As shown in Figure $1($a$),$ a clutch is used to connect a driven shaft to a rotating load.

The clutch consists of two closely spaced plates separated by a fitm of oil. This

allows slippage between the plates, so that the input shaft and output shaft can

rotate at different speeds. In this way, a constant speed drive can be connected to

the input shaft, and the load will not start abruptly, but will gradually speed up$.$

The system can be modeled with a combination of a rotational inertia, I, and a

rotational damper, with coefficient, c as shown in Figure $1($b$).$ Note this

simplified model assumes that the inertia of the clutch elements is negligible

a$)$ Find a transfer function, $T(s)=\frac{s}{{}_{in}}(s)$ relating the output angular velocity

of the load to the input, angular velocity applied to the input shaft

b$)$ The moment of the inertia for the load may be calculated by assuming it is a

solid cylinder with mass, $m$ and radius $R.$

Given; $m=2kg,r=0\mathrm{.}5m,c=3\frac{N-m}{ra\frac{d}{s}},$ Use MATLAB to plot the step

response if the load is initially at rest, and the drive speed is suddenly increased

from $0$ to $3ra\frac{d}{s}$

c$)$ Use the final value theorem to calculate the final value for a $3ra\frac{d}{s}$ step input.

Compare with results from $1$ b and check if they agree.

d$)$ From the root, call it $,$of the denominator of the transfer function T$($s$)$ found in

$1$a$,$ determine the system time constant, T $=1/$Re$(1)$and estimate the settling

time using ts $5$t$.$ Check

if the result agrees with your simulated step response.