International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 11 Issue: 06 | June 2024
p-ISSN: 2395-0072
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“The effect of exponentially thickness variation and thermal effect on vibration of visco-elastic non-homogeneous rectangular plate having clamped boundary condition” Amit Kumar*, Lalit Kumar1 *Department of Mathematics,Government Model Degree College,(UP) India
1Department of Mathematics, Kisan (PG) College, Simbhaoli, Hapur (UP) India
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Abstract - Vibration reduction is a major challenge
thermal effect on axisymmetric vibrations of an orthotropic circular plate of parabolically varying thickness.
better designing to these fields; especially in a nuclear power project, aerospace applications, such a reduction must be achieved with a minimal increase in weight. We have assumed Kelvin model for visco-elasticity that is the combination of the elastic and viscous elements in parallel. Frequency equation is derived by using Galerkin’s technique with a two-term deflection function. A plate structure is of key interest to space craft, telephone industry, nuclear power projects, civil and mechanical engineers. The vibration of plates is a common case of the more general problem of mechanical vibrations. In this paper to analysis the effect of exponentially thickness variation and thermal effect on vibration of visco-elastic nonhomogeneous rectangular plate having clamped boundary condition on two parallel edges and simply supported boundary condition on remaining two edges. Deflection and Logarithmic decrement at different points for the first two modes of vibration are calculated for various values of nonhomogeneity constants, thermal gradients, aspect ratio and taper constants.
Tomar and Gupta [5] have discussed on vibration of an orthotropic elliptic plate of non-uniform thickness and temperature. Several authors have studied the thermal effect on vibration of homogenous plates of variable thickness but none of the authors has so far considered thermal effect on vibration of non-homogenous rectangular plates of exponentially varying thickness. Bambill et al [6] discussed on the transverse vibration of an orthotropic rectangular plate of linearly varying thickness and with a free hole edge. Recently, Li and Zhou [7] have studied on nonlinear vibration and thermal bucking of heated orthotropic circular plate by shooting method. It is well known [8] that in the presence of a thermal gradient, the elastic coefficient of homogenous materials become functions of the space variables. Pronsato et al [9] have discussed on transverse vibration of rectangular membrane with discontinuously varying density. A simple model presented here is to study the vibration of non-homogenous visco-elastic rectangular plate of exponentially varying thickness subjected to thermal gradient has been discussed in the present paper. Deflection and logarithmic decrement at different points for the first two mode of vibration are calculated for various values of thermal gradients, non-homogeneity parameter, taper constant and aspect ratio for non-homogenous visco-elastic rectangular plate with exponentially varying thickness and boundary condition of plate is clamped on two parallel edges and simply supported on remaining two edges.
Key Words: thermal gradient, visco-elastic, exponentially thickness, rectangular plate, taper constant, aspect ratio.
1. INTRODUCTION During heating up periods the structure is exposed to high intensity, high fluxes and the material properties thus undergo significant changes, in particular the thermal effect on the modules of elasticity of the material cannot be taken as negligible. In the advance time the effect of thermal and variable thickness on vibration of non-homogenous viscoelastic plates are widely used in better designing in civil, aeronautical, mechanical, electronics and marine structures like jet engines designing, space crafts, gas turbines, nuclear power projects etc.
1.1 ANALYSIS AND EQUATION OF MOTION The governing differential equation of transverse motion of a visco-elastic non-homogeneous plate of variable thickness in Cartesian co-ordinates, as by Gupta and Kumar [11],
Fnconneau and Marangoni [1] have studied on the effect of thermal gradient on the natural frequencies of a rectangular plate. Tomar and Gupta [2] discussed on thermal effect on frequencies of an orthotropic rectangular plate of linearly varying thickness. Rao and Satyanarayana [3] analyzed on the effect of thermal gradient of frequencies of tapered rectangular plates. Tomar and Gupta [4] have studied on
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