Abstract
The First Part of this thesis is concerned with the theoretical formulation, critical examination and practical applications of the Finite Element approximation in the vibration problems of elastic beams idealising the actual turbine blades. The method is presented in a generality encompassing a large body of useful linear vibration theories of beams. The basic and refined element models are introduced, and the convergence properties of each model studied. A rigorous proof of the convergence of the Finite Element solutions to the exact values is included in the case of torsional vibrations. The use of refined models is advantageous, especially in cases where the modal curves are of some complexity, e.g., vibrations at higher modes and beams with certain boundary conditions. The general non-dimensional forms of element dynamic-stiffness matrices for linearly pre-twisted beams are given. numerical results obtained with polynomial approximation in bending-bending and bending-bending-torsion vibrations of beams, where the effects of shear and rotatory inertia may not be negligible and linear taper may be present, show good convergence characteristics. Satisfactory results are obtained with the use of only a few, number of elements. The application of the method to vibrations of beams carrying concentrated masses and supported on elastic springs is briefly demonstrated. Representative results show good agreement with the exact values. The Finite Element Method provides strong and stable convergence characteristics. It is simple to apply and ideally suited to digital computers. On the other hand, the versatility of the method, in the treatment of boundary conditions, alone is a considerable advantage over the other more conventional numerical methods. In the Second Part of the thesis the Finite Element Method is applied to investigate the vibrational characteristics of uniform and tapered slender beams with, or without, pre-twist, and having various boundary conditions. The first four frequency ratios and modal shapes are presented for the following ranges of problem parameters; Pre-twist = 0 to 90 degrees, ratio of flexural rigidities = 1 to 256, deptli and width taper parameters = -0.5 to 0.5. The boundary conditions considered are those obtainable from combinations of free, pinned and clamped conditions at the ends of a beam. Theoretical results agree closely with experimental results presented in the thesis.