# Linear approximation of the draw force curve



## Hank D Thoreau (Dec 9, 2008)

On several occasions I have discussed the linear approximation of the draw force curve. We call this an ideal bow. It is a useful device for understanding draw force and energy relationships at you change draw length. While it may not predict the exact draw force and energy, it is a useful tool in understanding how these properties change with changing draw length. On a plot of draw force versus draw length, a linear draw force curve is a straight line with a positive slope. In other words, it increases weight as you draw at the same weight per inch ratio. This is what you would expect for a perfect lever. It turns out there are no perfect levers. Even a longbow deviates from linearity since you are pulling the limbs at an angle. In the following posts I am going to use the linear approximation of the draw force curve to explain how energy changes with draw length. In particular, why it changes more than you might expect as you draw either less, or more than 28 inches due to the quadratic relationship between energy and distance. The charts below show the draw force curve, energy and energy per weight on the fingers for a 40 pounds ideal bow. The chart of the right is the first derivative of the draw force curve which measures pound/inch at each point along the draw cycle. It will be a straight line with a value of 2.35 pounds/inch. If we think of an idea spring, where F=-kx, where x is displacement and k is the force constant, k is also constant at 2.35.


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## Hank D Thoreau (Dec 9, 2008)

The first calculation uses me as a reference. I have a 32 inch draw length. Because of my long draw I store a lot of energy, more so than might be expected. What is shown below is the difference in stored energy for an ideal bow (linear DFC) each at a 40 pound holding weight at their respective draw length. This shows that I will store a whopping 24% more energy holding 40 pounds as someone holding 40 pounds at 28 inches. It also shows that an archer would have to draw 49.41 pounds at 28 inches to get the same stored energy as I get, or 56 pounds at 26 inches. The big difference is due to the quadratic (squared power) relationship between distance and energy. This is an important consideration when you are choosing arrows. That is why I have to shoot very stiff arrows.


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## Hank D Thoreau (Dec 9, 2008)

Let's use the linear DFC to look at the 2 pound rule, i.e., bows pick up 2 pounds for every inch pulled over 28 (or lose two pounds for every inch under). Let's look at a couple of examples using a linear DFC model.

Using the linear DFC approximation you can calculate it as follows:

Assume

40 pound bow at 28 inches. 70 inch recurve.

Brace height as measured from back of bow = 11 inches (9 1/4 BH + 1 3/4 AMO adjustment). This is the starting point for the draw.

At 28 inches the bow has been drawn 28 - 11 inches or 17 inches.

Pounds per inch (linear DFC) = 40 pounds/17 inches = 2.35 pounds per inch. (Extra significant figures added so you can see that the number is the same as the earlier calculation)

In this case a 34 pound ideal bow would be 2 pounds per inch. 

A 60 pounds bow would be 3.53 pounds per inch.

This is only an approximation. But it does demonstrate the principle. The two pound rule is specific to a particular weight bow. A heavy bow does not follow the 2 pound rule, because if it did, it would not be heavy.


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## Hank D Thoreau (Dec 9, 2008)

Using the linear DFC for my hypothetical 40 pound bow I get a 24% increase in energy from 28 to 32 inches (28.3 to 35 foot pounds). Using actual data for 31 bows in my database that have been tested out to 32 inches, I get 19%. This is a significant energy increase. Below is a plot showing the energy increase for the 31 bows. There is considerable scatter since this depends on bow design. There are bows of all types in this data.


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## Hank D Thoreau (Dec 9, 2008)

Energy increases as the square of the distance, distance in this case, being how far you draw the bow. You can see this in the parabolic shape of energy/draw curves. That is a consequence of force being proportional to distance.

I mentioned that I have to use much stiffer arrows than you might expect for the draw weight I am holding. I am not shooting stiff arrows, just much lower spine than someone with a 28 inch draw. That was from post 6. Did I mention somewhere that I shot stiff arrows? If so, what I meant was lower spine shafts. When I was competing in FITA target barebow I was shooting ACE 430 with 90 grain points out of a 40 pound bow. My arrows border on spines that are more common for compound bows, especially when you consider that I do not shoot heavy bows.


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## Hank D Thoreau (Dec 9, 2008)

Let's look at a particular example starting from the perspective of an ideal bow. The data below is for a D shaped Falco Trophy longbow. If it were an ideal bow the draw force curve would be a straight line. You might expect this for a longbow. While the DFC appears pretty straight you can tell from the first derivative curve that there are deviations, both above and below what a linear DFC would predict. Remember that the first derivative curve will be a straight line with a value equal to the slope of the linear draw force curve. To understand why the DFC has deviations you have to understand that an ideal beam must be pulled in a perpendicular direction for the force to be linear. When you start to pull a longbow, the string angle is much less than 90 degrees. That means you have to provide more force to get the limb moving, not unlike what you see when you start to pull a compound bow. As you approach a string angle of 90 degrees it takes less force since until you reach a minimum point. After that you start pulling at an obtuse angle, eventually pulling the limb tips together. This again, requires more force and eventually the bow reaches a point we would call stacking. You can see this in the slightly U shaped first derivative chart. This bow happens to have a fairly flat first derivative curve. This explanation works for a longbow because the lever length of the bow does not change as you draw. This is not the case for a recurve or a bow with a siyah, where the lever length changes as the string moves along the recurve until you reach the point where the string loses contact with the limb. A siyah has two levers, string on the siyah and string off the siyah. A recurve has a continuously changing lever length.


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## BarneySlayer (Feb 28, 2009)

You should have a class at camp. I would sign up.


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## oldmand (Aug 18, 2015)

Well, I think because of the.....or else it could be that.....if only the bow will... Ah, the hell with it.


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## Paul68 (Jul 20, 2012)

Hank - impressive presentation here. Makes me wish I paid more attention during calculus class. Clearly an advantage for you guys with a long wingspan. 

Just a quick question, but maybe not a quick answer. Does the limb length further exacerbate the difference? That is a guy with a 26" DL shooting a 58" longbow, vice your 32" DL pulling a 70" longbow? You get an extra 12" of lever to work with on your way to 40#s.


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## 3finger (Mar 29, 2018)

It might be helpful for the reader to state potential energy square relationship is the result of integrating the application of the variable force over distance. E=1/2kx^2. Examining the results for different values of k might also be beneficial as a simplified correlation of limb length to the slope of the DFC and potential stored energy. 

Good information though.


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## Brian N (Aug 14, 2014)

Hank - Thanks for the analysis (although it has been many, many years since college calculus and physics). Biology was my field, so perhaps this question is somewhat naive: In real world woods and limb materials, how much influence do variances in materials make in the curves? In other words, are some materials and combinations more "ideal" than others?


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## Hank D Thoreau (Dec 9, 2008)

Brian N said:


> Hank - Thanks for the analysis (although it has been many, many years since college calculus and physics). Biology was my field, so perhaps this question is somewhat naive: In real world woods and limb materials, how much influence do variances in materials make in the curves? In other words, are some materials and combinations more "ideal" than others?


Break the question into two parts, risers and limbs. Each wood will have a different modulus so stress/strain curves will be different. That would contribute to a difference is riser flexiblity. You could make up for that with riser design.

For limbs, wood is used as core material and serves as a spacer for the power generating outer layers. The wider the spacing, the greater the strength. 

I have not done any testing of the impact of different woods. It is an interesting question. When I bought the Falco Trophy that I discussed earlier, I asked the bowyer to select wood that he thought would look good. 

I have often wondered if there were optimum wood choices.


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## Hank D Thoreau (Dec 9, 2008)

3finger said:


> It might be helpful for the reader to state potential energy square relationship is the result of integrating the application of the variable force over distance. E=1/2kx^2. Examining the results for different values of k might also be beneficial as a simplified correlation of limb length to the slope of the DFC and potential stored energy.
> 
> Good information though.


My standard analysis package for bow testing includes a spring constant chart. Folks familiar with spring constants may like that. K changes as you draw the bow since real life bows are not ideal bows with a linear draw force curve.


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## Hank D Thoreau (Dec 9, 2008)

3finger said:


> It might be helpful for the reader to state potential energy square relationship is the result of integrating the application of the variable force over distance. E=1/2kx^2. Examining the results for different values of k might also be beneficial as a simplified correlation of limb length to the slope of the DFC and potential stored energy.
> 
> Good information though.


A key metric I started using several years ago is the first derivative of the draw force curve. I used it to explain how super recurves differ from conventional recurves. It is a key part of my analytical package I used when testing bows. I have a post explaining it that I am currently revising. It is very useful in quantifying smoothness. In fact, Sid Ball of Border Bows renamed it the Smoothness curve. It was also adopted by professor CK Liu of Cal Berkeley in his paper on optimized bow designs.


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## DDSHOOTER (Aug 22, 2005)

Hi Steve great job. As we have discussed in the past. Lift point or better string angle as you pointed out above would be key when choosing a bow length for your draw. Of a matter fact one could over draw a bow to maximize power vs durability. 
Dan


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## 3finger (Mar 29, 2018)

Hank D Thoreau said:


> A key metric I started using several years ago is the first derivative of the draw force curve. I used it to explain how super recurves differ from conventional recurves. It is a key part of my analytical package I used when testing bows. I have a post explaining it that I am currently revising. It is very useful in quantifying smoothness. In fact, Sid Ball of Border Bows renamed it the Smoothness curve. It was also adopted by professor CK Liu of Cal Berkeley in his paper on optimized bow designs.


My thinking is that the Liu paper on arrow design should be standard reading for anyone interested in arrow flight. Have not read the bow design work you mentioned but would like to give it a read.


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## Roof_Korean (Dec 19, 2018)

This is why in most cultures archery form for war involved anchoring past the ear. Those extra few inches of draw equate to as much as 20% additional energy, as you said yourself.


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## 3finger (Mar 29, 2018)

Roof_Korean said:


> This is why in most cultures archery form for war involved anchoring past the ear. Those extra few inches of draw equate to as much as 20% additional energy, as you said yourself.


And where the English Longbow man coined the phrase "Pluck Yew" and it's derivatives.


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## woof156 (Apr 3, 2018)

just a bump cause this is good information


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## deepwater3 (Aug 25, 2019)

What a fantastic read!


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## BarneySlayer (Feb 28, 2009)

Hank D Thoreau said:


> I have often wondered if there were optimum wood choices.


I was once talking to Ron Pitsley at Predator bows, asking about benefits of different cores.

He talked me out of a carbon core upgrade. Said that the glass is what does the work, the wood is just a spacer. Carbon core might get me a couple feet per second, maybe, because it was a little bit lighter, but the mass was dominated by the weight of the glass, and if it was him, he'd stick with just plain maple.

I think Sid said something similar, that the stored energy was more a feature of limb geometry as opposed to material elasticity limits, and that he could design a limb that had less stacking than another limb until it actually broke. I believe him.

Still, when I got my CH, I went with the green synthetic core, just because, I figured I wouldn't be throwing money into another bow for awhile, and aside from the 30# cheapie training rig with Axiom Plus limbs and a Morrison riser I got from Jim Castro (I think it was him who got his Mojo all over it), that has been the case.

Eventually figuring out the specifics to get a lightish Omega. I'm kind of jealous of the one my daughter got. Shoots very well, and I love the lines.


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## BarneySlayer (Feb 28, 2009)

Roof_Korean said:


> This is why in most cultures archery form for war involved anchoring past the ear. Those extra few inches of draw equate to as much as 20% additional energy, as you said yourself.


I suppose there is somewhat of a similarity to the 'Battle Rifle' concept. If you can drive a tack, that's very nice, but the priority is to get it there, and hit hard. Don't need minute of angle, just minute of man.

The other thing is that arrows are slow. 

At short range, precision is secondary to knowing your lead. Survival is enhanced if you can keep yourself moving (and out of range of people without range weapons)

At long range, you're not going to be hitting a moving target in battle at long range by increasing precision. You're going to hit targets when you and a whole bunch of buddies make it rain sharp sticks.


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## BarneySlayer (Feb 28, 2009)

Follow up question.

Why is the 'ideal' long bow draw force curve linear? Is it ideal simply in that it is an easy model? I can understand why you wouldn't want a bunch of stacking on the back end, though the little bump in the front loading up a little extra just after brace would be, I would think, of some benefit, all else being equal.


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## CFGuy (Sep 14, 2012)

BarneySlayer said:


> Eventually figuring out the specifics to get a lightish Omega. I'm kind of jealous of the one my daughter got. Shoots very well, and I love the lines.


Imperial or original? Debating selling my original for a slightly lighter Imperial, but this far into hunting season it may mean no hunting so I'm undecided.

Hank: For those not well versed in physics, does this translate into FPS gains? I.e. if you're getting 180fps out of a 10gpp arrow at 28", say at 40lb, would you get an increase in speed proportionate to that 20% energy gain? Not sure what that would look like specifically.


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## Hank D Thoreau (Dec 9, 2008)

Here is a table of calculated velocities based on the linear draw force curve. The holding weight at each draw length is 40 pounds, so the actual marked weights of the bows are different at 28 inches. Because of this, each arrow mass gives you the same GPP for all draw lengths in the table. Arrow mass is on top of the columns and velocities; GPP is at the bottom. You can see that at 32 inches of draw you will pick up 11% more speed over a 28 inch draw holding the same weight at full draw. I will be working on a part two of this thread where I extend the discussion to velocity. This is a preview. Note, you can see how much velocity changes at constant GPP. This is why GPP is a fairly week metric for judging performance and tuning. I did a test where I held Grains/Foot Pounds of Store Energy constant and I got a constant velocity at all draw lengths. This is because energy is a state function while force is not. This is why there is a conservation of energy law and not a conservation of force law.


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## deepwater3 (Aug 25, 2019)

Hank,

Great topic and research. Using the first derivitive of the dfc is an elegant way to standardize for comparison between bows, I like that. Thank you for posting!

This has been very enlightening as I have struggled a bit trying to find arrows that worked for me.... I have found that I'm always theoretically overspined even when getting good bareshaft and fetched groups/flight. I almost assumed I was doing something wrong at first as I needed heavier spine arrows than charts or forum folks would suggest.

My draw is 30.5 and I shoot shorter bows (60" and 62") in the 50 pound range... Labeled poundage, not on the finger poundage. I had never considered energy as the main driver in spine requirements but it makes perfect sense now. Thanks!

Any chance you'd be willing to share your dataset with me. It would be fun to play around with the last chart you shared exploring energy, speed, and arrow weight. If you happen to have my bow (hoyt dorado) as on of your 31 tested bows that would be even more interesting to me. If I can find a bow scale to buy or borrow locally (in a very rural part of Vancouver Island where shipping is cost prohibitive) I'll set up a "tillering tree" and get a dfc for you if you'd want it.

Either way, I hope you continue with this thread. I'm very much enjoying the analytical part of archery these days.


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