published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. Description based on print version record. Includes bibliographical references (leaves 106-108).

Thesis (M.S.)--Rochester Institute of Technology, 2007. / Typescript. Includes bibliographical references (leaves 67-68).

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The classical method, required for its solution, the application of boundary conditions to the solution of the beam equation. Except for the case cf the beam with one concentrated load at the center, it was not considered a practical solution. The transcendental equation obtained in the solution of the unsymmetrical case, considered in part B, was found too cumbersome to handle. It was not attempted in parts C and D. The Rayleigh Method proved to be a simple, accurate and reasonably rapid method for all cases considered. The Dunkerley Equation gave very satisfactory results for parts A, B, and C. It was rapid to use, accurate and in most cases the data could be found in prepared tabulations. Results were inaccurate for the two span beam, indicating the necessity for caution in its application to multi-span beams. The Ritz Method, which is a refinement of the Rayleigh Method, proved to be exceedingly accurate when applied to the beam with the single concentrated load. However, it was found, that as the number of terms in the assumed deflection equation increased, the work became more time consuming. It was used only in parts A and B. The Influence Coefficient Method and the application of D'Alembert's Principle, which methods are quite similar, proved to be simple, accurate, and rapid. However, as the number of degrees of freedom increased, the degree of the algebraic equation increased, which complicated the solution. The Iteration Method is probably the method to be used if the number of degrees of freedom exceeds three. As the number of modes increases the number of iterations would increase, but the individual operations in themselves would remain simple. This method proved simple and accurate to use. For the cases considered, it was more time consuming to use than either the Influence Coefficient Method or the application of D'Alembert's Principle. However, for higher degree situations, it should prove to be a more practical method. / Master of Science

C.T.G. Looney proposed a method of solving the problem of steady-state forced vibration of continuous frames. Starting with the analysis of a single span, simply supported beam, the method was extended to continuous beams and frames. An investigation of his method was undertaken comparing experimental deflection with those predicted by Looney’s method. A single span beam and a two span continuous beam was used in the experiment. A mechanical shaker was used to vibrate the beam. The experimental and nautical curves showed good agreement with the experimental values being consistently higher. Although the weight of the shaker's moving parts was less than one-tenth the weight of the beam it had considerable effect in lowing the resonant frequency and increasing amplitudes. / M.S.

The purpose of this thesis has been to investigate the influence of secondary effects for beams resting on elastic foundations and vibrating beams. Two problems have been presented for beams on elastic foundations and a general comparison has been made for an infinite vibrating beam. / Master of Science

As a suggestion for further work in deriving the equations by the Principle of Virtual Displacements, it might be advisable to write the equations based on the vertical displacements of the points of inflection of each column. The concentrated mass of each panel would be located at the center of the respective columns. This appears to be a better approximation than concentrating each mass at the lower joint. / Master of Science