(untitled) (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, exact vibration frequencies of circular, annular and sector membranes with a radial power law density are presented for the first time. It is found that in general, the sequence of modes may not correspond to increasing azimuthal mode number n. The normalized frequency increases with the absolute value of the power index |ν|. For a circular membrane, the fundamental frequency occurs at n = 0 where n is the number of nodal diameters. For an annular membrane, the frequency increases with respect to the inner radius b. When b is close to one, the width 1 − b is the dominant factor and the differences in frequencies are small. For a sector membrane, n − 1 is the number of internal radial nodes and the fundamental frequency occurs at n = 1. Increased opening angle β increases the frequency.