Exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders

Exact solutions are obtained for transient torsional responses of a finitely long, functionally graded hollow cylinder under three different end conditions, i.e. free–free, free–fixed and fixed–fixed. The cylinder with its external surface fixed is subjected to a dynamic shearing stress at the internal surface. The material properties are assumed to vary in the radial direction in a power law form, while keep invariant in the axial direction. With expansion in the axial direction in terms of trigonometric series, the governing equations for the unknown functions about the radial coordinate r and time t are deduced. By applying the variable substitution technique, the superposition method and the separation of variables consecutively, series-form solutions of the equations are obtained. Natural frequencies and the transient torsional responses are finally discussed for a functionally graded finite hollow cylinder. This project was supported by the National Natural Science Foundation of China (10872179 and 10725210), the Zhejiang Provincial Natural Science Foundation of China (Y7080298) and Zijin Plan of Zhejiang University.