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The Mathematics of FPS and ROF


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20 replies to this topic

#1 Guest_CheeseWiz_*

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Posted 23 July 2010 - 08:18 PM

Hey everyone, I was sitting around, mathematics and physics books in my lap, when I thought up some simple equations that would answer all the questions about FPS downgrading for a higher ROF. Thought it would be helpful to newbies who just want to have to change out a spring for higher ROF and wonder what spring will get what ROF.

Anyway, these equations were written to answer a specific question:

"Right now I have x FPS and y ROF, what will my ROF be if I lower my FPS to z?"

NOTE: In order to use these equations you need 2 measurements about your gun: The current FPS and the current ROF, there are many ways to get these measurements but if you can spare the cash a Chronograph that also has a ROF mode will be the easiest and most accurate method.

NOTE 2: if you have a good understanding of algebra and percents, you can just skim this to see what my equations are, but I go into detail on how to use them to better accommodate those who don't grasp mathematics as well as I do.

NOTE 3: Everything in here assumes a chrono with a .2g bb both at the beginning and the end

My Theory:

Given no changes besides a weaker/stronger spring, my theory is that FPS and ROF are directly related as the determining factor in both categories becomes the same. Namely, how hard it is to pull the spring.

Therefore, when changing springs, shouldn't the percent increase or decrease in spring tension [FPS] reflect the percent increase or decrease in ease to pull [ROF]?

Given this as true, we can now use simple percent calculations to predict the ROF at any given FPS.


How to run the numbers:


Step 1: percent change in FPS

What we are doing here is calculating how much the FPS [spring tension] changed in terms of a percent.

In these problems, the initial FPS and the final FPS are known/predicted, and are expressed as if and FF respectively.

Percent change in FPS = if / FF * 100
percent change in FPS is equal to Initial FPS divided by Final FPS multiplied by 100

EXAMPLE PROBLEM:

percent change in fps =

400 if / 300 FF * 100 =

4/3 * 100 =

.75 * 100 =

75%

Step 2: percent increase in ROF

This part is much more simple, all we need to do is take the percent found in the first problem, subtract it from 100% and add 100% to that value to set up the right number for the next problem.

Here the only needed data is the percent increase in FPS, which we got with the last equation, and will be expressed as %IIF

percent increase in ROF = [100% - %IIF] + 100%

percent increase in ROF = [100% - percent increase in fps] + 100%

EXAMPLE PROBLEM:

percent increase in ROF =
[100% - 75%] + 100% =
125%


Step 3: So what's my ROF CheeseWiz?

Okay, in this step we finally get to figure out what your [theoretical] ROF will be with your new FPS

Here the only things you need to know are you percent increase in ROF [which we figured out in step 2] and your current ROF [which can be measured by some chronos or an audio recording software, but you can find out how to do that somewhere else on ASF]

theoretical ROF = percent increase in ROF * current ROF

Now, a quick tip, when multiplying, you have to multiply the percent increase as a PERCENT, not as a normal number. Most calculators have a button with the % symbol, just hit that and it will figure out the percent for you, otherwise, take the percent increase in ROF and divide by 100 [it should be around 1.xx], then use the answer to this problem in place of the percent increase in ROF as a normal number.

EXAMPLE PROBLEM MULTIPLYING AS A PERCENT:

in this problem we use the same answers from our equations before, and we'll say you were getting 20 rps at the original FPS

Theoretical ROF =
125% * 20rps =
25 RPS


EXAMPLE PROBLEM MULTIPLYING AS A NUMBER:

Theoretical ROF =
125% / 100 * 20 RPS =
1.25 * 20 RPS =
25 RPS



Okay, so, now we know that if you have 20 RPS at 400 FPS, by doing nothing other than putting in a 300 FPS spring you should theoretically have 25 RPS. Fun little equations huh?

If anyone has any questions regarding the use of these equations feel free to PM me.

Edited by CheeseWiz, 23 July 2010 - 11:03 PM.


#2 Airsoftguy238

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Posted 23 July 2010 - 08:27 PM

Your have far too much time on your hands.
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#3 Guest_CheeseWiz_*

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Posted 23 July 2010 - 08:30 PM

EDIT:

I'ma think before I speak...

Edited by CheeseWiz, 23 July 2010 - 08:32 PM.


#4 Star_folder

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Posted 23 July 2010 - 08:40 PM

QUOTE (CheeseWiz @ Jul 23 2010, 09:18 PM) <{POST_SNAPBACK}>
Thought it would be helpful to newbies who just want to have to change out a spring for higher ROF and wonder what spring will get what ROF.

Calling me a newbie?

*edit*
Check my next post, re-trying the math.

Edited by Star_folder, 23 July 2010 - 08:57 PM.


#5 Guest_CheeseWiz_*

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Posted 23 July 2010 - 08:48 PM

QUOTE (Star_folder @ Jul 23 2010, 06:40 PM) <{POST_SNAPBACK}>
Calling me a newbie?

and I can't help but feel as if your logic might be a bit off, I'm pretty sure it's not quite right.

390fps at 14rps, ok, so changed that to 320fps
320/390*100=82%

So my increase in rate of fire should be 182% correct?

182% * 14rps = 25 rps

I think I've done all the math correctly, tell me if I'm wrong.

Problem is, that I got 19rps after downgrading.


lol, no, you just inspired me to actually put it up, I already had the theory.


and whoops, I forgot a step, correcting.


#6 Mr.Z@p

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Posted 23 July 2010 - 08:48 PM

interesting, but far off if your said example happens you have a motor the size of a brick

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#7 Guest_CheeseWiz_*

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Posted 23 July 2010 - 08:54 PM

QUOTE (Star_folder @ Jul 23 2010, 06:40 PM) <{POST_SNAPBACK}>
Calling me a newbie?

and I can't help but feel as if your logic might be a bit off, I'm pretty sure it's not quite right.

390fps at 14rps, ok, so changed that to 320fps
320/390*100=82%

So my increase in rate of fire should be 182% correct?

182% * 14rps = 25 rps

I think I've done all the math correctly, tell me if I'm wrong.

Problem is, that I got 19rps after downgrading.


okay, my equations have been fixed, run the numbers again and pay close attention to step 2, it now has the evil parenthesis in it. Thank you for notifying me of this, I knew that this step had to be done, I guess I forgot to type it :]

QUOTE (Mr.Z<AT>p @ Jul 23 2010, 06:48 PM) <{POST_SNAPBACK}>
interesting, but far off if your said example happens you have a motor the size of a brick


What are you trying to tell me?

Edited by CheeseWiz, 23 July 2010 - 08:56 PM.


#8 Star_folder

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Posted 23 July 2010 - 09:01 PM

Alright, lets try this again now that you’ve changed the equations a bit:

I was shooting 390fps at 14rps
And then switched to 320fps, so:
320/390*100=82$
And then
[100% - 82%] + 100% = 116%

So 116% * 14rps = 16rps.

Closer, but still not quite there, I had 19 rps after the downgrade.

a little more math that may help out. 14 to 19 rps is an increase of 35% after a spring decrease of 16% seeing this, the rps might increase twice the percentage of the spring decrease.

Edited by Star_folder, 23 July 2010 - 09:04 PM.


#9 Guest_CheeseWiz_*

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Posted 23 July 2010 - 09:08 PM

QUOTE (Star_folder @ Jul 23 2010, 07:01 PM) <{POST_SNAPBACK}>
Alright, lets try this again now that you’ve changed the equations a bit:

I was shooting 390fps at 14rps
And then switched to 320fps, so:
320/390*100=82$
And then
[100% - 82%] + 100% = 116%

So 116% * 14rps = 16rps.

Closer, but still not quite there, I had 19 rps after the downgrade.

a little more math that may help out. 14 to 19 rps is an increase of 35% after a spring decrease of 16% seeing this, the rps might increase twice the percentage of the spring decrease.


EDIT:

you have a few math errors:

100 - 82 = 18

so:
118% * 14 RPS = 17 RPS

Ha! 1 whole FPS closer!





As for the equations, please understand that these can only estimate your ROF, in real life there are countless other variables effecting it that I cannot account for. and with a 10.53% error [17/19 * 100], I think that the equations are pretty spot-on given the differences between theory and real life.

Edited by CheeseWiz, 23 July 2010 - 09:19 PM.


#10 Star_folder

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Posted 23 July 2010 - 09:20 PM

Very true.

Well, the test was done with an Ares Tavor. The gun has a custom mechbox allowing for quick spring changing. This eliminates variables within the gun as the spring and spring guide are the only things that moved.

I may start looking around to see what I can find on this topic, I've got a lot of different springs lying around to test.

#11 Guest_CheeseWiz_*

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Posted 23 July 2010 - 09:29 PM

QUOTE (Star_folder @ Jul 23 2010, 07:20 PM) <{POST_SNAPBACK}>
Very true.

Well, the test was done with an Ares Tavor. The gun has a custom mechbox allowing for quick spring changing. This eliminates variables within the gun as the spring and spring guide are the only things that moved.

I may start looking around to see what I can find on this topic, I've got a lot of different springs lying around to test.


Please do so, with testing the equations can only become more accurate.

A few variable I would like to list though:

-battery charge [this is almost impossible to control, especially as your battery goes through more and more charge/uncharge cycles]
-battey output [believe it or not, this isn't a constant]
-motor [especially as it is used, it won't always preform the same]
-airseal [especially with changing of springs, it won't always be put back the same way]
-temperature [colder air = less pressure = eaisier to compress]


There are infinite more like those, right down to how the electrons jump between the atoms, so the equations will never be truly exact.


#12 buppus

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Posted 23 July 2010 - 09:33 PM

I like the idea, but your math is way off. For one you've not provided a way to use your equation in reverse - if you're starting at 100% RoF and moving to a lighter spring, then what %RoF do you use when you want to move up to a heavier spring? It's unfortunately not as simple as calculating percent change in two different factors and then adding them together, because those numers have no correspondence to the actual physics at play (not to mention a host of other factors such as gear ratio, motor torque/rpm, electrical resistance, current and voltage supplied from the battery, etc. etc. etc. . .

What your equation suggests is that changing the 400fps spring to a 300fps spring is simply increasing the rpm of the motor by 75%, which is obviously not the case at all. It is true that the motor will not have to work as hard, but there is certainly not a direct relationship between the fps and rof as you suggest.

And to see how motor torque comes into play, and ignoring the shortcomings of the equation, but instead going with the general intent but in the other direction... say you started with your 400fps/20bps setup, which for your average setup means mostly stock parts, and a mediocre battery, and you dropped a 700 fps spring in there... I promise you it would not be able to pull it at all... let alone the 12.5 bps that if the equation worked would have produced. Why? A lot of reasons probably, but simply put, because the motor would not have enough torque to pull it (unless for some reason you were using a really nice lipo with infinite torque gears and a nice high torque motor for your original setup - unlikely).

I garauntee that if you dropped a 300fps spring into a gun that was already shooting 400fps <AT> 20bps, that it would not shoot at 35 bps. Realistically, you're looking at an increase of maybe 5 bps if you're lucky - definitely not 15 under any circumstances.
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#13 Star_folder

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Posted 23 July 2010 - 09:34 PM

Yeah, that's true. My goal is to just get to a point where an equation can provide an area answer. Like "it will be around 25rps +-2." as an answer.

#14 Guest_CheeseWiz_*

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Posted 23 July 2010 - 09:40 PM

QUOTE (buppus @ Jul 23 2010, 07:33 PM) <{POST_SNAPBACK}>
I like the idea, but your math is way off. For one you've not provided a way to use your equation in reverse - if you're starting at 100% RoF and moving to a lighter spring, then what %RoF do you use when you want to move up to a heavier spring? It's unfortunately not as simple as calculating percent change in two different factors and then adding them together, because those numers have no correspondence to the actual physics at play (not to mention a host of other factors such as gear ratio, motor torque/rpm, electrical resistance, current and voltage supplied from the battery, etc. etc. etc. . .

What your equation suggests is that changing the 400fps spring to a 300fps spring is simply increasing the rpm of the motor by 75%, which is obviously not the case at all. It is true that the motor will not have to work as hard, but there is certainly not a direct relationship between the fps and rof as you suggest.

And to see how motor torque comes into play, and ignoring the shortcomings of the equation, but instead going with the general intent but in the other direction... say you started with your 400fps/20bps setup, which for your average setup means mostly stock parts, and a mediocre battery, and you dropped a 700 fps spring in there... I promise you it would not be able to pull it at all... let alone the 12.5 bps that if the equation worked would have produced. Why? A lot of reasons probably, but simply put, because the motor would not have enough torque to pull it (unless for some reason you were using a really nice lipo with infinite torque gears and a nice high torque motor for your original setup - unlikely).

I garauntee that if you dropped a 300fps spring into a gun that was already shooting 400fps <AT> 20bps, that it would not shoot at 35 bps. Realistically, you're looking at an increase of maybe 5 bps if you're lucky - definitely not 15 under any circumstances.



Please re-read the equations, they have been edited since you started writing this post.

Also, the equations aren't meant to be used in reverse, I could provide the information to do so, but I didn't feel it would apply.

Also, the equations were never meant to be exact, I am well aware of the many other variables that I can't account for. Hence my frequent use of "predict" and "theoretical"

Finally, your 700 fps spring setup breaks the equation by making the the spring not cycle therefore the final FPS is 0 and the equation fails.

Edited by CheeseWiz, 23 July 2010 - 09:43 PM.


#15 Izak

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Posted 23 July 2010 - 09:49 PM

GUY STOP TROLLIN MY BROSIF!

Now the thing about his equasion is that it is probably almost spot on for a sertain base BPM.

If you have a diffrent motor/gears/batteries chances are that his equation will be off a little... but he did get around the actual equasion at least when you use airsoft guns... which is great considering the actual equasion is a BEAST.

Edited by Izak, 23 July 2010 - 09:52 PM.


#16 buppus

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Posted 23 July 2010 - 10:40 PM

It still stands that your basic premise is false - there is no direct relationship betwen FPS and RoF. Sure it happens to be in the ballpark in some instances, but those instances are severely limited. The accuracy of the predictions from your equation are no better than your average airsoft enthusiast's educated guesses. The perfect example is that while I was typing up my first response (before you edited the equation) my guess turned out to be exactly what it predicted. Hunterseeker5 commonly makes educated guesses about what kind of RoF one would expect from a given build, and I would take his word a million times over the equation.

I'm not trying to tear you down here - like I said, your equation may produce predictions that are similar to the real-world results in some cases. The problem is that it cannot do so reliably, and cannot even be used as a rule of thumb because the conditions for its accuracy are so limited. Your margin of error is probably, conservatively mind you, +/-3bps, and when the whole point of the equation is to calculate gains that are just that small, the margin of error destroys your result!

Again, no offense, but I'm more in favor of the guestimation method, because at least then it doesn't propogate a faulty understanding of the physics at play

Edited by buppus, 23 July 2010 - 10:41 PM.

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#17 xlizer

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Posted 24 July 2010 - 12:20 AM

Okay, here's the thing. the relationship between applied torque and rpm of a motor is rarely linear. In fact, I can say with reasonable certainty that it's almost never linear. In an industrial or scientific setting, a torque vs rpm graph is standard fare to look at when choosing a high-end motor. In airsoft, though, this is not an option because the data just isn't provided.

If you want to do this, you have two options. I'll start with the most interesting one: Build a setup that tests RPM vs. torque applied. It's a simple test really. Affix a cylinder to the axle of the motor and support both ends so that there is no torque on the axle. attach a weight to a string and the string to the cylinder. Run the motor at its intended voltage and measure how long it takes for the string to be pulled a given distance. Repeat for different weights until the motor stalls. post data on the internet.

If you wanted a more accurate setup, it's possible to wire a stopwatch up with two sets of latching limit switches in series. Set up the test with switch set A closed and switch set B open. By closing switch set B manually, you start the timer and begin the experiment. When the weight is at the end of its travel, have it press switch set A. This will open the circuit, stopping the measurement and the motor. There are many possible variations of this setup that you could employ to make this more accurate than doing it by hand. To make it simpler, you could set up the rig to run on a toggle and just have it pass by two individual switches a given distance apart.

You could try, as others have requested, actually swapping springs and taking measurements of BBPS and FPS. Graph those out, plot a best fit curve, and you have the data you're looking for and a rough idea of what the torque V RPM chart of your motor looks like.

EDIT: If you don't have the software to plot a best fit curve, just send me the raw data. I'll do it for you and send it back. I'm a physicist and I plot best-fit curves all the time, so it's really no trouble. I would never suggest anyone to do that kind of calculation by hand.

Edited by xlizer, 24 July 2010 - 12:32 AM.

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#18 buppus

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Posted 24 July 2010 - 12:36 AM

QUOTE (xlizer @ Jul 23 2010, 11:20 PM) <{POST_SNAPBACK}>
Okay, here's the thing. the relationship between applied torque and rpm of a motor is rarely linear. In fact, I can say with reasonable certainty that it's almost never linear. In an industrial or scientific setting, a torque vs rpm graph is standard fare to look at when choosing a high-end motor. In airsoft, though, this is not an option because the data just isn't provided.

If you want to do this, you have two options. I'll start with the most interesting one: Build a setup that tests RPM vs. torque applied. It's a simple test really. Affix a cylinder to the axle of the motor and support both ends so that there is no torque on the axle. attach a weight to a string and the string to the cylinder. Run the motor at its intended voltage and measure how long it takes for the string to be pulled a given distance. Repeat for different weights until the motor stalls. post data on the internet.

If you wanted a more accurate setup, it's possible to wire a stopwatch up with two sets of latching limit switches in series. Set up the test with switch set A closed and switch set B open. By closing switch set B manually, you start the timer and begin the experiment. When the weight is at the end of its travel, have it press switch set A. This will open the circuit, stopping the measurement and the motor. There are many possible variations of this setup that you could employ to make this more accurate than doing it by hand. To make it simpler, you could set up the rig to run on a toggle and just have it pass by two individual switches a given distance apart.

You could try, as others have requested, actually swapping springs and taking measurements of BBPS and FPS. Graph those out, plot a best fit curve, and you have the data you're looking for and a rough idea of what the torque V RPM chart of your motor looks like.

EDIT: If you don't have the software to plot a best fit curve, just send me the raw data. I'll do it for you and send it back. I'm a physicist and I plot best-fit curves all the time, so it's really no trouble. I would never suggest anyone to do that kind of calculation by hand.


Thanks for the explanation - I knew what I wanted to say but didn't feel like hashing out any more technicalities.. I'm glad someone could step in with some real engineering experience.

Also, have you suggested this to the guys that were trying to take on that comprehensive motor testing project? This is data that is sorely lacking when it comes to choosing airsoft motors and it would be great to have someone beginning to gather the data

Edited by buppus, 24 July 2010 - 12:40 AM.

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#19 ACE X-2

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Posted 24 July 2010 - 12:37 AM

QUOTE (Airsoftguy238 @ Jul 23 2010, 06:27 PM) <{POST_SNAPBACK}>
Your have far too much time on your hands.


Thank you for your contribution to the airsoft community. rolleyes.gif

#20 xlizer

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Posted 24 July 2010 - 12:57 AM

QUOTE (buppus @ Jul 24 2010, 01:36 AM) <{POST_SNAPBACK}>
Thanks for the explanation - I knew what I wanted to say but didn't feel like hashing out any more technicalities.. I'm glad someone could step in with some real engineering experience.

Also, have you suggested this to the guys that were trying to take on that comprehensive motor testing project? This is data that is sorely lacking when it comes to choosing airsoft motors and it would be great to have someone beginning to gather the data


No problem. This is what I do.

Anyway, I remember seeing that a while back, and I honestly don't remember if I advised them to do anything similar or not. I can only expect that either I was satisfied with their methods, or I advised them to do something like this. I'm not one to not jump in when there's inaccurate technical information being flung around.
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#21 Guest_CheeseWiz_*

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Posted 24 July 2010 - 12:16 PM

Well, it seems that there is a split opinion.

I feel that the 10% error is due to the inability to account for other variables, other feel that the basis for my theory [that fps and rof can be directly related] is incorrect.

As I lack the means to test my theory in multiple cases [a whole bunch of springs and a chronograph] we only have the test done by Star_folder to use as a basis.

As I have stated before, using the test done by Star_folder my theory holds at a 10% error, which I feel is pretty good given the unaccounted variables.

If you disagree with my theory, that's fine, it's only through constructive criticism that it will improve, but please state a test you preformed showing it as false, as saying what amounts to, "You're wrong because your theory is wrong." will not help me improve it.

If I can get enough data [rof given fps in the same setup] I will assemble a graph and put both my theoretical graph and the actual graph on it to see how my equations line up and how ROF can be better predicted.

Thank you.




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