The Vibration Characteristics of Nearly-Complete, ``Free-Free'' Circular Rings

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Abstract
The frequencies, frequency ratios, amplitudes of vibration, and mechanical constants of nearly‐complete circular rings of various diameters and cross sections were measured and analyzed. The first six modes of vibration parallel to the plane of the ring, and the first five modes transverse to the plane of the ring, were obtained for several rings. The positions of the nodal points are tabulated. For the parallel vibration the frequency ratio fn/f1 is found to be independent of the cross section of the ring and for modes higher than the second are given accurately by the equation fn/f1=0.628(n−0.200)2 where n is the mode. The frequencies of parallel vibration are found to be given by the equation, fn = (Kn/D2) (B/m)½, where K1=0.285, K2=0.630, Kn=0.1795(n−0.200)2 (for n>2), D is the mean diameter, B the bending stiffness, and m the mass per unit length. For the transverse vibration the frequency ratios depend upon the type of cross section, but are of the general form, fn/f1=K2(A/C) [n+k(A/C)]2, for n>2. The values of K2(A/C) and k(A/C) are given in graphical form. A/C is the ratio of bending to twisting stiffness of the cross section of the ring. The frequency of transverse vibration is given by the equation, fn = [ψn(A/C)/D2] (A/M)½. The values of ψn(A/C) for a wide range of A/C values are given graphically for the first six modes. The distributions of amplitude of vibration around the rings for the first three modes of both parallel and transverse vibration are shown graphically.

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