The rotational system shown in Figure $2$ is a simple model of a machine tool drive system. The

nonlinear friction effect NLD between inertia $J_2$ and ground is given by

$T_{NLD}={}_2(\frac{T_0}{|{}_2|})+C|{}_2|{}_2$

where $C$ and $T_0$ are positive constants. Let $T_K$ be the shaft torque in the torsional spring $K.$

The equations of motion are given by

$T_{}^{}=K({}_1-{}_2)$

$J_1{}_1^{}=-T_K+T_s$

$J_2{}_2^{}=T_K-T_{NLD}$

$($a$)$ Consider small perturbations of all variables for the case when ${}_2$ is always positive and

varies about a mean value $\frac{?}{\text{b}\text{a}\text{r}}({)}_2.$ Find the corresponding mean values of $T_{},{}_1,$ and $T_{}$ in

terms of ${}_2.$ Let the perturbations be denotod by widehat$({)}_1(t),$widehat$({)}_2(t),$widehat$(T{)}_K(t),$ and hat$(T{)}_s(t).$ Write

the corresponding linearized equations.

$($b$)$ Write the state space model of the linearized system with input $T_n(t),$ output ${}_2(t)$ and

the perturbed variables associated with the energy storage elements as state variables.