A report I completed back in 1998 has drawn lots of traffic to the website. You’ll find the report reposted below including a working model video! This report is strictly Copyright (c) 1998 Justin Hall. All rights reserved. Students – don’t attempt to copy.
Justin Hall | Dynamics, Spring | 4 May 1998
What used to be a weapon of war is now a weapon of mathematics. In past centuries, nations feared its massive size and destructive power. It was the deadly weapon in medieval times. Today, only its dynamics are feared by gawking engineers, physicists and mathematicians. Our knowledge of materials and structural support, along with basic physics principles have been advanced by this machine. It is the trebuchet. This report attempts to cover the trebuchet’s history, our fascination with the machine, and how technology is helping to model it.
Since the dawn of time man has attempted to build objects that can fire heavier, more deadly objects further distances. The goal is simple — to conquer the enemy. Mediterranean designs centered around catapults and trebuchets. To many, trebuchets are catapults. However, that is not nearly true.
A trebuchet, pronounced tray-boo-shay is a counterweight machine with a rotating mass. On one end lies a large mass and on the other a long sling. The catapult makes use of tension from twisting ropes or leather straps to propel the arm forward. The arm is propelled against a stop where the projectile is released. Catapults were “wild, finicky, and slow siege engines,” says Matthew King, a true trebuchet hobbyist (King). After each firing the machine needed repositioning. Catapults buckled and shattered when propelling objects much smaller than that which the trebuchet was capable of hurling. Trebuchets were more efficient and more stable. Using counterweights, a complete transfer of energy to the projectile was possible. They flung further and hurled harder.
Neither the trebuchet nor the catapult lives up to today’s military standards, however they can be analogously defined. The trebuchet would be likened to biological, nuclear and chemical warfare. The catapult would be defined as machine guns and tanks. It is fair to say that the trebuchet would win every battle when opposed to the catapult.
Figure 1.1:A graphical comparison between the three most basic trebuchet forms.
Before gunpowder (850AD to 1350AD) the trebuchet was the “siege weapon.” Efficiency was its key to success. From the early traction machine — where as many as 250 men pulled down on the short side of the arm then released it — to the advanced counterweighted machines, the trebuchets evolution was quick. First generation machines held fixed counterweights (boxes of rocks the sizes of peasant’s homes) and fixed projectiles. One-hundred yards was the average distance for a projectile to land. Then advancements allowed for rotating counterweights about the end, and projectiles in slings as long as the arm. With the later configuration 500-600 pound projectiles could be hurled 200 plus yards with accuracy! Here the mechanical advantage varied for the arm throughout the launch. More energy was transferred to the projectile. Further evolution saw props being placed on the counterweight — once again increasing the distance (King).
But the trebuchet added to more than just the militaries offense. Trebuchets advanced the concepts of pendulums and equations of motion. Leonardo Da Vinci used the trebuchet to tell time! However, the most ingenuous use came from scientist Jordanus of Nemore. From Jordanus came the concept of vectors. He gave direction and magnitude to a thing called a “force vector.” The development of work: “a vector proportional to the mass and vertical descent of an object” was also provided by Jordanus through his study of trebuchets (Valens pg. 38).
What has impressed me is the number of trebuchet enthusiasts that exist! It seems as if there are almost as many trebuchet fans as there are “Trekies.” And each with a unique approach to the military machine. Some like their device compact and small — able to throw only golf balls, cheese, skulls, and pumpkins. Others prefer the huge, 100 foot armed beasts capable of propelling grand pianos, commodes, and cars. The drive is to propel further and build more efficiently.
The fascination with trebuchets crosses all age and skill boundaries. Four years ago more than 20,000 enthusiasts embarked on Lewes, Delaware for the eighth annual Punkin’ Chunkin’ contest. Twenty-three hurling devices were entered in the competition. Numerous universities, including the United States Military academy are building trebuchets and catapults for class assignments. Richard Clifford and John Quincy of Texas are building a huge trebuchet, Thor, which will be able to propel a 1962 Buick about 1000 feet . National media from CNN, ABC, and CBS have been attracted raising interest internationally. Dave Barry has even poked fun at the trebuchet beast. With a 100 foot throwing arm and a 42 foot fulcrum, it is simply the world’s largest that exists today (Kvinta).
Trebuchet fascination has also been encouraged through the internet. If you would perform a search using any of the major engines, 3,000-5,000 different hits would be returned! Newsgroups allow trebuchet lovers to share stories while the world wide web links up users allowing interactive modeling and multi-media presentations. Software and sites to help maximize trebuchet performance are on the internet. Technology is surely advancing the trebuchet.
What technology can not do is change the physical principles that make the trebuchet work. For the rest of this report, I would like to focus on how to model a trebuchet.
Energy stored in a raised counterweight has potential energy, which is transferred to kinetic energy when rotated. When the counterweight is fixed to the short end of the trebuchet beam, as with first generation models, less of the potential energy was converted into kinetic. But, by simply hanging the counterweight by a chain more of the potential energy will be converted. As the weight falls, the chain and the beam form a bow shape able to move faster than the counterweight alone.
Second and third generation models implemented a more efficient sling. Here, the kinetic energy is transferred to the linked sling. As the trebuchet is released from rest, the sling moves close to the fulcrum. Great force from the lever is provided in this early stage. When the sling leaves the ground and passes by the long end of the arm less force is being applied but greater velocity is being achieved. Further and further from the fulcrum, the sling has more and more velocity. One author likens the move as an “automatic transmission” shifting gears.
Knowing when to release the sling is crucial. Using a fixed counterweight the mass will fall in a slight arc. When the arm and sling form a straight line, the falling counterweight has released all its energy. Any remaining energy is wasted. Therefore, the projectile should be designed to be released slightly before hitting this stall point. The projectile will begin flying a 45 degree angle to the earth — maximizing its flight distance (Catheral, pg. 55, 60-64). Almost as crucial is knowing how long to create the sling. Most agree trial and error and experimentation are the easiest ways to answer that question.
Experienced trebuchet builders have agreed upon a few rules of thumb. Novice trebuchet artists would be wise to start the design with these criteria:
Table 1.1:Sources such as Ken Peter and Donald Siano have found from experience these rules of thumb to be more than true.
Proving these rules of thumb is harder than it appears. Simultaneous differential equations of motion need to be applied. On March 12, 1998, Donald B. Siano published a report entitled “Mathematics of the Trebuchet.” Donald used Mathematica software to analyze the trebuchet. He began with a simple model — no sling, fixed counterweight — and advanced to a realistic model with a sling, hinged counterweight, and beam mass. After sifting through his findings, these results can be proven:
Table 1.2:4 variations of the counterweighted trebuchet were mathematically analyzed. These are the results.
Notice the huge difference between a trebuchet with no sling and fixed counterweight verses the trebuchet with a 3.5 foot sling and 1 foot hinged counterweight! The efficiency increases by 70%. It is also interesting to note that friction has little effect. Only 2% less efficient was the model with friction. Finally, when the beam mass was taken into effect, efficiency dropped 16% — resulting in a distance of 56 feet less.
Performing these calculations with each iteration by hand would be prone to error, lengthy, and tedious. Building small trebuchet models to measure the results can be very costly and lengthy. To create a table such as the one above, the author wisely choose computer simulation.
It is now the trend in engineering. Using dynamic modeling programs, finite element analysis software, and mathematical computational programs, tables such as the one above can be constructed in less time than ever. And optimizing a model can be readily completed. It was once said that a picture is worth a thousand words. Graphing the motion of the trebuchet using software allows a greater understanding of how the machine works. The bulk of analysis can be done by viewing the graphs.
The trebuchet is a great example of variations and variables. At least 15 different variables are present in the model (sling-no sling, fixed CW-hinged CW-propped CW, friction, arm length, fulcrum height, mass of projectile, etc). Each variable can drastically change multiple results. For example, the release angle is both a function of the sling length and the arm length. Software has progressed to the point where no data analysis even needs to be done. A good example is Working Model by Knowledge Revolution. By spending just twenty seconds one can change the sling length and find out the best release angle and range! And a new graph for the motion of the sling can be instantly viewed. Incredible.
Trebuchets are a pristine example of how progress works. First, in 850 AD an ingenator placed a large mass on one end and a small mass at the other rotating about some fulcrum point and called it a trebuchet. Physicists watched the machine propel its projectile time after time trying to grasp the laws of physics that made it operate. Over time ingenators then engineers added slings and propped counterweights, decreased friction and lubricated bearings to optimize the hurling distance. Today hobbyists share resources over the internet and use computer simulation to optimize all the variables present. They have made lighter machines that can propel further and with greater efficiency. Have we reached the plateau of optimized trebuchets? Only time will tell us. Until then, the trebuchet’s dynamics will remain feared only to be caged by simulations and real-life models.
Catheral E., Exploring uses of energy, Wayland Publishers Ltd, England, 1990
King, Matthew. The Holy Book of Trebuchets. http://www.andrew.cmu.edu/user/matthewk
Kvinta, Paul. It’s a Bird! It’s a Plane! It’s a Case of Spam! Outside Magazine. August 1995.
Peter, Ken. Ken’s Trebuchet: how does it work. http://www.mpsi.net/~kenpeter/trebfaq.jsp
Siano, Donald. The Algorithmic Beauty of Trebuchets. http://members.home.net/dimona/
Valens E.G., Motion, Longman Group Ltd, London, 1965