Mathematical Model Piston Pump 1

[tomm0005]

Good morning, I would like to ask you to model this problem, as I have some doubts. A constant force F acts on a hydraulic piston which is connected to a second piston that must lift a mass M. Inside the pipe, there is a constriction that causes a pressure drop. How can I write the differential equation that governs the motion of the piston lifting the mass M?

 

[goutam_freelance]

Make a schematic of the system for clearer understanding.

 

[tomm0005]

Make a schematic of the system for clearer understanding.

This is the system:

 

[LittleInch]

This is something that impacts speed of the lift and the maximum force available to lift the mass M

The pressure drop across the restriction will impact flowrate.

 

[Snickster]

I believe I may have come up with a solution as shown below, which appears to be a second order non-linear differential equation:

 

[tomm0005]

I believe I may have come up with a solution as shown below, which appears to be a second order non-linear differential equation:

View attachment 874

thank you so much

 

[LittleInch]

Lets just go back to basics here.

Your mass M is just a vertical force which is M x g, so lets call it F1.

At rest / static, the pressure in the whole system is the same, so the equation F2 x A2 = F1 x A1

Now to get this mass to move upwards, you need to apply more force at F2, lets call it Fm.

The amount of Fm is variable depending on the velocity Vm of the Mass. This movement of the fluid creates a pressure drop in your constriction.

I think it should be easy to create an equation whereby movement of the F2piston creates a volumetric change, assuming this is a fixed F2 velocity, which results in a pressure increase in the F2 chamber. There will be an increase in the F1 chamber, but it is much lower and only really the piston friction so you can probably ignore it or add 10%.

So then solve fr different velocities of F2.

I still don't really know what is your variable that you are trying to find?

 

[BEMPE16524]

"Fluid Mechanics" by Frank M. White:

 

[Snickster]

Well I was close. I forgot to include the weight force of the mass acting down, and I did not realize that you can reduce the restriction effect to a linear function R*Q = R*A*V and then use an approximate value of R as shown. I think my equation is still correct if I include the Mg factor although it will be hard to solve as a non-linear equation with dx/dt to a power of 2.

 

[BEMPE16524]

Well I was close. I forgot to include the weight force of the mass acting down, and I did not realize that you can reduce the restriction effect to a linear function R*Q = R*A*V and then use an approximate value of R as shown. I think my equation is still correct if I include the Mg factor although it will be hard to solve as a non-linear equation with dx/dt to a power of 2.

yes considering you are using your brain while I'm just copy and paste from external resources, you are not wrong at all.

 

[Snickster]

I did my analysis on my own without having tried to do a differential equation set up like this in about 20 years so I don't know what you are talking about.

 

[BEMPE16524]

I did my analysis on my own without having tried to do a differential equation set up like this in about 20 years so I don't know what you are talking about.

I fully appreciate your talent. that's all.

 

[goutam_freelance]

These are excellent derivations:

The following can be used to introduce a further generalization with respect to laminar and turbulent flows

.

 

[tomm0005]

thank you guys, now everything makes perfect sense