Two pendulums, coupled by a spring, are to be controlled by two equal and

opposite forces $,$ which are applied to the pendulum bobs as shown in the figure.

$($a$)$ Derive the differential equations of motion.

$($b$)$ Linearize the equations using small angle linearization.

$($c$)$ Provide the equations in the second$-$order matrix form.

$($d$)$ Determine the transfer functions $1()/()$ and $2/().$ All the

initial conditions are assumed to be zero. $1()=\{1()\},2()=\{2()\},$

and $()=\{()\}.$

$($e$)$ Determine the state$-$space representation with the angular displacements

$1$ and $2$ as the outputs, $$ as the input.

$=+$

$=+$

Show the details of the state variables and all matrices.