SOME PROBLEMS ON FREE VIBRATION AND. FORCED MOTION OF BEAMS AND PLATES OF NON-UNIFORM THICKNESS

Abstract

The work presented in this thesis is an attempt to study some problems on free vibration and forced motion of rectangular beams and circular plates of non-uniform thickness. The whole range of the subject of study is covered in nine chapters. Chapter I is an introduction which presents a survey of the literature on the problems considered here. The rest of the thesis is divided in two parts : PART-I FREE VIBRATION OF RECTANGULAR BEAMS AND CIRCULAR PLATES Free vibration takes place when a system oscillates under the action of forces inherent in the system itself, and when the external forces are absent. The system when given an initial disturbance will vibrate at one or more of its natural frequencies, which are properties of the dynamical system determined by its mass and stiffness distribution. The resulting motion will be the sum of the principal modes in some proportion, and will continue ad infinitum in the absence of damping. Thus the mathematical study of free vibration yields information about the dynamic properties of the system, relevant for evaluating the response of the system under forced vibration. In this part, free vibration of rectangular beams and circular plates of non-uniform thickness is analyzed. It comprises of the following four chapters: In Chapter II, the free transverse vibration of a rectangular beam having two linear variation in thickness along the length is analyzed by classical theory of beams. The beam is assumed to be made up of two beam elements joined end to end having in general different linear variation in thickness. The solution of the equations of motion is obtained by Frobenius (power series) method. The arbitrary constants arising in the solution are solved by the end and continuity conditions. Numerical results for natural frequencies and normalized mode shapes are computed for first four normal modes for a clamped-clamped and a cantilever beam. The variation in thickness is taken in such a way that the average thickness of the beam remains constant. In Chapter. III;' the free transverse vibration of a rectangular beam having two linear variation- 'in thickness along the length is analyzed by shear theory of beams. The beam is assumed to be made up of two beam elements joined end to end and having in general different linear variation in thickness. The solution of the equations of motion is obtained by Frobenius method. The arbitrary constant arising in the solution are solved by end and continuity conditions. Numerical results: for first four normal modes for a clamped-clamped and cantilever beam are compared with those of classical theory. The variation in thickness is taken in such a way that average thickness of the beam remain constant. In Chapter IV, the free axisymmetric. vibration of a circular plate whose thickness, density and: elastic properties, along the radial , direction, vary in 'any: number of steps, is analyzed by classical theory of plates. Numerical results for natural frequencies and normalized mode shapes for first four normal modes of vibration for clamped, simply-supported..........