For the hydraulic system given,

Develop the dynamic equations as $q_{mi}$ is input and piston displacement $x$ is output. Assume that

there is no leakage between the cylinder and pipe connection. $(,$ fluid density$)$

Tank Capacitance, $C_1$

Mass, $m$

Piston area, $A$

Hint:

Write elemental equations between tank $1,$ tank $2,$ piston$-$spring system $6$ piece.

Get last equation in the form of $q_{mi}=f(C,R,m,A,,g,k,x)$