$3\mathrm{.}7($Motor drive$)$ Consider a system consisting of a motor driving two masses that

are connected by a torsional spring, as shown in the diagram below.

where $1$ and $2$ are the angles of the two masses, i $=$ d$$i$/$dt are their velocities,

Ji represents moments of inertia, c is the damping coefficient, k represents the shaft

stiffness, kI is the torque constant for the motor, and Td is the disturbance torque

applied at the end of the shaft. Similar equations are obtained for a robot with

flexible arms and for the arms of DVD and optical disk drives.

Derive a state space model for the system by introducing the $($normalized$)$

sptate variables x$1=1,$ x$2=2,$ x$3=$ $1/$$0,$ and x$4=$ $2/$$0,$ where $0=$

k$($J$1+$ J$2)/($J$1$J$2)$ is the undamped natural frequency of the system when the

control signal is zero.