# The Algorithmic Beauty of the Trebuchet

On the physics and mechanics of a medieval trebuchet, a type of catapult, its design, working and optimization through improved simulation and mathematical analysis.

### Trebuchet Mechanics.

is a free manuscript that provides an extensive description of the mechanics of trebuchets. It starts with elementary black-box and "see-saw" models, and builds gradually to the full model that includes a sliding, constrained sling. Geometry, physics, detailed derivation of equations. Energetics; definition of efficiency; tips on design; analysis of the finger mechanism; a monte carlo design method. Plenty of material for the mathematically inclined, but also lots of insight for the guy that just wants tips on efficient designs and to explore how to make his design perform efficiently.

# TrebStar Simulator Application

Trebstar is a free app that allows you to compute the range and range efficiency for your design. Input the parameters for the treb, and watch the animation. It allows propped, hinged counterweights, includes the constrained, sliding sling, and gives the range, range efficiency of the design. Various plots and tables allow one to examine the effect of variations in the trebuchet design on the efficiency, range, and forces exerted on the parts of the mechanism.

## Some Features of Trebstar Simulator

1. An optimum sling length search for the trebuchet is included, greatly easing your search for an optimum design.
2. A CW diagram plots the position of the CW as a function of time. How straight is the fall of the CW? Does it matter?
3. Range efficiencies with and without the beam as an energy source are both shown.
4. Trebstar provides a table of 15 variables as a function of time, enabling you to make up your own plots of just about any combination of variables. It gives the kinetic and potential energies of the parts as a function of time and as a function of beam rotational velocity, etc.
5. The maximum force on the axle normalized to the weight of the CW is shown, to be used in calculating a safe diameter of the axle.
6. Nonuniform beams are handled by input of the radius of gyration and the center of mass of the beam.
7. Defaults for the parameters may be saved, so variations on a nominal design can be readily made.
8. One can more systematically explore the parameters by using a text input file to generate multiple runs, and save the results to a text file.
9. It has lots of help files attached. Read the help files!

TrebStar should be particularly useful for those elementary physics/science classes that are studying energies and forces. Plots of these as a function of time or configuration parameters can be used as a basis for a variety of projects in the classroom. What things other than range efficiency can be used to determine how good a design is?

### The Free Macintosh System X, intel Version of Trebstar

For System 10.6, thru 10.12 (Snow Leopard to Sierra)

This is a .zip file, 2 MB.

### The Free PC, Windows 10 Version of Trebstar

This is a .zip file, 3.1 MB.

### Warnings:

Both of these versions will warn you that you are downloading an application from the internet, which can be dangerous because of malware and viruses. I have checked both versions for viruses, and perhaps you should too. These warnings can be easily skirted in both platforms. On the Mac, go to the System’s preferences/security and privacy file and check the box allowing the download to proceed. There is something similar on the PC side. You may have to deal with similar warnings from your antiVirus application.

### A Bit of History about Trebstar

I was inspired to work on the simulation of the trebuchet by the article in July 1995 of Scientific American, back when it was a pretty decent publication. As I recall, it stated that a team of men at the Naval Academy had tried to write one, but achieved only partial success.

My work started with a desire to learn Mathematica by applying it to this problem. This eventually succeeded by using Lagrangian mechanics to derive the three differential equations of motion for both parts of the movement (the initial sliding phase, and the movement of projectile at the end of the sling). The equations were long and complicated, but were easy to transfer to other programming languages and solved by conventional numerical methods. Solutions using Fortran, Javascript and even Visual Basic soon followed.

Eventually I determined to wrap it around a user friendly interface using what was then called RealBasic, a new language at the time, published by Real Software. The experience of learning how to write an interface provided a pleasant and rewarding experience, and I soon had one good enough and by 1999 eventually took on the form that is available today on this website. Although a few competitors eventually emerged, TrebStar was the first.

Over the 18 years since that time, many changes occurred in the internet with various browsers coming and going, and the constantly updated and changing generations of computer chips. The professionals at RealStudio did an outstanding job of making the maintaining the code smooth for me, for which I will always be grateful. For about 18 years I sold it for a nominal sum (\$15) and had many satisfied customers. The company and language have changed over the years, and they are now known by Xojo, and they continue to offer a great product.

The message is that the code is reliable and useable by hurlers of all ages and determinations. The champions at Punkin Chunkin, boy-scouts and hobbyists, and whole classrooms of students got some enjoyment out of it, and I am proud to now offer it to the world for free. Happy hurling!

### Thermodynamics of the Trebuchet

The first law of thermodynamics of the trebuchet is that the efficiency of the mechanism, whatever its configuration, cannot be greater than 100%. The corollary is that you can’t do better than putting all the potential energy of the counterweight into the kinetic energy of the projectile, releasing it at the optimal angle for range. Thus, as shown in the Manuscript, “Trebuchet Mechanics,” the greatest range possible is Rm = 2*(m1/m2) h, where h is the distance the counterweight of mass m1 falls, and the mass of the projectile is m2.

The second law of thermodynamics of the trebuchet says you can’t even get really close to that ideal limit. The real world imposes friction and losses of various kinds, so the efficiency of the machine will be something significantly less than 100%. it will be, in the best of worlds, something like 60 to 80% or so. This means the treb you build will go R ≈ 0.7 * 2*(m1/m2) h.

The third law of thermodynamics says that there is the state at the end where all motion ceases and there is no energy to extract. Obviously, this is at the end of the throw, where the beam is pointed straight up, and the counterweight is closest to the ground. Hopefully, your design will have it a little above the ground!

### Will It Break?

"Will It Break?" is a manuscript for serious trebuchet builders that will help you to design and construct your dream. It provides methods for understanding and predicting the forces and stresses in your design during the hurl. Includes easy-to-use formulas for choosing dimensions and materials for all of the parts. Is the axle diameter large enough? Is the CW box strong enough? Will the beam break? How about the sling and the sling release prong? Forces tending to break the truss.

It includes a discussion of the forces acting on all the parts, their handy scaling rules, and many completely worked examples. Provides easy guidance for the novice, and leads for the experts. Tips to improve the safety of your operation. This is a sequel to my popular, free "Trebuchet Mechanics" manuscript, and has a more "practical" emphasis. 42 pages, 21 figures.

Will It Break?.

### Other material about Trebuchet Mechanics

• An animation
• A mathematica program
• A trebuchet javascript calculator
• A fortran montecarlo simulator

## My Own Trebuchet!

This babie throws bowling balls. Over the trees!

It works!

### Onager Physics

"Onager Physics?" is another free manuscript that analyzes the physics of the Onager (Mangonel). It concentrates on the properties of the rope bundle, or skein, that generates the torque on the beam, but has some more general remarks on the scaling properties, efficiency and energetics as well. Download the file Onager Physics.

This site was created by Donald B. Siano