Understanding pneumatic springs

[mishaparem]

Hello all,

I'm working on a DIY project, I know I need to use pneumatic springs because it allows me to be flexible with the spring rates, but I forgot quite a bit about compressible air, and from what I remember, in my courses we mostly dealt with compressible air for aerodynamics, and skipped over pneumatic springs (maybe in a different major, we would have gone over this stuff)

Can someone give me a quick run down, or suggest some reading? I'm having a difficult time finding info in Google.

Part of my question is how to determine what geometry of pneumatic springs I need. For instance - I have a physical limitation of a 0.5m stroke, but what diameter bore? Obviously a 0.016m diameter will spring less than a 0.1m diameter at the same maximum pressure of the cylinder, that's intuitive for me, but how do I calculate how much exactly the spring will... well... spring?

Thanks!

Misha

 

[jrmichler]

An air spring will spring as much as you design it to spring. Start with the following search terms:
Ideal gas law.
Isothermal compressibility.
Adiabatic compressibility.
Air cylinders. When you look up air cylinders, note the allowable working pressures.

Have fun.

 

[mishaparem]

Sweet, thank you! I started remembering bits and pieces of the Ideal gas law after I posted yesterday. Here's where I'm at right now, posting for proof reading, and for anyone else who goes looking for how to determine the ballpark specs of pneumatic springs that they need for their project:

P₁*V₁ = P₂*V₂
F₁*A₁ = F₂*A₂
P=F/A

Solving for force exerted by the rod (with the cylinder being fixed at the other end)
F_rod(x) = P_rod(x)/A_rod
P_rod(x) = P_cyl(x)*A_rod/A_cyl
P_cyl(x) = P_comp*V_comp/V(x)
V_comp = (pi/4)*d²*L_comp
Vext*Pext = Pcomp*Vcomp
L_comp = L_ext*P_ext/P_comp
L_comp = Stroke*P_ext/P_comp

Substituting
P_cyl(x) = P_comp*(pi/4)d²*L_comp/((pi/4)d²*x)
P_cyl(x) = P_comp*L_comp/x
P_rod(x) = P_comp*L_comp*A_rod/(x*A_cyl)
F_rod(x) = P_comp*L_comp/((pi/4)x*d_cyl²)

F_rod(x) = P_comp*L_comp/(x*A_cyl) , L_comp < x < Stroke

Which should help me ballpark the specs for the pneumatic spring I need to order based on:
P_comp is the maximum pressure of the cylinder
d_cyl is the diameter of the cylinder
L_comp is the position of the pistol at P_comp
L_comp = Stroke*P_ext/P_comp, where:
Stroke is the stroke of the spring

Also, without a separate reservoir, the effective stroke (how much it can actually compress without failing) of the spring is:
Stroke_Effective = Stroke-Lcomp
Stroke_Effective = Stroke*(1-P_ext/P_comp)
If the spring compresses any further, the cylinder or seals may fail from excessive pressure

With a reservoir, Stroke_Effective = Stroke, and L_comp is the length of the reservoir (same diameter as the cylinder) A reservoir of a different diameter, but same volume may be used:
Vres = Vcomp
L_res*(pi/4)*d_res² = L_comp*(pi/4)*d_comp²

Does that sound about right so far?

 

[jrmichler]

The crowd here at PF likes to derive a general equation, then solve it for specific cases. Air cylinders, however, are only available in discrete diameters. Because of that, a design table is the approach that I use to solve this type of problem. You have three things to deal with, and they are not independent. First, the static force exerted by the cylinder. Second, the spring constant of the cylinder. Third, the position of the cylinder (because it affects the spring constant).

The static force is easy. Force = pressure times net piston area. You already know what static force you want, just make a list of air cylinder bore diameters and the required air pressure.

The spring constant is the change in force from a small displacement divided by the displacement. This is where it gets interesting. For a slow displacement, the compression is isothermal and the volume of air that is compressed is the volume trapped between the piston and the valve/regulator/whatever is controlling the air pressure. For a fast displacement, the compression is adiabatic and the volume of air that is compressed is the volume between the piston and the nearest restriction. Note that there is a dead volume of compressed air with the cylinder bottomed out. You may need to consider the effect of air expansion on the opposite side of the piston in addition to the effect of air compression on the first side of the piston. Do this for each cylinder diameter. And for at least three piston positions - each end of stroke and middle of stroke. The spring constant will vary with the piston position in the cylinder.

Another variable is air pressure. You can pressurize both sides of the piston to increase the spring constant without changing the static force.

If you have a "large" reservoir, your spring constant will approach zero. This is a good data point for a sanity check of your calculations. If you calculate the case of zero dead volume and piston almost bottomed out, the spring constant should get very large. The other good point for a sanity check.

Normal air cylinders have friction that may need to be considered. That friction is variable (lesson learned the hard way).

 

[mishaparem]

Ugh, I've just realized how much I've forgotten about math and physics, working dead end jobs in the office