Figure $4$: A mechanical filter

$4.$ The concepts of frequency response function, impulse response function, and Duhamel's integral are not restricted only to oscillatory systems. In fact, they are valid for any linear systems. Consider the mechanical filter as shown in Fig. $4.$ The mechanical filter consists of a cart of mass $m,$ a spring of stiffness $k,$ and a dashpot of damping $c.$ The input of the filter is the motion of the cart $x(t)($not the force acting on the cart$),$ and the output is the displacement $y(t)$ of the hinge connecting the spring and the dashpot.

$($a$)$ Apply equilibrium condition at the hinge to derive the input$-$output relation

$T{y}^{}+y=x(t)$

where

$T=\frac{c}{k}$

$($b$)$ Determine the frequency response function of the system.

$($c$)$ Find the impulse response function of the system by letting $x(t)=(t)$ and solving the differential equation $(11).$