.K., Khanna A.., 2007, Vibration of visco-elastic rectangular plate with linearly thickness variations in both directions, Journal of Sound and Vibration 301 (3-5): 450-457.
[2] Gupta A.K., Singhal P., 2010, Thermal effect on free vibration of non-homogeneous orthotropic visco–elastic rectangular plate of parabolically varying thickness, Applied Mathematics 1 (6): 456-463.
[3] Khanna A., Sharma A.K., 2012, Mechanical vibration of visco-elastic plate with thickness variation, International Journal of Applied Mechanical Research 1 (2): 150-158.
[4] Khanna A., Bhatia M., 2011, Study of free vibrations of visco- elastic square plate of variable thickness with thermal effect, Innovative System Design and Engineering 2 (4): 85-90.
[5] Leissa A.W., 1969, Vibration of Plates, NASA, SP-160.
[6] Huang C.S., Leissa A.W., 2009, Vibration analysis of rectangular plates with side cracks via the Ritz method, Journal of Sound and Vibration 323 (3-5): 974-988.
[7] Singh B., Saxena V., 1996, Transverse vibration of rectangular plate with bi- directional thickness variation, Journal of Sound and Vibration 198(1): 51-65.
[8] Fauconneau G., Marangoni R.D., 1970, Effect of a thermal gradient on the natural frequencies of a rectangular plate, International Journal of Mechanical Sciences 12(2): 113-122.
[9] Wu L.H., Lu Y., 2011, Free vibration analysis of rectangular plates with internal columns and uniform elastic edge supports by pb-2 Ritz method, International Journal of Mechanical Sciences 53(7): 494-504.
[10] Lee H.P., Lim S.P., Chow T., 1987, Free vibration of composite rectangular plates with rectangular cutouts, Composite Structures 8 (1): 63-81.
[11] Kuttler J.R., Sigillit V.G., 1983, Vibrational frequencies of clamped plates of variable thickness, Journal of Sound and Vibration 86(2): 181-189.
[12] Daleh M., Keer A.D., 1996, Natural vibration analysis of clamped rectangular orthotropic plate, Journal of Sound and Vibration 189(3):399-406.
[13] Jain R.K., Soni S.R., 1973, Free vibration of rectangular plates of parabolically varying thickness, Indian Journal of Pure and Applied Mathematics 4(3): 267-277.
[14] Lal Roshan., 2003, Transverse vibrations of orthotropic non-uniform rectangular plates with continuously varying density, Indian Journal of Pure and Applied Mathematics 34(4): 587-606.
[15] Malhotra S.K., Ganesan N., Veluswami M.A., 1988, Vibrations of orthotropic square plates having variable thickness (linear variation), Composites 19(6): 467-72.
[16] Alijani F., Amabili M., 2013, Theory and experiments for nonlinear vibrations of imperfect rectangular plates with free edges, Journal of Sound and Vibration 332(14): 3564-588.
[17] Johri T., Johri I., 2011, Study of exponential thermal effect on vibration of non-homogeneous orthotropic rectangular plate having bi- directional linear variation in thickness, Proceeding of the World Congress on Engineering, London .
[18] Sakata T., Hosokawa K., 1988, Vibrations of clamped orthotropic rectangular plates with C-C-C-C boundary conditions, Journal of Sound and Vibration 125(3): 429-39.
[19] Quintana M. V., Nallim L.G., 2013, A general Ritz formulation for the free vibration analysis of thick trapezoidal and triangular laminated plates resting on elastic supports, International Journal of Mechanical Sciences 69 (2013) :1-9.
[20] Sakiyama, T ., Huang M., 1998, Free vibration analysis of rectangular plates with variable thickness, Journal of Sound and Vibration 216(3): 379-397.
[21] Xing Y.F., Liu B., 2009, New exact solutions for free vibrations of thin orthotropic rectangular plates, Composite Structures 89(4): 567-574.