# Pneumatic Spring

## Main Question or Discussion Point

How do you calculate the spring constant of compressed air? I know force = k * integral(dx). How do you relate volume to the force?

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it is not a constant

pressure, volume and temperature are all related with a single formula
as volume goes down, pressure goes up, hence the non-constant comment from Curl

sophiecentaur
Gold Member
But you can say that, if the temperature is constant, the pressure times the volume is constant (Boyle's law). So, if the piston of the gas strut is in a certain position (corresponding to a certain volume) for a certain load plus atmospheric pressure, it will go to half that volume if the total load is doubled.
Starting with 1 atmosphere in the unloaded strut, if the area of the cylinder is A (in m2), then the force to compress it to half that volume (twice atmospheric pressure) will be about 10e4A N.

It doesn't follow Hooke's Law, of course, but you can predict the force needed for a given displacement. Actually, because of the Law involved, you can measure a bigger range of forces using an 'air spring' than a steel coiled one.

You can estimate it by using the bulk modulus.
If you have a gas of volume V and want to compress it by $$\Delta V$$, you need an increase of pressure
$$\Delta p = -B \frac{\Delta V}{V}$$
B can be found for both isothermal or adiabatic processes and for air is of the order of $$10^{-5}$$ Pa.
If you apply this to a cylinder of length L and area A,
then
$$\Delta p = F A$$
and
$$\frac{\Delta V}{V}=\frac{\Delta L}{L}$$
Then
$$\frac{F}{\Delta L}=- \frac{B A}{L}$$
and assuming a constant B for small compressions, you could say that the term on the right hand side is the elastic constant of the cylinder.
It depends on the geometry too, not only on the properties of the material. Same as for a real spring.