The more and more I read about the physics of a trebuchet, the more I realize how little I actually know about the incredibly complex dynamics and the forces involved in accelerating an eight to ten pound object. If the motion of the machine were not so damn quick, I think we would all have a better intuitive understanding and a deeper fundamental understanding.
I have spent the last two months with two treb simulators (WinTreb and A-Treb). These simulators are a great aide in developing a good understanding of the "split second" motion of these machines. The ability to slow down time is invaluable to see what is actually occurring. What one thinks ones "sees" in real time, is often quite different from reality.
These simulators appear to be quite accurate. We have inputted the Yankee Siege parameters into these simulators and they have predicted very close to our actual distances thrown (within + or - 2%). As everyone knows, these machines must obey the laws of physics. (Although sometimes they appear to defy these laws).
Variations in the distances from throw to throw appear to be related to the time of release. (Excluding, of course, wind). There is a very small window of time in which to release for maximum distance. Plus or minus one tenth of a second can make a huge difference. (Small machines are even more critical - in a small machine everything happens quicker, making timing more difficult).
I've had to use these two simulators to explore as many ways as I can, to redesign Yankee Siege. One of the limitations of A-Treb is that it does not allow very high counterweight to missile ratios. Yankee Siege is now running a ratio of 1400 to one, (14,000 pound counterweight to 10 pound pumpkin). WinTreb will allow higher counterweight to missile ratios.
I do like A-Treb's graphs of forces on the machine. One very interesting note is that Yankee Siege has a reaction force on the axle of 100,000 pounds! (The force on the axle experienced when the counterweight reaches bottom).
A-Treb's graphs of acceleration and velocity of the throwing arm, really help one understand what happens as the counterweight "bottoms out" at the lower part of it's travel, causing the throwing arm to greatly accelerate. It also helps one visualize a transfer of energy from counterweight to throwing arm to sling. Each part of the machine will momentarily "stall" and as a consequence of the principle of conservation of energy, that energy present in the one moving part will have to be transmitted to another part of the machine (conservation of energy says that throughout the cycle the total energy of the machine will remain constant). Energy is conserved - you can't get rid of it! If one part of the machine "stalls" (no kinetic energy), then another part of the machine must possess the rest of the energy (either in kinetic or potential energy). Yankee Siege, for example, has 247,000 joules of energy stored in its counterweight. (14,000 pounds raised 12 feet). Throughout the cycle, that entire 247,000 joules must be accounted for. A-Treb has some nice graphs that show the relative partitioning of energy between the counterweight throwing arm and missile throughout the cycle.
In addition, angular momentum is conserved. As the sling and projectile swing around (due to centripetal acceleration) and finally overcomes the throwing arm, there is a dramatic increase in the moment of inertia of the beam/projectile combination. The projectile, as a consequence of its change in radius relative to the axle, "steals" some of the angular momentum and energy from the throwing arm, causing a slowing of the angular velocity of the throwing arm with a dramatic increase in the tangential velocity of the projectile. (Analogous to the skater who slows her spin by extending her arms).
As they say, you don't have know how a car works to drive it. But, if you want to know how to design a better car then you had better know every detail of how it functions. Yankee Siege is a "dumb design". We win only by brute force. It's time to re-design in a "smart way". We have to find a new way to more efficiently transmit a greater portion of the 247,000 joules of energy available to the projectile.
The single biggest "consumer" of energy is the throwing arm. Our throwing arm is a tapered steel built up I-beam weighing 2600 pounds! It takes a lot of energy to rotate such a large mass. Yankee Siege was designed to throw larger (300 pound) projectiles. Yankee Siege is very inefficient at throwing small (10 pound) projectiles. If we can design a less massive throwing arm, that won't break, we could throw significantly farther. Simulations with WinTreb show that with significant reduction of the throwing arm mass we could be throwing over "4000 feet"! Only one problem, we only have 2500 feet at our test site! 4000 feet seems almost too good to be true. Could the simulator be wrong? Where would we practice? Could a pumpkin take the g-force? The goal of the trebuchet division should be to beat the air cannons. Could such a machine be built? Should I wake up and smell the coffee?
Perhaps we should be a little bit more conservative and not go for the "ultimate machine" and aim for a more realistic 2500 foot machine. This would not require re-designing the whole machine but perhaps re-designing only a portion of the throwing arm. We have cut our existing long end of the throwing arm off about three feet from the axle and will be bolting on a new long end.
The new long end of the arm will be fabbed from ASTM A-572 steel, 3/16 inch thick plate, starting with a cross section of 12 inches by 12 inches and tapering to 3 inches by 3 inches over 23 feet. This will give a total weight of approximately 500 pounds. (Compared to over one thousand pounds for the old arm). 500 pounds is a huge savings in weight!
A note about A-572, grade 50, steel: this is a nobidium/vanadium low alloy structural steel with a minimum yield strength of 50,000 psi. Compare this to A-36 (mild steel) with a minimum yield strength of 36,000 psi. It (A-572) is easily cut and welded with no reduction in strength. (This is not a heat treated steel and will not degrade in strength in the heat affected zone of the weld).
The last nine feet of the tip of the arm will be a carbon fiber cantilever (clad with Kevlar for impact resistance). The carbon fiber tip be a "bolt on item", with the option of going to an aluminum 9 foot extension if the carbon fiber should fail. The carbon fiber will weigh less than 10 pounds! (A savings of over 35 pounds over the old extension).
We have decided on this "hybrid" arm for several reasons.
First, a total carbon fiber arm would be prohibitively expensive and perhaps not as forgiving as steel with regards to impact resistance. (When carbon fiber fails, it fails catastrophically without warning).
Second, steel, although heavy, is very stiff and easily fabricated and repaired in the field. An A-572 tapered steel arm, would make a relatively light strong arm. I have worked the numbers for a cabled stayed steel arm and came out with very little weight savings over a cantilever, with a lot more complexity. We have decided to keep it simple, with a pure cantilever. (I like the "looks" better, also).
Third, the part of the arm closest to the tip is the most important area of the arm to keep light. (Moment of inertia increases with the radius squared). It takes a lot of energy to rotate a mass that is a great distant from the axle. It is absolutely critical that the last nine feet of the throw arm be extremely light and stiff. What better way to solve this problem than to use a material that has the highest strength to weight ratio of all common structural materials (carbon fiber). We have decided to put the most effort where it counts the most.
Fourth, we have decided not to go with wood because of maintenance problems and reliability.
Fifth, aluminum: Yankee Siege is seriously considering going to an aluminum arm. Aluminum has a more favorable strength to weight ratio. A 23 foot extension to Yankee Siege's base arm could be fabbed out of 1/4 inch 7075-T6 aluminum plate with a yield strength of 75,000 psi! The main drawback of aluminum is that it can't be welded without a significant reduction in strength (40 percent reduction). Fabrication would require bolted connections (austenitic stainless steel bolts). Now I finally understand why airplane wings are riveted, not welded! The heat of welding destroys the tempering of the aluminum. There would be significant reduction in the weight of the 23 foot arm. (From 1000 pounds to 330 pounds, a 670 pound savings)! Aluminum is a material to seriously consider.
Nim-Cor Company in Nashua, NH. Phone 1(888)464-6267 will be fabbing the 9 foot carbon fiber extension. They have a carbon fiber automated filament winding mandrel system with the epoxy cured in an autoclave under pressure. Nice people to talk to, very helpful. They can only fab a uniform (prismatic) beam (no hand lay-up or tapering). They suggested wrapping the arm in Kevlar to give the arm some impact resistance. (An errant steel slip ring on the end of the sling could destroy the arm).
We will start gathering materials and start fabbing as soon as good weather comes.
Last but not least, perhaps the hardest part of the whole project is telling your wife how expensive the new carbon fiber extension will be!
I would appreciate your feedback! If you think this line of reasoning to totally foolish I would like to know. Do you have any suggestions? I would like to have an open discuss. We are all facing similar problems with strength and weight.
The Yankee Siege team is very excited over the new arm. Will it work? Failure would be o.k. Just not in competition! Looking forward to your feedback.
P.S. Next posting mid-May
Steve Seigars, YS