I found a great article on the physics of a trebuchet. It is entitled "Siege Engine Dynamics" written by Mark Denny and published in the 2005, European Journal of Physics pages 561-577. It can be downloaded for a fee or if you have access to a major university library it can be downloaded for free. This paper is a bit hard to digest, unless you have a strong math background, it may not be worth reading. Some of the symbols are not very clearly defined, but by reading it several times most readers would be able to decipher the meaning. (I had to read it about five times before understanding it fully).
I have spent the past two months trying to design a new throwing arm for Yankee Siege. The team has narrowed down choices of shapes and materials to three or four different materials/configurations. (More about this later).
Before going ahead with a new lighter throwing arm, I first of all, had to determine if there would be any benefit to a lighter throwing arm. How far is Yankee Siege away from the "speed limit of a trebuchet"? I call this the SPLAT law. What is the SPLAT law? There is an upper limit of speed that no trebuchet can exceed (catapults do not have this limit!). This limit, is the speed due to the acceleration of gravity. The center of gravity of the counter weight can NEVER fall faster than the speed due to the acceleration of gravity (32 feet per second per second). This is the speed of a free falling body. The counter weight will, of course, always fall a little bit slower than a free falling body because it has to overcome the inertia of the throwing arm.
IF Yankee Siege has added so much weight to the counter weight that the counter weight speed is very close the speed of a free falling body, then making a lighter arm will only have a very slight marginal effect on the throwing arm angular velocity (speed of rotation) and therefore, will not throw significantly further. I am still searching for that mass less throwing arm which would offer no resistance to rotation and would allow the counter weight to free fall and therefore attain the maximum angular velocity.
The other major factor that limits the angular velocity (speed of rotation of the throwing arm) is the length of the short end of the throwing arm. The greater the length of the short end (counter weight end) of the throwing arm, the slower the maximum limit of the angular velocity (rotation) of the throwing arm. (It takes longer for the counter weight to drop because it has a longer distant to fall). Another way to look at this is if you increase the length of the short end to infinity, then the counter weight would have an infinite height and would never complete its fall and therefore the throwing arm would rotate infinitely slow. This is the main reason why large scale trebs have relatively slow moving (rotating) throwing arms. Their angular velocity is limited by the SPLAT law and the SPLAT law is reached at a lower angular velocity. Conversely, a small scale treb with a very small, short end of the arm, can rotate very quickly because the counter weight doesn't have very far to drop and can complete the cycle very quickly.
Yankee Siege has a 12 foot drop of the counter weight. If we are close to the ultimate speed limit, the counter weight should take .86 seconds to fall. (The time it takes a free falling body to fall 12 feet). Yankee Siege takes a little over 1 second to fall 12 feet (timed with a stop watch off of Pumpkin Hammer's website showing Yankee Siege firing). Thank you Pumpkin Hammer-by the way you have a great website.
The other consideration is the fact that the counter weight does not drop straight down. And we would expect it take a little longer to drop if it doesn't go straight down but takes another (non-vertical) path. We may be even closer to the limit than we originally thought.
The conclusion from this discussion is that Yankee Siege is approaching the speed limit and that making a lighter arm will help but not dramatically increase the angular speed of the arm because we are so close to free fall.
More on the new throwing arm details next month. Hope to hear you feed back!
Steve Seigars, YS
Sunday, January 20, 2008
I found a great article on the physics of a trebuchet. It is entitled "Siege Engine Dynamics" written by Mark Denny and published in the 2005, European Journal of Physics pages 561-577. It can be downloaded for a fee or if you have access to a major university library it can be downloaded for free.