An electric motor drives an air compressor through two gears with speed reduction ratio $2$:$1.$

The moments of inertia of the motor rotor, the pinion and the gear wheel are $0\mathrm{.}015kg*m^2,$

$0\mathrm{.}001kg*m^2$ and $0\mathrm{.}01kg*m^2$ respectively. The compressor has rotating masses $1\mathrm{.}7kg,$

reciprocating masses $1\mathrm{.}6kg$ and crank radius $60mm.$ The shaft connecting the motor to the

pinion is $509mm$ long and has a diameter $30mm,$ while the shaft connecting the gear wheel

to the compressor is $1260mm$ long and has a diameter $38mm.$ diameter. Both shafts have

modulus of rigidity $G=8{10}^{10}\frac{N}{m^2}.$ The resisting torque of the compressor is given by$,$

$T=-200-40sin1000t,$ N$.$m$,$

where $t$ is time in seconds.

a$)$ Determine the steady state amplitudes of torsional oscillation of the motor rotor and the

compressor crank.

b$)$ Plot the torques on the motor and compressor shafts versus time.

c$)$ Determine the maximum torsional stress in each Shaft.