A $22kg$ compressor impeller wheel is driven by a $13kg$ turbine mounted on a common steel shaft, as shown in Figure $1$ below. E $=207$ GPa. Assuming the bearings to be stiff and acting as supports.

Calculate:

a$)$ The reactions of the supports at A and $C$

$[4]$

b$)$ The shaft deflection at B and D using Macaulay's method.

$[11]$

Figure $1$: Turbine shaft