# Linearized ordinary differential equation for Hydraulic Mill

The hydraulic mill is consists of two main part; servo valve and hydraulic cylinder.

We will consider flow-control servo valve. The dynamics of the servo valve is $$$$\label{1}\tag{1} Q_s=C_Q\pi dx\sqrt{\frac{2}{\rho}(P_s-P_1)}$$$$ Here,

• $$Q_s$$=supply flow,
• $$C_Q$$=flow coefficient,
• $$\rho$$=oil density,
• $$x$$=servo-valve displacement,
• $$P_s$$=supply pressure,
• $$P_1$$=output pressure of the valve.

The equation for the flow of oil to the hydraulic cylinder, $$$$\label{2}\tag{2}Q_s=a\dot{y}+\frac{V_1}{\beta}\dot{P_1}$$$$ Here,

• $$a$$=area of the cylinder,
• $$y$$=hydraulic piston displacement,
• $$V_1$$=volume of the primary side of the cylinder,
• $$\beta$$=bulk modulus of the oil,
• $$P_1$$=cylinder pressure on primary side.

Schematic: Hydraulic servo system

Equation \eqref{1} needs to be linearized for both input-flow & output-flow to the cylinder according to valve displacement $$x$$.

Help me find the ODE for rate of change in cylinder Pressure $$\dot{P_1}$$ and $$\dot{P_2}$$.

Assuming an operating point at $$x_0,P_{s_0},P_{1_0}$$ we have
regarding $$K_0x\sqrt{P_s-P_1}$$
$$K_0 x \sqrt{P_s-P_1} = K_0 x_0\sqrt{P_{s_0}-P_{1_0}}+K_0\sqrt{P_{s_0}-P_{1_0}} (x-x_0)-\frac{K_0 x_0 (P_1-P_{1_0})}{2 \sqrt{P_{s_0}-P_{1_0}}}+\frac{K_0 x_0 (P_s-P_{s_0})}{2 \sqrt{P_{s_0}-P_{1_0}}}+O_1(|x-x_0|^2)+O_2(|P_s-P_{s_0}|^2)+O_3(|P_1-P_{1_0}|^2)$$
$$a\dot{y}+\frac{V_1}{\beta}\dot{P_1} = K_0 x_0\sqrt{P_{s_0}-P_{1_0}}+K_0\sqrt{P_{s_0}-P_{1_0}} (x-x_0)-\frac{K_0 x_0 (P_1-P_{1_0})}{2 \sqrt{P_{s_0}-P_{1_0}}}+\frac{K_0 x_0 (P_s-P_{s_0})}{2 \sqrt{P_{s_0}-P_{1_0}}}$$