What are the conditions necessary for the peak efficiency of a trebuchet?

1. What are the conditions necessary for the peak efficiency of a trebuchet? Matt Keller, Chase Kernan, Calvin Ward, and Simon Monley

2. Background Information The counterweight trebuchet was the most advanced piece of siege artillery in the world of c.1200. It has been estimated that a counterweight trebuchet from c.1200 could propel a 14.96 kg casting-stone about 200 meters, which was much more effective in mass destruction as compared to the previously used bow and arrow, which ranged 140 meters max and had no wrecking force at all. Trebuchets were operated by as many as 25 people at a time (due to how heavy the counterweight could be and the difficulty to restrain the arm from firing). The trebuchet was utilized to throw missiles other than casting-stones such as rotting carcasses (an primitive form of biological warfare). They were employed by Saladin during his victorious campaigns of 1187 and 1188, but the Malkum sultan Baybars (1260-77) gained the maximum advantage from them by carrying the ready to assemble materials for the trebuchet for speedy construction when laying siege to a castle. The trebuchet of those days used a counterweight on the back of a long arm with a sling on the end that would hold the object that was being fired. The sling in which the projectiles were placed added to the velocity with which they were flung at a high arc. The Trebuchet was not only more powerful than all previous mechanisms but also more accurate, due to the fact that the range could be changed by changing the counterweight mass and the pivot length could be easily altered. The introduction of the counterweight trebuchet shifted the balance of castle siege in favor of the besieger.

3. Materials

4. Procedures 1. In order to build our trebuchet first buy several pieces of wood, 4 1”x1”x8’, 2’x2”x4’, and a .5”x31”x31”. Also gather two metal threaded rods, one that has a 1/4th inch radius and one with a 1/8th inch radius each measuring 20”. 2. Follow the blueprints below which we found online using a scale where 1/8th of an inch on the picture was approximately 1 inch. 3. First Build the arm of the trebuchet 4. Start by measuring all the pieces, 5. Cut the longest part of the arm and the shorter 2 wooden pieces and finally the smaller 3 blocks. 6. Bolt the longest pieces together as illustrated below. The sling will be the last component you construct. 7. Using a hand held drill and a drill press drill the holes through where you can place the axel for the counterweight and the arm.

5. Procedures Continued 8. Next build the base. This is probably the most difficult part of the trebuchet that you will build. 9. Start by building the structure shown below in a 2-dimensional form. 10. Lay a .5 inch thick board that is 31 inches between the two vertical beams and screw it in. 11. Lay 2 10 inch 1”x1” beams under each side of the board and screw them on. 12. Cut the support beams and screw them into each side of the horizontal board and to the top of the vertical beams. 13. Add two pieces of trim on their sides to the board that is lying across parallel to the horizontal, this creates an ally which the sling can travel down and not catch on anything. 14. Notice that we did not build the cross beams to support the support beams.

6. Procedures Continued 15. After this Construct the box for the counterweight using the following parameters and a board that is .5 inches thick. 16. Drill a hole in the top of the base structure for the axel that would hold the firing arm. 17. Line up the holes in the axel and the frame and stuck a rod through the holes and measured how long we needed to cut the rod. 18. Cut the rod using a special metal saw, put the rod into the wholes and screwed on the nuts. 19. Repeat step 15 with the axel that connects the counterweight to the throwing arm only connecting the counterwight bucket to the throwing arm. 20. Use a metal grinder to sand down the ends of the rods so that they did not cut anyone. 21. The last step was to build the sling.

7. Procedures Continued 22. Cut two peaces of string 20 inches long and a third piece of string 45 inches long. 23. Cut a piece of denim 4”x8” and cut a hole 1 cm from each corner. 24. Hammer a nail into the top of the throwing arm angled in the direction in which we wanted the projectile to eventually go. 25. Securely tie the two 20 inch strings to the nail. And loosely fasten the 45 inch string to the end of the nail. 26. Tie the 20 inch strings to the holes on one side of the so that they are approximately 3” apart. 27. Thread the 45” string through the other two holes so that it doesn’t scrunch the fabric and then make it so the length protruding from each whole is even. 28. Loosely tie a the two strings together at the end, then 2 inches above that tie the strings together again creating a loop which can fit around the nail

8. Procedures Continued 29. Load the counterweight with a specified amount of weight, and have the arm secured so that it will not fire 30. Place a golf ball in the middle of the denim and fold the denim around it so that the golf ball is between two sheets of denim with someone holding it. 31. Clear firing area 32. Release 33. Measure to the point where the golf ball hits the ground. 34. Repeat steps 30-33 four times with the same amount of weight 35.Repeat steps 29-34 until you have the desired variations of counterweight masses.

9. Blueprints

10. Cutting

11. Drill Press

12. The Group and the Finished product

13. Touching Up

14. Trials

15. The Motions

16. More Trials

17. Terms and Definitions Used • Time- The amount of time from the moment the projectile is launched to when it first hits the ground • Distance- The distance from the middle of the trebuchet to the point at which the projectile first hits the ground

18. Results

19. Results cont.

20. Graphs

21. Graphs cont.

22. Graphs cont.

23. The Math

24. Conclusion • In our investigation of the efficiency of a trebuchet, we found the peak efficiency to be 28.1% (with an uncertainty of 51.7%) at a counter-weight mass of 9.98 kg. With a theoretical input of 29.81 J from the falling counter-weight, the trebuchet launched the projectile at 19.83 m/s, giving an output of 8.38 J. However, the efficiency quickly declined as more weight was added, leading to an efficiency of only 13.6% at a counter-weight mass of 22.91 kg

25. Conclusion cont. • A similar pattern occurred with the distance of the projectile. While the trebuchet did throw the projectile a maximum of 38.40 m at the highest counter-weight mass, the rate of change in distance gained per mass of counter-weight added was maximized at the same counter-weight mass, 9.98 kg. For every kg added, approximately 2.57 m of distance was gained. This was almost triple the value (0.92 m/kg) at 17.01 kg. An interesting correlation occurred here between the launch angle and the rate of change. An optimum angle would be 45ｰ (see side note). The peak rate of change of distance occurred at the peak rate of change of angle. At 9.98 kg, the launch angle was only 18.6ｰ but it was increasing at a rate of 9.20ｰ/kg. The angle at the lowest rate of change of distance was 41.97ｰ and only increasing at a rate of 0.93ｰ/kg. So while the efficiency was optimized at 9.98 kg, the angle of fire approached the optimum value with much higher counter-weight masses. This means that, while the extra weight did decrease the efficiency, the best distance for the given amount of output energy was achieved at the highest counter-weight masses.

26. Conclusion cont. • By extrapolating the distance graph, we found that a counter-weight mass of kg is required to overcome the forces of friction and actually launch the projectile. These high values of friction likely arose from the contact between the wood and the very rough, ridge-covered metal screws that were used as pivots on the trebuchet. This can serve as a possible explanation of the decrease in efficiency at higher counter-weight masses. The extra weight bent the metal rod and thus drastically increased the friction around the pivot. At very high masses this was especially evident as the rod bent several centimeters. This not only increased the friction but also made the counter-weight bucket move more erratically and would at times hit the sides of the trebuchet. All of this would contribute to energy losses and therefore decrease the efficiency of the catapult.

27. Conclusion cont. • We recorded the highest efficiencies in the range of 10 kg, but looking at the graph of efficiency, it appears that an even higher efficiency would be achieved with a counter-weight mass less that 9.98 kg. The graph of efficiency would have an x-intercept at kg when the trebuchet no longer fires, and the graph is falling from a maximum point at 9.98 kg. Thus the maximum has to lie between kg and 9.98 kg and the shape of the graph indicates it would lie around kg. We therefore reached the conclusion that the conditions necessary for peak efficiency was a counter-weight mass of around kg. However, if a maximum distance is desired, a much higher counter-weight mass should be used.

28. Error Analysis • There is no literature value for our specific catapult and so an exact error value is impossible to find. However, our uncertainty values (as high as 54%) show that there is severe variations in our data. One of the most prominent sources of error was our method for recording the distance the projectile traveled. First, we had to approximate where the projectile first landed and then, since our tape measure was only 25 ft long, we had to use a series of cones to mark off distances greater than 25 ft. Since the projectile never went exactly in the intended direction, the cones measured a distance that wasn't exactly parallel to the actual path of the projectile. This would have lead to distances that were shorter than the actual distance traveled.

29. Error Analysis cont. • Another possible source of error is our method of timing the flight of the projectile. Besides just the human error in reaction time, it was often difficult to determine the exact instant that the projectile left the trebuchet due to its high rotational velocity. Combined with very small values, and a low-precision stopwatch, the uncertainty values for the time recorded (max. 46.2%) were primary cause of the high overall uncertainty values

30. Error Analysis cont. Finally, the sling design itself had to be refined several times. In some of our earlier designs, the flight path would be too unpredictable to record relevant data. While our final design was still slightly unpredictable, it was a vast improvement over our initial designs. It incorporated a releasable sling so that the projectile would be released at roughly the same location every time (however, as stated earlier, this depended on the counter-weight mass). One end of the sling was attached to the beam, while the other was looped over a nail sticking out of the end of the beam. So when the beam reached the appropriate angle, that end would slide off and release the projectile (see diagram below).

31. Bibliography • Author Unkown. Index of Ingenium pics. Sept. 6 2005. Treb1.gif, Treb2gif, Treb3.gif, Treb4.gif, Treb5.gif http://www.tasigh.org/ingenium/pics/ • Wicked How-To’s. How To Build A Catapult / Trebuchet : Plans and Instructions. August 17th 2007. http://wickedhowtos.com/index.php/2007/08/17/how-to-build-a-catapult/ • Wayne Cambell, Hila science camp. The Hila Trebuchet. Parts and Overview, Build the counterweight basket, Attach uprights and build base, Construct the sling, Final assembly, Using your trebuchet, Video discussing the science behind this trebuchet. http://hila.webcentre.ca/projects/trebuchet/index.htm • Wikipedia. Trebuchet. December 3, 2008. http://en.wikipedia.org/wiki/Trebuchet • Trebuchetstore.com. History and Mechanincs of the Trebuchet. Trebuchetstore.com. Date modified unknown. http://www.redstoneprojects.com/trebuchetstore/trebuchet_history.html • Trebuchet. Glass Giant. http://www.glassgiant.com/geek/trebuchet/ • Medievil-castle-seige-weapons. Trebuchet Blueprint- how to build your own siege engine. 2008. http://www.medieval-castle-siege-weapons.com/trebuchet-blue-print.html • Medievil-castle-seige-weapons. Trebuchet Physics. 2008. http://www.medieval-castle-siege-weapons.com/trebuchet-physics.html • Connor Gerdes et al at Rose-Hulman Institute of Technology. Trebuchet. July 26, 2008. http://www.rose-hulman.edu/Catapult2008II/reports/group26.pdf • Wikidhowtos.com. How to build a catapult. August 17, 2007. http://wickedhowtos.com/index.php/2007/08/17/how-to-build-a-catapult/

32. Bibliography cont. • Wayne Campbell. The Hila Trebuchet. http://hila.webcentre.ca/projects/trebuchet/index.htm • MonsterGuide.net. How to build a trebuchet. http://www.monsterguide.net/how-to-build-a-trebuchet.shtml