The Nemoid foot pump is a pretty ingenious contraption -- it can pump up your airbeam to 9 psi (not too shabby), weighs less than 4 ounces, packs small and flat, and is made almost exclusively of fabric-y flexible components. So how does it work?
The basic premise of the pump is that it uses one way valves to intake and expel air. The big red valve (which is grey in the latest version of the pump) sucks in air when it is opened, but the air can't escape back through the same opening. Inside the pump, a chunk of reticulated open cell foam traps the air inside, until you are ready to step on it.
Once you step on the pump, you exceed the "crack" pressure of the second one way valve in the exhaust tube. The valve opens and allows air to flow (again, one way) from the pump tube into the airbeam. As your foot lifts off the pump, the action of the foam rebounding and decompressing sucks in more air into the pump so you can keep pumping.
It is a simple mechanism to deliver the air, but it is not trivial to execute. The intake valve needs to be big enough to take in enough air volume so that it doesn't take a ridiculously long time to pump up the airbeam. The diameter and height of the pump influences this too. Notice that our latest version of the pump (the one closest to the bottom in the picture below) goes from 3 "pancake" layers to 2 layers. Even though the air volume being delivered was smaller in the 2 layer version compared to the 3 layer, we found that the 3 layer version was a little bit too unstable when people were stepping on it. Our fabric has also changed a bit over the years as you can see below.
In addition, the reticulated foam needs to be the right density so that it is not too hard or too easy to step on the pump. If anyone wants to see what a mushy pump feels like, let me know and I can see you a few unsuccessful prototypes.
One of the reasons this pump works so well for our airbeams is that it self-limits the air pressure you can deliver to the beam, without the need for an additional pressure relief valve. The way this works is that the diameter of this pump is 4.75" (area is about 17.7 sq inches). As you're stepping on it, you're delivering a stepping force onto this pump area, which equates to pressure (pounds per square inch). Once the airbeam reaches the target pressure, say 8.5 psi, the force that it would take to further step on the pump area exceeds what an individual can realistically generate. For an airbeam pressure of 8.5 psi, this is over 150 lbs of stepping pressure that would have to be applied by a single foot directly normal to the area of the pump. And that's why we tell you to stop pumping up the airbeam when it gets hard to step on the pump.