SOME VIBRATION PROBLEMS OF ELASTIC PLATES

Abstract

The present study is an attempt at the investiga tion of some vibration problems of elastic plates. The whole range of the subject of study is covered in eight chapters, out of which the first seven deal with the free transverse vibrations of rectangular and circular plates of linearly or parabolically varying thickness with or without elastic foundation. The derivation of frequency equations and displacement functions for the plates of variable thickness on elastic foundation and their solutions for different combinations of boundary condi tions comprise the details of these chapters. Using high speed digital computer, numerical results for frequencies, deflections and moments corresponding to the thickness variation parameter and the foundation stiffness parameter have been presented both in tabular and graphical forms. The flexural vibrations of a circular plate according to Mindlin's theory resting on elastic foundation go in for the formation of the last chapter. The results obtained in this study have generally been compared with the published ones. For a survey of literature on vibration problems of elastic rectangular and circular plates, especially the plates of variable thickness with or without elastic founda tion taking into consideration the effects of shear defor mation and rotatory inertia on the frequencies of vibration, in the 'Introduction* the local libraries, National Docu mentation Centre, New Delhi, the various centres of advanced studies in Mathematics and the relevant published papers have been the provision sources. Chapter-wise summary of the thesis is given below. Chapter I Free transverse vibrations of an elastic Infinite plate with parabolic thickness variation in one direction resting on Winkler's elastic foundation have been studied on the basis of classical theory of plates. The governing differential equation of motion is solved by the method of Frobenius. The transverse displacement of the plate has been expressed as a power series and the frequencies, deflections and moments corresponding to the first two modes of vibration are computed for various values of found ation modulus and taper constant for two combinations of boundary conditions. Chapter II Effects of transverse shear deformation and rotatory inertia on the frequencies of vibration of an infinite plate of variable thickness on elastic foundation (Winkler's type) are undertaken for the present study. The governing differential equations of motion of the rectangular plates derived by Mindlin have been extended to the plates with parabolic thickness variation in one direction and are solved by Frobenius method. Transverse displacement and the LL UL local rotation of the plate are obtained as power series. Numerical results for the frequency parameter have been computed for the two combinations of boundary conditions and for various values of foundation modulus as well as taper constant. Some of the numerical results are compared with the results obtained on the basis of the classical plate theory. Chapter III This chapter is the study of free transverse vibra tions of a rectangular plate with parabolic thickness variation in one direction on elastic foundation satis fying Winkler's assumption on the basis of classical theory of plates. One pair of parallel edges of the plate is consiered as simply supported and the other one is taken as clamped-clamped and clamped-simply supported and Frobenius method is employed for the solution of the governing diff erential equation of motion. The transverse displacement of the plate represents the product of an infinite series and the function satisfying boundary conditions at the parallel edges, considered as simply supported. Frequencies for the rectangular plate (with one edge clamped other three edges simply supported and one pair of parallel edges clamped other simply supported) have been computed for various values of foundation modulus and taper constant corresponding to the first two modes of vibration while two ratios of length to breadth of the plate have been taken into account. Chapter IV Free axlsymmetric vibrations of an elastic circular plate of linearly varying thickness on an elastic founda tion on the basis of classical theory of plates form the subject matter of the study in this chapter. Frobenius method is applied solving the resulting differential equat ion of motion. The transverse displacement of the plate is expressed as an infinite series in terms of the radial coordinate. The frequencies, deflections and moments corresponding to the first two modes of vibrations are computed for the circular plate with clamped and simply supported edge conditions for various values of taper cons tant and foundation modulus. Chapter V Effects of transverse shear deformation and rotatory inertia on flexural vibrations of circular plates of linearly varying thickness resting on the Winkler's elastic foundation are studied here. The Mindlin's equations of motion have been extended to the plates of variable thickness, and solved by the method of Frobenius, The transverse displacement and the rotation of the plate with thickness variation in one direction are expressed as an infinite series in terms of the radial coordinate. Frequency parameters have been computed for the circular plate with clamped and simply supported edge conditions for various values of taper constant and «f foundation modulus. Comparison between the results so obtained and the corresponding results obtained on the basis of the classical plate theory is made. Chapter VI The present chapter deals with effects of transverse shear deformation and rotatory inertia on free transverse vib rations of circular plates of parabolically varying thickness on elastic foundation. Frobenius method is used to solve the governing differential equations of motion. The transverse displacement and the rotation of the plate with thickness variation in one direction are expressed as a power series. Numerical results for frequency parameter are computed for the circular plate with clamped and simply supported edge conditions and are compared with the results of the linearly varying thick plate. Chapter VII Rayleigh Ritz method is used for the study of the lowest natural frequency of the square plates of parabolically vary ing thickness point supported at the corners. Frequency parameters and amplitude ratios have been computed for various values of taper constant. The results thus, obtained are compared with the published results, pertinent to the uniform square plate-point supported at the corners. Chapter VIII The natural frequencies and corresponding mode shapes for the free axisymmetric vibrations of an isotropic elastic circular plate according to Mindlin's theory resting on Winkler's foundation are investigated by employing finite difference method. The governing differential equation of motion as given by Mindlin has been transformed into polar coordinates corresponding to the first three modes of vibration. Frequency parameters and mode shapes for a clamp ed circular plate have been computed. Matrix method has been employed to solve the set of homogeneous linear equations obtained after using finite difference equations. The computations reported in the present work have been carried out on an IBM-1620 Computer at the Computer Centre, Structural Engineering Research Centre, Roorkee.