Arm design is an issue that comes up a lot. This page has information about several popular designs and tips for building them.
Arm mass is very important. It is easy to make an arm strong enough, what is hard it getting the arm light. In the case of a arm, light does not refer to its mass, but its moment of inertia. The moment of inertia ($I$) is a measurement of how hard it is to turn something (ignoring friction). It is calculated by taking the sum of all the parts of the arm times their mass times the distance from the center of rotation squared.(1)
Where ($m$) is the point's mass, and ($r$) is the radius (distance from the axel).
What this means is that removing a pound from the arm 10 feet from the axle is 100 times better than removing a pound one foot from the axle. This combined with the fact that the bending load on the arm during the throw (but not necessarily during cocking) decreases the further you get from the axle means that all arms should be tapered to minimize weight. There are cases where this does not apply completely such as when cocking by pulling the arm tip down, the load on the arm can be greater that it would ever be during operation. An extreme case of this is the king Arthur design which is why alternate cocking methods were designed for it. Also mass near the axle can have a larger effect than suggested by the moment of inertia if the axle moves, as is the case with floating axle designs, or designs on wheels.
When deciding how to taper your arm, also keep in mind the area moment of inertia, which shows that strength increases with the cube of the height but only linearly with the width. This means that a 2x6 is 9 times stronger on edge than it is on its side, for the same weight. Because of this, removing material from the side of your throwing arm will reduce it's strength less than removing the same amount of material from the top and bottom.
Although an indepth discusion of area moment of inertia is too much for this article, please view this list of formulas for deciding how different arm shapes will provide different weights and strengths.
Torsion arms have two main differences from other arms. First they are subjected to large compression loads from the bundle, and second they often strike a hard stop.
The base of a torsion arm must withstand the compression loads applied, and the immense bending force right on the edge of the bundle. Failing to do this is one of the most common causes of torsion arms breaking. The compression from the bundle weakens the arm to bending, and it just breaks off at the bundle. When trying to avoid this effect remember that mass at the point of rotation slows the arm down very little, so adding heavy reinforcements does not hurt performance significantly. Adding steel bars to the top and bottom of the arm is a common and effective approach. They should extend all the way from the end of the arm, through the bundle, and a significant distance up the arm.
In the case where a torsion arm will strike a hard stop it needs to be protected. Padding the stop, often with leather, can help, but wrapping the arm with string/rope also helps.
Around the axle hole is not only the point of highest bending load, but also a huge point load. Hanger axles also impose a huge point load on both the hangers and the arm. Because of these large loads the arm should be made the strongest at the axles. The most common approach to handling this bending load is the Simple Tapered Sandwich, see below. While the bending reinforcements do help handle the point loads, several other things can be done to help. Because the biggest problem from these point loads is splitting wooden arms, adding reinforcements that will not split is usually a good choice. These reinforcements go on the side of the arm and the axle goes through them. Attaching some wood with the grain going in a different direction, or attaching metal plates are common methods. This is especially important in the case of hanger axles where the axle hole is near the end of the arm. Even with these extra rienforcments leaving a significant length of material between the axle hole and the end of the arm is crucial.
Each sling has two ends, one permanently attached to the arm, and the other placed on the pin so that it can release and open the pouch. Although it can sometimes be difficult, the ends of the sling should be attached as close together as possible. This helps prevent "rolling" of the projectile. Think about taking a piece of rope and tying both ends to the same eye bolt in the ceiling. Now place a pipe in between the two pieces, slide it down until the rope it tight, and swing it back and forth. Notice how the pipe stays in the same location relative to the string. This is good for projectiles. Now think of the opposite extreme. Tie both ends far apart so that the rope it tight. When you move the pipe it moves relative to the string, obviously. If your sling travels 180 degrees relative to your arm during launch, i.e. starts touching the arm and releases straight off the end of the arm, your projectile will roll twice as far as the two ends are apart.
This rolling causes two problems. First, and most detrimental, is that it can cause the projectile to roll out of the pouch sending it who knows where and causing a dry fire on your machine. The second is a result of the first. In order to keep the projectile in the pouch it has to be made larger. A larger pouch weighs more, costs more, creates more drag, and is more complex.
The following design elements can be combined or used individually.
Simple Tapered Sandwich
The tapered sandwich arm is made from a bunch of thin boards laminated together. One end of all the boards usually line up at the high stress end of the arm (The short arm for trebuchets, and the bundle end for torsion). The boards in the middle are full length and the ones on the sides are shorter, often extending just past the axle on trebuchets. The other board lengths are in between to produce a tapered arm. The boards do not have to be the same thickness or width. This design can be taken to the extreme to produce a wide arm to reduce the unsupported axle span.
Bridged arms are arms where the tension load is removed from the main member and transferred to strictly tension members such as steel cables. The cables are held away from the arm by rods protruding from the arm, usually one near the axle and occasionally more further down the long arm. Sometimes arms are bridged on just the top/from which only helps during cocking and acceleration. These arms have been known do break on deceleration after release, which is why many arms are also bridged (but usually to a lesser extent) on the bottom.
Many arms recently have been extended with aluminum bars or tubes.
Open web beams work on a similar system to bridging except the outer members can often accept compression loads, so rigid bars are used instead of cables, and more bracing is usually included between them and the rest of the arm.
Structural Steel or Aluminum Beam
Some arms are made out of a single metal beam such as an Aluminum tube or a steel I-beam. These arms can be hard to taper.
Various composite materials can be very strong and light and therefor great for arms. Composites arms often consist of the composite shell over a light core to hold the shell material where it is most effective. A great example of composite arms can be found at Siege-Engine.com's Mista Ballista Arms page.
The core material is often a light foam. Is some cases the core can just be hollow.
Steel sheet metal on wood for example.
Draw Down Point
The draw down point is typically placed at the end of the throwing arm. This allows for the greatest leverage when pulling the arm down. Although this places more weight at the end of your arm, it is generally not enough to cause problems.
Some machines will have a self adjusting draw down point. A cable or heavy rope is attached at the end of the throwing arm and near the axle. A pulley is placed on the cable so that it can slide back and forth(think zip-line). The draw down line is then attached to the pulley. When the arm is in the vertical position the pulley is closest to the axle. Although this provides less leverage, less is required since the counterweight is below the axle. What it lacks in leverage, it makes up for with increased arm travel per foot of line pull. As the arm comes further down, the pulley slides along the cable towards the end of the arm, increasing the leverage.
The two advantages of this design are its increased efficiency and lowered height for initial attachment, reducing the need for ladders or fancy re-hooking procedures.
For some machines, such as FAT or its cousins, the draw down point cannot be attached to the arm. It must instead be attached to the counterweight. In this configuration it would be more aptly called the lifting point.
On some machines the load is reduced due to leverage gained by the arm, on others the entirety of the load must be bore.
Attachment to Arm
Most commonly an eye bolt is used due to its strength, light weight, and ease of attachment.
Attachment to Draw Down Line or Trigger
One of the biggest decisions to make when thinking about draw down points, or lifting points, is what to do with the line during the launch. If you remove it, then you have to develop some sort of procedure for reattaching is after the launch. If you leave it in place, it is an extra line that can become tangle in any number of undesirable locations.
For fixed axle trebuchets the problem is in reattaching the line. For small machines the attachment point can often be reached from the ground. On large machines a ladder is required. Sometime the ladder is built right into the arm. Although the coolness factor goes way up when you climb 25 feet (7.5 meters) to the top of your machine, your legs will become very tired after only a few launches, plus the arm wiggles back and forth as you climb, exhilarating for some, terrifying for others.
On a FAT the problem is releasing the line after the counterweight is raised. Climbing up is an option, but this places you in the danger zone of a cocked treb, not very safe. On many machines the trigger is tied in to the line so that when the machine is triggered to launch, it is released from the line.