Given the dynamic balance unit below:

$($a$)$ Derive the equations of motion for M$1$ and M$2.$

$($b$)$ Find the transfer functions X$1($s$)/$F$($s$)$ and X$2($s$)/$F$($s$).$

$($c$)$ Let f$($t$)$ be a sine function f$($t$)=$ sin$($sqrt$($k$12/$M$2)$t$).$ Show that x$1($t$)$ is zero as t approaches infinty. hint: use the final value theorem.

lim t$-->$ infinity $($f$($t$))=$ lim s$-->0($sF$($s$))$