A flywheel with inertia $J$ is driven by an input torque and is connected to a lead screw that advances a

mass $m.$ The flexibility of the lead screw is modeled using a torsional spring. The lead screw and ball nut

form a transducer that converts rotational to translational motion similar to a rack and pinion but uses an

$N($threads $\frac{?}{cm})$ rather than a radius $r.$ First, define the transducer equations, then find the transfer

function for the system relating the input torque and the velocity of the mass.

For the problem above with $J=0\mathrm{.}05kg-m^2,K=50N-\frac{m}{r}ad,B=1,000\frac{N}{\frac{m}{s}},m=4kg,N=2$

threads$/$cm$,$ design a controller having the desired position of the mass as the input, and that uses

feedback to control the torque on the flywheel so that $T_2=K(x_{DCS}-x).$ Hint: use the kinematics $x^{}=v$ to

add an extra block in your feedback diagram to convert $v(s)$ to $x(s).$ Find $K_1$ so that the system has the

shortest settling time and $K_2$ to achieve the shortest peak time without the damping ratio decreasing

below $0\mathrm{.}6.$