On the lateral vibrations of bars of conical type

This paper contains a discussion of the lateral vibrations of a thin conical bar of circular section, which has its tip free. By means of a discussion of the roots of certain equations containing Bessel functions of the second and third orders, both of real and of imaginary argument, the frequencies and nodal arrangement associated with the first three tones are investigated, in the case when the base of the bar is clamped. The lateral vibrations of conical bars of circular section were first treated by Kirchhoff, in 1879. In his investigations, Kirchhoff was concerned with the case of a bar with its tip free and its base clamped, and he limited himself to a discussion of the period associated with the gravest tone, and considered neither the higher periods nor the positions of the nodes associated with them. J. W. Nicholson, in the course of an investigation of the lateral vibrations of certain types of bars of variable section, discussed the case of a double cone (consisting of two equal cones placed base to base,) vibrating with both tips free, and discussed the periods and nodal ratios associated with the first three symmetrical tones. His results, however, throw no light on the question of the vibrations of a conical bar with a clamped base, owing to the peculiar nature of the conditions at the centre.