Digitized by Kansas Correctional Industries

Digitized by Kansas Correctional Industries

published_or_final_version / Civil Engineering / Master / Master of Philosophy

Thesis--Bonn. Extra t.p. with thesis statement inserted. / Includes bibliographical references (69-72).

Thesis--Bonn. Extra t.p. with thesis statement inserted. / Includes bibliographical references (69-72).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1965. / Typescript. Vita. Includes bibliographical references.

A knowledge of the natural frequencies of a vibrating plate is of great importance if an effective design is to be made which will prevent critical conditions of heavy vibration from occurring. Those frequencies which are associated with the symmetric modes are especially important. Many approximate methods have been devised to determine these natural frequencies. In this dissertation a method of frequency determination is suggested through an integral equation approach. The plate vibration problem is formulated as a problem in the solution of a homogeneous, linear Fredholm integral equation of the second kind in which the kernel is either symmetric or can be made so by a convenient transformation. The integral equation, as formulated, satisfies the boundary conditions in that it includes Green's function of the plate which is a solution of the isolated force problem. Three approximate methods for solving the integral equation are described mathematically and then applied to three elementary examples. The three methods used ares 1) method of successive approximation, 2) method of collocation and 3) the trace or the kernel. It is shown that using the trace of the kernel always gives a lower bound to the frequency and is particularly useful for the determination of the fundamental frequency. After solving the three elementary problems the integral equation approach is made to the uniform circular cantilever plate where the frequency is approximated both by collocation and by the use of the trace of the kernel. The first and second approximate mode shapes are then derived and shown graphically. The results are seen to compare favorably with results obtained from the Rayleigh-Ritz method. Finally, the fundamental frequency is determined for the circular, stepped cantilever plate and the clamped elliptical plate. For the stepped plate fundamental frequency curves are drawn for various positions and magnitudes of the step. The fundamental frequency curve of the clamped elliptical plate is drawn as a function of the eccentricity of the ellipse. A frequency obtained from experiment is reported along with a calculated value determined from the Rayleigh-Ritz method. It is seen that the integral equation approach is about 19% below the experimental value whereas the Rayleigh-Ritz method gives a fundamental frequency about 27% above the experimental value. / Ph. D.

Whereas analytic and experimental investigations of plates subject to lateral vibrations have been rather thorough, the present study is an analytic determination of the various critical frequencies of vertically cantilevered thin elastic rectangular plates vibrating freely within their own planes. Within the restrictions imposed by excluding any motion perpendicular to the face of the plate, the upright edges are free to move in the other two directions, as is the top horizontal edge. Three different base conditions are imposed: • A clamped lower edge; • A lower edge which is freely vibrating transversely in the plane of the wall where the vertical fibers of the wall are fixed at their roots; and • A horizontally freely pulsating lower edge where the vertical fibers of the wall are fixed at their roots. The first two conditions are considered in relation to plate vibrations which are essentially vertical while the first and third conditions are each employed with essentially horizontal plate vibrations. In every case the effect of a uniform load placed along the upper edge is studied. Critical frequencies and associated amplitude coefficients are obtained for various ratios of base length to wall height. The solution, which is presented in tabular and graphic forms, is obtained by using the method of iteration on the Rayleigh-Ritz energy procedure. It is concluded that, for a wall with a clamped base vibrating in accordance with the given stipulations, the fundamental period is proportional to the square root of the face area of the wall. When the base of the wall is vibrating there is only one critical period, and it varies with the height of the wall. The factor of proportionality should take into account the material of which the wall is composed. For designing unframed walls, subjected to dynamic loads in their plane, where the applied shear is to be taken as some constant times the dead load at the base of the wall, the recommended lateral force requirements of the Seismology Committee of the Structural Engineers Association of California, as set forth in 1959, seem adequate as modified above. / Ph. D.