PROBLEM'S OK VIBRATION OF BEAMS, PLATES AND SHELLS

Abstract

This thesis is an attempt to study some of the problems on vibration of uniform isotropic beams, plates and shells. It is divided into three parts. The first part deals with the flexural vibrations of beams which include the effect of internal viscous damping in addition to the effects of rotatory inertia and shear. Viscoelastic beams with the standard linear element type of internal damping are studied. Numerical solutions are obtained for vibrations which are purely harmonic in time and for those which are purely harmonic in distance. The numerical solutions are obtained also for forced and free vibrations of a cantilever beam. The second part deals with flexural vibrations of elastic moderately thick circular plates. The equations of vibration given by Mindlin are taken and their solution is obtained in terms of Bessel functions with the aid of three auxiliary variables. The natural frequencies and relative displacements are obtained for torsionless axisymmetric vibrations with free and clamped edge conditions and for non symmetric vibrations with clasped edge condition. Comparison is made with the corresponding frequencies of a circular plate given by the classical plate theory. The third part of the thesis deals with vibrations of elastic spherical shells and comprises chapters III to VIII. In chapter III, the equations of vibration of moderately thick non shallow spherical shells are derived from the three dimensional equations of elasticity by integration over the shell thickness. These equations include the secondary effects of rotatory inertia and radial shear, and their validity is verified by energy considerations. It is shown that these equations satisfy several fundamental theorems of elasticity. ii In chapter IV, the equations governing torsionless axisymmetric vibrations of non shallow shells are given when the surface loads are absent and their solution is obtained in terms of Legendre functions. Numerical results are obtained for the natural frequencies, mode shapes and strain energies due to stretching, bending and shear strains of a hemispherical shell with free edge. Chapter V deals with"torsional and pure thickness-shear vibrations of non shallow shells. Numerical results are obtained for shells of various opening angle. In chapter VI, a system of five auxiliary variables is employed to obtain the solution for non symmetric vibrations of non shallow shells in terms of associated Legendre functions. Both cases when the secondary effects of rotatory inertia and radial shear are taken Into account and when they are neglected, are included. Numerical results are obtained for the natural frequencies and mode shapes of a hemis pherical cap with clamped edge both for torsionless axisymmetric and non symmetric vibrations. Chapters Til and VIII deal with vibrations of shallow spherical shells, in chapter V-i, the equations governing vibrations of moderately ick shallow shells are derived from those given in chapter III and their solution is obtained in terms of Bessel functions, in tal results arc obtained for the natural frequencies and mode shapes of a shallow ith cj edge. Numerical results for free edge condition are also included. In chapter VIII, the equations of vibration of thick shallow shells are derived. These equations include the effec s of variation of normal stress «- and radiel -iisplacement w over the thickness of the sheil in addition to those of rotatory inertia and radial shear. A system of six auxiliary variables is employed in deriving the sep arable solution in terms of Bessel functions. Numerical results ar<- iii obtained for natural frequencies for torsionless axisymmetric vibra tions of a shallow cap with clamped edge. The numerical work presented is chapter I Is done on a des calculator, and that presented in chapters II and IV to VIII is done on IBM 1620 Computer in about 80 hours duration. The whole work presented in this thesis is original research by the author except sections 1.1, 1.2, 2.1 which have been put in to present a connected account of the whole. The material of chapters I and III has been published in University of Roorkee Research Journal. The material of chapter IV was presented by the author in Ninth Congress of Theoretical and Applied Mechanics held at Kanpur In December, 196lt. A note may be added about nomenclature. It will be seen that axisymmetric motion may be of three types, namely torsionless, torsional and pure thickness-shear motions. As we deal largely with torsionless axisymmetric motion, we have often used for brevity the terms 'torsion less' or'axisymmetric' motion for this type of motion. When we refer in the text to any equation occuring in another chapter, we generally mention the section number first and then the equation number next within brackets. For example, eqn. 3.U (^+.1) stands for eqn, (^.1) of chapter III section 3.*r.