Dynamic analysis of pneumatic cylinder

[nekojita]

Hi,

First I'll say that I am not a mechanical engineer (electrical )so please bear with me as I have been assigned to investigate the problem below.

We have a pneumatic cylinder configured with an inlet port, a piston attached to a rod, and an outlet vent. The inlet port is connected to a valve which provides a burst of air pressure. (not constant)

The rod drives a sharpened component into an external second material. The constituent equations that I have used to study this are:

F= A *P; Eq1
where A=area and P= pressure

Alternatively, F= m*a; Eq2
Where m= mass and a = acceleration

Using a high speed camera, we were able to determine the velocity by applying the following:

v=d/t; Eq3
Where d= displacement and t = time

Acceleration was determined using the following:

a= (v-u)/t; Eq4

Where v= final velocity, i.e., 0m/s, Initial velocity, t= time

Finally, momentum, p = m*v; Eq5
Where m= mass and v = velocity


The above calculations are all static, however, to understand this dynamically, I would like to calculate the force, velocity, and momentum as a function of displacement of the rod in the cylinder as well as the calculation of force, velocity, and momentum as a function of time.

It would also be good to factor in the friction of the piston and cylinder wall as well if possible.

There are two cases:

1) In the first case there is a small separation distance (gap) between the rod and the driven slug when the rod has reached its full stroke travel.

2) In the second case, the rod drives the slug of material into the second material and the rod and slug are in contact at the end of the rod's stroke. (no gap)

The contention is that the loss in momentum (the rod and piston no longer in contact with the slug) is significant because the mass of the slug is very small compared to the mass of the rod and piston. The counter argument is that the slug's velocity is sufficient to cause it to continue to drive into the external material after loss of contact.

This is an interesting problem but I am stuck. Any help from the Forum would be greatly appreciated.

Thanks,

Art

 

[hydtools]

Take the high speed camera data and create a plot of d vs t. Do a curve fit to derive an equation for d as a function of t. Differentiate with respect to t the equation of d as a function of t to derive v = dd/dt. V is the slope of the d vs t curve. Differentiate with respect to t the equation for v to derive the equation for a = dv/dt.

Ted

 

[nekojita]

Hi Ted,

Thanks for your response and suggestions. What you suggest would provide a partial answer, but I really need a more comprehensive analytical foundation for this problem.

Already getting the time/distance information was incredibly arduous because it involved counting thousands of frames from the high-speed video. I really want a mathematical representation of the physical system so that I can play the "what if" game.

Any chance that you could help with that? I'll help you with an electrical engineering problem in exchange.

Thanks,

Art

 

[3DDave]

There is software available that analyzes video, making measurements, so that may simplify things. You might also look at using an LVDT (linear variable differential transformer) for making the measurements electronically.

Other than that you are looking at a complicated system.

For example, when the valve opens, the gas in the supply line expands with a negative pressure wave at the speed of sound in the supply line; the gas accelerates, and the conditions in the supply line change so the speed of sound changes. The supply line itself will contract as the pressure initially drops, possibly until choke flow develops somewhere in the supply.

The gas in the cylinder is compressed by the incoming gas,; a complicated jet forms that transitions from laminar to turbulent flow, also at the speed of sound, possibly with the development of a shock wave at the inlet. The volume of the cylinder will change as the force on the piston increases beyond the ability of static friction to hold the piston in place and then the mass of the piston and rod will begin to accelerate, allowing the volume to increase, with an attendant negative pressure wave and it's effects upstream to the choke.

There will be various temperature related interactions with pressure and volume.

I'd suggest sticking with the experiments or hire a high-end analysis company.

 

[jlnsol]

Also, do not assume the air pressure on the rod-side of the piston to be atmospheric: it is not! The outlet vent is most likely screwed into a port which forms a restriction in air flow. According to Bernoulli the thrust is proportial to the air speed squared. Therefore the air pressure on the rod-side of the piston will also be proportional to the piston speed squared.
Your Eq.2. above will change in something like:
F = Ap * p1 - Apr * p2 - Fw = m * a
wherein:
Ap = surface area of piston
p1 = air pressure on piston
Apr = surface area of piston on the rod-side
p2 = air pressure on piston rod-side (dependent of v squared)
Fw = friction of piston seals en rod bearing
m = mass of all moving parts

 

[nekojita]

3DDave,

Thanks for your response and suggestions. One problem with components like the LVDT is that the cylinder is 5mm in diameter with the piston proportionately smaller. We were looking into some some software to analyze the high speed camera fottage.

Art

 

[nekojita]

jlnsol,

Thanks for pointing out the additional areas to consider in this calculation. Any chance that you would be able to provide a time dependent model?

Thanks again,

Art

 

[israelkk]

You need to write a set of differential equations that takes into account the instantaneous gas flow rate chances into the piston and the instantaneous pressure change in the cylinder as a result of it, the instantaneous pressure decrease/change in the gas supply source (vessel ), the heat transfer and thermodynamics of the whole process inside the cylinder and inside the vessel. Then numerically solve the equations.

You need to take into account that the flow rate through the valve orifice can be choked at the beginning and changes to uncooked when the pressure inside the cylinder rises. Therefore, you are dealing with a quite complicated problem which needs an assistance from an expert with theoretical and practical experience. Such expertise is rarely exists in the aerospace and military industry and is not common in the commercial industry.