Date of Award

Spring 1970

Document Type

Thesis

Degree Name

Master of Science in Mechanical Engineering - (M.S.)

Department

Mechanical Engineering

First Advisor

Benedict C. Sun

Second Advisor

Arnold Allentuch

Third Advisor

Aaron Deutschman

Abstract

The natural frequencies of the transverse vibration of a thin, isotropic, circular plate with free, clamped, and simply supported edge conditions were studied extensively. The frequency equation for each edge condition was derived from the classical partial differential equation of plate vibration. These equations, which are in terms of Bessel functions, were then solved numerically to find the natural frequencies. Since the accuracy of the Bessel function values is very important in evaluating these frequencies, a comprehensive digital computer program was devised to calculate these values to eleven digit accuracy. In this Bessel function program four different methods were required to insure a rapid convergence. They are: (a) Infinite Series; (b) Asymptotic Series; (c) Recursion Formula; (d) Approximate Numerical Method.

The nodal patterns are known from the form of the solution to the fourth order partial differential equation of the vibrating plate. The order of the Bessel functions in the frequency equation corresponds to the number of equally spaced nodal diametral lines. The eigenvalues of this equation determine the number of concentric nodal circles which are present in the various nodal patterns. For each edge condition, twenty-six frequencies were computed for each of the first twenty-six orders of the frequency equation. The accuracy of these computations has been carried out to ten significant figures. Methods to be used in computing the radii of the nodal circles corresponding to these frequencies were also discussed. However, these values were not obtained.

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