An Approximate Formula for the Flexural Vibration of a Ring of Rectangular Cross Section

TLDR: In this article, the authors proposed an approximate formula which is applicable over the entire range of length-to-diameter ratios, obtained by combining the two previous formulae appropriately on the assumption of an analogy to a single-degree-of-freedom system of a mass and two springs in series, which depends not so much on a mathematical procedure as on an intuitional consideration.

Abstract: Concerning the out-of-plane twist-bending vibration of a ring of rectangular cross section, an analytical expression of the natural frequency is given by the theory of a circular arc bar for the region of small length-to-diameter ratios, whereas for the region of large ratios another one is given by the Rayleigh's theory of inextensional vibrations of a shell. However, those two formulae are both invalid in the middle range of the ratios. Especially from a practical point of view, the present paper proposes an approximate formula which is applicable over the entire range of length-to-diameter ratios, This formula is obtained by combining the two previous formulae appropriately on the assumption of an analogy to a single-degree-of-freedom system of a mass and two springs in series, which depends not so much on a mathematical procedure as on an intuitional consideration.