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Introduction Edit

When dealing with any man-powerer projectile weapons it's important to utilize the energy stored in them effectively. This is especially important with hand-held weapons that don't have a winch to assist pullback. If efficiency of the weapon is low, i.e. lots stored energy goes to waste instead of to the projectile, the rate of fire, range and power of the projectiles is reduced.

This article describes the basics of effective ballista design from a theoretical perspective. For further details look at the bow design article on the Crossbow building wiki: much of it is applicable here. There's also lots of similar discussion on the personal torsion weapons page.

The basics Edit

Use proper spring cord material Edit

If you're looking for a modern material, use twisted nylon. While there are several types of nylon, any kind will probably do. Braided nylon will not cut it, according to Nick Watts' early tests. If you are going for authentical materials, use high-quality sinews. Deer or moose sinews will probably work well. Cattle sinews are probably worse, because in modern days the cattle is mostly kept indoors and their sinews may thus be weak. Horsehair or hair probably works, but the ancient authors said that sinew is superior, and they are probably right.

Interestingly polyester rope has very similar properties to sinew, according to Clift et al (2004). The static properties of a polyester rope do not necessarily make it a good candidate for use in torsion springs; after all, the cords need to strike fast, not just stretch similarly to sinew. So far I have not heard of anyone using polyester in ballista springs, nor have I tested it myself.

Increase the energy storage capability of the torsion springs Edit

The primary method of increasing the energy storage capability of a torsion spring is to pretension the cords in it to as high degree as possible. Linear pretensioning is generally accomplished with a stretcher, but it can also be achieved with various types of wedge systems. In any case, increased linear pretensioning increases the amount of energy required to stretch the cord any given amount, which in turn increases the cord's return speed. The return speed can be heard from the sound pitch when the cord is plucked. Of course, when the cord is stretched, it also becomes thinner and lighter, which in part improves it's return speed with any given amount of stored energy.

Energy storage capability of the spring can also be increased with rotational pretensioning, that is, by rotating the washers in the torsion spring frame. This step is in practice necessary, but going overboard with it will cause the torsion spring to stack. In the cheiroballistra a reasonable amount of washer rotation seems to be around 180 degrees, but hard data is still lacking. Measuring the effects of washer rotation would require plotting the force-draw curves (energy input) and bolt energy (energy output) at various washer rotations. While this would be a quite tiresome process, it would yield interesting insights into when rotating the washers stops making sense, and how the stack at the end of the draw is increased with increased rotation.

In an optimal, theoretical scenario we wouldn't need any pretensioning at all; rather, we'd start with cords that are just barely tight, then pull back the arms until the cords have stretched near their breaking point and then release the bowstring. This would, in theory, allow us to store and release the maximum amount of energy with any given amount of cord. However, in practical implementation this approach is doomed due to two reasons:

  • It implies significant torsion spring stack
  • There is lots of easy stretch (slack) in the cords

So practice a significant amount of pretensioning - linear and rotational - is required for a correctly operating ballista.

Increase the efficiency of energy transfer Edit

When the bowstring is released, the energy stored in the springs is transmitted to the projectile via the arms and the bowstring. The ratio between the kinetic energy of the projectile and the energy fed in the springs is called effeciency, usually expressed as a percentage. The higher the percentage, the less energy gets wasted.

The primary reason for energy wastage in mechanical weapons of any kind is moving mass. For example, in a bow the limbs and the bowstring are - by necessity - still moving when the arrow has left. Thus the limbs and the bowstring still have some amount of kinetic energy left, and this residual energy is not transmitted to the projectile.

The same thing happens with ballistas. When the a ballista's bowstring is released, the torsion springs start rotating, which forces the arms to move, which force the bowstring to move, which in turn forces the projectile to move. As the cords in the torsion spring cords move only a very small distance - a few centimeters - any residual energy in them is neglible and can be ignored. Overly heavy arms, however, will waste tons of energy, especially if their mass placement is wrong, i.e. the mass is concentrated near the tips: this is because the tips have to move a long distance during a shot. Mass in the arm near the spring bundle is not nearly as critical, as that part moves fairly little. The bowstring is a major energy waster, as it's middle has to move at the same velocity as the projectile. Thus having a bowstring that's only as strong and heavy as necessary is very important.

Efficiency can be increased by increasing projectile weight. A heavy projectile will by necessity move slower than a light projectile, given the same amount of stored energy. A slower projectile also means that the arms and the bowstring move slower, and thus have less residual energy in them after the projectile is on its way. Effeciency can also be increased by increasing the amount of bowstring movement for any given arm rotation. For details, look at the personal torsion weapons article.

The efficiency of a well-made and fully-tuned cheiroballistra has not yet been determined, but it seems fairly high. Until fairly accurate measurements are taken, this statement is not of much value. However, it will be interesting to see how the cheiroballistra compares to similar weapons. In the author's tests a well-made crossbow with a steel bow can achieve ~70% efficiency with fairly heavy bolts. Reducing bolt weight causes a fairly steep drop in efficiency and increases the loudness of the shots considerably. Modern compound bows can allegedly achieve efficiency levels of 86-88%.

Increase the energy fed into the springs during draw Edit

More energy can be fed into the springs by increasing the draw length (i.e. arm rotation). A long draw is particularly important in weapons that are cocked by hand, i.e. without a winch, because the maximum draw weight of the weapon can't go beyond a certain point or the weapon can't be cocked at all.

Constraints Edit

Endurance of the spring cord Edit

All cordage has it own, characteristic limits on how much it can stretch before breaking. Sinew and nylon can apparently stretch 15-20% before breaking. The cord stores energy when it's stretched, and returns the energy when it's relaxed. It's important to realize that the energy spent pretensioning the cords is lost, and that the benefits of pretensioning are indirect: this is because we can't feed energy into the spring once (i.e. during pretensioning) and release it many times (during shooting). For details on various spring cord materials see the article by Clift et al (2004).

Torsion spring stack Edit

The "torsion spring stack" is rather hard to explain in text only, but I'll try. At rest, the spring cords are straight, forming an "|" you will. As the washers are rotated, the will "bend" sideways, progressively moving towards a ">"-shape. Doing this increases the draw weight of the ballista, as the cords are forced to stretch more as the arm is rotated. However, this comes at the cost of reducing the geometrical advantage provided by cords in a "|"-shape. Basically displacing a "|" cord sideways for, say, 1cm, requires very little energy. Moving it another 1cm requires proportionally more energy. As we move the cord more and more sideways, the more energy we need for each additional centimeter, simply because the cord needs to stretch more and more at each step. The theoretical worst case would be that of a "=" or "-", where the cord is stretched directly, with no geometrical advantage whatsoever, and where huge amount of energy would be used to move the cord a tiny amount. Obviously we don't want to go too far into that direction.

So, a "|"-shaped cord moves a longer distance sideways than a ">" cord, given the same amount of energy fed into both. The fact that washer rotation increases the frequency of the spring cords compensates for some of the lost cord velocity in the ">" case, but the basic idea is to stay as close as possible to the "|" case.

The torsion spring stack manifests itself in two ways:

  • Going beyond a certain draw length is simply not possible because of a "wall". The wall is obviously partially caused by increasing bowstring/limb angle (i.e. archery stack), but increasing washer rotation makes this wall much steeper.
  • Increasing draw weight beyond a certain point does not increase bolt velocity at all or increases it very little. I've noticed this phenomenom several times, but the evidence is still a bit anecdotal.

TODO: Add a diagram explaining this phenomenom

Archery stack Edit

The "archery stack" is also a result of simple geometrics, and applies to bows as well as ballistas. Basically it's a product of loss of leverage during the draw. During early draw drawing the bowstring moves the limbs only a small amount, meaning the bowstring feels light to pull. As draw length is increased, this geometrical advantage is progressively lost, because the angle of the bowstring and the limb increases. Going much beyond 90 degrees bowstring/limb angle is not usually worth the effort, because the draw weight starts to rise sharply: this is especially true in weapons which are cocked by hand without a winch.

TODO: Add a diagram explaining this phenomenom

Summary Edit

In retrospect the physics involved in torsion ballistas are so simple that most of the "magic" is lost:

  1. Use as much linear pretensioning as possible without breaking the cords
    • Increases the frequency (return speed) of the spring cords
    • Reduces torsion spring stack at high arm rotation
  2. Use a minimal/conservative amount of washer rotation
    • Using an excess amount causes torsion spring to stack
  3. Rotate the arms as much as possible, as long as...
    • ...the torsion spring cords do not stack
    • ...the bowstring/arms do not stack (a.k.a. archery stack)
  4. Minimize dead weight
    • Limbs should be as light as possible, in particular in quickly moving parts (i.e. limb tips)
    • The bowstring should be as light as possible
  5. Optimize the force-draw curve
    • The geometrical advantage of the bowstring at the start of the draw should be as big as possible. Use an inswinging design rather than outswinging if you're not interested in reconstructing a particular historical model.
    • The draw weight should not climb up too steeply at the end of the draw, especially if the weapon is cocked by by hand (or stomach).
      • This is affected by both kinds of stacks described above
  6. Use good spring cord materials

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