The first symmetrical mode of vibration of a conical shell

Abstract

An aid in the design of loudspeakers would be a solution of the vibration of a conical shell. The general solution is exceedingly complex, but a partial solution can be obtained for the lowest symmetrical modes, the ones which appear to be most important in loudspeaker performance. The conical shell can be considered to be divided into identical radial lamina. The vibration is thus found to correspond to a beam whose width varies linearly with the distance from the apex, and whose stiffness varies inversely as the square of the distance from the apex. The problem, therefore, reduces to a fourth-order differential equation. Power series solutions of this equation are given in this paper. However, for the truncated cone the necessity of using four boundary conditions leads to very complicated calculations to obtain the eigentones. A more convenient method for obtaining the first symmetrical mode is the Rayleigh-Ritz method. For the free-free vibration, a simple function is found to fit the boundary conditions. This leads easily to the calculation of the resonant frequency. McLachlan has measured the fundamental frequency of such a metallic cone obtaining 4000 c.p.s. for its first symmetrical mode. The calculated frequency, using the method outlined above, is 3700 c.p.s. The first symmetrical mode in loudspeakers occurs at 500 to 900 c.p.s. and is controlled greatly by the boundary conditions at the skiver, spider, and dust cap.