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 nonlinearity 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
Release Bar
Sling Length
LOW SETTING
2
0
HIGH SETTING
5
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
1
2
3
4
RELEASE
BAR
2
2
5
5
SLING LENGTH
DISTANCE
0
3
0
3
167, 172
196, 201
167, 171
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
i
s
t
a
n
c
e
200
180
160
140
120
100
2(-)
5(+)
Rel Bar(A)
0(-)
3(+)
S Length(B)
-1(-)
AB
1(+)
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
s
t
a
n
c
e
300
200
S Length(-)
100
0
S Length(+)
2(-)
Rel Bar(A)
5(+)
Statistical analysis performed on the experimental data was as follows:
Dependent Variable:
Number Runs(N):
Multiple R:
Squared Multiple R:
Adjusted Squared
Multiple R:
Standard Error of
Estimate:
Variable
Constant
Rel Bar
S Length
AB
Distance
8
0.998703
0.997408
0.995463
2.89396
Coefficient Std Error
156.875
-27.125
-12.375
-26.875
1.02317
1.02317
1.02317
1.02317
95% CI
± 2.84133
± 2.84133
± 2.84133
± 2.84133
Tolerance T
1
1
1
P(2 Tail)
153.323
-26.511
-12.095
-26.266
0
0
0
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:
S
Contour Plot
3
100
2.4
L
e 1.8
n
1.2
g
t 0.6
h
0
120
180
140
Predicted
distance is 165
2
160
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 nonlinearity (for a quadratic standpoint) for Throwing Arm, Release Bar, and
Sling Length. The factors studied were:
FACTOR
Throwing Arm
Pivot Point
Weight
Release Bar
Sling Length
LOW
2
1
2
1
1
Mid
3
HIGH
4
2
3
5
3
3
2
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 twofactor linear interactions in a near orthogonal matrix. Three Replicates were
conducted at each combination.
Run
Throw
Arm
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Pivot Point Weight
2
2
4
4
4
3
4
2
4
3
2
2
3
4
4
4
2
2
2
2
1
2
1
2
1
2
1
2
1
1
2
1
2
2
1
2
2
1
1
1
Rel Bar
3
2
3
3
3
3
3
3
2
2
3
3
2
2
2
3
2
3
2
3
Slength
1
1
5
5
1
5
1
3
1
3
1
5
3
5
5
1
5
5
1
5
Distance
2
3
2
1
1
3
3
3
2
3
1
1
1
3
1
2
2
3
1
1
189
166
56
249
180
237
166
231
72
71
80
182
170
90
21
284
184
145
107
187
Distance
195
164
64
250
183
233
163
236
71
69
87
184
173
92
23
287
187
141
110
188
Distance
189
162
63
252
180
235
161
241
73
69
84
185
173
87
21
285
186
146
107
187
Analysis of the experimental data using DOE Wisdom software resulted in
the following outputs:
Dependent Variable:
Number Runs(N):
Multiple R:
Squared Multiple R:
Adjusted Squared
Multiple R:
Standard Error of
Estimate:
Variable
Constant
Th Arm(A)
Pivot Pt(B)
Weight(C)
Rel Bar(D)
Sling(E)
AB
AC
AD
AE
BC
BD
BE
CD
CE
DE
Throwing
Arm**2
Release
Bar**2
Sling
Length**2
Distance
60
0.999558
0.999116
0.998727
2.50284
Coefficient Std Error
95% CI
187.343
1.38873 ± 2.80460
-13.2023 0.438988 ±
0.886555
39.021 0.391495 ±
0.790639
36.9847 0.373246 ±
0.753786
-14.1588 0.438988 ±
0.886555
-3.34534 0.412283 ±
0.832623
32.1995 0.409803 ±
0.827614
14.756 0.438988 ±
0.886555
-26.8993 0.398814 ±
0.805421
-20.7431 0.479787 ±
0.968950
-2.06236 0.391495 ±
0.790639
18.2449 0.409803 ±
0.827614
18.1843 0.420623 ±
0.849465
-1.95045 0.438988 ±
0.886555
6.73799 0.412283 ±
0.832623
-26.7014 0.479787 ±
0.968950
-22.5124
1.34958 ± 2.72554
Tolerance T
P(2 Tail)
0.639
134.902
-30.074
0
0
0.688
99.672
0
0.781
99.089
0
0.639
-32.253
0
0.822
-8.114
0
0.734
78.573
0
0.639
33.614
0
0.821
-67.448
0
0.756
-43.234
0
0.688
-5.268
0
0.734
44.521
0
0.811
43.232
0
0.639
-4.443
0
0.822
16.343
0
0.756
-55.653
0
0.45
-16.681
0
1.34958 ± 2.72554
0.45
-8.86
0
-11.3403 0.824447 ± 1.66500
0.819
-13.755
0
-11.9569
Analysis of the above output table suggests all linear, quadratic, and twofactor interactions studied are significant. Pivot point and weight appear to
be the most important individual factors. The contour plot generated from
the above experimental data was as follows:
Prediction is 175
S
Contour Plot**Throwing Arm(A)=4.00000,Pivot Point(B)=2.00000,Weight(C)=2.00000
l
3
110 99
i
132
121
n
143
g
154
L
e
n
g
t
h
2
165
176
1
165
1
2
3
4
5
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 represents 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
Weight
Release Arm
Pivot Point
CONSTANT VALUE
2
4
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
Release Bar
LEVELS
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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
S Length
2
5
2
4
5
2
2
3
4
3
4
4
5
3
5
Dist
0
1
3
3
2
2
1
3
0
0
1
2
3
1
0
Dist
167
162
201
148
130
202
187
180
183
181
185
173
91
193
171
172
164
196
147
129
200
188
178
178
183
182
173
90
192
167
Analysis was conducted with RSDiscover Software. Statistical Analysis
from the above experimental data was as follows:
Lea st Sq uare s Co effi cient s, R espo nse D , Mo del DESI GN
Term
C oeff .
Std . Er ror
T- valu e S igni f.
---- ---- ----- ---- ---- ---- ----- ---- ---- ----- ---- ---- ---- -1 1
1 88.9 73423
1. 1242 97
2 ~R
- 20.7 24299
2. 7064 48
3 ~L
-9.2 44451
2. 7064 48
4 ~R*L
- 27.1 99766
0. 8605 61
5 ~R** 2
- 17.1 39165
1. 1341 53
6 ~L** 2
- 14.8 89165
1. 1341 53
7 ~ R**2 *L
- 4.20 8499
1.4 5656 9
-2.8 9
0.00 91
8 ~ R*L* *2
- 3.55 4001
1.4 5656 9
-2.4 4
0.02 41
9 ~ R**3
- 2.86 3756
2.6 4002 2
-1.0 8
0.29 09
10 ~ L**3
0.89 5006
2.6 4002 2
0.3 4
0.73 81
No . ca ses = 30
R- sq. = 0. 9938
Re sid. df = 20
R- sq-a dj. = 0. 9909
~ in dicat es f acto rs ar e tr ansf orme d.
RMS Err or = 2.6 98
Con d. N o. = 10. 21
> 1 SUM MARY Ano v
> 2 CO MPON ENTS A
3 VAR IANC ES
4 FIX ED E ffec t
5 RAND OM E ffec
6 M IXTU RE P ool
7 FULL Fac tori
8 INTE RPRE TATI
9 RESP ONSE /MOD
10 O PTIO NS
11 NE XT
12 MA IN
The above analysis suggests only the cubic terms for the two factors are not
significant.
Pooling the insignificant terms resulted in the following reduced
L east Squ ares Coef fici ents , Res pons e D, Mode l DE SIGN __CO PY
model:
Ter m
C oeff .
Std . Er ror
T- valu e S igni f.
--- ---- ----- ---- ---- ---- ----- ---- ---- ----- ---- ---- ---- -1 1
1 89.2 61924
1. 0607 53
2 ~R
- 23.4 44778
1. 0218 00
3 ~L
-8.3 17722
1. 0218 00
4 ~R*L
- 27.2 29441
0. 8441 89
5 ~R** 2
- 17.3 24630
1. 0959 68
6 ~L** 2
- 15.0 74630
1. 0959 68
7 ~ R**2 *L
- 4.31 9778
1.4 2500 1
-3.0 3
0.00 61
8 ~ R*L* *2
- 3.44 2722
1.4 2500 1
-2.4 2
0.02 44
No . ca ses = 30
R- sq. = 0. 9934
Re sid. df = 22
R- sq-a dj. = 0. 9913
~ in dicat es f acto rs ar e tr ansf orme d.
1
2
3
> 4
5
> 6
7
8
9
10
11
12
ST EP
Obey HIE RAR
K EEP In
Di spla y DA T
A ll S UBSE TS
Sh ow C OEFF I
HIS TORY /PRE
PO OL M ixtu r
CO LLIN EARI T
RESP ONSE /MO
NE XT
MA IN
RMS Err or = 2.6 49
Co nd. No. = 4. 31
Since the data is coded to an orthogonal scale, the coefficients are indicative
of the magnitude of each effect. The R*L interaction appears to be the
biggest hitter followed by R (Release Bar). Note the higher order
interactions as well as the linear and quadratic factor effects are also
significant.
DI ST
3*
*
*
1 80
1 90
1 10
1 60
*
1 30
1 70
L 2*
E
N
G
T
H 1*
1 40
*
*
1 50
1 60
*
1 90
*
*
1 80
1 90
1 70
0*
2 .0
2 .2
2 .4
1 80
2 .6
2 .8
*
3 .0
3 .2
3 .4
3 .6
RELA RM
D
3 .8
*
4 .0
4 .2
4 .4
4 .6
4 .8
*
5 .0
DI ST
DI ST
1 80
1 40
1 00
0
.
0
0
.
8
1
.
6
2
.
4
LENG TH
2
2 .
3 .
0
4 .
8
. 6
4
RELA RM
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 favorably with an actual distance obtained (from
confirmation) of 194 in.
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:
250
200
L= 0
150
L =1
L =2
100
L =3
50
0
2
3
Release Bar Setting
4
5