What are the conditions necessary for the peak

efficiency of a trebuchet?

Matt Keller, Chase Kernan, Calvin Ward, and Simon Monley

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.

Materials

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 trebuchet4. 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.

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.

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.

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

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.

Blueprints

Cutting

Drill Press

The Group and the Finished product

Touching Up

Trials

The Motions

More Trials

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

0 = Vy

t _ 4 . 9 t

2

0 = t _ Vy

_ 4 . 9 t _

Vy

= 4 . 9 t

Vx

= d i s t a n c e / t i m e _ = a r c t a n _

Vx

Vy

_

Angle – Speed

_ V _ = s p e e d = _ Vx

2

_ Vy

2

Work Done

w o r k =

1

2

m V

2

where m is the mass of the projectile.

Theoretical Work w o r k = 9 . 8 m h where m is the mass of the counter-weight and h is the maximum distance the counter-weight falls

Efficiency

e f f =

w o r k d o n e

t h e o r e t i c a l w o r k

_ 1 0 0

ResultsMass Counter Weight (± 0.01kg) Trial

9.97903214 1 2 3 4 5 Average Uncert.Distance (±0.1 m) 20.1 24.4 24.4 19.2 21.3 21.9 23.7% 2.5656997Time (±0.1 s) 1.10 1.40 1.20 1.30 1.10 1.22 24.6% 0.4248909Vx (m/s) 18.288 17.4171429 20.32 14.7710769 19.3963636 18.0385167 30.8% -2.207341Vy (m/s) 5.39 6.86 5.88 6.37 5.39 5.978 24.6% 2.0819654Angle (deg) 16.4217627 21.4977359 16.1388614 23.328033 15.5299279 18.5832642 42.0% 9.1982659Speed (m/s) 19.0657558 18.7194141 21.1536474 16.0860689 20.1313443 19.0312461 26.6% -0.854503Work Done – Output (J) 8.34784741 8.04731413 10.2763047 5.94246045 9.30704903 8.38419514 51.7% -0.726092Theoretical Work – Input (J) 29.8077682Efficiency 28.0% 27.0% 34.5% 19.9% 31.2% 28.1% 51.7% -4.2%

Mass Counter Weight (± 0.01kg) Trial12.47379018 1 2 3 4 5 Average Uncert.

Distance (±0.1 m) 32.004 32.6136 27.432 24.9936 24.384 28.28544 29.1% 1.1647462Time (±0.1 s) 2.60 2.70 2.40 2.00 1.70 2.28 43.9% 0.0489916Vx (m/s) 12.3092308 12.0791111 11.43 12.4968 14.3435294 12.5317343 23.2% 0.2520258Vy (m/s) 12.74 13.23 11.76 9.8 8.33 11.172 43.9% 0.2400589Angle (deg) 45.9852129 47.6036347 45.8152795 38.1035692 30.1458635 41.530712 42.0% 0.1239621Speed (m/s) 17.7150998 17.9147377 16.3994665 15.8811212 16.5869146 16.8994679 12.0% 0.3554144Work Done – Output (J) 7.20698566 7.37033716 6.17626451 5.79200289 6.31826253 6.57277055 24.0% 0.2848505Theoretical Work – Input (J) 37.2597102 4.48056Efficiency 19.3% 19.8% 16.6% 15.5% 17.0% 17.6% 24.0% -0.6%

Trebuchet Efficiency

28.13%

17.64%

16.50%

14.99%

13.59%

-4.20%

-0.56% -0.61%-0.24%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

0.00 5.00 10.00 15.00 20.00 25.00

Counter-Weight Mass (+- 0.01 kg)

Efficiency

Efficiency d Efficiency / dMass

Results cont.Mass Counter Weight (± 0.01kg) Trial14.51495584 1 2 3 4 5 Average Uncert.

Distance (±0.1 m) 29.2608 35.3568 27.1272 28.6512 32.9184 30.66288 26.8% 0.9285389Time (±0.1 s) 1.90 3.00 2.30 2.40 2.30 2.38 46.2% 0.0320672Vx (m/s) 15.4004211 11.7856 11.7944348 11.938 14.3123478 13.0461607 27.7% 0.1841424Vy (m/s) 9.31 14.7 11.27 11.76 11.27 11.662 46.2% 0.1571295Angle (deg) 31.1541756 51.2794017 43.6974438 44.569649 38.2180255 41.7837391 48.2% 0.0035943Speed (m/s) 17.995807 18.8411881 16.3132336 16.7574892 18.2169207 17.6249277 14.3% 0.2278531Work Done – Output (J) 7.43719386 8.15235379 6.11148236 6.44888074 7.62107764 7.15419768 28.5% 0.1854004Theoretical Work – Input (J) 43.3567537 4.48056Efficiency 17.2% 18.8% 14.1% 14.9% 17.6% 16.5% 28.5% -0.6%Mass Counter Weight (± 0.01kg) Trial

17.00971388 1 2 3 4 5 Average Uncert.Distance (±0.1 m) 28.6512 28.6512 36.2712 34.7472 36.576 32.97936 24.0% 0.9200806Time (±0.1 s) 1.90 2.60 2.70 2.60 2.50 2.46 32.5% 0.0033917Vx (m/s) 15.0795789 11.0196923 13.4337778 13.3643077 14.6304 13.5055513 30.1% 0.3801914Vy (m/s) 9.31 12.74 13.23 12.74 12.25 12.054 32.5% 0.0166195Angle (deg) 31.6908188 49.1412434 44.5621258 43.6299817 39.9393601 41.792706 41.8% -0.617976Speed (m/s) 17.7220146 16.8446199 18.8546887 18.4638111 19.0816955 18.193366 12.3% 0.3173803Work Done – Output (J) 7.21261299 6.51611708 8.16404109 7.82905143 8.36181151 7.61672682 24.2% 0.2857396Theoretical Work – Input (J) 50.8086957 4.48056Efficiency 14.2% 12.8% 16.1% 15.4% 16.5% 15.0% 24.2% -0.2%Mass Counter Weight (± 0.01kg) Trial

22.90641469 1 2 3 4 5 Average Uncert.Distance (±0.1 m) 32.3088 39.9288 42.3672 41.148 36.2712 38.40 26.2%Time (±0.1 s) 2.70 2.40 2.10 2.90 2.30 2.48 32.3%Vx (m/s) 11.9662222 16.637 20.1748571 14.1889655 15.770087 15.7474264 52.1%Vy (m/s) 13.23 11.76 10.29 14.21 11.27 12.152 32.3%Angle (deg) 47.8714028 35.2548938 27.0234361 45.0424377 35.5512733 38.1486887 54.6%Speed (m/s) 17.8388165 20.3736931 22.6474935 20.0811066 19.3832026 20.0648624 24.0%Work Done – Output (J) 7.31 9.53 11.78 9.26 8.63 9.30 48.1%Theoretical Work – Input (J) 68.4223769Efficiency 10.7% 13.9% 17.2% 13.5% 12.6% 13.6% 48.1%

Trebuchet Distance

21.88464

28.28544

30.66288

32.97936

38.4048

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

0.00 5.00 10.00 15.00 20.00 25.00

Counter-Weight Mass (+- 0.01 kg)

Distance (+- 0.1 m)

Trebuchet Distance

2.565699723

1.164746224

0.928538947 0.920080597

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00

Mass of Counter-Weight (+- 0.01 kg)

dDistance / dMass (m / kg)

GraphsTrebuchet Efficiency

28.13%

17.64%

16.50%

14.99%

13.59%

-4.20%

-0.56% -0.61%-0.24%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

0.00 5.00 10.00 15.00 20.00 25.00

Counter-Weight Mass (+- 0.01 kg)

Efficiency

Efficiency d Efficiency / dMass

Graphs cont.Trebuchet Distance

2.565699723

1.164746224

0.928538947 0.920080597

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00

Mass of Counter-Weight (+- 0.01 kg)

dDistance / dMass (m / kg)

Graphs cont.Trebuchet Distance

21.88464

28.28544

30.66288

32.97936

38.4048

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

0.00 5.00 10.00 15.00 20.00 25.00

Counter-Weight Mass (+- 0.01 kg)

Distance (+- 0.1 m)

The Math

Vi n i

= p

_c o s _ _ _

s i n _ _ _ _

0 = p s i n _ _ _ t _ 4 . 9 t

2

0 = t _ p s i n _ _ _ _ 4 . 9 t _

t =

p s i n _ _ _

4 . 9

d i s t = p c o s _ _ _ t

d i s t =

p

2

s i n _ _ _ c o s _ _ _

4 . 9

c =

p

2

4 . 9

d i s t = c s i n _ _ _ c o s _ _ _

d i s t ' = c _ c o s

2

_ _ _ _ s i n

2

_ _ _ _

0 = c o s

2

_ _ _ _ s i n

2

_ _ _

t a n

2

_ _ _ = 1

_ = 4 5

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

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.

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.

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.

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.

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

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).

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/

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