8.22 A torsional system consists of a shaft with a disc of mass moment of inertia I mounted at its center. If both ends of the shaft are fixed, find the response of the system in free torsional vibration of the shaft. Assume that the disc is given a zero initial angular displacement and an initial velocity of 8.23 Find the natural frequencies for torsional vibration of a fixed-fixed shaft. 8.24 A uniform shaft of length / and torsional stiffness GJ is connected at both ends by torsional springs, torsional dampers, and discs with inertias, as shown in Fig. 8.31 State the boundary conditions. x = 0 k F x = 1 101 k12 102 F. Figure 8.31 Shaft connected to torsional springs, torsional dampers and discs at ends. 8.25 Solve Problem 8.23. Lif one end of the shaft is fixed and the other free.