The influence of a mass on the free flexural vibrations of a circular ring

The general solution was obtained for the free flexural vibrations in the plane of a thin circular ring containing a point mass. As a degenerate case of the general solution, the solution for a uniform ring alone was derived from the general solution by taking the point mass to be zero. Numerical calculations of the frequencies and mode shapes of the first and second flexural modes were made for values of the point mass in the range from zero to infinity. The results are presented in graphical form.

The predominant feature of the investigation was the difference in frequency and mode shape found in the symmetrical and antisymmetrical modes, and the particular orientation of the nodes with respect to the point mass. It was noted that similar phenomena were observed experimentally for vibrations of imperfect bodies of revolution. In conclusion, it was brought out that a ring with a point mass offers a convenient mathematical model for a preliminary theoretical investigation of the vibrations of imperfect bodies of revolution.