Q$\mathrm{.}2.$

The power$-$transmitting system shown in above figure consists of a helical gear, a bevel gear, and a shaft that rotates at $400$ rpm and is supported by two roller bearings. The load on the bevel gear $($at right end$)$ is $P_x,P_y$ and $P_z$ with the magnitudes of $3409N,2795\mathrm{.}3N,$ and $3000$ N respectively. All length dimensions are in millimeters.

$($a$)$ Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports.

$($b$)$ Draw shear$-$force and bending$-$moment diagrams for the shaft.

$($c$)$ Find the shaft diameter rounded up to the nearest $5$ mm $.$and assuming no step throughout the shaft for ground AISI $1040$ Steel $(S_{ut}=620$MPa,$E=200$GPa, density$=7800k\frac{g}{m^3},$ specific weight$==76518\frac{N}{m^3})$ considering infinite life. Take reliability factor as $97\%$ and factor of safety as $3\mathrm{.}5.$

$($d$)$ Find the shaft's critical speed in rpm without the loads.

$($e$)$ Select identical direct mounted single row tapered roller bearings for $10,000-$hour life at $98\%$ overall reliability.

Use application factor of $1\mathrm{.}2.$