During the Summer of 2000 PBS Featured the Recreation of One of the Most Monstrous Trebuchets

TREBUCHET During the summer of 2000 PBS featured the recreation of one of the most monstrous trebuchets ever built (named “Warwolf” by its creators). This massive device was able to fling boulders (weighing several hundred pounds) hundreds of feet at castle walls causing massive destruction. The Warwolf trebuchet was used King Edward I of England in his attack on the Scottish castle, Stirling, in 1304. Trebuchets were originally invented between the 3rd an 5th centuries in China. The trebuchet was instrumental in the rapid expansion of both the Islamic and Mongol empires in the years following its invention in China. Trebuchets were the ultimate super weapons of destruction during the 12th and 13 Th centuries in Northern Europe.

Based upon the above and other efforts, we are pleased to offer a miniature version of “Warwolf” for use in the classroom. This device is a wonderful hands- on tool for training in problem solving, designed experiments, and six sigma black belt training.

“ Mini-Warwolf” can be used as a follow-on to the catapult or as a first experimental challenge. As many have learned, the catapult serves as an excellent mechanism to demonstrate and sell the use of simple factorial experimental design techniques. Unfortunately, not all mechanisms in real life are this simple (linear). Sometimes non-linear experimental design approaches are required. Because of the complexity and inherent non-linearity of this device, it serves as an excellent challenge for those who wish to go beyond the relative simplicity of the catapult. The trebuchet includes detailed instructions of use and setup of this exciting device (purchase from Launsby Consulting at 1-800-788- 4363).

This paper demonstrates the use of simple and complex design of experiments. In Experiment One a simple two-level factorial experiment was conducted. Experiment One provides useful information about linear factor effects and simple interactions. We found that it did not provide a very good prediction, unfortunately, at the point we selected for confirmation testing. Experiment Two is a more complex design in which several factors are evaluated at three levels and linear two-factor interactions are evaluated. Predictions are better from this experiment. Finally, in Experiment Three, we evaluate two key factors at 4 levels and assess both simple and complex interactions. The prediction from Experiment Three confirms extremely well.

Experimental Design One: The following factors were held constant during experiment one:

FACTOR CONSTANT VALUE Weight 2 Release Arm 4 Pivot Point 2 Our team conducted numerous experimental designs on the trebuchet. In this experiment we conducted a two-factor, two-level full-factorial design with one replicate.

Factors and levels for this experiment were:

FACTOR LOW SETTING HIGH SETTING Release Bar 2 5 Sling Length 0 3

The response to be measured was flight distance of the projectile (in this case a tennis ball…not a 200 pound rock). The orthogonal array used and resultant data were:

RUN RELEASE BAR SLING LENGTH DISTANCE 1 2 0 167, 172 2 2 3 196, 201 3 5 0 167, 171 4 5 3 90, 91

Analysis of the experimental data was conducted with DOE Wisdom software. Some graphical and statistical output from the package were:

Main Effects 220

D 200 i s 180 t 160 a n 140 c e 120

100 2(-) 5(+) 0(-) 3(+) -1(-) 1(+) Rel Bar(A) S Length(B) AB Factors The above main effects plot suggests a substantial interaction between Release Bar (A) and Sling Length (B). The interaction can be further evaluated with use of an interaction plot.

Interactions 400

D i 300 s t 200 a n S Length(-) c 100 e S Length(+)

0 2(-) 5(+) Rel Bar(A) Factors

Statistical analysis performed on the experimental data was as follows:

Dependent Variable: Distance Number Runs(N): 8 Multiple R: 0.998703 Squared Multiple R: 0.997408 Adjusted Squared 0.995463 Multiple R: Standard Error of 2.89396 Estimate:

Variable Coefficient Std Error 95% CI Tolerance T P(2 Tail)

Constant 156.875 1.02317 ± 2.84133 153.323 0 Rel Bar -27.125 1.02317 ± 2.84133 1 -26.511 0 S Length -12.375 1.02317 ± 2.84133 1 -12.095 0 AB -26.875 1.02317 ± 2.84133 1 -26.266 0

Both factors as well as the two-factor interaction are significant in this experiment. The square multiple R of near one suggests the experimental data is well explained by the model. The contour plot for this experiment was as follows: Contour Plot 3 100 S 2.4 120 180 L 140 e1.8 n 1.2 Predicted g 160 distance is 165 t0.6 h 0 2 3 4 5 Rel Bar Distance

A confirmation run was conducted with the release bar positioned at 3 and the sling length set at 2 (we just selected this point for convenience. We could have tested other points as well). The actual measured distance was 194, substantially different from the prediction from the above contour plot (we missed by approximately 30 inches). Based upon articles written about the trebuchet and its physical complexity, we were somewhat surprised the results from the confirmation were this good. From the above, it was decided to conduct a more complex experiment so as to model additional complexity in the trebuchet.

Experimental Design Two: In this experiment we decided to evaluate additional factors as well as non- linearity (for a quadratic standpoint) for Throwing Arm, Release Bar, and Sling Length. The factors studied were:

FACTOR LOW Mid HIGH Throwing Arm 2 3 4 Pivot Point 1 2 Weight 2 3 Release Bar 1 3 5 Sling Length 1 2 3 The Response was (again) distance. The experimental design used was a 20 run computer generated design (D-Optimal). This design allowed for the evaluation of linear and quadratic factors effects as well as all possible two-factor linear interactions in a near orthogonal matrix. Three Replicates were conducted at each combination.

Run Throw Pivot Point Weight Rel Bar Slength Distance Distance Distance Arm

1 2 1 3 1 2 189 195 189 2 2 2 2 1 3 166 164 162 3 4 1 3 5 2 56 64 63 4 4 2 3 5 1 249 250 252 5 4 1 3 1 1 180 183 180 6 3 2 3 5 3 237 233 235 7 4 1 3 1 3 166 163 161 8 2 2 3 3 3 231 236 241 9 4 1 2 1 2 72 71 73 10 3 1 2 3 3 71 69 69 11 2 2 3 1 1 80 87 84 12 2 1 3 5 1 182 184 185 13 3 2 2 3 1 170 173 173 14 4 2 2 5 3 90 92 87 15 4 1 2 5 1 21 23 21 16 4 2 3 1 2 284 287 285 17 2 2 2 5 2 184 187 186 18 2 1 3 5 3 145 141 146 19 2 1 2 1 1 107 110 107 20 2 1 3 5 1 187 188 187

Analysis of the experimental data using DOE Wisdom software resulted in the following outputs: Dependent Variable: Distance Number Runs(N): 60 Multiple R: 0.999558 Squared Multiple R: 0.999116 Adjusted Squared 0.998727 Multiple R: Standard Error of 2.50284 Estimate:

Variable Coefficient Std Error 95% CI Tolerance T P(2 Tail)

Constant 187.343 1.38873 ± 2.80460 134.902 0 Th Arm(A) -13.2023 0.438988 ± 0.639 -30.074 0 0.886555 Pivot Pt(B) 39.021 0.391495 ± 0.688 99.672 0 0.790639 Weight(C) 36.9847 0.373246 ± 0.781 99.089 0 0.753786 Rel Bar(D) -14.1588 0.438988 ± 0.639 -32.253 0 0.886555 Sling(E) -3.34534 0.412283 ± 0.822 -8.114 0 0.832623 AB 32.1995 0.409803 ± 0.734 78.573 0 0.827614 AC 14.756 0.438988 ± 0.639 33.614 0 0.886555 AD -26.8993 0.398814 ± 0.821 -67.448 0 0.805421 AE -20.7431 0.479787 ± 0.756 -43.234 0 0.968950 BC -2.06236 0.391495 ± 0.688 -5.268 0 0.790639 BD 18.2449 0.409803 ± 0.734 44.521 0 0.827614 BE 18.1843 0.420623 ± 0.811 43.232 0 0.849465 CD -1.95045 0.438988 ± 0.639 -4.443 0 0.886555 CE 6.73799 0.412283 ± 0.822 16.343 0 0.832623 DE -26.7014 0.479787 ± 0.756 -55.653 0 0.968950 Throwing -22.5124 1.34958 ± 2.72554 0.45 -16.681 0 Arm**2 Release -11.9569 1.34958 ± 2.72554 0.45 -8.86 0 Bar**2 Sling -11.3403 0.824447 ± 1.66500 0.819 -13.755 0 Length**2

Analysis of the above output table suggests all linear, quadratic, and two-factor interactions studied are significant. Pivot point and weight appear to be the most

Prediction is 175 important individual factors. The contour plot generated from the above experimental data was as follows:

S Contour Plot**Throwing Arm(A)=4.00000,Pivot Point(B)=2.00000,Weight(C)=2.00000 l 3 99 i 110 132 121 n g 154 143

2 165 L e 176 n g 165 t 1 1 2 3 4 5 h Release Bar Distance The prediction of distance from the above plot for Release Bar = 3 and Sling Length =2 is approximately 175. As you recall from experimental design one (where a simple linear model was fit to the data, the Prediction at this point was approximately 165). This appears to represent an improvement over the results from experimental design one, but is still relatively far from the confirmation value of 194 inches. A third experiment was then conducted with even greater complexity.

Experimental Design Three: In this experiment we held the following factors constant at the prescribed levels:

FACTOR CONSTANT VALUE Weight 2 Release Arm 4 Pivot Point 2

Only two factors were varied but each was evaluated at four levels as to evaluate linear, quadratic, and cubic factor effects as well as simple and complex interactions. The Factors with applicable levels were:

FACTOR LEVELS Release Bar 2,3,4,5

Sling Length 0,1,2,3

A 15-run computer generated design (D-Optimal) was conducted with two repetitions. Experimental data was as follows: RUN Rel Bar S Length Dist Dist 1 2 0 167 172 2 5 1 162 164 3 2 3 201 196 4 4 3 148 147 5 5 2 130 129 6 2 2 202 200 7 2 1 187 188 8 3 3 180 178 9 4 0 183 178 10 3 0 181 183 11 4 1 185 182 12 4 2 173 173 13 5 3 91 90 14 3 1 193 192 15 5 0 171 167

Analysis was conducted with RSDiscover Software. Statistical Analysis from the above experimental data was as follows: Least Squares Coefficients, Response D, Model DESIGN > 1 SUMMARY Anova > 2 COMPONENTS Anova Term Coeff. Std. Error T-value Signif. 3 VARIANCES ------4 FIXED Effects 1 1 188.973423 1.124297 5 RANDOM Effects 2 ~R -20.724299 2.706448 6 MIXTURE Pooling 3 ~L -9.244451 2.706448 7 FULL Factorial 4 ~R*L -27.199766 0.860561 8 INTERPRETATION 5 ~R**2 -17.139165 1.134153 9 RESPONSE/MODEL 6 ~L**2 -14.889165 1.134153 10 OPTIONS 7 ~R**2*L -4.208499 1.456569 -2.89 0.0091 11 NEXT 8 ~R*L**2 -3.554001 1.456569 -2.44 0.0241 12 MAIN 9 ~R**3 -2.863756 2.640022 -1.08 0.2909 10 ~L**3 0.895006 2.640022 0.34 0.7381

No. cases = 30 R-sq. = 0.9938 RMS Error = 2.698 Resid. df = 20 R-sq-adj. = 0.9909 Cond. No. = 10.21 ~ indicates factors are transformed. The above analysis suggests only the cubic terms for the two factors are not significant. Pooling the insignificant terms resulted in the following reduced model:Least Squares Coefficients, Response D, Model DESIGN__COPY 1 STEP 2 Obey HIERARCHY Term Coeff. Std. Error T-value Signif. 3 KEEP In ------> 4 Display DATA 1 1 189.261924 1.060753 5 All SUBSETS 2 ~R -23.444778 1.021800 > 6 Show COEFFICIENT 3 ~L -8.317722 1.021800 7 HISTORY/PRESS 4 ~R*L -27.229441 0.844189 8 POOL Mixture 5 ~R**2 -17.324630 1.095968 9 COLLINEARITY 6 ~L**2 -15.074630 1.095968 10 RESPONSE/MODEL 7 ~R**2*L -4.319778 1.425001 -3.03 0.0061 11 NEXT Since8 ~R*L**2 the data is coded -3.442722 to an orthogonal 1.425001 scale, the -2.42 coefficients 0.0244 are indicative of 12 MAIN the No.magnitude cases = of 30 each effect. R-sq. The= 0.9934R*L interaction RMS appears Error = to 2.649 be the biggest hitterResid. followed df by= 22 R (Release R-sq-adj. Bar). = Note0.9913 the higher Cond. order No. interactions = 4.31 as well as the~ indicates linear and factors quadratic are factor transformed. effects are also significant. DIST

3* * * 110 * 160 190 180 130 170 140 L 2* * * E 150 N G 160 T H 1* * * * 190 180 190 170

0* 180 * * * 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 RELARM D

The above contour plot indicated a Release of 3 combined with a Sling Length of 2 should produce distance of approximately 193 inches. This compares quite DIST favorably with an actual distance obtained (from confirmation) of 194 in.

100 2 2 . 0 0 3 1 . 0 . . 2 4 . . . 8 0 8 . 6 6 4 4 LENGTH RELARM

SUMMARY:

The preceding analysis suggests better and better predictions can be obtain in a highly non-linear technology when additional runs (defined in an orthogonal array) are conducted. In experiment one, a simple linear model (with interaction) was fit to the data. The second experiment allowed for the use of a quadratic model. In the third experiment we fit a cubic model to the data. The parsimonious model indicated that several higher-order interactions were significant. At the test point of verification, the third experimental design provided the best predictions.

Miscellaneous:

What does a non-linear interaction look like? Using the data from Experiment Three, we graphed the statistically significant quadratic/linear interaction between Release Bar and Sling Length. The graph is as follows:

200 L= 0 150 L =1 L =2 100 L =3 50

0 2 3 4 5 Release Bar Setting