Problem $1$:

$($a$)$ The shaft shown is part of a material$-$handling system aboard a ship. All power $(50$ hp $)$ comes into the shaft through a chain sprocket at point C$.$ The shaft rotates at $480$ rpm $.$ The V$-$belt sheave at A delivers $30$ hp to a hoist. The gear pair at E delivers $20$ hp to hydraulic pumps. The sprocket, sheave, and gear are located axially by retaining rings. The shaft is made from SAE $3140$ OQT $1000$ steel. Using a design factor of $3$ and assuming there are no frictional losses, determine the minimum acceptable diameters $(D_1-D_7)$ at each section on the shaft based on fatigue analysis.

$($b$)$ Select bearings for the above$-$mentioned shaft. $[$Note: as mentioned earlier, the shaft is part of materialhandling system, $480$ rpm $]$