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# Formula for Gas Damper?

## Formula for Gas Damper?

(OP)
Hallo everybody

I am hoping somebody could possibly help me with a formula to estimate the damping force generated by a piston moving down a cylinder with a closed end; in doing so it compresses an ideal gas but at the same time, gas is vented through a small hole. I would like to express this force as a function of piston speed, gas properties, piston and vent hole diameters, and the initial chamber length (i.e. stroke length)?  Both isotropic gas compression and gas outflow will contribute to incremental piston movement... I would be happy to assume average density and temperature values for calculating the orifice flow (followed by a re-run). Another assumption would be to assume choked flow through the venting hole for relative high piston speeds.

I gave this a shot, have the basic expressions, but am battling (rusted math's!) to bring the two effects together in one formula - as required for use in my Working Model 2D motion simulation software. The aim is to damp/tame the force exerted by a simple coil spring actuator. Surely this must have been done before?

Thanks / regards

### RE: Formula for Gas Damper?

Gert,

There are available Pyrex cylinders with monolithic graphite pistons that function as dampers.  These pistons leak around the diameter though.  Do a search under dampers and "graphite piston" and you should find them.

You realize that this gadget might get HOT, right?

The graphite/pyrex dampers company will send you a cute little sample - it's unbelievably smooth acting.

Larry

### RE: Formula for Gas Damper?

Gert,

Once I have designed dampener (thee step - two stroke in one direction it had variable speed).

I have started off with air and ended using much much less compressible fluid. My second problem was variable and relatively high environmental temperature.

I suggest you create very simple set of functions where you will define your ultimate conditions. If you define max pressure for flow through the ventury (compressible flow) and working from there you go back to piston speed. Make couple of iterations for different temperatures and you are almost there. If you have suction part you have to repeat all over again with completely different pressure difference. Since vacuum is so limited you may realize that you will need either very large piston or hole that almost can not be manufactured.

Testing helps but if you are not on right track with air go for oil. At least you can specify different dynamic viscosity.

Hope this helps,

Darko
darkov@onramp.ca

### RE: Formula for Gas Damper?

Let me try to clarify the problem.
If you have a gas cylinder at rest at atmospheric pressure and you start to move the piston and continue at constant speed, after an initial unsteady behaviour, you'll get a steady condition, where pressure will stabilize at a certain value and, if the piston is adiabatic, all other conditions will remain constant (until you reach the stroke end!).
I suppose you want a formula for such a steady condition: otherwise I don't see what kind of formula you are looking for, as only a solution with a time step procedure can be used to predict piston's behaviour in a general situation (any piston movement, effect of heat exchange through piston walls...). Of course this is the big difference with respect to a liquid damper, where the damping force is simply proportional to the istantaneous piston speed  (though of course also this requires some simplifying assumptions).
If your goal has been correctly understood by me, here is the principle for arriving at a solution:
-by equating gas density times piston area times piston speed to the mass outflow you can determine the pressure in the cylinder (as both gas density and mass outflow can be expressed as functions of pressure);
-you have also to assume a gas temperature, and this will depend on the previous history of the cylinder; an assumption can be that the cylinder was initially at ambient temperature and that the so far unknown pressure has been reached with a known transformation (adiabatic for example), so you can express the gas density as a function of pressure only, discarding the temperature dependency
-now you have a single equation from which you can calculate the pressure in the cylinder for any given speed of the piston rod: p=f(v)
-as the force exerted on the cylinder is p x A, the relationship between damping force and speed is A x f(v) (and may of course be far from being a linear relationship!).
To make a sample application of the above principle let's assume that the pressure will be quite high so that we can take the absolute pressure as equal to the gauge one, also that the gas mass outflow is proportional to pressure (this may of course be far from reality), and also that gas density is simply inversely proportional to pressure (this corresponds to neglecting the effect of temperature): under such conditions you get p (and damping force) proportional to the square root of piston speed.

prex
motori@xcalcs.com
XcalcS
Online tools for structural design

### RE: Formula for Gas Damper?

Addition/correction to the preceding message: of course gas density will be more proportional than inversely proportional to pressure!
Now assuming that the gas outflow is proportional to the square root of the pressure (this is closer to reality, except for critical flow) your damping force will be inversely proportional to the square of the speed.

prex
motori@xcalcs.com
XcalcS
Online tools for structural design

### RE: Formula for Gas Damper?

(OP)
Thank you very much to everybody that responded!

I realise that computing the dampening effect would be easier if one takes a time step approach.  Problem is, that because I want to simulate the damper together with the rest of my system, it would be beneficial to make use of my commercial motion simulation software - that however, requires the dampening force to be defined as a single formula (with variables like piston speed, distance moved, etc).  I have expressions for an oil damper, but for compressible air, we know it is not so easy!  However, I believed somewhere somebody might have established such a formula to estimate the dampening force - even if it is empirical.  Some pneumatic cylinders use similar dampening at stroke ends.

Prex, if the piston moves at constant speed, too fast for a small choked (sonic) orifice to keep up with, I think the air might compress even more and the pressure will keep rising.  In my application, the actuating (spring) force and therefore speed, will vary...  If one is not talking about estimating calcs, incorporating several assumptions, I tend to agree that this problem is history dependant - to be solved by time steps only.

Regards

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