Formulas

[drow13f]

Hey folks, I'm new to the forum. I've been shooting a bow since I was a kid, but just started traditional about 5 years ago.

My wife and I have a private school and I'm wanting to start an archery program for our high school students. We're a small school so I'm talking about 5-10 students in a normal year.

I figured this would be a great way to use a fun activity to encourage them to learn some math and science.

I've found a lot of the formulas I need; Finding bow efficiency, calculating speed, kinetic energy and momentum, FOC. I'm still looking for a few formulas and I haven't been able to find them anywhere.

I've been looking for a formula for determining spine needed based on point weight, draw weight and length. Also looking for a good formula for penetration based on the density of the target and the energy of the arrow.

I've found several trajectory and drag formulas for working out max range and drop per set distance, but they've all been a bit too complicated for high school science. So if anyone knows of an easier formula for these it would be appreciated as well. All of the ones I've found are more on the level of college physics and what I'm looking for wouldn't have to be as accurate.

We'll probably start with calculating draw length and bow length. Then we'll test our bows for efficiency. We'll be building and shooting these bows

 https://worldarchery.org/news/131733/30-pvc-bow-thats-growing-archery-americas

and building wood arrows. We'll also be building targets out of different materials to perform our penetration tests.


I know some of this may seem ridiculous to some of you but I just figured it would be a good way to mix something fun with some education, because there is always that one student that says "why do I have to learn this stuff, I'm never going to use it" and I want to show them that it can be used to do fun stuff too.  And I realize there are tons of reference charts and calculators out there we could use, but I'm more interested in working out the formulas, coming up with a calculated prediction and then fine tuning with trial and error from there.

I'll take any and every formula you can come up with
Thank you

[SportHunter]

Stu Miller has created a great arrow spine calculator that takes into consideration many of the variables you mentioned and has a good reputation of accuracy if the data input is accurate. You may want to look him up for some insight. A version of his calculator has been adopted by 3rivers archery.

[drow13f]

That sounds great, you wouldn't happen to know the formula that he used would you?

[SportHunter]

Here is a link to the calc, his contact email is on there. Good luck with the project.

 http://heilakka.com/stumiller/

[drow13f]

Thanks SportHunter, I'll try and get a hold of him.

[drow13f]

I just realized that I should share the formulas I already have for critique and for anyone that wants to look at them.

I plan on starting with calculated draw length which is (arm span / 2.5)

I couldn't find a formula for bow length so I'll have to use a reference chart or come up with a formula but here is what I'm going on.  I realize this has more to do with preference, finger pinch, smoothness of draw and stacking than anything else.

DRAW LENGTH……………..BOW LENGTH
14-16 inches……………….48 inches
17-20 inches……………….54 inches
20-22 inches……………….58 inches
22-24 inches……………….62 inches
24-26 inches……………….64-66 inches
26-28 inches……………….66-68 inches
28-30 inches……………….68-70 inches
31 inches  ++…………........70-72 inches

We will then build strings for our bows, and shoot for a brace height ratio that will give us 3.5 inches/inches drawn as a starting point to be adjusted later.  (Example 24 in draw = 24/3.5= 6.85).  

We will then get a calculated string length for our bow to get our BH.  The equation (c=2 (sqrt (r2 -t2)))  A 24in DL  will build a 64in bow so to reach our BH we plug in (2(sqrt(32x32) – (6.85x6.85)= 62.5)

We will then string the bows and fine tune from there.  Then we will map out the draw force curve for each bow.  To do this we mount our bow in the vice and draw our bow an inch at a time attached to a bow scale and pulley recording the draw force at each inch of draw to 32 inches.  We then take and measure the difference in #s between each inch.  We then take our starting weight difference and divide by two.  We then take every subsequent difference and average it with the previous difference  and then add to our running total of inch#s  (example chart below: at 10 inches its 11# the fourth column records the difference between 10 and 9 inches.  The fifth column then takes the average of those two numbers and adds it to the number before it.  At 11 inches the 5th column shows the average of 11# and 5# in the fourth column (8#) and adds it to the previous 5.5#.  The sixth column is the running sum of the 5th column as Stored Energy in inch#.  The seventh column is the sixth column divided by 12 for foot#)

  http://www.buildyourownbow.com/wp-content/uploads/2010/10/brents_recurve_FDcurve2.jpg  

To get the calculated Stored Energy per Pound of Draw Force simply divide the seventh column of SEft# by the corresponding #DF to get the SE/PDF.  

I have this worked out on a 25# target bow.  I will then find my draw length of 29 inches on the #DF column which is a measured 27#.  Moving over to the SE ft# column, my 27# draw only has 20.2# of SE.  So rather than try and match a spine to 27# I will match the spine to 20# because the extra 7# will be wasted in the limbs and string.

I still need to find a descent formula for determining the spine weight for the deflection needed for this bow, but for now I'll just use reference charts.  

To keep things safe and the math simple we will build our arrows around 10 grains per pound of draw weight. So with our 20# draw will shoot for a 200 grain finished arrow.   Knowing this we will try and calculate a Front of Center between 10 and 15%.

The formula for FOC is

( 0.5 * Nock Wgt ) + Fletch Wgt + ( 0.5 * OAL * Shaft Wgt ) + ( OAL – 1 ) * Point Length ) / ( Nock Wgt + Fletch Wgt + Shaft Wgt + Point Wgt ) – 0.5 * OAL) / OAL)

With this formula we can plug in the known numbers of Nock Wgt, Fletch Wgt, Point Wgt, Point Lgth, and OAL (Overall Length determined by our Draw Length or spine formula if I find one) to solve what shaft weight we need to determine how many grains per inch we want.

My current set up gave me a calculated FOC of 11.8% my confirmed FOC is 11.3%.  

Now that we have an arrow plan for our bow we will make a prediction of what kind of speed we can get out of this.  


SQRT ( Draw lgth – BH ) * SEft# * 32.17 * 7000 / ( 12 * ( Arrow Wgt + String Wgt + Nock Wgt + Silencer Wgt)

My current set up with 20.2# SE# ; 29 in DL;  8 in BH; 200 gr arrow and 116 gr combined string weight gives me a calculated speed of 158.75 fps.  And this is confirmed with chronograph between 150 and 160.  I plan on building a shooting machine in the future for more precise measurements.

Next we find our Kinetic energy so that we can find the efficiency of our bow.  (FpsxFps) * Arrow wgt) / 450240.  With our speed of 158.75 and Arrow weight of 200 gr we get a KE of 11.19.  To calculate momentum is (Fps * Arrow wgt) / 225400) (multiplied by 100 for a perctentage).    To calculate the bows efficiency just divide the KE by the SE#, my set up is 55.4% efficient.

That is about all I have at the moment, but I'm looking for formulas to find spine and arrow length.  And I also have a trajectory formula that I haven't worked out yet, but I'm looking for a simple drag formula to incorporate into the trajectory to give a more accurate reading for drop and max distance.  All of the formulas I have found so far for drag are too complicated for high school level so if you find a simple one let me know.  

I would also like to find a fletch recovery formula that will help us determine how many square inches of exposed fletching we need to recover the arrows flight at X amount of distance.   Someone just sent me a good penetration formula but I'm still looking for others to compare penetration based on target density.      

Like I said I plan on building these pvc bows with the students and make the predictions and then go out and shoot to see how close our predictions are.  I have my doubts about the efficiency of the pvc bows but that's what is in our budget for now.    After this we'll be working on form and getting proficient with the bows, maybe throw the dartboard target up and play some games  But through the year I plan on throwing in experiments here and there.

I'll give the students the materials to play with and we'll have competitions on fastest arrow within a certain g/# and competitions for deepest penetration or once I get the shooting machine built we will do one with least amount of drop of x amount of distance.   I just want to get them excited about the math and science behind it while learning a skill.

[drow13f]

and I forgot to mention that the calculations that are worked out are not a pvc bow.  I haven't built one of those yet, I worked these out on my 25# Samick Journey and they were very accurate predictions.

[KenH]

If you're serious about building PVC Archery gear, I serious suggest you check out the Google Plus group on the subject.  Folks here know wood and wood/glass, and a little wood/hon/sinew, but they mostly don't do PVC.

Send me a PM for details.

I think your draw length to bow length table is 'way out of line.  I shoot almost exclusively bows between 48" and 50" n-t-n strung, and some as small as 40".  My draw length is 27".

Depending on the design of the bow, there is NO stacking or finger pinch at those lengths, and the draws are smooth.

I think your brace height is somewhat skewed, as well.  Straight bows or straight bows with recurve tips want a brace height of 6"-6.5", where seriously reflexed/recurved bows, like the Scythian designs want a 7" to 7.5" brace.  To blindly say a 24" draw bow needs a 6.85" draw seems out of line with general practice.

[Mark R]

Every bow will be different, they all have a sweet spot on where they like brace height according to the draw length, arrow specs, style, ect ect of each individual. If you are just using a vise type of  shooting machine your results can be different then the individual results.

[Mark R]

I must add keep up the good work,you are an inspiration.

[LittleBen]

[drow13f]

Hey Ken,

Definitely interested in the group.

You shouldn't look at these formulas as anything more than predictions and starting points.  I know there is no way possible to account for all of the variables and personal preferences out there.  

All I'm trying to do is lay out all of the starting points in a logical order but even the order is depends on the goal.

But the order that I've laid out here is for beginning students that are about to learn the basics of target shooting. And since I'm more concerned with showing them the importance of knowing how to do algebraic equations and how to think critically, I don't want them just to pick an arrow out of a box and try to hit the target. I want them to think about what their goal is and then plan to achieve it.

If we are looking for a certain spine weight and deflection I want them to know why. The formula will only get us close but it will be a good starting point and hopefully help them understand the concept, but then I want them to know how to adjust it. Not just because I told them cut some off or adjust your BH, add a heavier point, I want them to know why.

I realize the BH ratio to inches drawn may not work, I've never shot with anyone that had a draw length less than 26 inches so I have no idea how that will work, but I've researched the idea a lot and found that for youth bows BH can be set shorter because of the short DL.   Also we're all going to be shooting bows that are made of the exact same material somewhere in the 25# range.  For a student with a 28 inch draw they will have a definite advantage over a student with a 24 in draw and a BH between say 7 and 7.5.  The 28 in draw will draw 21 inches and reach a full 25 pounds.  The 24 in draw will draw 17 inches and probably only reach 20#.  

We could definitely make it work if we had to, but I was going to try the idea that I researched first being that with a 3.5 ratio the 24in would still draw 17 in but the 28 would draw 20 bringing them a little closer.  Plus the 24 would be using a shorter bow so it would be a little snappier having a stiffer static spine.  

I may even do some test before hand to see what the # is for my 29 inch draw on different bow lengths and maybe different string lengths to play with BH just to get an idea.

All of this is just an attempt to make the bows as equal as possible.  I know they won't be perfectly equal but they will be closer.  maybe it will work maybe it won't, we'll just have to see.

All of this is done in the name of fun and learning with the emphasis on learning. There is a scientific formula for everything, and if there isn't one we can make one, there may be too many variables for the solution to be viable, but the solution can at least give us a place to start.

I really appreciate all of your advice and I'm trying my best to explain my intentions without writing a book.