Problem A double reduction gearbox design has developed to the point that the general layout and axial dimensions of the countershaft carrying two spur gears have been proposed, as shown in Figure $1($a$).$ Figure $1($b$)$ shows that the gears and bears are located and supported by shoulders and held in place by retaining rings. The gears transmit torque through keys. Gears have been specified as shown in Figure $1($c$),$ allowing the tangential and radial forces transmitted through the gears to the shaft to be determined as follows.

$W_{23}^t=2000N,W_{54}^t=10000N$

$W_{23}^r=1000N,W_{54}^r=4000N,$

where the superscripts $t$ and $r$ represent tangential and radial directions, respectively, and the subscripts $23$ and $54$ represent the forces exerted by gears $2$ and $5$ on gears $3$ and $4,$ respectively. Proceed with the next phase of the design, in which a suitable material is selected, and appropriate diameters for each section of the shaft are estimated, based on providing sufficient fatigue and static stress capacity for infinite life of the shaft, with minimum safety factors of $3\mathrm{.}0.[$Guideline: Follow the steps presented in Example $7-2,7-3,$ and $7-4,$ and the report should include the FEA results $($the Von$-$mises contour and deflection, Refer to Figure $19-7$ on Page $968$ of our Textbook$)$ for the final design.$][$Changed diameters: $d_3=300mm$ and $d_4=60mm]$