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
Yankee Siege Latest
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.
Tuesday, December 25, 2007
Late December 2007
Spent most of the time this weekend pouring over books on beam theory and strength of materials. I found two great books by J.E. Gordon written in the 1960 & 70's entitled "Strength of Materials and Structures" and "The New Science of Strong Materials". These books are essential reading for anybody interested in designing any structure. Mr. Gordon has quite a talent of clearly and concisely explaining the theories of bending and elasticity. These books are quite readable and anyone with a basic understanding of algebra and geometry can easily digest. These are great books for the layman and are a good introduction to further engineering studies. After reading these books, you are left with a more intuitive understanding of why materials behave as they do.
As I am wandering through these books, I am constantly thinking of how these principles apply to a trebuchet. There is a good explanation on the strength of a material and, its elasticity and its density and how that relates to its strength to weight ratio.
Mr. Gordon points out that the specific modulus of elasticity (that is its stiffness per weight) is virtually the same for all the common structural materials. (With the possible exception of carbon fiber composite). This explains why there are several choices of materials to construct a throwing arm with no clear cut winner in stiffness to weight. Whether one uses steel, aluminum, wood, fiberglass, titanium, etc... there is not a lot of differences.
I want to make one important distinction between strength and stiffness, they are not the same! When it comes to strength and weight there is quite a difference from material to material. Titanium is a clear winner of all the metals. Carbon fiber is a clear winner of all the non-metals in regards to strength to weight.
That comes to the next question. What is more important in designing a throwing arm? Is stiffness or strength more important? The obvious answer is that both are important. The throwing arm must be strong enough not to break and at the same time be stiff enough not to deflect too much on the pull down as to produce too much of a recoil upon release of the trigger.
Note: Yankee Siege has developed an annoying recoil of the arm upon release of the trigger. We have added so much weight to the counter weight that upon release, the throwing arm snaps back and creates a "hop" of the pumpkin off the ground and a subsequent oscillation of the arm that leads to a quite unpredictable release of the projectile. We are trying to time the release of the projectile to an arm that is oscillating in its own mode of frequency.
This brings home the point of the difference between strength and stiffness. Stiffness relates to elasticity. A throwing arm may be quite strong and at the same time quite flexible (elastic), leading to greater deflections when cocked. (Note: an increase in the cocking angle will cause less deflection, with no deflection if cocked to 90 degrees).
For example: two identical size throwing arms made out of different materials will exhibit different amounts of deflection even if they had the same tensile strength. Mild steel and some aluminum alloys have identical tensile strength but aluminium will deflect three times as much upon pull down (aluminium has one third the elasticity of steel).
If deflection is deemed an important criteria, then choice of an appropriate modulus of elasticity is of critical importance.
Yankee Siege is trying to figure out a way to take advantage of the elasticity of the beam and figure out the frequency of the elastic oscillation and to take advantage of the "forward snap of the arm".
Perhaps a very elastic beam would be a help not a hindrance! If we could only time it right and design an arm to oscillate at the right frequency so that the release would coincide with the forward "snap". Cellulose is very elastic-maybe Trebarbaric knows something we don't! I'm still looking for the ultimate material!
A simpler approach would be to design an arm that is very stiff (steel would be a good choice as it has the highest modulus of elasticity of all the common structural materials). A very stiff arm would deflect very little and would not be a nightmare to time the release because the deflections are insignificant (they would be very small but high frequency).
We are still in the design phase for the new throwing arm. We have not made any final decisions as to the materials or combination of materials or configuration. I have now finished re-reading Gordon's books for the third time and I am still learning something new each time.
As you can probably tell from these musings, the Yankee Siege team is somewhat obsessed with designing an energy efficient arm. I spend the majority of my waking hours thinking and reading and imagining. I keep thinking that maybe if read just one more book, one more page, one more article that the ultimate design will suddenly become obvious.
In reality, there are probably a dozen ways to construct an efficient arm. We may end up with two or three different arms that can be bolted on and see which one works best.
I think the hardest part of the whole design process is making a final decision! Committing to a final design means that you have said to yourself that this is the best design that I can come up with at this time, given my level of knowledge at this time. Perhaps I should wait a little more time, gain some more knowledge, and redesign. At what point do you say enough is enough. At what point do you pull the trigger and let gravity take over!
I'll try to keep everyone posted monthly. I enjoy hearing from you all. Yankee Siege team would like to hear your feedback. If you think we are proposing some utterly foolish idea, we would like to hear from you. I hope some of these discussions will help other teams, as well as Yankee Siege. I would especially like to hear from Pumpkin Hammer, Trebarbaric and Merlin. I think that we should keep in touch throughout the year. I call these teams, members of the "1000 Foot Club". We are the members of an exclusive club that have broken the 1000 foot mark (no easy accomplishment). I hope I haven't left anyone out. Anyone may join the club-you just have to throw more than a thousand feet in any competition. By the way, Yankee Siege fully recognizes Trebarbaric's throw of 1866 feet!
Yankee Siege hopes to start a new club in November, (the 2000 Foot Club)! Wishful thinking!!! Hope to have several new members!
Steve Seigars, YS
As I am wandering through these books, I am constantly thinking of how these principles apply to a trebuchet. There is a good explanation on the strength of a material and, its elasticity and its density and how that relates to its strength to weight ratio.
Mr. Gordon points out that the specific modulus of elasticity (that is its stiffness per weight) is virtually the same for all the common structural materials. (With the possible exception of carbon fiber composite). This explains why there are several choices of materials to construct a throwing arm with no clear cut winner in stiffness to weight. Whether one uses steel, aluminum, wood, fiberglass, titanium, etc... there is not a lot of differences.
I want to make one important distinction between strength and stiffness, they are not the same! When it comes to strength and weight there is quite a difference from material to material. Titanium is a clear winner of all the metals. Carbon fiber is a clear winner of all the non-metals in regards to strength to weight.
That comes to the next question. What is more important in designing a throwing arm? Is stiffness or strength more important? The obvious answer is that both are important. The throwing arm must be strong enough not to break and at the same time be stiff enough not to deflect too much on the pull down as to produce too much of a recoil upon release of the trigger.
Note: Yankee Siege has developed an annoying recoil of the arm upon release of the trigger. We have added so much weight to the counter weight that upon release, the throwing arm snaps back and creates a "hop" of the pumpkin off the ground and a subsequent oscillation of the arm that leads to a quite unpredictable release of the projectile. We are trying to time the release of the projectile to an arm that is oscillating in its own mode of frequency.
This brings home the point of the difference between strength and stiffness. Stiffness relates to elasticity. A throwing arm may be quite strong and at the same time quite flexible (elastic), leading to greater deflections when cocked. (Note: an increase in the cocking angle will cause less deflection, with no deflection if cocked to 90 degrees).
For example: two identical size throwing arms made out of different materials will exhibit different amounts of deflection even if they had the same tensile strength. Mild steel and some aluminum alloys have identical tensile strength but aluminium will deflect three times as much upon pull down (aluminium has one third the elasticity of steel).
If deflection is deemed an important criteria, then choice of an appropriate modulus of elasticity is of critical importance.
Yankee Siege is trying to figure out a way to take advantage of the elasticity of the beam and figure out the frequency of the elastic oscillation and to take advantage of the "forward snap of the arm".
Perhaps a very elastic beam would be a help not a hindrance! If we could only time it right and design an arm to oscillate at the right frequency so that the release would coincide with the forward "snap". Cellulose is very elastic-maybe Trebarbaric knows something we don't! I'm still looking for the ultimate material!
A simpler approach would be to design an arm that is very stiff (steel would be a good choice as it has the highest modulus of elasticity of all the common structural materials). A very stiff arm would deflect very little and would not be a nightmare to time the release because the deflections are insignificant (they would be very small but high frequency).
We are still in the design phase for the new throwing arm. We have not made any final decisions as to the materials or combination of materials or configuration. I have now finished re-reading Gordon's books for the third time and I am still learning something new each time.
As you can probably tell from these musings, the Yankee Siege team is somewhat obsessed with designing an energy efficient arm. I spend the majority of my waking hours thinking and reading and imagining. I keep thinking that maybe if read just one more book, one more page, one more article that the ultimate design will suddenly become obvious.
In reality, there are probably a dozen ways to construct an efficient arm. We may end up with two or three different arms that can be bolted on and see which one works best.
I think the hardest part of the whole design process is making a final decision! Committing to a final design means that you have said to yourself that this is the best design that I can come up with at this time, given my level of knowledge at this time. Perhaps I should wait a little more time, gain some more knowledge, and redesign. At what point do you say enough is enough. At what point do you pull the trigger and let gravity take over!
I'll try to keep everyone posted monthly. I enjoy hearing from you all. Yankee Siege team would like to hear your feedback. If you think we are proposing some utterly foolish idea, we would like to hear from you. I hope some of these discussions will help other teams, as well as Yankee Siege. I would especially like to hear from Pumpkin Hammer, Trebarbaric and Merlin. I think that we should keep in touch throughout the year. I call these teams, members of the "1000 Foot Club". We are the members of an exclusive club that have broken the 1000 foot mark (no easy accomplishment). I hope I haven't left anyone out. Anyone may join the club-you just have to throw more than a thousand feet in any competition. By the way, Yankee Siege fully recognizes Trebarbaric's throw of 1866 feet!
Yankee Siege hopes to start a new club in November, (the 2000 Foot Club)! Wishful thinking!!! Hope to have several new members!
Steve Seigars, YS
Subscribe to:
Posts (Atom)
