The lateral vibrations of sharply-pointed bars
1920
In a preceding paper, a discussion was given of the lateral vibrations of bars of circular cross-section formed by the revolution of the curve y = A xn —when n is between the values zero and unity—about the axis of x . The matter arose in connection with the siliceous deposits found upon a certain type of sponge spicule, as discussed in a joint paper by Prof. Dendy and the present author. It is of some interest to obtain a more extended knowledge of the vibrations of solids belonging to this class, with a view to further applications. The phenomena presented change in a curious manner with the value of n , and, in certain respects, could not be foreseen in an elementary way. A discussion of the subject, in numerical terms, for an exponent n between 1 and 2 is very laborious, and in the present paper we confine attention to the case n = 2. This is a limiting case, which presents very exceptional features, and gives rise to a period equation of an unusual type. It illustrates clearly, at the same time, the effect of sharpening the ends of the rod beyond the point at which they are conical ( n = 1). The rod is a free-free bar, symmetrical about its axis, and each half is obtained by the revolution of a portion of a parabola about the tangent at its vertex.
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