The trebuchet is one of the most famous siege engines of all time, and designing and optimizing the powerful machine involves some tedious work.
To understand how to optimize the design of a trebuchet, first we need to understand exactly what designs and principles make up one. A trebuchet in its most pure form involves a counterweight that falls completely along the vertical axis and a swing arm. Attached to the end of the swing arm is typically something like a sling to increase rotational velocity, but it isn’t essential to trebuchet design. Potential energy is stored in the lever mechanism as the large counterweight is lifted. Once dropped, the potential energy stored in the counterweight is transformed through linear motion into rotational kinetic energy for the projectile.
It’s important to note that no compound motion or mechanism is involved in the trebuchet, rather it is one simple lever arm along with a counterweight. Once the mechanism is understood, the variables to controlling flight begin to appear. Being such a simple design, the only things you can vary are the lever arm length, the height, the counterweight weight, and where to position the fulcrum along the lever arm. While the constraints may seem limiting, optimizing all of these factors can produce a highly efficient machine.
The physics behind a trebuchet are more complicated than the mechanism would suggest. By placing the fulcrum much closer to the counterweight end, a higher linear velocity can be achieved by the projectile. This becomes simple lever physics, but you also don’t want to have the fulcrum too close to the counterweight, or your mechanical advantage will begin to dwindle. There will also come a point where increasing the weight of the counterweight will do no good other than putting extra strain on the trebuchet. No matter the weight of the counterweight, it is limited by its free-fall velocity.
The sling on the end of the trebuchet releases upon an angle α, at which point the counterweight does no more work on the projectile. One can change this release angle by adjusting the angle of the holding pin at the end of the lever arm. The more in line with the lever arm the holding pin is, the smaller the release angle, α, is.
You may be beginning to see how calculating the design of a trebuchet can be a little difficult, but luckily we have engineers who have widely studied the design of trebuchets and already determined optimized designs. Dr. Donal B. Siano is one of those engineers. He published a paper in 2001 titled Trebuchet Mechanics detailing an intensive study into optimizing all variables in the war machine’s design. Here are some of the key facets of design he found to be true throughout his testing:
- The length of the payload arm should be 3.75 times longer than the counterweight arm.
- The length of the sling should be the same length as the payload arm.
- The initial angle between the counterweight arm and the support should be 45 degrees at launch.
There is a whole slew of other equations and optimization techniques included in Siano’s paper, including many page-long derivations and differential equations. If you want to relive your past college differential equations woes, I would highly recommend giving it a read.
To get deeper into some of the optimization physics, we need to understand the ratio of payload arm to counterweight arm, called the beam ratio. There have been many trebuchet designs that used ratios closer to 4:1 or even 5:1. Increasing the ratio increases the launch velocity of the projectile, but it also means the counterweight must be much larger to provide the necessary force. When you start to grasp the idea of the ratios in relation to the moment arm of the trebuchet, this might become a little easier to understand. To determine what ratio you need to use, you have to take into account your available material strength, as well as considering the counterweight weight, as we will discuss below.
Next, we need to examine how heavy to make the counterweight. The generally accepted ratio is 133:1, or the counterweight needs to be 133 times as heavy as your intended projectile. If you want to determine your specific counterweight ratio, the math is fairly simple. Called Phsstpok’s Rule, the equation states that the mass ratio of the counterweight to projectile is equal to the beam ratio times twenty. It is expressed in the equation below:
MR=BR x 20
I don’t know about you, but that’s the kind of math I like to see when doing any sort of designing, simple math. Using these ratios, optimizing your counterweight weight and launch arm length shouldn’t be too difficult.
To tie up a few loose ends in the design process, the height of your support should be high enough to keep the counterweight freely moving throughout all of its rotation. Sizing of the trebuchet is going to come down to the distance you need to launch the projectile. This is a little beyond trebuchet optimization, but in order to size a trebuchet based on range, you would need to back calculate the launch velocity of the projectile using projectile motion equations. From here, it would be a simple matter of plugging that launch velocity into trebuchet sizing equations, and you can derive your sizing constraints from there.
On a final note, unless you are designing a trebuchet of great scale, it generally a good idea to build components that are adjustable out in the field. Everything can look good on paper, but as the all-knowing engineer that you are knows, everything is different in the real world.