The computer evaluation of the natural frequencies of vibrating circular plates with free, fixed and simply supported edges
Department of Mechanical Engineering
Master of Science
Sun, Benedict C.
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.
njit-etd1970-002 (86 pages ~ 4,472 KB pdf)
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Created June 27, 2005