Draw Torque – An Explanation to the Bow/Crossbow Performance Gap

As I was working on the Armalogy, one of my pet projects, I wanted to be able to classify each weapon by how much potential power it has and how efficient it is at delivering it, in a single standardized axis that fits all weapons types. This is where I stumbled onto the age-old debate people familiar with Archery have probably already heard:

How come a crossbow with a 500kg Draw Force delivers the same energy as a 50kg bow?

To illustrate this, I’ll give you a common-tongue example by comparing two classics:

- Our first contender, a 73kg longbow (or 160lb, meaning the force needed to draw it), from Tod’s Workshop channel. This is a very high power warbow, which requires a lot of training to be drawn. It delivers arrows with 122 Joules of energy.

- Our second contender, a 567kg crossbow (1250lb), Tod’s Workshop again. If you’re not familiar with medieval crossbows, that’s an astonishing number: it’s close to 8 times the power of our longbow! No bow in the world even compares, so you would expect energy equivalent to nearly a 1000 Joules, which is twice the energy of a modern pistol.

Well, the actual answer for the latter is... 145 Joules. Pretty pathetic, honestly. Where did all that power go?

And it gets even worse! Let see this example: a 95kg Balestrino, a type of small crossbow. It delivers... 5 Joules?! Something’s amiss, clearly.

Let’s recapitulate the efficiency of each weapon in a chart:

I’ve translated Kilograms into Newtons, as it is a more proper unit of force, and added the ratio of the two, which I’m calling Final Efficiency, as it is the end result when all factors combine.

So, clearly, the longbow totally outclasses the other weapons in terms of efficiency, even though it’s far weaker and of a relatively similar design. What gives?

We must be missing an important factor here. Most archers at this point will tell you that’s the Draw Length, i.e. the distance between both hands at full draw, and they will give you rules of thumb on how much energy each inch of it adds.

It’s a very useful metric to know which bow fits which user arm length, but it’s not really intended for calculus, as it includes a meaningless distance between where the bow is held and where the string stops its travel, which has no impact on energy. As such, most rules based on it are at best approximations that won’t apply effectively to our situation.

What we want is the actual distance through which the bow gives energy to the arrow. This is called the Power Stroke, and is the distance between the string at rest and the string when drawn.

So, let’s compare them in our weapons: Our Longbow is at 58cm, our Crossbow 16.5cm, and our Balestrino a mere 5cm. Clearly, there is a correlation between our previous efficiencies and the Power Strokes, so we’ve got something.

Let’s try multiplying the two values we’ve got:

As our new value is using the same units as torque, I’ve decided to call it Draw Torque.

This value seems to scale a lot better for our energy outputs. Let’s make a second ratio with it, and call it Torque Efficiency:

Now, at first glance, this doesn’t look substantially different from before. Truth is, we’ve gone from 1:7 and 1:34 discrepancies between our most and least efficient weapons to 1:2 and 1:3, and the remaining gap actually fits very well with the expected efficiencies of each design. For example, our crossbows use lighter projectiles per Draw Force, have higher friction, and are made of metal instead of wood, all of which contribute to a lesser efficiency.

What can we learn from this?

- Draw Torque, in N·m, exactly like rotational torque, tells you a bow’s potential power is not just defined by its Draw Force, but also by the “leverage” its Power Stroke applies on its projectile. If you halve the Stroke, you must double the Force to keep the same energy output.

- Final Efficiency, in J/N, tells you how efficient your weapon is, excluding Draw Force.

- Torque Efficiency, in J/(N·m), tells you how efficient your weapon is, excluding both Draw Force and Power Stroke.

Draw Force is meaningless without Power Stroke, just like a lever without length. Both values are critical to define the potential power of a bow or crossbow. As such, I find it weird that for bow performance, Power Stroke is never used nor communicated, but Draw Length is.

PS: I’m using Metric units only, because I’m a proud frenchman, honhon.

– Notes –

Obviously, I’m not the first to note a relationship between Draw Force and Power Stroke. I’m basing myself mostly on Tod’s explanations, which are frankly amazing. I’ve learning practically everything about crossbows from him.

However, I’ve not come once against an explanation on bow dynamics that describes this relationship as multiplicative and linear, and gives a more exhaustive solution to the bow/crossbow efficiency gap, and I’ve certainly never encountered the term Torque applied to efficiency for those.

I’ve also compiled more than 40 tests to test my theories here: Bows and Crossbows Tests

It is hard to be certain that we’re talking about a truly linear relationship, I was in fact stuck for a long while thinking on an equation like N·m^(1.2). To settle the argument, the best is just to compile as much data as possible, especially of weapons identical on every aspect but Draw Force and Power Stroke.

With what I’ve collected, we can already see relatively stable results between similar designs: We’ve got metal crossbows at around 0.16, horn composites at 0.23, fiberglass recurves at 0.40, and compounds at 0.55 (.475 to 0.625). For bows, we’ve got longbows at 0.29, fiberglass recurves at 0.43, and compounds at 0.60 (0.5 to 0.74).

Between crossbows and bows of the same type, the efficiencies are again relatively stable: composite crossbows are at 0.23 vs self longbows at 0.29, recurves at 0.40 vs 0.43, and compounds at 0.55 vs 0.60, which shows a somewhat constant energy loss incurred by crossbows. The composite crossbow vs self longbow comparison might be unfair, but the chart lacks test results for true self-to-self and composite-to-composite comparisons.

I also know I didn’t make a single reference to Stored Energy and Draw Curve. Those metrics are really useful to get an idea of how much energy each specific design stores, and how much it looses through shooting. However, it doesn’t really inform much on how designs compare at their peak, and it’s an information that’s very rarely given on test results, making it very hard to build substantial data from it.

This is of course not the end of the story. This chart needs more test results, especially for self crossbows and longbows, and it totally lacks tests for steel and horn composite bows. Another interesting next step I’m researching is an efficiency formula that also excludes the factor of arrow weight, which has a big impact on the energy output. Most weapons tend however to be used with projectiles that gives them energy pretty close to their peak.