Dynamic Analysis of Rings by Finite Differences

The solution for the free vibrations of initially disturbed thin elastic rings is obtained by converting the differential equations of motion to two sets of finite difference equations: a conventional set, and a modified set based on a technique first presented by Chuang and Veletsos in their investigation of the static equilibrium of cylindrical shells. In the conventional set, both of the displacement variables are defined at the same location on the finite difference network, but in the modified set the two displacement variables are defined at staggered locations. The analytical frequency equation associated with both sets of difference equations is derived and shown to be correct by an actual computer solution. A sample ring is considered and the superiority of the modified difference equation is shown by comparing the natural frequencies of both sets of difference equations with the exact natural frequencies.