Abstract
The transverse vibrations of rigid elastic flexible turbine discs, centrally clamped, are investigated theoretically by using the Rayleigh-Ritz method, and experimentally by applying the techniques of Holographic Interferometry to determine their mode shapes patterns. This thesis deals with discs of a uniform constant thickness, and with discs whose thickness is variable either hyperbolically or linearly at any radius r. It also deals with discs when they are rotating about their polar axis with constant angular speed, and when they are not rotating and clamped at the centre. The discs are considered as annular plates, the boundary conditions being free at the outer radius and rigidity clamped at the inner radius. Throughout the thesis a and b will define the outer and clamped radii respectively. Frequency values are obtained under the condition of various clamping ratios b/a, that is to say the ratio of the clamped inner radius to the free outer radius of the disc. The effect of the ratio b/a, the variation in thickness, and rotation on the frequency of vibration of the discs are considered and discussed. The thesis consists of three parts. The first part is the analytical work. The equation of kinetic and strain energies of rotating uniform and variable thickness disc rigidly clamped at its centre and free at its outer edge are derived. The object is to obtain the frequencies eign values by the Rayleigh-Ritz approximate method using the strain and kinetic energies in the form of integral functions and under the Axisyrametric, without nodal diametre, and Asymmetric, with nodal diametres, conditions. The eign values representing the frequences p of vibration in rad/sec. in terms of the dimensionless frequency parameter A are obtained theoretically for a number of cases as follows: a) The umbrella mode using a single term in the functional series used to represent the lateral displacement y. b) The first twelve nodal diameter mode, using a single term solution, that is to say nodal diameters n - 1,2, 3,...12 and nodal circles s = 0. c) A two term solution is used to obtain the first five modes of vibration with one nodal circle, that is to say n = 0,1,2,3,4 and s = 1. Note that this solution will also give better approximations for the cases n = 0,1,2,3,4 and s = 0. Frequency results are obtained for both discs of constant thickness and discs of variable thickness. The frequencies obtained above by the Rayleigh-Ritz method are compared with the exact values given by Southwell and by Raju and Govel for case of discs of constant thickness. The results obtained in case of variable thickness discs are compared to the values obtained by Kovalenko. The theoretical results are also compared to the results obtained experimentally. The second part of the thesis is the experimental analysis which is concerned with determining the frequencies and mode shapes of vibration of a number of circular discs, clamped at the centre making use of a holographic interferometer incorporating a Helium-Neon continuous wave laser. The theoretical principles of holography are explained briefly with reference, in particular, to the vibrational measurement of engineering components. The advantages of holographic interferometry over the classical sand pattern method, and the disadvantages and limitation of both methods are discussed. The time-average fringe holographic technique is used to obtain the nodal patterns of the stationary discs. The principle of the double-exposure holographic method is mentioned. However in order to use the method of double exposure which could be applied to a rotating disc, a pulsed laser is required. Making use of the time-average technique, nodal patterns are obtained for nodal circles modes, nodal diameters modes, and combinations of nodal circles and diameters modes for the case of a nonrotating disc. The sand pattern method is also used to determine the mode shapes of the discs. The practical requirements for a holographic interferometer in terms of the equipment available are examined. The arrangement finally adopted is limited to measurements on an object up to 10 inches in diameter. The third part of the thesis deals with the numerical computation and the experimental results. Selected tables and graphs are presented with a discussion of the results. The theoretical and experimental results are compared. The nodal patterns obtained by the time-average holographic technique and by the sand pattern method are also compared. The integral functions of the energy expressions are solved by Simpson's rule numerically using Fortran IV computer language. The effects of thermal stresses on the disc deflections and shearing stresses acting on the normal section of the disc are considered to have a negligible affect on the disc vibrational behaviour.