Calculating FvD for Hydraulic Cylinder with Air Bubble

[randall016]

I want to calculate the force-displacement curve for a damper that would have hydraulic oil with an air bubble in it, so very similar to an air spring with a limited amount of volume. I came up with the following solution:

F(x)=Po*A((Vo/Vo-A*x)-1)

where F would be the external force applied to the rod, Vo is the initial volume of the air in the hydraulic cylinder, Po is the initial pressure of the air in the hydraulic cylinder, A is the area of the piston, and x is the displacement. When I first calculated this I got:

F(x)=Po*A*Vo/(Vo-A*x)

the problem with this is F(0)=Po*A so I subtracted this off of the function which accounts for the atmospheric pressure on the external portion of the piston.

Here is my initial derivation:
Po*Vo=P1*V1 (Boyles law for isothermal case, assuming compression is slow)
when the piston is displaced a distance x, there is a change in volume dV and dV=A*x
so V1=Vo-dV=Vo-Ax
also P1=F/A (external force on rod/area of piston)
substituting these into boyles aw gives:
F(x)=Po*A*Vo/(Vo-A*x)
then the atmospheric pressure needs to be subtracted

I think the error is when I use P1=F/A. Should P1=(F+PoA)/A)? This would account for all the external forces on the piston and then it gives me the solution of:
F(x)=Po*A((Vo/Vo-A*x)-1)

I also attached a sketch to clarify. Does this look correct?

 

[gtoguy97]

Yes, that looks correct. The P1=(F+PoA)/A equation is the correct one to use, as it takes into account all the external forces acting on the piston. With this equation, you can calculate the force-displacement curve for a damper with an air bubble in it.