By Harry Marx

The following article may be highly theoretical, but happily has a very useful and practical outcome.

After various and long discussions and investigations with fellow archers, and some investigation on my part, I have come to realise how little we know about dynamic spine. In case you didn’t know, there are two types of spine: static and dynamic. Essentially, spine is about how stiff an arrow is or acts. Static spine is measured simply by how far 28 inches (71.12 centimetres) of the shaft bends if a lateral force of 1.94 pounds (880 grams) is applied. If the arrow bends 0.300 inches, its static spine (s-spine) is 300 (it’s a bit confusing the first time, but the stiffer an arrow is, the “bigger” the spine, but the smaller this number.) Dynamic spine (d-spine) is how the arrow bends when accelerated by the bow.

But how do you measure how far the arrow bends when you shoot it? This is very difficult. You need a high-speed camera, which I don’t have. So instead we will hypothesise, but first let’s look at the huge and many differences between d-spine and s-spine before we try putting numbers to them.
The first difference is the source of the force that bends the arrow. For s-spine it is simple: it is applied laterally by the spine tester – a weight that is hung on it. For d-spine, most would argue that it is the draw weight of the bow, but this is not the case.

The arrow bends during release because it is being pushed from the front by its inertia (or weight) and not because the string is pushing at its nock end. If the arrow had no mass and was shot even from a 100 pound bow it would stay as straight as... an arrow. Intuition tells us that this buckling force must be the same as the draw weight, but it is much less. The reason is that the only weight, and therefore the only resistance/inertia that is in front of the shaft of the arrow, is the broadhead and insert. It follows therefore that it is only these two items that are trying to keep the tip of the shaft stationary. As you go further back along the shaft, let’s say halfway towards the nock, half the weight of the shaft also adds to the inertia that pushes back. But now the part of the shaft that receives this force is half as long and therefore much stiffer. At the nock, all the pile weight and shaft weight, but not the nock, is presenting inertia and trying to buckle the last inch of the arrow. So the total weight that provides the buckling force is a more complex formula than just the sum of the item’s weights multiplied by the acceleration.
The second difference between the two spines is that whereas when measuring s-spine we wait for the shaft to bend and become stationary before we measure the extent of the bend, with d-spine we don’t wait. In fact, the very nature of d-spine is that the buckling force is only felt for a very short time, mostly less than 20 milliseconds. That is how long the arrow is pushed by the string before it leaves the bow. So the force that bends the arrow only has a very short time in which to bend it.

The third difference is the length of the shaft. Making the shaft longer increases the weight of the shaft and arrow, as well as the lever this force has to bend the arrow. Shaft length would have affected s-spine as well, were it not for its definition limiting it to exactly 28 inches. The d-spine is therefore dependent of shaft length, while s-spine is not.

The fourth difference is added weights, such as the tip, insert and pile weights. These, of course, play no role in s-spine, but are the very essence of d-spine. They provide the biggest influence in providing the inertia or buckling force. At the same time, you will notice I made the qualification – “at the tip of the arrow”. If you add any weight at the nock, it will have no direct effect on the buckling force, since its inertia is behind the shaft and pushes against the nock, not the shaft.

Note, however, that the increased total arrow weight will decrease acceleration, which will decrease the buckling force. However, the shaft will be exposed to it longer. If we add weight at the tip, the acceleration is less, but the inertia pushing the shaft is more, so the buckling force is about the same, but the resistance is less (the extra mass was not on the shaft). Again, not so simple.
The fifth difference is a bit sneaky. The d-spine is dependent on the s-spine and shaft weight, while the s-spine is dependent on the shaft material and dimensions only. There are two forces that resist the arrow being bent during release – the inertia of the shaft and its stiffness. The second, more obvious one is the stiffness or elasticity of the shaft. For any force, the shaft will bend only so far, no matter how long you apply the force – this is s-spine. For d-spine, an additional constant inertia is also resisting the displacement.

The sixth difference is dimensions. S-spine is a single number, but d-spine actually consists of two numbers. If you think about it, d-spine is a measurement of how an arrow vibrates and not how it bends, and all vibrations have two dimensions: amplitude (how far it bends) and frequency (how fast it bends). These two quantities are independent of each other. If, for example, you increase the pile weight, you increase how far the arrow will bend (as we saw above), but not directly, and only slightly, how fast it will bend. So we will have to investigate the two dimensions separately. So far we have only looked at the amplitude and will continue to do so for now. I will get back to frequency later.

Amplitude
How far the arrow bends (D), is given approximately by:
D ≈ K.mp / mt / [ 1/L + 1/(FdYs) ]
where mp is the pile weight, mt is the total weight, L the arrow length, Fd is the draw weight, Ys is the static spine, and K = 0.1.
From this we can see the following:
* if the draw weight (Fd) is increased, all else being equal, the arrow will bend more,
* if the s-spine is increased (but the numeric value decreases), the arrow will bend less,
* if the total arrow weight is increased without an increase in pile weight, the arrow bend will less,
* if the pile weight is increased, the total weight must increase also, but the ratio will increase, and therefore the arrow will bend more.

(Note that “time” disappeared from this formula. In the equation “time” was replaced with an approximation thereof, derived from the draw force and arrow weight, assuming a constant acceleration.)
Sadly, this equation tells us nothing we did not know already, except for one surprise. What will happen if we add a weight tube to the shaft, where the weight tube is very limp, and therefore does not alter the s-spine? I have seen several experts fall over this one. The incorrect answer is that the added weight causes the arrow to bend more. The correct answer is that the added weight slows the arrow down and therefore decreases the buckling force – and also, it adds weight to the shaft that resists the bending as inertia. Therefore, the arrow bends less, and behaves as though it had a stiffer spine.
But there is another approach that is more informative. In 1757 Euler described the critical loading of a column – the force under which it will buckle and collapse. This equation is used today in a wide range of engineering areas, but is literally the corner pillar of architecture. If it is applied to an arrow being accelerated, the equation simplifies to:
Fc = k/(YsL) x Wt / ∑(Widi)
where k=5,000,000; Ys is s-spine, L is arrow length, Wt is the total weight of the arrow without the nock, Wi is the weight of each component (again excluding the nock), and di is the distance between its centre of gravity and the point where the string pushes against the nock. (The “∑” means that for each Wi and di you must multiply them with each other and then add the results together.) The final result Fc is in pounds.

This equation tells us what the maximum constant force against the nock is, with subsequent acceleration of the arrow, that will cause the arrow to buckle. The answer, in pounds, can be directly compared to draw weight, creating an index of how close the arrow is to failing under the load. I did this for Gold Tip arrows, using the company’s spine charts, and saw that whenever an arrow’s Fc drops below 38 per cent of the draw weight of the bow, a stiffer arrow is suggested, which is then entered onto the charts at about 41 per cent. This is a very small window, from 41 to 38 per cent.
We will come back to the amplitude later. The next item we must discuss is frequency of vibration.

Frequency
Frequency of vibration, the second aspect or dimension of d-spine, plays a huge role for traditional archers. More specifically, it is important for all archers using bows that do not have a “cut out” riser. These archers are shooting around the riser, and not through it. This is where the so-called archer’s paradox is applicable. However, that is another story. For now, we want to use the vibrations of the arrow to make it “slither like a snake” around the riser. For these bows (and all finger releases) the arrow vibrates horizontally during the release.

The arrow, once released, starts bending at a certain frequency. If the arrow is released at zero milliseconds, it should be about halfway past the riser in ten milliseconds, and its fletching should pass the riser at about 15 to 20 milliseconds. Of course, the exact time depends on the draw weight (and the draw-weight curve, to be more specific) of the bow and the total weight of the arrow.
If you want total riser clearance of the arrow, it should of course be bent to its maximum (to the left if the arrow is passing to the left of the riser) when it is halfway through the acceleration process. When the fletching passes the riser, the arrow should be bent to its maximum in the opposite direction (to its right if the arrow is to the left of the riser).

This means that the arrow, being accelerated with the string still in the nock, should vibrate at a particular frequency. Exactly what this frequency is, however, is not an easy answer, and one which I am not tempted to guess. However, I believe it will be proportional to the natural vibration frequency of the arrow. If you could measure the fundamental frequency of an arrow with good clearance and compare it with one that doesn’t have good clearance, for a particular bow, you could predict any arrow’s ability to clear the riser on its frequency alone.

One way to measure the vibration frequency of your arrows would be to record the sound they make when tapped and to subject the recording to a Fourier analysis. This will tell you the presence of all the frequencies at which the arrow vibrates. But that is a story for another day.

Instead, I have hybridised two equations with each other to provide some theoretical floor for our discussion. The first is the fundamental frequency for a bar free at both ends, and the second is for a bar fixed at one end. The equation contains a few constants, for which I only have provisional values. This means that the precise result of the equation is not very accurate, but the relative values, for the purpose of comparing arrows, is very useful.
Fn ≈ 7400000 * (3.5596 - 3*(1-1/Wp)) x 1/L2/(S.Ws(gpi))1/2
(Yes, it’s a beauty isn’t it?) Wp is the weight of the pile, Ws is the shaft’s weight per inch, Ys is the s-spine, and L the length of the shaft. The idea behind the constants is to compensate for the arrow being somewhere between free at both ends, and anchored (with a field point or broadhead – inertia anchor) at one end.

Some of the implications of this equation are:
* If the pile weight increases, the frequency decreases – that is, the arrow will take longer to flex from one side to the other (flex time = 1/Fn).
* If the arrow is made longer, the frequency decreases and the flex time increases.
* If the s-spine is increased (the arrow is stiffer, but the numeric value decreases), the frequency increases and the flex time decreases.
* Lastly, when you increase the shaft weight, the frequency decreases and the flex time increases.
You may also notice that if spine is increased (the arrow is made stiffer) and weight also increases due to material limitations, the changes can cancel each other out and frequency can stay the same.
For compound bows, and all other bows with cut-out risers, frequency is not very important. For these bows, the ability of the arrow to flex serves to receive uneven nock travel. The vibration is mostly vertical during the release (unless you finger-release).
Of course, once the arrow is free of the string and starts rotating, the vibration is rotated in all directions. Also, I do not know the significance of the frequency relative to the arrow’s speed – in particular, how it affects air friction. I would imagine that it does.

Practical outcome: a case study – building heavy arrows
I use Gold Tip’s Big Game 100+ arrows with a weight of eleven grains per inch. The pile weight I use is 215 grains. The reason for this heavy pile is, of course, to increase the FOC. Anyway, when I calculated the critical buckling force for these arrows, 32 pounds, as a percentage of the bow’s current draw weight, 72 pounds, I saw it was 40 per cent, right in the bucket, as by Gold Tip’s standards. (By the way, the FOC is 14 per cent.) But these are 600 grain arrows and I want to build an 800 grain arrow. I use a compound bow with a cut-out riser, so I do not worry about frequency. There are three options:
Option 1: Adding additional weight at the tip to get to 800 grains. This will decrease the load percentage to 36 per cent, which would be too low according to Gold Tip’s charts.
Option 2: Using a weight tube of eight grains per inch. The result would be 807 grains, with a critical load of 43 per cent. Sounds good, but the FOC is now only 10,4 per cent.
Option 3: Using a combination of a weight tube of 5,22 grains per inch, and an extra pile weight of 200 grains. This gives 838 grains, an FOC of 15,3 per cent, and a critical load percentage of 40,5 per cent. Now we are talking business.

The table shows a number of typical arrows and set-ups. See if you find them acceptable and whether the indexes work or not. The CL percentage should be 38 per cent or more (for Gold Tips). D should ideally be less than one, but remember it’s an approximation. (See table 1.)
The first seven rows are adaptations of a very marginal arrow (CL%=38.0), where I changed a single parameter each time in order to show you the influence of each:
row 2: Adding a weight tube makes the arrow bend less and flex more slowly.
row 3: making the shaft stiffer decreases bending and makes it flex faster.
row 4: extra shaft weight, same as row 2.
row 5: fletching that is only 20 grains makes the arrow bend MORE.
row 6: making the shaft shorter makes the arrow bend less, and flex faster.
row 7: a combination of changes, greatly reducing bending and pushing CL% up to 47 per cent – almost too stiff?

The second group of arrows are medium-weight hunting arrows. Here I selected components for a 600 to 800 grain range that looks acceptable as regards CL% and FOC. In general, these two values oppose each other.

The last arrow is 1000 grains. It combines double shafts, pile weights and weight tubes to get to a satisfactory design. You must have noticed by now that even though weight tubes have no s-spine they do play a major role in d-spine.

Table 1: The extra weight comprises brass weight rods, screwed into the insert from within the shaft. Wp is the pile weight, and Wt is the total arrow weight. Gpp is the grain per pound draw weight. D is the deflection, or how far the arrow bends, calculated using the following formula:

D ≈ k.mp / mt / [ 1/L + 1/(FdYs) ] ; k=0.1 The critical loading of the arrow was calculated with Fc ≈ k/(YsL) x Wt / ∑(Widi) ; k=5,000,000 The frequency of vibration was calculated using Fn ≈ 7,400,000*(3.5596 - 3*(1- 1/ Wp)) x 1/L2/(YsWs(gpi))1/2 For me CL% is the critical one – the percentage of the critical load over the draw weight And lastly, the FOC was calculated using FOC ≈ ( ∑(Widi) / (WtL) - 0.5 ) x 100

One last piece of useful information: you have may noticed that the super-heavy arrow uses a spine of 220. There are no commercially available arrows like these in the Gold Tip range. In fact, this is a combination of two shafts, one inside the other. So to round of this discussion on dynamic spine, an equation to compute the s-spine of such a double shaft: total spine = spine1 x spine2 / (spine1 + spine2).

Dynamic spine is not something we can easily measure and calculating it is not trivial either. But I hope you have a better understanding of it.

If you don’t understand it, feel free to read this article many times, until you do (wink-wink). If anybody wants more information, please contact me at marxh@unisa.ac.za, or look me up on www.AnchorPoint.co.za.

Updated: Wednesday, February 18, 2009 12:57 PM