When a bow is drawn, it stores energy. The farther it's drawn, the more energy is stored. When the bowstring is released, this stored (potential) energy is converted into kinetic energy of the projectile (among other things). The amount of energy stored in a bow can be calculated by plotting it's force-draw or F/D curve. A force draw curve is plotted by measuring the draw weight of the bow at several points along the entire draw length. The area that's left below the curve approximates the energy that's stored in the bow. For bow design purposes it's the shape of the curve that's most important; the actual units or numbers are not as relevant.
In simple straight bows with no special features such as recurved tips the force-draw curve is more or less concave. A theoretical best-case bow's F/D curve is a straight line and is illustrated below. The bow in question would store 14000 units of energy:
In real bows, however, the concavity of the curve is apparent. This is because more energy is required to draw the bowstring back the same amount at the end of the draw than at the beginning.
The concavity of the curve depends on the ratio between bow's length and it's draw length. Very long straight bows will have a relatively flat F/D curve: this is because the string angle does not change as much as with shorter bows. This means that a long bow (such as the English longbow) stores a lot of energy. Similarly, a short bow will have a very concave F/D curve and thus it stores much less energy. The difference is illustrated with the two (made-up) F/D curves shown below; the shorter bow stores 8019 units of energy, whereas the longer bow stores 59% more energy (12741 units):
If bow needs to be very powerful, keeping the final string angle small by making the bow long is a good idea. However, long bows tend to recoil slowly and transfer energy efficiently only to relatively heavy arrows (or bolts). In flight shooting where dry-fire speed is most important, a short bow with a very long draw is a much better choice, as it recoils faster.
The shape of the force-draw curve is very important in handbows because the bow's maximum draw weight is limited by the human physique. Or in other words, the faster the maximum draw weight is reached and the longer we stay close to it, the better. The force-draw curve of an imaginary modern round wheel compound bow illustrate this point nicely:
This bow would store more energy (14743 units) than the best case straight bow. With more aggressive cam configurations the energy storage could be increased further, as described on this excellent page. So, essentially a modern compound bow allows storing more energy without tripping over the draw weight limitations of the shooter.
In medieval-style heavy crossbows the F/D curve is not nearly as relevant as in handbows because the final draw length is not a big issue due to mechanical cocking and release. Also, as a bow is made thicker (=more powerful), it's not generally made longer in proportion. This means that it's draw length has to be reduced or the bow will break. So, the shorter the crossbow's draw length gets, the flatter it's F/D curve will be, meaning it's energy storage characteristics approach the optimum. However, this comes at the cost of reducing the increase of dry-fire speed as discussed above. This phenomenom explains why even the very heavy steel crossbows of the medieval times had relatively limited maximum range, around 400 yards, regardless of their very high draw weight and energy storage (e.g. Payne-Gallway 1990: 21)
The F/D curves also illustrate another important point: a shorter draw stores less energy than a longer draw, all other things being equal. For example, a typical handbow has 30" (inch) draw length, whereas crossbows usually have 10-20" draw. This means that if the poundage and other factors are kept the same, a crossbow will store 1/2 - 1/3 of the energy of the handbow. So, unlike people tend to think, the draw weight of a bow (or a crossbow) does not matter much unless draw length is known, too.
The whole energy storage / force-draw curve is discussed in detail elsewhere on the Internet, as understanding it is especially important for modern compound bows:
According to Baker (2000d:44) there are only two things that affect how good performance one gets from a bow:
- The amount of energy put into the bow while drawing it
- Loss of energy while transferring it to the arrow
The former is simple to plot using F/D curves (see above). The latter depends on a number of factors, but most importantly the speed and mass of the bow's limbs and the bowstring right after the projectile is on it's way. In a nutshell, the more moving mass there is, and the quicker it moves, the more energy is wasted. This translates to two things:
- Heavier projectiles increase efficiency of energy transfer as they make the bow limbs and bowstring move slower, meaning that less energy is wasted.
- The amount of energy bow's limbs can store per unit of mass is very important factor in determining energy transfer efficiency. This especially true when using light projectiles.
The energy storage capacity of different materials varies wildly (Koii, B.W. 1991b: 9 -> Gordon J.E. 1978)):
|Material||Energy storage per mass (kgf cm/kg)|
From this table it's easy to see why fiberglass is used in modern bows, why sinew and horn were used in old composite bows, and why steel bows have the reputation for having slow cast. That said, the very low number for steel seem suspicious: my own spring steel and wooden bows (elm and maple mostly) suggest a significantly higher number for (spring) steel. In fact, the above number, 1300, may belong to some generic construction steel which has a very low yield point compared to good-quality spring steel. Unfortunately getting real numbers for commercially-available spring steels seem to be difficult.
In any case, if a material is capable of storing fairly small amount of energy per unit of mass, it only means that:
- The bow has to be physically heavy to store the same amount of energy
- Projectiles have to be fairly heavy to reduce energy losses
- It's best to focus on utilizing leverage as much as possible.
Traditional bowyer Tim Baker has done remarkable job outlining basic bow design principles in his article in The Traditional Bowyer's Bible (TBB), Volume 1 (Baker 2000d: 43-116). He made some corrections to that in TBB Vol.4 (Baker 2008: 113-158). Although Baker is a specialist in making traditional handbows, his high-performance bow design principles apply as well to crossbow design. With handbows high performance is only a small part of the efficiency of the bow, meaning how well it works overall (Baker 2000d: 44). With a crossbow this only true only to a lesser extent, as all parts of the operation can be mechanized. Below I'll outline Tim's bow design principles as I see them apply to efficient crossbow bow design. Many of the of the real issues with handbows are non-issues with crossbows:
- Archer's paradox: Traditional handbows are not (usually) centershots. This means that the arrow has to bend around the handle during launch. To obtain good accuracy the arrow spine (=stiffness) has to be matched to the bow and the weight of the arrowhead. This is not as big an issue with crossbows as their bolts don't have to bend around a handle and have short, stiff shafts.
- Draw length limitations: Draw lenght of most handbows is pretty similar, around 30 inches (or 75cm). Bows with really short draw length would be difficult to use efficiently. Bows with a really long draw length would be impossible to draw fully. Crossbows are not limited by draw length, which allows them to scale both down (pistol crossbows) and up (various siege engines).
- Side view profile: Side-view profile is very important in handbows, as it has big effect on the energy storage curve (see Baker 2000d: 51-52). High energy storage in the early phases of the draw is essential, so that the draw weight at the end of the draws stays with reasonable limits. For example, static recurves increase leverage at the end of the draw, reducing string angle, stack and thus the final draw weight. The length of an English longbow serves the same purpose: it's long limbs give high leverage to the draw by keeping the string angle small. Thus the longbow stores lots of energy while keeping the final draw weight at reasonable limits. With crossbows the final draw weight is less an issue and high initial energy storage much less important - especially if a mechanical cocking device such as a windlass is used.
There are a few design principles that are just as valid with crossbows as with handbows:
- Draw weight: This should be as high as comfortably possible (Baker 2000d: 45). This is limited only by the cocking device used - or lack of it.
- Draw length: At a given draw weight, we should have as long draw length as possible (Baker 2000d: 46). This means it's better to have a thinner bow with long draw than a thicker bow with short draw. This applies especially with flight-shooting bows.
- Bowstring/brace height: Bowstring should be as low as possible to reduce wasted energy (Baker 2000d: 48). With a crossbow the margins are smaller than with the handbow, so making a string that's just right takes a little experimenting, especially if the bowstring stretches. So, make the bowstring as long as possible without making it slack.
- Limb mass: A bow's limbs should be as light as possible. Everything else being equal, heavier limbs are slower (Baker 2000d: 65-66).
- Mass placement: Mass placement is very important when designing a bow. Mass in outer limbs - especially tips - reduces the cast much more than mass in the inner parts. Therefore we should aim at making the tips as light as possible.
- Minimal dead mass: Dead mass means a portion of a bow that does not work (=bend) or does not bend enough. Dead mass can be used as a lever "attached" to the working portion of the bow, though, resulting in increasing the cast. This lever effect is based on the fact that outer limbs of a bow store much less energy than the inner limbs (Baker 2008: 119).
- Bowtring weight: The bowstring should be as light as possible, as it serves no other purpose but to transfer energy from the bow to the bolt (Baker 2000d: 73-74).
- Bowstring stretch: The bowstring should stretch as little as possible to avoid wasting energy at the final stages of the shot (Baker 2000d: 74).
The so-called stack requires special mention in relation to crossbows. Stack means means the loss of leverage caused by increasing angle between the bow and the bowstring during draw. For example, at the very beginning of the draw the ratio between string and bow tip movement may be 1:5, whereas at the end of the draw it may be 1:1 or even more. This also means that at the end of the draw each centimeter of bowstring movement adds vastly more energy to the bow than at the beginning. This causes the sensation of a "wall", a rapid increase of draw weight, also known as stack. Because of this "wall" stack was (and is?) often misinterpreted by archers as a sign of imminent bow failure. This is not the case, which is easily verifiable: the bow will be much harder to pull if you use an extra long, slack string which has a high initial string angle.
In handbows avoiding stack is a good practice as the final draw weight should be kept manageable or accuracy will suffer. Or, in worst case, the bow can't be pulled back at all. With crossbows the final draw weight is a non-issue, especially if a separate cocking device is used. In fact, some cocking devices counter the effects of stack as a side-effect of the way they operate (e.g. correctly designed wippe cocking levers). Also, with crossbows a good lock design negates the effect of increased draw weight.
In crossbows stack - or more correctly high initial string angle - can also be beneficial. See the "Adding deflex" section for more about this.
Bow and bowstringEdit
Matching the bowstring to the bow is very important to produce durable bows that are also good performers. There are three reasons for this:
- Bowstring's only purpose is to transfer energy to the projectile (e.g. a bolt or an arrow). Using a lighter bowstring results in enhanced projectile velocities. The common rule of the thumb is that reducing one unit of weight from the bowstring allows shooting projectiles 1/3 units heavier without any reduction in velocity.
- Too strong bowstring can break the bow, especially in case of accidental dry-fires. One of my crossbows (codename Delta) suffered this fate.
- Prestressing the bow by using a short bowstring (=bracing it high) reduces bolt velocities and/or stresses the bow unnecessarily. This is because the bow can at most output only as much energy as one feeds into it and this feeding only starts after the bow has been braced. In other words, a bow braced high does not store any extra energy that could be fed into the projectile. Some very stretchy materials such as sinew may benefit from prestressing, but in most cases it's disadvantageous, as it reduces the bow's safe draw length. Note that bracing the bow high is not the same as using a bow with lots of initial deflex.
This last reason is surprisingly important, especially for crossbows. One of my crossbows (a.k.a. Gamma) is a prime example of this (see File:Bolt velocities.ods)
|Step||Bowtips, braced(cm)||Bowtips, cocked(cm)||Brace height (cm)||Bowstring weight (g)||Power stroke (cm)||Bolt velocity (m/s)||Bolt energy (J)|
The test bolts were relatively heavy for this crossbow, 81 grams, so the figures roughly represent maximum energy output of this crossbow. The longer power stroke increased efficiency of lighter bolts significantly. The original bowstring was way too short and heavy, meaning the bow was braced high. By replacing it with a longer and lighter one energy output was improved somewhat (1,25%) and bow stress reduced significantly. As bow stress was decreased, it became possible to increase draw length, first by 1cm, then by additional 0,7cm. At this point energy output had increased by 15%. Around 5% more could be obtained without any greater stress on the bow than originally.
The method used above to estimate the relative stress levels of a bow - measuring the distance of the bow's tips when cocked - is crude but probably accurate enough for most purposes.
Even in the most basic bow the bowstring moves much longer distance than the bow's tips: the bowstring acts as a kind of a lever. In addition to this basic characteristic of all bows, there are two main methods of increasing leverage further:
- Increase bow tip movement itself. The stiff and light tips of Andaman/Holmegaard -type bows are a prime example of this.
- Increase bowstring movement. Examples of this include adding deflex and adding a stiff middle section. Also most or all modern compound bows utilize this principle.
In crossbows high initial string angle or stack can be beneficial, unlike in handbows. This applies especially to very heavy steel crossbow bows which can bend only a few degrees before breaking. By increasing the natural bend of the bow and thus initial string angle we get:
- Longer power stroke (for any given bend angle)
- Faster string acceleration at the final phases of the shot
- Shorter and lighter string
An example of this is shown here:
In this particular case the bow is 800mm long and is bent 60 degrees from it's rest position. Here "to bend N degrees" means bending the bow so that angle of the sector (of the circle) which bow's bend follows increases N degrees. This assumes that the tiller of the bow is circular, not elliptical. The principle applies with elliptical tiller, though.
If the bow's natural bend (or deflex) follows a 10 degree sector (bow #1) we get:
- Bow tip movement: ~104mm
- Total power stroke: 236,93mm
- Power stroke after first 10 degrees of bend: 91,34mm
On the other hand if the natural bend follows a 90 degree sector (bow #3) we get:
- Bow tip movement: ~101mm
- Total power stroke: 284,03mm (19,88% more than in bow #1)
- Power stroke after first 10 degrees of bend: 144,07mm (57,73% more than in bow #1)
Although energy storage of the bow is not increased at all by this, the potential bolt speed is. This means the bow is capable of shooting lighter bolts faster. It should also be noted that the bow is stressed the same amount when bent the same amount (in degrees) whether it's natural bend (deflex) is 10 or 90 degrees. This is easily verifiable with relatively simple calculations implemented here: File:Stretch compression calculator.ods.
This increase in leverage is almost certainly part of the reason why heavy steel bows for medieval crossbows were very aggressively deflexed. That said, increasing initial string angle (or adding deflex) only makes sense for bows with relatively limited tip movement; lighter bows can provide enough bowstring movement without extra deflexing.
Adding stiff middle sectionEdit
Another simple way to increase draw length is to add a stiff middle section to the bow. This lengthens the bowstring and thus draw length. This is easily demonstrated with the example bow shown in above picture. If the bow's natural bend (or deflex) follows a 10 degree sector and has 200mm long stiff middle section (bow #4) we get:
- Bow tip movement: ~104mm
- Total power stroke: 253,45mm (6,97% more than in bow #1)
- Power stroke after first 10 degrees of bend: 98,07mm (7,36% more than in bow #1)
On the other hand if the natural bend follows a 90 degree sector and has a stiff 200m middle section (bow #5) we get:
- Bow tip movement: ~101mm
- Total power stroke: 313,43mm (32,28% more than in bow #1)
- Power stroke after first 10 degrees of bend: 159,50mm (74,62% more than in bow #1)
So, in this case by simply adding deflex and a stiff middle section we can increase draw length by 32,28%.
NOTE: this technique is not especially useful for crossbows: the draw length does not increase enough to justify making the bow longer and thus more cumbersome to use.
The words tiller or tillering are used rather freely in traditional archery literature. For example, the act of removing wood from a stave in order to extract a bow from it is called tillering. The side-view profile shown when the bow is drawn can look elliptical or circular. In these cases it's said that the bow has an elliptical or circular tiller. A bow can also have uneven tiller, meaning that one side bends less during draw than the other. Or the tiller of a bow can simply be bad, meaning that it's bending too much in some places and too little in others. This confusing parlance is easy to understand given that traditional wooden bows are tillered (Hamm 2000b: 257-286). You can't use set of dimensions and just make a traditional wooden bow without further thought, as wood is not uniform enough as a material. However, you can make a steel bow - there's really no need for tillering.
When designing bows it's important to understand how leverage affects the bow limbs when they are bending. The center of the bow gets the full leverage from the rest of the limbs. Halfway up the leverage is halved and at tips there is no leverage. This means the limbs need to be increasingly flexible as the tips are approached. Otherwise the inner parts of the bow are overstressed (=bend too much) and/or the tips understressed (=bend too little) if the draw length is kept the same. If we had a bow that's of uniform width and uniform thickness - a really bad idea - the center would bend a lot, the middle half of that and the tips not much at all. So, one key factor in getting best possible performance from the bow is to stress all parts equally and as much as safely possible. There are some exceptions to this rule, especially when dealing with handbows (see Baker 2000d).
Now, there are two basic ways to even the strain in the limbs:
- Thickness tapering: Progressively reducing the limb thickness as you move towards the tips. The stiffness of a limb is proportionate to cube root of it's thickness. For example, if limb thickness is halved, stiffness of the bow drops to 1/8. This mean that thickness tapering should never be linear; rather it should follow the cube root curve, relatively most material being removed from the tips.
- Width tapering: Progressively reducing the limb width as you move towards the tips. Halving the width of the bow halves the draw weight.
In the case of wooden handbows both methods are usually used simultaneously. However, it's beneficial to use as much width tapering as possible: this allows removing as much mass from the tips as possible, increasing the bow's dry-fire speed (=maximum speed at which the bowstring can move). It's simply not possible to reduce as much mass with thickness tapering alone without overstraining the tips, as is shown in this graph:
If pure elliptical tillering (no width tapering) is used, maximum amount of material that can be removed while maintaining even strain throughout the limb is 25%. If pure circular tiller (width taper only) is used, twice as much (50%) material can be removed safely. This is the underlying reason why well-made aggressively width-tapered bows shoot faster than bows which are tapered mostly in thickness. However, the angle of the bowstring and the nocks also affect how the limb behaves and what parts are stressed most at any given draw length. For example, if the bow is drawn so far that halves of the bowstring and the nocks are parallel, the limb tips can't possibly bend more; all further bending has to take place in the center part of the bow.
Below are some calculations on how a steel bow's tip mass is reduced by progressively more aggressive width tapering. They are based on standard density steel (7,8g/cm^3). The mass of the nock extensions is not taken into account:
|Length (mm)||Thickness (mm)||Center width (mm)||Tip width (mm)||Total mass (grams)||Mass of tipmost halves||Mass of tipmost fourths|
|900||6,2||40||17||1240,4 (-0,00%)||495,1 (-0,00%)||216,3 (-0,00%)|
|900||6,2||40||10||1088,1 (-12,27%)||380,8 (-23,09%)||149,6 (-30,83%)|
|900||6,2||40||6||1001 (-19,30%)||315,5 (-36,27%)||111,5 (-48,45%)|
As can be seen, the closer to the theoretical optimum (taper to nothing) we get, the more limb mass decreases, especially at the tips where it matters the most.
The amount of thickness and width tapering also determines how the bow bends, also called it's tiller. An evenly strained bow with width tapering only should look circular when drawn: it has a circular tiller. An evenly strained bow with both width and thickness tapering, on the other hand, looks elliptical and thus has an elliptical tiller (Baker 2000e: 32-33). These tiller shapes are illustrated below, a circular tiller on the left and an elliptical tiller on the right:
The reason for this difference in side-view profile is simple: the neutral plane of the bow is (usually) in the center, meaning it not under stress when the bow is drawn. The back of the bow stretches and the belly compresses - more and more as we move away from the neutral plane. Each of these are visualized below:
The back of the bow is at the top, the neutral plane in the middle and the belly at the bottom. The picture at the left represent the bow at rest. At left the same bow is shown fully drawn. The pictures are exagerated, but illustrate the point: when a bow is drawn, the back stretches and the belly compresses. Also, the farther away from the neutral plane we are, the more material stretches. Therefore thicker portions stretch and compress more at any given angle than thinner parts. Or to put it the other way, they can't bend as sharply as thinner parts without being overstrained. Therefore a bow with thickness taper has to have elliptical tiller: it's thick center simply can't bend in as aggressive angle as the thinner tips.
Practical bow designsEdit
In practice bow designs deviate from theoretically "perfect" designs for a variety of reasons:
- Limitations of the materials
- Bow's intended use
A few of these practical designs are presented below.
Simple pyramid profile bowEdit
The simple pyramid front-view profile (see Baker 2000d: 67) is one of the most efficient bow designs out there. This kind of bow is of even thickness and tapers linearly and aggressively in width towards the tips. The tips should be as narrow as possible, but never narrower than they are thick, or they become unstable. This bow design is especially well suited for relatively thin (up to 6mm thick) steel bows that are cut (as opposed to forged) from steel, e.g. leaf springs. This design is also relatively easy to execute in wooden bows which are made from planks or laminated from veneer. To get best results from this design, make sure the center of the bow is as wide as possible compared to the tips, the idea being to stay as close to the theoretical optimum (taper to zero) as possible.
If the bow is relatively thin and wide, it's easy to stay close to the theoretical optimum (dashed line) without unstabilizing the tips:
However, if bow's thickness is increased without also increasing the width at center, the design will start deviating noticeably from the optimum:
For this reason the enhanced pyramid profile makes more sense for thick and/or narrow pyramid-profile bows.
Enhanced pyramid profile bowEdit
As the name implies, the enhanced pyramid profile bow is slightly advanced version of the simple pyramid profile bow. Unlike the simple version, it allows near-optimal tapering even on thick bows. The underlying idea is very simple:
- Follow the theoretically optimal width taper (taper to zero) as far towards the nocks as possible, i.e. until the limb is as wide as it's thick. Stopping width tapering at this point avoids most tip stability problems.
- Make the rest of the limb of even width.
- Optionally taper this outermost part of the limb in thickness. Depending on bow thickness this may give small performance gains, increase stability of the tips somewhat and allow a slightly longer draw.
This enhanced design is visualized in the below pictures. Theoretical optimum is shown as a dashed line on the first two pictures. In the last picture the dashed line marks the difference to a simple pyramid profile bow of same dimensions:
As can be seen, this design is a big improvement on the simpler one especially on thick bows. On thin bows the simple pyramid profile is more than adequate. This design also works really well in real life:
Care should be taken not to create too acute an angle between the tapered part and the tip part.
Bow from loose laminatesEdit
This design is effectively an extension of the simple pyramid profile bow even though it looks superficially different. As each laminate is loose, they are all stressed individually. In other words, it does not really matter whether the laminates are layered on top of each other, or attached to each other side-by-side. This idea is visualized in the picture below:
As can be seen from the picture above, both bows (laminated and pyramid profile) are of the same total width at any given point. The only major difference between these two bows is that the laminated bow is narrower and consists of several, loose layers. When the four laminates are tied to each other, the performance of both bows will be similar.
The need for additional width tapering (for each individual laminate) can be reduced by increasing the number of laminates. The below picture shows the necessary width taper at each part of the bow when it consists of 8 laminates:
As can be seen, there's probably little benefit in starting to width taper the laminates until 2/8ths or 3/8ths away from the nocks. Before this point the bow's bend will close to optimal even without any width taper.
If optimum performance is not sought for, this type of bow is relatively easy to make by tying together a bunch of good quality, even thickness and even width laminates. Width tapering as described above can be omitted without big performance penalties.
There are several materials that work with this design:
- Bamboo: window shades
- Thin steel: old saw blades (for light bows)
- Thick steel: spring steel (for heavy bows)
In fibrous materials (e.g. wood or fiberglass) the simplest possible design with uniform width laminates is probably the safest, as the fibers are not cut through. With steel the more fancy design with width-tapered laminates can be followed, if you're so inclined.
The main advantage of this loose laminate design is that it allows reaping the benefits of aggressive width tapering without making the bow excessively wide. Also, it allows ridicuously long draw lengths without compromising draw weight too badly.
This design also has some disadvantages:
- Increased complexity of construction
- Energy losses through friction caused by laminates sliding along each other during draw and release
- Requires uniform quality laminates with very few flaws
- Making the nocks can be challenging, especially if working with laminates that are prone to splitting (e.g. wood or fiberglass)