Exact solution for forced torsional vibration of finite piezoelectric hollow cylinder

An exact solution is obtained for forced torsional vibration of a finite class 622 piezoelectric hollow cylinder with free-free ends subjected to dynamic shearing stress and time dependent electric potential at both internal and external surfaces. The solution is first expanded in axial direction with trigonometric series and the governing equations for the new variables about radial coordinate r and time I are derived with the aid of Fourier series expansion technique. By means of the superposition method and the separation of variables technique, the solution for torsional vibration is Finally obtained. Natural frequencies and the transient torsional responses for finite class 622 piezoelectric hollow cylinder with free-free ends are computed and illustrated.