Is there any method to develop an ODE for a hydraulic system?

Question

If I have a hydraulic system like this:

• $d(t)$ is disturbance in force
• $p(t)$ is the pressure in bar
• $y_1(t)$ is the length in meters
• $y_2(t$ is the other length in meters

Which method should I use to develop an ODE for a hydraulic system?

I have tried diffrent ways but I allways end up in wrong solution. Here is what I have been tried.

• Lagrangian mechanics - How should I apply pressure-energy $p(t)$ and disturbance-energy $d(t)$ ?
• Newtonian mechanics - How should I take regard of pressurefall and non-stiffness(as you see, the system have no spring, which means that the cylinder can go to infinity in theory. There is no limit in this system).
• Analogy with RLC circut - How should a write down a RLC circut of this system? I know that the mass would be the coil(Inductance), the resistor whould be damping and the cacaptitor whould be stiffness

$$L\ddot{y} + R\dot{y} + Cy = 0$$

Is the classic formula from:

$$L\dot{y} + Ry + C\int y = 0$$

So what do you think? Which method should I use? I'm not asking you to solve this for me. I'm only asking for guidance.

Answer 1

Write down the forces acting on each piston and add them up. The net force is what is accelerating your piston. This gives you second-order differential equations, which you then can translate to first-order differential equations and solve.

A few hints:

• Do you assume the blue liquid to be incompressible? If yes, what does this mean for $y_1$ and $y_2$?
• How would the forces change if $y_1$ and $y_2$ were different?
• If you ignore hydrostatic pressure, what’s the pressure of the green liquid?
• Does it matter where the mass $M$ is attached?