Torsional Vibrations of Conical Frustum

This paper examines the effect of conicity, cone thickness, and cone completeness on the free axisymmetric torsional vibrations of a thick, hollowed, conical frustum cantilevered at the small end. The solutions for eigenfrequencies and eigenmodes are obtained by finite difference approximations of the governing basic equation of elastic motion, written in terms of a set of orthogonal conical coordinates. For a cone angle of 180°, the results agree with known frequencies for the case of an annular circular plate fixed along the inner radius and free along the outer radius. The results show that the frequencies increase with an increase in conicity, this rate being greater for thick cones than for thin cones. The effect of an increase in thickness is to decrease the eigenfrequencies. The frequencies, as expected, are found to decrease as the cone approaches completeness, and this effect is very strong. In fact, in comparison the dependence of frequency upon thickness and conicity is slight.