An integral equation approach to vibrating plates

Abstract

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.