The Vibration Characteristics of ``Free-Free'' Circularly Curved Bars

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Abstract
The positions of the nodal points and the values of the frequencies of vibration have been determined for the first five or six modes of parallel and transverse vibration of circularly curved bars of uniform cross section with central angles ranging from 356° to 60°. For vibration in the plane of curvature (parallel) the relative positions of the nodal points are the same for the same central angles, regardless of the type of cross section. Also, for the parallel vibration the frequency in the nth mode is given by the expression, fn=θ2K(n,θ)L2(Bm)12,where θ is the half central angle in radians, L is the length, B is the bending stiffness of the cross section about an axis perpendicular to the plane of the ring, m is the mass per unit length, and K is a ``frequency constant'' which depends upon n and θ. For n>1 or 2, K may be expressed accurately as, K(n,θ)=S2(θ)[n−g(θ)]2K(1,θ). Graphs of S(θ), g(θ), and K(1, θ) are presented. For vibrations transverse to the plane of curvature the relative positions of the nodal points depend only upon the central angle. The frequency in the nth mode of transverse vibration is given by the expression, fn=θ2ψ(n,θ,A/C)L2(Am)12,where θ, L, and m are the same as defined above, A is the bending stiffness of the cross section about an axis in the radial direction, and ψ is a ``frequency constant'' which depends upon n, θ, and the A/C ratio. For n>2, ψ may be expressed quite accurately as, ψ(n,θ,A/C)=Q2(θ,A/C)[n−q(θ,A/C)]2ψ(1,θ,A/C). Graphs of Q(θ, A/C), q(θ, A/C), and ψ(1, θ, A/C) are presented.

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