Vibrations of beams and helices with arbitrarily large uniform curvature

An analytical, numerical, and experimental study of the vibrational modes of beams with constant curvature, ranging from small values up to helices with large numbers of turns, is presented. It is shown that, after an initial stage at low curvature in which extensional symmetrical modes hybridize so as to become inextensional, all modes show a decrease in frequency with increasing beam curvature. The frequency reaches a minimum at a value of the curvature which is a function of mode number and successive minima are separated by steps of π in the opening angle of the beam. For large values of curvature it is shown that, for both symmetric and antisymmetric modes, there are two types of vibrational modes with comparable frequencies. Modes develop into one or the other of these types in a way that is precisely defined but that has the appearance of being random. Physical descriptions of the processes involved are given, and the modes of the two types are described.