$(10$ pts$)($Pendulum problem with control$)$ Consider a controlled linear pendulum system

${}^{}(t)+g(t)=u(t)$

where $u(t)$ is a control input $($may be related to an applied torque$).$ Set all initial condi$-$

tions to zero.

$($a$)(2pts)$ Find the transfer function of the system, i$.$e$.$ find $\frac{(s)}{U(s)}$ for all zero initial

conditions.

$($b$)(4$ pts$)$ Suppose the control is given by $u(t)=sin(t),\sqrt[\phantom{2}]{g}.$ Obtain the output

$(t)$

$($Hint: You may want to use the result from Problem $2)$

$($c$)(2pts)$ Set $g=9\mathrm{.}81\frac{m}{s^2},=1ra\frac{d}{s}.$ Then simulate the control system $(3)$ using

Matlab $($ode$45)$ and plot the output $(t)$ in the time interval $0,20$ seconds.

$($Hint: See lecture$4.$ m for a similar example of the mass$-$spring system with control.$)$

$($d$)(2$ pts$)$ Finally, set $=\sqrt[\phantom{2}]{g}.$ Repeat $($c$).$

You should observe that the system is destabilizing. This phenomenon is called

resonance.

Note: This is a very basic introduction to resonance. Interested students are en$-$

couraged to read about the Millennium Bridge incident in London. Students who

have already taken or are taking the Vibrations course would learn more about this

topic.

$($bonus: $2$ pts$)$ Repeat Problem b$)$ with $=\sqrt[\phantom{2}]{g}.$