Saturday, April 9, 2016

Reverse engineering the bow: II

Reverse engineering the bow: a simple dynamic model

Abstract. This paper is the second one by the author dedicated to simple models for the physical description of bows. The first paper "a simple static model", dealt with the force vs. draw length curve by means of an approximation describing the bow as a rigid arm moved by a torsion spring (Hickman's model). In the present paper, the same model is used to develop a treatment which describes the motion of the arrow. This calculation is not meant to be a quantitative description of a real bow, but as a simple exercise in modeling designed to help us to understand the physics of bows. 

For many thousands of years, humans have used their intuition to build bows which were effective and also beautiful. Over that span of time, our brains may have evolved – at least a little - in such a way to be especially fit to "model" the behavior of such a complex system as an arrow shot by a bow. Of course, after so many thousands of years we have also developed mathematical methods which we can use for the same purpose. Still, the mathematical description of the behavior of a bow is a very complex matter. Here, what I'll be trying to do is use a mathematical model which is still simple enough that we can manage it easily and that we can use as a supplement to our well trained (but limited) mental modeling. The model was developed first by Hickman and others in 1930 and still today known as "Hickman's model". In another paper ("a simple static model") I have shown the results of the model in approximating the static behavior of a bow. Here we'll do the same for a dynamical simulation, that is one which describes the motion of the arrow. Here, as in the previous work, I am not trying to get quantitative results; for an elaboration of the model to take into account other factors you can see the excellent article by Tuyn and Koot published in 1992 on the European Journal of Phisics and available on the internet. What I am doing here is just to try to get a "mind sized" grasp (to use a term coined by Seymour Papert) of the behavior of a bow as an example of how the combination of human intuition and simple physical models can be used to optimize a technological system.
Onager models
Let's see first of all, as an exercise, how we can model a machine much simpler than a bow: a torsion catapult. One of the simplest models, the classic "spoon" catapult, was the one called "onager" by the Romans and "monankon" by the Greeks. The first term refers to the kickback of the machine which evidently reminded to the users that of a kicking onager, the other has its root in the Greek word for "single" (monos) which obviously refers to the single arm. The Romans seem to have been much more intensive users of these machines throughout the classical period, so we'll call them simply onagers here. They were also used for much of the span of the middle ages, until they were ousted by gunpowder, as all those machines which were called "torsion artillery". Here is a picture of an onager (a 19th century replica):

Onagers are real world machines which are very well described by a simple model which was developed in the 30s by Hickman to describe bows. Here it is:

The force acting on the arm is that of a torsion spring which at the time was made by twisting such things as sinews, but also human hair or whatever suitable material came at hand. We can assume that the force generated by the spring is linearly proportional to the torsion angle and it is was applied at a point at a distance r0 from the center of rotation of the arm. Note that most ancient onagers had a secondary bow midway of the arm. It seems that the purpose of this secondary bow was mainly to stabilize the arm and to make it go straight, the force associated with it was much weaker than that of the torsion spring, so that here we'll consider only the latter.
Now, the purpose of an onager catapult, as of any piece of artillery, is to shoot a projectile at the highest possible speed. In making a mathematical model, we want to know what are the factors which maximize this speed. Intuition does tell us that the stronger the spring the faster the speed, but other factors are not so simple to modelize in "mind sized" bits: for instance how long should be the arm?
As a first approach, what we would need to do is to solve the motion equation of the system. For a rotating system as this one, we have a simple differential equation:
As we know, is the torque and is defined here as a force times the radius r0. In turn, the force is defined as a constant times the angle that we write as , so I is the momentum of inertia, defined as: . Here, I is easy to calculate and is simply the sum of the momentum for the projectile (mr2) and for the arm, which can be shown to be proportional to Mr2, with the proportionality coefficient depending on the shape of the arm.Considering a cylindrical arm of constant section, the coefficient is c=1/3. It is smaller for a "tapered" arm, i.e. one which gets smaller towards the end. For a triangular or conical arm it turns out to be c =1/6
The formula above is a differential equation which is rather easy to solve, we all know that its solutions are sinusoidal functions, for instance in the simple case in which qe=0 we have as a solution for the angular velocity 
Another way to consider the system is to work on the conservation of the energy of the system. We assume that the system is elastic, i.e. fully conservative and that all the mechanical energy released by the spring is transformed in the kinetic energy of the arm and of the projectile. The energy of the spring is given as a force that moves its point of application along a circle of radius r0, therefore (inglobating the constant radius in the spring constant K):
If we assume that the mass of the arm is negligible with respect of that of the projectile (m>>M), we have an especially simple case. Since the energy must be conserved, as the arm is released, at any moment, the sum of the elastic energy and of the kinetic energy must be a constant
For q = qmax the arm is at rest, and so the kinetic energy is zero, , so we have that .
For the more general case, with a non negligible mass of the arm, we can use the formula for the rotational energy with . Since I is proportional to r2, then the result is simply the following, with c the constant for the arm shape, and v the speed at the arm tip.
And we see that, as it is obvious, the speed of the projectile is proportional to such things as the square root of the spring constant, whereas it is inversely proportional to the mass of the projectile itself, again as you would expect. Less obvious it is that the speed does not depend on the arm length. It would seem therefore that the best onager catapult should have a very short arm. This may be true, but only within specific limits. Here the limit lies in the performance of the torsion spring. If we make the arm very short, the spring must accellerate at a speed that gets close to that of the projectile itself. This is not possible since the spring itself has a mass and plenty of internal anelastic stresses which would dissipate a lot of energy. In practice, the optimal length of the arm is a compromise between these factors, a compromise that could only be found by trial and error by catapult builders.
Notice that the formula that we found gives an expression for v as a function of the angle, but not as a function of time, which is what we would have found if we had solved the motion equation. The two expressions are anyway equivalent, as you may verify by substituting the result for the simple case of qe =0,We can also use the formula to find the accelleration of the system, something that we can do by derivating the equation for the speed with respect to time. For simplicity, we'll do it for the special case of qe =0 and M<<m.

We find that the maximum accelleration in an onager catapult is for q=qmax, that is at the start of the action. This is an expected result, and from here we could go on finding other relations. However, the onager is really not such an interesting system, and the procedure we have developed will turn out to be useful for the more complex case of the hand held bow.
Modeling the bow
Under several respects a bow is similar to an onager catapult, even though, of course, the bow has two limbs instead of a single arm. What does make a big difference is the presence of the string which produces a very different geometry which causes almost all the elaestic energy available to be transferred to the arrow. Bows are very efficient machines and the only reason why sometimes things like onagers were used is that they were simpler to build. Bows use cantilever beams as elastic elements rather than torsion springs as onagers. The mathematical description of a cantilever spring is much more complex than that of a torsion spring/rigid arm system. However, already in the 30's Hickman had shown that the behavior of a bow could be approximated with a simple model (Hickman's model) that assumes that the bow limbs are rigid and operated by torsion springs. In practice, what we are doing is to consider the bow as something like a double onager (actually in ancient times there were bow-like machines operated by torsion springs, these were the most commonly used catapults). So, let's make a geometric bow model based on this approximation.

Notice that the model shown here has no "riser", i.e. nothing of the rigid part to which the bow limbs are attached. The effect of limbs can be included in the calculations with reasonable ease, but the formulas become considerably more complex and for the purposes we are interested in the results of the calculations do not change much. So in the following we'll always consider bows with "zero length" risers. As I said earlier on, the idea is to keep things as simple as possible.
Now, we could calculate forces and write a differential equation. However, as we did for the onager, we'll take a road based on the conservation of the energy. Just as for the onager, the energies must be conserved in a bow according to :
where the symbols are as before, but here m is the mass of the arrow and M the mass of the limbs. We consider the mass of the string as negligible. Notice also that the string is supposed to be infinitely rigid, something that will turn out to be a poor approximation later on. So, first of all we want to find a relation between v and w, which takes some geometrical work.
From all this we have:
taking into account that we can also write
We can now derivate with respect to time to obtain a relation between the angular and the linear velocity (the speed of the arrow):
Now, we can get back to the formula for the conservation of the energy and substituting we get

and, finally:

These formulas look rather complicated, but they are not terribly so. Note, incidentally, that if m = 0 (no arrow) the equation for w reduces to the same formula found for the onager. In such case, the limbs accellerate freely until the maximum q is reached. In the onager, the arm is stopped by the arm rest in the bow by the string. In both cases the kinetic energy of the arm (limbs) must go somewhere and it appears as the "kick" of the onager or by an equivalent "kick" that may break down a bow shot without an arrow (or with too light an arrow)
Now, what we are interested in is how the speed of the arm (w) and that of the arrow (v) vary as a function of the arrow movement along the x axis. To do that we should explicit q as a function of and substitute, but this procedure leads to very cumbersome formulas. What we can do instead is to use a simple program to calculate a series of values for x, v and from the formulas above. We can then plot v and w as a function of the x vector. We can also calculate the accelerations using the same method described before for the onager catapult.
We can go on and make the calculation, of course we need parameters. We are not trying to make anything exactly quantitative, but we may as well use values taken from actual bow, to remain within the right order of magnitude. So, in the MKS system, we can set r = 1m, M= 1 Kg (mass of the limbs) and for m (mass of the arrow) we are in a range from approximately 10 g to 40 g). Regarding K, it has the dimension of an energy divided by an angle. We can calculate it from the formula: assuming that E is the elastic energy stored in the drawn bow, we can take it as approximately 100 J. Assuming that the bow is fully drawn for q= 0.5 radians and for the sake of simplicity qe=0, we have K= 200. For c, we use 1/6 as we discussed before.

If we just plug in these values for a 25 g arrow, a bracing height of 15 cm and a "draw" equal to the limb length (a bit optimistic, a normal value is around 0.9), we get a final arrow speed of ca 55 m/sec, which is a very reasonable value. This shows that the calculation does reproduce more or less the operation of a real bow. And here is a plot of the calculation
This is a very fundamental plot. It shows how the tip velocity goes through a maximum along the bow movement, and then goes to zero rather smoothly. From the figure you can see that the ratio of the arrow speed to the tip speed goes to infinity as the arrow gets close to leaving the string. This ratio is often called t(tau) and this behavior is called the "tau effect". Notice how efficient is the bow in comparison to the less refined onager. In an onager, the speed of the projectile is the same as that of the arm. In the bow, the tau effect causes the speed of the arrow to be several times faster than that of the tip of the limb. Of course, we do not get something out of nothing, what we do is to trade mass for speed maintaining the energy fixed. Unlike the onager, the bow is optimized for shooting light projectiles at high speed, which is very good – for instance – for hunting. If you want to throw big stones at your enemies in a war situation you are probably better off with an onager type catapult, but you can't use it to hunt!.
For a better understanding of what goes on during shooting, we can now plot the accelerations of tip and arrow for the same case as above, obtained using the same method we used fot the onager. Here are the results:

We see how the tip accellerates first then starts deceleating about midway, while it transfers kinetic energy to the arrow. The arrow has a positive accelleration during the whole trajectory within the shooting action, until the end point, when it leaves the string
Now that we have a working model, we can use it to see how the effect of the various parameters. Of these, some are rather obvious. For instance the spring constant (i.e. the "weight" of the bow) and the "recurving" effect have both the same effect in increasing the final speed of the arrow, but change very little the shape of the curves above. The advantage of recurving is best understood in terms of a static model. Other parameters have a less obvious effect and deserve a specific discussion

Arrow weight
The arrow weight is one of the major preoccupations of archers. Let's see its effect, here we change the mass of the arrow from a light 5 grams one to a heavy 45 grams one in steps of 10 grams. We are still considering a non-recurved bow (a "long bow") the effect of recurving are simply of increasing the final speed

As we'd have expected, light arrows go much faster than heavy ones. We also see, however, that the lightest arrow has a very strong velocity variation which, intuitively, we would see as a cause for instability. To visualize this point, let's plot now the accelerations for ths same case as above

We see that we pay the higher speeds obtained for much larger accelerations of both the arrow and the limbs. At the limits, shooting a too light arrow with a strong bow may lead to break the arrow or even the bow, another effect weel known in practice.
The phisics of the "bracing height", i.e. the distance between the string and the bow for a strung bow is one of the least understood points of archery. Empirically, no archer would ever shoot a "loose" bow and it is known that there is an optimum bracing height, but this point is not easy to determine, nor it is clear why there is such a point. Normally, the manuals will tell you to refer to the manufacturer's specifications. The manuals may also tell you to try different heights and some will tell you to seek the best bracing point as the one which gives the "most satisfactory noise". More rigorous tests require either sophisticated instruments, or a very long series of trial and error experiments. Among several problems, here, one is that the bracing height is normally changed by twisting or un-twisting the string, something that changes its elastic properties and its stiffness. Furthermore, changing the string length also changes the degree of tension of the limbs and hence the weight of the bow. Hence, for each height the whole bow would need to be "retuned" in terms of arrow weight, spine, nocking point, etcetera..... No wonder that this point is not easy and not so well understood.
We can use Hickman's model to try to get some insight on how bracing affects the speed of the arrow. So, let's try with the same parameters as before, a "longbow" with an arrow weight of 25 g, total limb weight of 1 Kg and a draw length equal to the limb length, assumed to be constant with varying bracing heights. Here are the results for speeds:

Somewhat surprisingly, we observe that the final arrow speed changes very little for different bracing heights. The small variation may be explained taking into account that – as we said – with larger heights we pull more the limbs so what we lose in draw length we gain in higher weight of the bow. This is easily seen in the plot of the accelerations:

Where we see that for higher bracing heighs the arrow moves for a shorter path but it is accelerated more, in the end attaining about the same speed.
The speed does increase a little for lower heights, a result which seems to agree with the calculations reported by Tuyn and Kooi. However, a surprising result is that the model gives no "optimal" bracing height: the highest speed for the arrow and the lowest accleration for the tips is for a bracing height equal to zero. Here, clearly, the model does not reproduce reality. It takes little effort to find out which is the bad approximation: the model assumes not only perfectly rigid arms, but also an infinitely rigid and zero mass string. Obviously, on practice this is not the case.
Let's see what happens as the arrow moves along the x axis. The model assumes no mechanical action of the string on the tips and this makes sense. Indeed, as you release the arrow, there is no mechanical pull of the string to the limb. All the force at the tips is generated by the acceleration of the arrow and this is the result of kinetic energy being transferred from the limbs to the arrow. (it is the "tau effect" we had mentioned before). The model tells us that in an "ideal" system the tau effect would bring the limbs to a complete stop exactly at the moment when the arrow leaves the nocking point, independently of the bracing height. Actually, the stopping action would be the smoother the smaller the bracing height.
In practice, reality takes over when the arrow leaves the nocking point. No string is infinitely rigid and all – obviously – do "give in" little. In addition, the string has a non negligible mass, this means that not all the elastic energy of the limbs goes to the arrow as kinetic energy, part of it goes to the string as both elastic and kinetic energy. As the arrow leaves the nocking point this string energy reappears as vibrations and as a strong "kick" to the limbs, forcing them to stop their motion. For a very small bracing height, the slack of the string may lead the limbs to vibrate all the way to the other side, as if the bow had no bracing at all, this may the cause of the bad noises and sudden increase in vibrations that archers report when they reduce the bracing beyond a certain point.
These effects are very difficult to include in Hickman's model, and even if it were easy, it would force us to introduce a number of variable parameters that would destroy the simplicity of the model. But the fact that the model does not reproduce reality in a perfect way does not mean that it is useless. On the contrary it, helps us to understand the physical phenomena going on. In telling us about an ideal system, it has permitted us to isolate those effects which are far from ideal and which eventually have to do with the behavior or real world materials. From these considerations we can conclude that the most rigid and lighter strings are the best for archery, which is again in agreement with the conventional archer's wisdom. I've also the impression that in many modern bows the bracing height could be reduced without any loss of performance. Some modern bows may be somewhat "overbraced", perhaps because the use of fiberglass and other modern materials does forgive a deformation of the limbs that would be dangerous for older materials. But, as we saw, there are several effects that go on and which tend to compensate each other, so that the bracing height turns out to be a very forgiving parameter.


The optimization of a multi-parameter technological system is always a difficult task, one that turns out to be actually impossible in a rigorous sense if the number of parameters exceeds a certain limit. All this means that the optimization of complex systems can be only obtained with a judicious application of human intuition and experience as well as of the "good enough" principle. The bow, even though apparently simple, is actually one of those systems. The calculations reported here were not meant to be rigorous, but just as an example to illustrate how the coupling of simple models and human intuition can lead to a better understanding of the physics of a technological systems.