My role in the project:
• Designed five factorial experiment for analysis of maximum distance travelled in catapult.
• Identified the nuisance’s variable which helped to reduce the noise in the system.
• Analysed the output of experiment using ANOVA method to find significant factors.
• Performed enough replications to create a representative body of data.
• Summarized the results and methodology in a report. (Tool Used-Minitab).
1_Introduction + EAM Vocabulary + how to navigate in EAM.pdf
Design of Experiments & Regression Analysis
1. IEE 5160: Design of Experiments & Regression Analysis
PROJECT
Submitted by:
Kunal Dalal
Rohan Ghorpade
Sagar Sable
Yuva Balaguru
Department of Industrial and Entrepreneurial Engineering
Western Michigan University
2. INDEX
CONTENT
1.INTRODUCTION………………………………………………………………………………………………….
2. RECOGNAZATION OF STATEMENT PROBLEM…………………………………………………………….
3.SELECTION OF RESPONSE VARIABLE………………………………………………………………………
4.SELECTION FACTORS, LEVEL AND RANGES………………………………………………………………
5.FACTORIAL DESIGN……………………………………………………………………………………………
STEP-1 HALF FACTORIAL DESIGN……………………………………………………………………
STEP-2 CENTRAL COMPOSITIE DESIGN…………………………………………………………......
STEP-3 FULL FACTORIAL DESIGN…………………………………………………………………….
6.RESULT AND DISCUSSION…………………………………………………………………………………….
7. REFERENCES…………………………………………………………………………………………………….
8. APPENDIX……………………………………………………………………………………………………….
3. INTRODUCTION
A Catapult is a ballistic device used to launch a projectile without the aid of explosive devices—
particularly various types of ancient and medieval siege engines [1]. As part of the course project,
we have assembled a prototype of a Catapult used in ancient warfare. This prototype is a stationary
catapult and its purpose is to attack the enemy at the furthest distance and try to keep them away
from the empire walls. In order to do so, it is essential to determine the factor which contribute to
throw the projectile, called as payload to the farthest distance.
RECOGNITION AND STATEMENT OF PROBLEM:
The problem recognized in this scenario is to determine the longest distance the catapult can throw
the projectile in order to hit the target to the farthest distance. The problem can be stated as finding
the factor levels and their interaction which leads to throw projectile the farthest distance. One
combination of a particular level for each factor will throw the projectile the longest distance. This
distance can be measured and the catapult can be stationed at that distance away from the enemy
to hit the target and keeping the catapult at the safest position.
4. SELECTION OF RESPONSE VARIABLE
We selected the ‘distance travelled by the projectile’ as our response variable because it easily
defines our purpose on using the catapult. Shorter distance travelled by the projectile shows the
catapults ability to attack its closest target and longer distance shows its ability to attack targets at
longer distances, which is its basic purpose.
Our null hypothesis H0 is that the factors do not have any effect on the change in the response
variable. Therefore, increasing or decreasing the factor levels does not change the distance
travelled by the projectile significantly.
Measurement of the response variable:
The response variable i.e. the distance travelled by the projectile (ball) was measured from the
front base of the catapult to the point where the ball first hits the ground. This distance was selected
as the point where the projectile hits is the point of impact and the projectile will explode in case
of the real life catapult. This distance was measured using a measuring tape and the distance was
measured in centimeters.
5. SELECTION OF FACTOR LEVELS AND RANGES
There are five factors that have a direct impact on the response variable. These factors are:
1. Tension mount position [A]: The position of the band increases or decreases the pull on
the band. This effects the force with which the ball is thrown which directly effects the
distance covered by the ball. Three positions of the bands form the 3 levels.
2. Number of rubber bands [B]: The band are used to retract the arm to its original position
to throw the ball to its intended target. A pair of bands is low level, pair of 2 bands is
medium level and pair of 3 bands is high level.
3. Throw arm stopping position [C]: The arm holds the bucket. The stopping position of
the arm determines the release position of the ball. There are 3 levels for the stopping
position.
4. Projectile holding position [D]: The bucket holds the ball to thrown. There are 3 levels
of the bucket, high (1) – uppermost position; low (-1) – lowest position and medium (0) –
central position.
5. Throw arm launch position [E]: The arm is pulled in order to generate torque. After
releasing the arm, it hits the stopping point to release the ball in trajectory. There are three
levels depending on the distance the arm is pulled.
Environmental factors such as wind is a nuisance uncontrollable factor which affects the
distance travelled by the projectile. But as the projectile is our case is very heavy, its effect
on the distance travelled is negligible and hence it was controlled during the experiment by
conducting the experiment indoors.
The elasticity of the bands decreases with time. This effect is controlled by new set of
bands for each combination of factors. It is assumed each new set of bands has equal
elasticity.
6. Table No.1
Factor
Abbreviation
Factors Levels
A Tension mount position
Low
Center point
High
B Number of rubber bands
Low
Center point
High
C Throw arm stopping position
Low
Center point
High
D Projectile holding position
Low
Center point
High
E Throw arm launch position
Low
Center point
High
7. Figure 1: Actual catapult representing factors
FACTORAL DESIGN
A half factorial design is generated using MINITAB with 4 center points. Half factorial design
was done in order to limit the number of combinations to be experimented and four center points
combinations are done to check for if the model exhibits curvature. A 2-level 5 factorial design is
created with 4 center points in MINITAB. The factors are input as shown:
8. The following experimental design (25
factorial) was generated by Minitab. The order was
randomized to avoid interdependence between observations. Also, data copied directly from
Minitab is difficult to understand, so it has been presented in a better, readable format as shown in
the Fig-2.
Flow of Procedure.
Step 1: Performing half factorial analysis 2^5-1 factorial design with 4 center points.
A) Setting up the levels of catapult.
Setting all the high and low points for all the five factors.
Defining the high and low quantitative for factors.
Factors and their resemblance
A – Throw arm stopping position
B – Projectile holding position
C – Tension mount location
D – Throw arm launch position
E – Number of rubber bands
Fig 2. Factor level input to generate factorial design
9. B) Performing experiments.
Creating a fractional factorial design of 16 experiments with 4 center points (i.e. where
factors are at center points).
With the obtained set of experiments order, perform experiments.
There is only one block since the generator is E=ABCD which is a high order interaction.
c) Results and Discussion:
Excel sheet of P-value, sum of squares error and regression equation.
Table no.2
Factors P-value Result
Constant 0.000 Significant
A 0.001 Significant
B 0.00 Significant
C 0.000 Significant
D 0.001 Significant
E 0.002 Significant
AB 0.037 Significant
AC 0.013 Significant
AD 0.008 Significant
AE 0.679 Insignificant
BC 0.001 Significant
BD 0.010 Significant
BE 0.006 Significant
CD 0.023 Significant
CE 0.025 Significant
DE 0.005 Significant
Curvature 0.004 Significant
Using MINITAB and performing the Half factorial design with four center point, the above table no.2
show that results which show the significant and insignificant factors and interaction affecting on the
response (distance).
10. Figure – 3. Half Normal Plot of all the factors and interaction effects
The P-value observed from the table no 1 clearly says that all the mains effect i.e. A, B, C,
D and E are significant. The 2-way interaction AE is the only insignificant term of all other
two order interaction effects.
The adjusted sum of squares of curvature is high with significantly low p-value explains
strongly that this model has curvature response surface.
The Regression Equation of this ½ factorial design is:
Response = 202.88 + 27.87 A + 74.37 B + 36.00 C + 32.38 D + 22.75 E + 7.87 A*B
+ 11.75 A*C
+ 13.88 A*D + 1.00 A*E + 27.75 B*C + 12.63 B*D + 15.75 B*E + 9.50 C*D
+ 9.13 C*E
+ 16.50 D*E + 41.12 Ct Pt
Discussion:
From figure-3 it can be seen that, except interaction factor AE all other effects are
significant.
The center points has been taken into consideration where all the continuous factors are set
mid-way between high and low.
Since the average mean response of the center points is significantly greater than
average mean response of the factor at their high and low settings, the model shows
curvature response surface.
11. Also there are many large interaction effects in the model which confirms the
curvature of the response surface.
Now since the detection of curvature has been found, the next step is to estimate the model
of curvature because modelling curvature on the basis of the available 20 observations from
experiment 1 is not possible.
To model the response curvature, some method should be used which could project the
behavior of the model at its extreme points in either direction.
Step 2:
A. Using Central composite design:
Checking the need for using Central composite design because of the occurrence of
curvature.
Although center points detect the curvature, the available information is not enough to
model it.
So to model a curvature response, square terms are needed which can be obtained by adding
more points to the design which are star points.
These additional points convert the design to a face-centered central composite design.
This is a form of response surface design, which makes it possible to fit a quadratic model
that has linear main effects, all 2-factor interactions, and square terms of all continuous
factors.
Thus Central composite design is used in this model further to model its curvature by
adding center points to the previously model.
B. Performing Central Composite design (CCD):
Performing CCD on the available model by adding more axial points along the factors and
more center-point readings of the factors.
By creating a structured randomized order of 30 runs from surface response option of
Minitab, 10 more experiments are performed to fit into the model of CCD.
These 10 experiments are performed in the given order generated from Minitab. In this
experiments all the factors are set at high and low once individually where all other factors
are set at center points. These runs are done to understand the behavior of a factor at high
and low condition when everything else is at center position.
After collecting the responses from CCD, the measurements from the first experiment and
the measurements from the CCD are put into together to fit into the model of CCD to get
it analyzed in the same order it was generated by Minitab.
The table no 1 and 2 below shows the figures observed after running the CCD model.
12. Table no.3
Linear P-value Result
A 0.004 Significant
B 0.000 Significant
C 0.001 Significant
D 0.001 Significant
E 0.02 Significant
Square P-value Result
A*A 0.363 Insignificant
B*B 0.832 Insignificant
C*C 0.702 Insignificant
D*D 0.871 Insignificant
E*E 0.775 Insignificant
Table no.4
2-Way Interaction P-value Result
A*B 0.331 insignificant
A*C 0.16 insignificant
A*D 0.104 insignificant
A*E 0.899 insignificant
B*C 0.006 significant
B*D 0.134 insignificant
B*E 0.07 significant
C*D 0.247 insignificant
C*E 0.264 insignificant
D*E 0.06 significant
13. Figure – 4 Surface Plots of Responses
C. Results and Discussions:
It can be observed from the above table no.3&4 that all squares terms are insignificant
whereas main effects are significant.
By adding the center points to the model, curvature can be seen in the figure – 4.
Figure – 4 represents the behavior of response surface between two factors.
The two interactions BE and BC comes significant.
The regression equation obtained from this CCD model is quadratic in nature which is used
further to find the responses for full factorial design in the next step.
The regression equation in uncoded unit is:
responses = 214.15 + 28.17 A + 74.06 B + 32.72 C + 32.94 D + 20.44 E - 18.7 A*A
+ 4.3 B*B - 7.7 C*C + 3.3 D*D + 5.8 E*E + 7.87 A*B + 11.75 A*C + 13.87 A*D + 1.00 A*E
+ 27.75 B*C + 12.62 B*D + 15.75 B*E + 9.50 C*D + 9.12 C*E + 16.50 D*E
Step 3: Optimal design for the catapult model.
A. Generating a full factorial design from Minitab.
With 4 center points runs, 10 CCD runs and a factorial design of 32 runs, a full
factorial model is generated from the Minitab.
This full-factorial design is used to find the optimal factor settings of the catapult
in-order to achieve the maximum distance travelled by the ball.
A 0
B 0
C 0
D 0
E 0
Hold Values
001
02 0
-1- 0
1
0
-1
1
0
1
300
SER
B
A
150
020
--1 0
1
0
-1
1
025
SER
C
A
51 0
002
--1 0
1
0
-1
1
0
1
250
SER
D
A
150
2 00
--1 0
1
0
1-
1
250
SER
E
A
001
002
-1-
0
1
0
-1
1
0
1
003
SER
C
B
001
002
-1-
0
1
0
-1
1
0
1
03 0
SER
D
B
150
002
052
-1-
0
1
0
-1
1
0
1
052
003
SER
E
B
51 0
200
052
-1-
0
1
0
-1
1
0
1
052
300
SER
D
C
200
240
-1-
0
1
0
-1
1
1
240
280
SER
E
C
020
225
052
-1-
0
1
0
-1
1
1
052
572
SER
E
D
sesnopserfostolPecafruS
14. B. Using the quadratic equation from the CCD design to find the optimum settings of
catapult.
The quadratic equation from CCD model is:
responses = 214.15 + 28.17 A + 74.06 B + 32.72 C + 32.94 D + 20.44 E - 18.7 A*A
+ 4.3 B*B - 7.7 C*C + 3.3 D*D + 5.8 E*E + 7.87 A*B + 11.75 A*C + 13.87 A*D + 1.00 A*E
+ 27.75 B*C + 12.62 B*D + 15.75 B*E + 9.50 C*D + 9.12 C*E + 16.50 D*E
The effect of interaction BC, BE and DE is significant as its P-value is very less.
C. Using quadratic equation in excel to find the responses for full factorial
Using the combinations of factors obtained from the Minitab to find the response
from quadratic equation by eliminating the significant factors.
The next important step in finding optimized setting is choosing significant
factors to provide the correct the combination for achieving maximum distance.
Table No.5
All Factors main effect, interaction and Squares
S= 30.6641 R2
= 95.81%
Table No.6
BE, BC& DE Removed
S= 49.2516 R2
= 85.58%
Table No.7
BE, BC& DE kept and Remove the all other factors
S= 32.3156 R2
= 89.14%
As seen from the table no 5,6&7 the standard error of the regression increases
significantly while the coefficient of determination decreases when factors BC,
BE and DE are removed from regression model.
But the value of S and R-square remains approximately same while all other
factors except the mains effect and interaction BC, BE and DE are removed from
the regression equation.
Thus the mains effect A, B, C D, E and interaction effect BC, BE and DE are
considered for regression model to find the optimized setting of the catapult as
they have significant effect on the responses.
Result and Discussion:
By solving the regression equation in excel for full factorial design runs, following
responses are obtained.
15. Table No.8
A B C D E response
High Low High Low High 128.48
Low Low Low Low Low 85.82
Low Low High High Low 128.64
High Low High High High 227.36
center Low center center center 140.09
High High High Low High 363.6
High High Low Low Low 203.28
center center center Low center 181.21
center center center center center 214.15
center center High center center 246.87
High High Low High Low 236.16
High Low Low High Low 175.04
Low High Low High Low 179.82
High High High Low Low 324.22
center center center center Low 193.71
Low High High Low High 307.26
center center Low center center 181.43
High Low Low Low High 118.54
Low High High Low Low 267.88
Low High High High Low 300.76
High High Low Low High 242.66
High Low Low Low Low 142.16
High High High High Low 357.1
Low High High High High 406.14
Low Low Low Low High 62.2
High High Low High High 341.54
High center center center center 242.32
center High center center center 288.21
Low Low High High High 171.02
Low High Low Low High 186.32
High Low High High Low 184.98
High Low Low High High 217.42
Low center center center center 185.98
Low Low Low High High 161.08
Low High Low Low Low 146.94
High High High High High 462.48
Low Low High Low Low 95.76
Low Low Low High Low 118.7
center center center center High 234.59
Low Low High Low High 72.14
center center center center center 214.15
Low High Low High High 285.2
High Low High Low Low 152.1
16. The maximum value of 472cm distance is observed from the full factorial runs.
The factors combination for this particular run where maximum distance is
achieved is checked once again to see if the catapult with the same configuration of
factors achieved the same response.
Now, the catapult is set to the same configuration of all high setting where optimum
response was observed.
The response obtained from this new run comes closer to the response obtained
from regression model of factorial design.
Thus the optimized design is achieved by setting all the factors to high levels as
observed from the table no.8 above.
17. REFERENCES
1. (2016, June 11). Retrieved June 27, 2016, from https://en.wikipedia.org/wiki/Catapult
2. http://support.minitab.com/en-us/minitab/17/topic-library/modeling-
statistics/doe/response-surface-designs/should-i-include-or-exclude-a-term/
