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I'm trying to mathematically model an actuation system and it comes down to relating the pressure across the pump and the actuator as shown in the picture. Assuming no friction in the pipes and the inlet and outlet flow rates are the same (both equal Q3), how can I relate P1, P5 to P2, P4?

I tried considering the hydraulic analogy and use Kirchhoff's voltage law but didn't know what to do with the actuator. Would really appreciate it if someone could point me in the right direction.

enter image description here

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  • $\begingroup$ Unless I am missing something in this setup, if no friction in pipes, then no pressure drop due to flow, and P1=P2 and P4=P5 ... The pressure difference on the actuator (double acting cylinder??) connects to the force the cylinder is exerting or receiving. The volume displacements connect to the linear motion of the cylinder. $\endgroup$
    – Pete W
    Feb 1 at 5:03
  • $\begingroup$ @PeteW I thought the inlet and outlet would cause an increase and decrease in pressure, making them not equal? They are going into different lines just have the same flow rate. And yes it is a double acting cylinder. $\endgroup$
    – Lin
    Feb 1 at 5:15
  • $\begingroup$ If the connections between 1/2 or 4/5 are long and skinny, and you have the tube dimensions and density viscosity etc, you can calculate pressure drop from flow, using Poiseuille law (works if flow is laminar... can tell laminar vs turbulent from Reynolds number. If >2000 then turbulent, slightly more complicated). This pressure drop might be negligible in a slow moving hydraulic system, don't know. $\endgroup$
    – Pete W
    Feb 1 at 14:25
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If inlet flow = outlet flow, then your system will quickly reach a state of equilibrium. The pump spins to transfer fluid from node 5 to node 1. There would be some minor pressure difference due to pipe flow resistance. Probably not enough to break the static friction on cylinder seals (approx 100 psi for a typical 3000 psi cylinder). Even if you did have enough pressure difference to start the cylinder moving, your actuator would just move towards node 4 then remain stationary. From that point on, you would be continuously flowing with no actuator motion.

Then it's easy to find the difference between P1 and P2. Pete's comment is correct for most practical hydraulic systems. Use Poiseuille's equation for laminar flow: https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation. Turbulence in hydraulic lines is rare. Typical working fluids like ISO 46 grade oil are far more viscous than water. Also flow rates are relatively low due to tyipcal component sizes for positive displacement pumps and valves.

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