Problem $7-6$

$7-6$ Bevel gears transmit torsional, bending, and axial loads to rotating shafts, such as shown in Figure $13-36.$ For this problem, the free body diagram for a shaft ABC is shown in the figure, where points A and $C$ are at the bearings. Line $BG$ is on the bevel gear extended to the shaft center from point $G,$ the contact point with the mating gear. Transmitted from the mating gear are tangential$,$ radial, and axial forces $F_G^b=500lbf,F_G^z=58lbf,$ and $F_G^x=173lbf,$ respectively. The bearing at $C$ supports the axial thrust load. For the shaft material, $S_u=90k,S_y=65k,$ and has a fully corrected endurance limit of $40$ kpsi $.$ Let $(K_f{)}_a=2\mathrm{.}3,(K_f{)}_b=2\mathrm{.}0,$ and $(K_{fs}{)}_t=1\mathrm{.}5,$ where the subscripts a$,$ b$,$ and t represent axial, bending, and torsion, respectively. The diameter of shaft ABC is $1\mathrm{.}5$ in $.$

$($a$)$ Review the "Shaft Design Formulae Handout" and develop an equation for calculating the fatigue factor of safety for the case: completely reversed bending, steady torsion, and a constant axial force.

$($b$)$ Determine the value of the design factor. Do not ignore the axial force effect

$($c$)$ Determine the value of the design factor ignoring the axial force. What is the contribution from the axial force?

Prob