Thermal Effect on Vibration of Tapered Rectangular Plate

A mathematical model is constructed to help the engineers in designing various mechanical structures mostly used in satellite and aeronautical engineering. In the present model, vibration of rectangular plate with nonuniform thickness is discussed. Temperature variations are considered biparabolic, that is, parabolic in x-direction and parabolic in y-direction. The fourth-order differential equation of the motion is solved by Rayleigh Ritz method for three different boundary conditions around the boundary of plate. Numerical values of frequencies for the first two modes of vibration are presented in tabular form for different values of thermal gradient, taper constants, and aspect ratio. 1. Introduction The structures are designed to support the high speed engines and turbines subjected to the vibration. Due to faulty design, there is unbalance in the engines which causes excessive and unpleasant stresses because of vibration. In the field of science and technology, it is preferred to design large machines and structures for smooth operations with controlled vibration. Recent studies in the field of vibrational behavior create a huge interest for scientists and engineers in the designs and constructions of complex systems or structures such as ships, submarines, aircrafts, launch vehicles, missiles, and satellites. Tapered plates or plates of nonuniform thickness are commonly used in many engineering applications such as nuclear engineering, aeronautical engineering, and chemical plants, under the influence of elevated temperature to control high vibration. A collection of research papers on vibration of plates with different shapes and boundary conditions is given by Leissa [1] in his monograph. Leissa [2] discussed different models on free vibration of rectangular plates. Jain and Soni [3] analyzed the free vibrations of rectangular plates with parabolically varying thickness. Tomar and Gupta [4] studied the effect of thermal gradient on the vibration of a rectangular plate with bidirectional variation in thickness. Leissa [5] investigated the effect of thermal gradient on the vibration of parallelogram plate with bidirectional thickness variation in both directions. Singh and Chakraverty [6] studied the transverse vibration of circular and elliptical plates with variable thickness. Sharma and Chand [7] analyzed the vibrations in transversely isotropic plates due to suddenly punched hole. Leissa [8] discussed the historical bases of the Rayleigh and Ritz methods for the vibrations of plates. Chakraverty et al. [9] studied the flexural vibrations of