NATURAL VIBRATIONS OF CANTILEVERED TRIANGULAR PLATES I.
CARNEGIE INST OF TECH PITTSBURGH PA DEPT OF MATHEMATICS
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In this report the Rayleigh and Rayleigh-Ritz methods are applied to the natural vibrations of uniform triangular plates clamped on one edge and free on the other two. Two shapes of plate are investigated an isosceles plate clamped on the base and a right triangular plate clamped on one leg. Upper bounds are obtained for the fundamental modes and in some cases for the second and third modes as well. The following classes of functions are used to represent the displacement of the plate 1 One-parameter polynomials with integral exponents 2 Two-parameter polynomials with integral exponents 3 Threeparameter polynomials with integral exponents 4 Fiveparameter polynomials with integral exponents and 5 Monomials and binomials with variable exponents. In the absence of an exact solution of the problem the frequencies obtained are compared with experimentally determined frequencies for plates of similar shape. For this purpose a formula is developed which relates the frequencies of plates of similar planform but differing in size, thickness, and materials. The comparison between the analytical and experimental results is summarized in a table. While several of the simpler functions yield a satisfactory approximation to the fundamental frequency, only the five-parameter function of the functions thus far examined gives reasonable accuracy in the second and third modes, and at the same time it offers a good approximation to the fundamental. Author