The impossibility of spring CPS

[Ben]

Today I crunched the numbers in an attempt to design a good spring CPS system. To get any decent amount of pressure (40 PSI and up) with a good pressure ratio the dimensions of the spring will be unreasonable or simply impossible.

I've discarded the old %drop idea for measuring the constancy of pressure and am going to use a simple pressure ratio from now on. The pressure ratio R equals the pressure when empty (or near empty) divided by the pressure when full. For springs R simplifies to the ratio between the displacement of the spring when the PC is empty and the displacement of the spring when the PC is full. An R value of 1 is ideal.

The biggest limit to this system was the shear stress in the spring. Spring-tempered steel (1065 steel) is the most common type and has a yield strength (the maximum stress before deformation occurs) of about 71000 psi. Stronger steels have yield strengths of up to 132000 psi. Even without a safety factor, none of the dimensions are acceptable with high pressure ratios.

(A side note for cantab: I remember you said earlier that compressing springs fully might overstress them. That assumption was correct in many cases when I looked at some of McMaster-Carr's stock springs.)

If anyone's interested, I used these spring equations to approximate spring performance for certain dimensions. Some quick math showed that these equations are very accurate compared against given spring constants.

Good pressure can be achieved with worse pressure ratios. But I see no reason to design such a system because air pressure is cheaper, more versatile (with the ability to choose a pressure), and can be stored in an unstressed state. The only reason I looked into this was because I thought spring water guns could potentially be shorter, but I was wrong.

Also, some of you probably have noticed that the physics forum is now "Engineering water guns". This is a more accurate description of the forum in my mind.

Edit: I've kept investigating and found some acceptable values with a decent safety factor of 1.5. However, as the springs here are completely custom, they may not be cheap. Especially if a stronger steel is used, these springs probably won't be cheap. But it's worth knowing about if someone is considering a spring water gun.

Let me note that the values I've found are only an approximate range that returns acceptable values, and what is acceptable to me may not be acceptable to you. These pressure chambers are going to be about 3 feet long on average. A pressure chamber with a maximum of 45 psi pressure and a pressure ratio of 0.75 has only about 675 mL of water capacity, which is terrible for the length of the pressure chamber. Air pressure seems much more reasonable still, but these designs are interesting nonetheless.

[Ben]

The website I linked to above with equations suggests that the point of maximum shear stress is on the outside of the wire. After speaking to one of my instructors about this he said that is incorrect and it occurs on the inside of the wire.

Also, my failure criteria was incorrect. The Tresca failure theory states that when the shear stress is higher than half the yield strength, yielding occurs. I should have realized this because we were just taught it a few weeks ago, but I guess it didn't make an impression (It will now). This makes designing a working spring system much more difficult, especially if you want a safety factor on top of that.

I also hadn't considered systems of multiple springs in this analysis. After looking at them, I can see that they're not going to help more than an acceptable wider spring would.

To summarize the reasons to NOT make a CPS spring water gun:
- the analysis involved in designing a CPS spring water gun is too advanced for most people
- the construction is difficult
- the system would be stored in a stressed state
- the most reasonable designs would use custom springs, which may be expensive
- stronger steels make better systems, but they cost much more
- the most reasonable designs still aren't too reasonable when compared against air pressure

The only advantage I see when comparing spring water guns against air pressure is that air pressure's pressure-volume curve follows the inverse of the volume, while spring water guns have a linear curve. However, with the same pressure ratio, the two should be indistinguishable from each other, so the point is moot.

[cantab]

The main advantage of a spring-powered water gun I can see is it would make it possible to pressurise mechanically - by pulling a lever or something that compresses the spring, drawing the piston back and pulling in the water. Sort of like a water crossbow.

In fact, that brings to mind something. A recurved bow uses elastic deformation to obtain close to constant force. A compound bow also achieves constancy of force, in that case by pulley mechanisms. If one connects to bowstring to a piston, one can then make a constant pressure water bow.

[Captain Galaxy]

There was a Super Soaker exactly like this called the SS Bow and Arrow. Because of how rare it is, almost nothing is known about it. Looks like it would be totally worthless in a water war.

[aEx155]

http://forums.sscentral.org/t4847/

I'm just going to say that while it is a very bad idea to do it without proper equipment, doing it with the right stuff can be a very impressive feat (if you can do it).

Too bad we don't know how it performs yet...

[Ben]

For general use I don't think single action priming is a good idea. Such a system would be beneficial only when either the pressure is low, the volume is low, or both. You simply can't transfer enough energy is one action.

For example, to pressurize 500 mL to 2 bar (with constant pressure), you need to transfer 100 J for the pressurization alone (I don't know what it takes to move the water but I assume it's rather small and depends on how fast you do it). With a 0.6 meter throw you'd have to pull 165 newtons of force. This is actually a reasonable system and I'd like to make it, mainly because you probably could go through pulls like that quickly. But that's about the limit of practicality for the idea.

After examining the math again, I've decided that if you're going to use a spring in a water gun, don't pre-stress it. Pre-stressing is when the spring is not unstretched in the system's equilibrium state. This will make the gun much easier to design, but it'll also mean you won't have constant pressure.

Let me also say to NOT build a spring water gun unless you've done the math to make sure the spring can handle the stress. The formula on the page I've linked to does this. Make sure the stress doesn't exceed a good safety factor over half the yield stress and you and your spring will be fine.

It'd also be necessary to make sure the pressure chamber, piston, and other components of the system can handle the stress, but that'd take too long to explain and others could (and have) explained it better on the internet before. Basic solid mechanics isn't too difficult.

After futzing around with the numbers for a bit, here's a reasonable system that could work:
Spring:
Material: 1065 steel
Outer diameter: 2.75 inches
Wire diameter: 0.3772 inches
Coils per inch: 2.117
K: 29.36 pounds/inch

System:
Maximum pressure: 30 psi
Pressure ratio: 0.5229
Piston diameter: 3 inches
Deflection from maximum: 3.453 inches
Maximum deflection: 7.222 inches
Water volume: 13.3 ounces
"Dead space": 28.78 inches
Stored energy: 557.2 inch-pounds
Safety factor: 1.2

Let me note that the safety factor is pretty low and I haven't even looked at pressure chamber material strength, so I wouldn't suggest building this. That's not saying anyone really could though--it uses custom springs.

The bow idea sounds interesting, but too complicated. Though, if it works and works well it'd be worth trying. I know nothing about this though... but I'm intrigued so I'm taking a look.

[aEx155]

For the bow idea, are you guys thinking of using some kind of plunder/piston in place of an arrow i a regular pow, or something like the Bowgo where a bow acts as a piston's spring?

[cantab]

First I've heard of the bowgo. So yeah, I was considering piston in place of an arrow.

You have a quite long piston, in a tube the rear of which has slits in the side, which the bowstring passes through, being fastened to the piston. It needs to be long enough such that at full draw the front of the piston is forward of the end of the slits, or the water comes out through them. This means you need a piston the length of the draw, in a tube twice that length, the rear half slitted.

A regular bow would likely require a lot of skill to use, like it would shooting arrows. A crossbow would be simpler to use, but more complex to build. Also crossbows have shorter draws.

It would be somewhat hazardous. Especially if air shot, or using a large nozzle, the piston will move quite fast. Whatever stops it coming out the barrel needs to be very solid. I'd also make the piston as light and unaerodynamic as possible, so that if it did escape the gun, it wouldn't go far or cause much damage.

[Ben]

After doing the math for some springs I know can compress an appreciable distance, I'm not sure the stress formula I've been using is correct. For example, one spring commonly used in homemade Nerf guns apparently can only be compressed about 2 inches before it will yield when I've seen it compressed 8 without yielding.

I'm going to write the stress equation I've been using on paper and ask my mechanics instructor if it is correct. This'll have to wait about a week because of Thanksgiving break, but that's fine as long as I remember. Remind me if I don't.

[isoaker_com]

While I cannot report on performance, the new Pulse Series from Buzz Bee Toys available presently in Australia all use springs. From Fox's insights, the blasters perform quite well, though PC sizes are small. Either they are doing the impossible or something is awry with the initial calculations that a spring-based system isn't feasible (unless, of course, one doesn't consider the Pulse line truly CPS-like performance).

[cantab]

Indeed. 'Constant pressure' seems to have become a bit of a buzzphrase, yet the APH design proves that it's perfectly possible to get good performance despite having pressure drop.

[Ben]

Yes, let me reiterate that it's not difficult to get good pressure with low, but acceptable pressure with my current model. However, the pressure ratio will be zero more often than not.

As a reminder, the pressure ratio is how I measure constancy of pressure. A pressure ratio of 1 is constant, a pressure ratio of 0.5 means the pressure at the end of the shot is half the initial pressure, and a pressure ratio of 0 means the pressure drops completely. The pressure ratio is simply the initial pressure divided by the final pressure.

At the moment I'm at my parents house for thanksgiving, and luckily my father seems to have an old mechanical engineering design book with an entire chapter on springs. So perhaps this could clear some things up. The equations I've been using are correct, but perhaps my failure criteria is not.

Also, the maximum shear stress does occur at the edge of the spring wire, but on the inside of the spring. Perhaps my instructor misunderstood what I was saying when I said "outside of the spring" and thought I was referring to the outside of the spring wire. I should have been more clear.

Edit:

Okay, I've changed my failure criteria to what my father's book suggests, which is substantially different from what I was using earlier. Using McMaster-Carr springs I can easily design a system with >400 mL of water capacity, a pressure ratio of 0.7, a maximum pressure of about 35 psi, and only about 23 inches of "dead space". However, it would use three springs, at a total cost of about $30. Again, it seems air pressure is better, but custom springs might be more useful.

I'm going to try to design a system where the total amount of energy is limited to see if I can make a water gun with only one priming method.

[Ben]

I'm not sure anyone read my edits so I'll summarize my updated findings.

Spring water guns with decent pressure ratios are possible. However, their capacity and pressure will be limited and they will have higher costs than air pressure. The most reasonable systems would use custom springs, which can become expensive, but costs can be lowered by using common gauges of wire and a common inner diameter.

Most importantly, the math involved with designing such a system is beyond the scope of this forum. For this reason alone I do not suggest designing a spring water gun. One thing to keep in mind is that my analysis so far has only involved the stresses of the springs. The PVC pipe might not be able to take the combined pressure and tension stresses. I have not done the analysis to determine this yet. Something stronger like sch. 80 pipe or aluminum might be necessary.

For single action systems (like a water gun with a pump linked mechanically to the piston), springs are more reasonable, but perhaps not much more reasonable than air pressure. Air pressure is still a possibility in such systems if you can seal the location where the shaft exits, but this is more difficult than spring systems. Spring systems however offer you the possibility of hiding a pulley system inside of the spring, which'll give you the force multiplication needed to pull the spring.