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Guide to Comparing Bows and Crossbows

Posted November 1st, 2017 at 04:55 PM by HackneyedScribe
Updated June 10th, 2018 at 11:50 AM by HackneyedScribe

This paper is meant to as a guide to make proper comparisons for the power of bows/crossbows

Here are the terminologies.
1) Dry Fire Speed: Speed that the string of a bow/crossbow travels when there is no projectile.
-Normal projectiles cannot travel faster than the dry fire speed, but lighter projectiles can shoot closer to dry fire speed than heavier projectiles. This is because the weight of the projectile slows down the string upon release. How much the string slows down is dependent on the weight of the projectile.
2) Draw Weight: The amount of force required to pull the string back to the bow/crossbow’s full draw length.
-Having a heavy draw weight means that the weight of the projectile slows the string down less. All else being equal, projectiles can be shot closer to dry fire speed this way.
3) Powerstroke: The distance from the string at rest to the string at full draw
-A long powerstroke gives a longer distance for the string to accelerate the projectile. So like draw weight, a longer powerstroke decreases the effect of projectile weight in slowing down the string. Much like how a racing car or sprinter can achieve a faster final velocity if they are allowed to a longer track.

Click the image to open in full size.

The Traditional Bowyer's Bible compared bows of the same draw weight shooting the same projectile weight, but with different draw lengths (and hence different powerstroke). One can see how velocity increases with longer powerstroke.

Click the image to open in full size.

4) Brace Height: The distance from the middle of the bow/prod to the string at rest

5) Draw Length: The distance from the middle of the bow/prod to the string at full draw
-Draw Length = Powerstroke + Brace Height.
6) Prod: The bow of the crossbow
-Heavy prods and bows slows down the string of the bow/crossbow upon release, because a bigger portion of the stored energy is used to push the bow or prod forward. Like how a tank, despite its massive horsepower, cannot drive faster than a race car due to the former's massive weight.

Click the image to open in full size.

What increases prod weight? A high draw weight means you need more material for the prod to gain the necessary stiffness to achieve that draw weight. A lengthier prod means you need more material to achieve the same draw weight as a shorter prod. Adding a tiny amount of prod thickness means a massive increase in draw weight, whereas a wider prod would only increase the draw weight by a little in comparison. So all else being equal, a thicker prod is lighter than a wider prod.
7) Potential Energy: Maximum amount of energy that can be transferred into a projectile without breaking the laws of physics. It is the area under a Force Draw curve, the equation used to estimate potential energy is the same as that used to measure the area of a triangle. If you want potential energy to be shown in units of joules, make sure you input draw weight as pounds and powerstroke as inches, and then convert the result by dividing by 8.85 because 1 joule = 8.85 inch lbs.

Potential Energy = = Draw Weight * Powerstroke/2

8) Force Draw Curve: Graph showing the force (draw weight) required to pull the string back a certain distance or vice versa.

Click the image to open in full size.

9) Dynamic Efficiency: The amount of potential energy stored in a bow/crossbow that can be transferred to the projectile
-Energy of projectile = Potential Energy * Dynamic Efficiency
10) Cross Section: The shape of the bow limbs. Generally speaking, increasing width of the limbs by twice means increasing the draw weight by twice. But increasing thickness by twice would increase draw weight by 8 times. So thick bows are efficient because the limbs weigh less for the same draw weight.

11) The energy of a projectile can also be calculated by its weight and by its velocity. The equation is:
-Energy of projectile = projectile weight * velocity * velocity/2
Some common misconceptions:

1) People commonly compare the power of the weapon by only looking at how far it shoots. This completely ignores the fact that it takes more power to shoot a heavier projectile to the same distance as that of a lighter projectile. For example, the machine that can launch a piano 100 meters is far more powerful than a machine that launches a golfball 200 meters. Without knowing the weight of the projectiles that were shot, comparing range is useless. There are also other factors such as FOC (forward of center) or cross-sectional area, but given that the shape of the two projectiles are more or less the same, weight is the primary factor.

2) Having heavier and heavier draw weights becomes more and more useless unless if the projectile increases in weight accordingly. The reason is due to a weapon’s dry fire speed. No matter how high the draw weight is, dry fire speed is the limit to how fast it can shoot. Once the projectile velocity approaches dry fire speed, adding more draw weight or longer powerstroke will not make the projectile fly any faster.
The main advantage to high draw weights (or long draw lengths) is that it can shoot a heavier projectile without sacrificing much in velocity. So when shooting weapons with high potential energy, the projectile should increase in weight accordingly. Otherwise after a certain point there is certainly no advantage to increase potential energy.

3) Due to reason number 2 above, the “Dynamic Efficiency” of two weapons should NOT be compared by having them shoot projectiles of the same weight (which is unfair for the heavier weapon), BUT by having them shoot projectiles of the same velocity. Take the second picture of this paper. One can see that as powerstroke increases, velocity increases. Because each shot used the same 470 grain arrow, projectile energy likewise increased alongside the increased velocity. But the amount of increase in projectile energy was smaller with each additional inch of powerstroke. This meant that dynamic efficiency have a negative correlation with powerstroke:

Click the image to open in full size.

But is this comparison fair? No, of course not. This is because despite the increase in powerstroke, the trial is still shooting an arrow of the same weight! The third column of the graph shows that although dynamic efficiency is decreasing, the "arrow weight/potential energy" ratio is decreasing even faster! Hence, if arrow weight increased in ratio to powerstroke, then dynamic efficiency would actually increase. This means that as long as the weight of the arrow increased relative to an increase in Potential Energy, then having a longer powerstroke might even help increase dynamic efficiency!

4) Know what you are comparing.

For example, the equation I gave above for calculating potential energy is just an estimate for the area under the force draw curve. The equation is derived for the same equation used to calculate the area of a triangle, but this isn't the total area under the force draw curve. Such an equation fails to estimate the area shaded in yellow:

Click the image to open in full size.

So why do I still use the equation? Because it's simple to do. If you want to measure the area under the force draw curve exactly, then that takes a lot of work because you'll have to measure the force of the bow inch by inch. That type of information is rarely available. Measuring under the assumption of a linear curve makes it easier, though it would result in underestimating the potential energy, leading to overestimating the dynamic efficiency. Which leads to another point, make sure the method of measurement is the same. Comparing the actual potential energy of one bow to the linear potential energy (calculated using the simplified equation) of another bow is highly inaccurate. For the sake of fairness, one should use the linear estimation for all cases, or the actual estimation for all cases, but not mix them together. As shown in the table provided below, the estimation for dynamic efficiency is used in which all estimations are using the equation of a linear draw force curve. Now if I instead calculated the actual area of the draw force curve for some of them, but only used the linear estimation for others, then the comparison becomes meaningless.

5) The most important factor: Compare under All Else Being Equal!

This is the most important part. People tend to compare two different crossbows in which multiple attributes of each crossbow are different, and judge that one singular attribute is the determining factor for why the two weapons vary in performance. This is highly presumptuous. There is no guarantee that all the other different factors will not affect the performance as well. To ACCURATELY judge whether one single attribute affects the performance of a weapon, comparisons should be made in an all else being equal situation. For example, the following chart is a combination of the Medieval steel crossbow replicas made by Tod Todeschini and Medieval composite crossbow replicas made by Andreas Bichler:

Click the image to open in full size.

Those highlighted in green are composite crossbows, organized by Projectile Weight to Potential Energy ratio. Those not highlighted in green are steel crossbows. If we are to ask whether composite prods are more efficient than steel prods, the chart at first glance may say that there is no affect at all. But this is because each crossbow being compared have OTHER attributes that affect efficiency too, which muddies the comparison. The only two crossbows in the list that have an "all else being equal" condition, is the 1250 lb steel crossbow and the 1200 lb composite crossbow. The differences between draw weight, powerstroke, potential energy, projectile weight, and 'projectile weight to potential energy' ratio of these two crossbows are all fairly small. So through this we can see that when given an "all else being equal" situation, prods made of composite are more efficient than prods made of steel.

*Those of the same draw weight are performances by the same crossbow, but using a different quarrel.
** The 1276 lb composite crossbow shows the absolute MINIMUM efficiency, because the projectile velocity was measured too close to the chrono. Meaning the projectile haven't finished accelerating when it entered the chrono. Actual efficiency should be higher than 41-46%
*** Andreas Bichler also confirmed that longer powerstrokes help with efficiency, which is why composite bows have higher efficiency than the composite prods shown in the table. He also mentions that his earlier replica crossbows (the 617 lbs one) weren't a very good imitation of actual medieval crossbows. The long powerstroke combined with the short prod actually managed to break the horn. This suggests that the prod suffered from stacking, meaning that after pulling the string a certain length, the force of the pull is no longer storing additional energy for the prod, but pulling the limbs of the prod apart.
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