Exercise $6$

Hydraulic elevators are often used in low buildings where the elevator only services two or three floors. The elevator can be modeled as a mass connected to a piston$-$cylinder, as seen in the figure below. The pressure $\backslash ($ p $\backslash )$ is a gauge pressure measured relative to atmospheric pressure. The supply pressure $\backslash ($ p$\_\{$s$\}\backslash )$ is the system input.

$($a$)$ Draw a free body diagram and write a differential equation to describe the vertical dynamics of the elevator car, assuming that $\backslash ($ m $\backslash )$ and $\backslash ($ b $\backslash )$ are known. At this point, do not make any substitutions involving parameters or components from the hydraulic system.

$($b$)$ Write a differential equation to describe the height of the fluid in the cylinder, assuming that $\backslash ($ P$\_\{$s$\},$ R$,$ A $\backslash ),$ and the fluid density $\backslash (\backslash$rho $\backslash )$ are known. Do not assume that the pressure is the same at the top and bottom of the cylinder.

$($c$)$ Combine the differential equations from above into a single expression for $\backslash ($ H$($s$)\backslash )$ in terms of $\backslash ($ P$\_\{$s$\}\backslash )$ and the other constants associated with the system. Since there are two "forcing terms," you won't get a transfer function. Instead, just isolate $\backslash ($ H$($s$)\backslash ).$