A schematic of a clutch-testing machine is shown. The steel shaft rotates at a constant speed . An axial load is applied to the shaft

A schematic of a clutch-testing machine is shown. The steel shaft rotates at a constant speed ω. An axial load is applied to the shaft and is cycled from zero to P. The torque T induced by the clutch face onto the shaft is given by
T = f P (D + d)
4 Where D and d are defined in the figure and f is the coefficient of friction of the clutch face. The shaft is machined with Sy = 800 MPa and Sut = 1000 MPa. The theoretical stress concentration factors for the fillet are 3.0 and 1.8 for the axial and torsional loading, respectively.
(a) Assume the load variation P is synchronous with shaft rotation. With f = 0.3, find the maximum allowable load P such that the shaft will survive a minimum of 106 cycles with a factor of safety of 3. Use the modified Goodman criterion. Determine the corresponding factor of safety guarding against yielding.
(b) Suppose the shaft is not rotating, but the load P is cycled as shown. With f = 0.3, find the maximum allowable load P so that the shaft will survive a minimum of 106 cycles with a factor of safety of 3. Use the modified Goodman criterion. Determine the corresponding factor of safety guarding against yielding.