Design of Experiments and
Taguchi Experimental Design
Professor Joe Greene
CSU, Chico
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Design of Experiments
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Need for Experiments
Factorial Experiments
Two-Factor Factorial Experiments
Statistical Analysis for Two-Factor Experiments
Other Factorial Experiments
– General Factorial Experiments
– Randomized Complete Block Design
– 2k Factorial Design
• Taguchi Designs
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Need for Need for Experiments
• Need to establish cause and effect relationships
• Home
– Car repair- Trouble-shooting starting, noise, and braking problems
– Home repair- Electrical and mechanical problems, cooking, etc.
– Gardening and lawn maintenance- watering and pesticide use
• School
– Studying versus grades performance
– Attendance versus grade performance
• Industry
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Maintenance and trouble-shooting of equipment
Effects of moisture, line rate, operators on productivity and quality
Trouble shooting production problems for incoming Materials
Trouble shooting production problems on Target values for performance or
appearance
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Experimental Goals
• Statistical Accuracy
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Proper selection of the responses to be measured
Determination the number of factors that affect a response
The interactions between the factors
The number of repetitions per run
The form of analysis to be completed
• Cost
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Minimize the cost
Reduce the number of experiments to the minimum
Study the main factors
Thoroughly understand the process under study
Choose the minimum number of experiments
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Factorial Experiments
• Study the effects of 2 or more factors with factorial
experiments
• Each factor and each combination of factors are studied
• Example
–
–
–
–
Factor A has 2 levels (high, low)
Factor B has 2 levels (on, off)
Then the total number of experiments is 2x2=4, or
high-on, high-off, low-on, low-off
• Experiments measure the difference of the response from
one level of the factor (high for A) and another level (low
for A).
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Factorial Experiments- Design
• Example
– Factor A- 2 levels- A1, A2
– Factor B- 2 levels- B1, B2
– Measured values are
• 10, 20
• 30, 40
B1
B2
A1
10
20
A2
30
40
– The effect Factor A has on the
experiment is the average
difference between the levels
A= 30+40 - 10+20 = 20
Conclusion
2
2
Changes in Factor A causes more
B= 20+40 - 30+10 = 10
of an effect than B. Factor A is
more significant than Factor B
2
2
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Factorial Experiments- Interaction
• Example- No Interactions
B2
40
30
B1
B2
20
Series1
Series2
10 B1
0
A1
A2
Factor A
Observation
Observation
50
Interactions
40
30
B1
20 B2
10
Series1
Series2
B1
B2
0
A1
A2
Factor A
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Two-Factor Factorial Experiments
• Problem- Molding problems on Injection Molder has caused
defects to sky rocket. The problems started when the PP
resin was switched to a different supplier. Additionally, a
new operator was added
• Experiment design to determine the cause of the defect
– Factor A- 2 levels: Resin 1= PP_old and Resin 2 PP_new
– Factor B- 2 levels- Operator 1= Tom and Operator 2= Bob
• Experiment run to see what is the cause- Operator or Resin
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Two-Factor Factorial Experiments
• Experimental Layout
Resin
Operator
1
PP_old
Bob
Molded
Quality
Points
10
2
PP_old
Tom
30
3
PP_new
Bob
20
4
PP_new
Tom
60
Observation
Experiment #
50
40
30
20 Tom
10 Bob
0
PP_old
Tom
Bob
Series1
Series2
PP_new
Factor A
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Significance of Difference
• Level Averages
– Sum Differences from Table
– Graph Results
• Analysis of Variance
– Statistical Measurement Method
– Measures the total variability in the data measured by the Sum of
Squares
– Separate out the the differences caused by the individual factors
– Calculates the differences caused by the error
– Uses the F statistic to calculate the significance
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ANOVA Example
• Analysis of Variance
Source of
Variation
Sum Of
Squares
Degrees of
Freedom
Mean
Square
F statistic
Factor A
SSFactor A
#Levels -1
SS/Df
MSA/MSE
Factor B
SSFactor B
#Levels -1
SS/Df
MSA/MSE
Error
SSError
n-a
SS/Df
Total
SSTotal
Sample size
n-1
SSFactor A = Page 334
SSFactor B = Page 334
SSTotal = Page 334
Note: F Statistic determines
significance. If F is greater than a
specified value than the factor
is significant.
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ANOVA Example Calculation
• Analysis of Variance
Experiment
1
2
3
4
Resin
PP_old
PP_old
PP_new
PP_new
Operator Impact Results
Bob
10
20
Tom
30
40
Bob
20
30
Tom
60
70
SST= Sum (results)2 - y..2
N
= (10)2 + (20)2 + (30)2 +…+(70)2 - (1280)2
8
Squares of Results Sum
SSResin = Sum (Resin y.)2 - y..2
100
400
500
n
N
900
1600
2500
SSOperator = Sum (Operator y.)2 - y..2
400
900
1300
n
N
3600
4900
8500
SSError= SST - SSResin - SSOperator
12800 = y..
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Other Factorial Experiments
• General Factorial Experiments
– Involves experiments with more than 2 factors
– Requires many experiments to run
• Randomized Complete Block Design
– Special design of experiment that blocks out certain
extraneous effects
– Used to investigate the effects of one ore more factors
when entire experiment cannot be run under
homogeneous conditions
• 2k Factorial Design
– Special design for 2 levels and k factors
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Taguchi Experimental Design
• History of Dr. Genichi Taguchi
– After WWII, the Japanese initiated a major effort to participate in
the world market.
– The first products were inexpensive, but of poor quality.
– The Japanese government set up government agencies modeled
after US companies (Bell Labs). One such company, Electrical
Communication Laboratories of Japan (ECL), hired Dr. Taguchi to
reduce the cost of experimentation.
– Dr. Taguchi developed a series of experiments that resembled
partial factorial designs and featured orthogonal (balanced) arrays.
– The experimental method is called “The Taguchi Approach”
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Comparison: Taguchi vs.
Conventional Experimental Design
• Traditional experimental designs were introduced by R.A.
Fisher in 1920’s in England
• Limitations of traditional design
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Limited variety of layouts and difficult data analysis
Limited number of variables with many required repetitions
Passive approach to interactions. Difficulty in resolving them
F statistic only recognized as fully significant. Partial effects are
not calculated
• Taguchi has
– Multiple layouts and designs and efficient data analysis
– Minimum number of experiments
– Active approach to interactions and calculates partial contribution
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Features of Taguchi
• Orthogonal Arrays
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–
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Efficient data collection
Separated effects from one another
Balanced, separable, or not mixed
Minimum number of experiments
• Experimental Designs
– Two Level- L8, L16, L32 have 8 experiments, 16 experiments, and
32 experiments, respectively
– Three Level- L9, L27 have 9 experiments and 27 experiments.
• Data Analysis- Software available
– Level Averages
– ANOVA
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Examples Taguchi
• Design of Experiment for thermoplastic composites
• Objectives
– What is the best combination of Twintex composites and GMT?
– What are the optimum process conditions?
• Paper for SAE
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Improving Performance of BMC
Bumper Beams
DOE Study
• Evaluate Effectiveness of Prepreg Technology
to Selectively Increase Stiffness & Impact
Performance
• Find Optimum Combination of the 2 Materials
• 4 Variables were Compared vs Static Load :
– Prepreg Type
– BMC Glass %
– Weight Fraction of Prepreg
– Tonnage of Press
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Improving Performance of BMC
Bumper Beams
DOE Study
• 3-Point Loading Test (ASTM D790), with
FMVSS 581 Pendulum Impactor head
– Typical mid-sized Vehicle Bumper was Used
– Test loaded Beam at Centerline @ 51 mm/min
– Load/Deflection Response Measured &
Recorded
• Test’s Measurable was Beam’s Static Load,
Recorded at 25 mm of Deflection
• Specific Load = Static Load/Beam
Weight
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Static Test Setup with a Pendulum
Face Moving at a Constant Speed
into a Rigidly Mounted Beam
constant
speed
Pendulum Face
F
Bumper Beam
a. Cross Car
Pendulum Face
Bumper Beam
b. Cross Section
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DOE Study
• TP-BMC Glass Weight Percentage:
 20%,
 30%, and
 40%.
• .Weight percentage of prepreg:
 25%,
 50%, and
 75%.
• .Press Tonnage (metric):
 450 t,
 675 t, and
 900 t.
• .Prepreg type:
 satin weave (1:1),:
 twill weave (4:1), and
 unidirectional (uni);
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DOE Study
Figure 2: Continuous Extrusion System for Thermoplastic
TaguchiBMC Technology
22
Select Material Properties of
Test Products
Material
Density
(g/cm3)
1.11
Glass
Volume
(%)
13
Tensile
Strength
(MPa)
101
Tensile
Modulus
(GPa)
6
TP-BMC30
TP-BMC40
GMT+
Satin
Twill
Uni
1.19
19
114
7
1.19
1.47
1.47
1.64
19
35
35
50
130
240
400*
675*
6
13
22*
37*
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Improving Performance of BMC
Bumper Beams
DOE Study
• Materials Processed on conventional BMC and
GMT Equipment
• BMC logs were extruded. Prepreg Plates Cut to
Shape and heated in GMT oven
• Projected Area of Part was 370 x 1520 mm, with
Nominal Thickness of 8 mm
• GMT & Prepreg added in Combinations of
Fractions of Prepreg to Total Beam Weight
• 3 Beams in each Combination Molded for
Experiment
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Process Conditions for
Experiments
PARAMETER
SETTING
Oven Type
Zone 1 Temperature
Zone 2 Temperature
Zone 3 Temperature
Zone 4 Temperature
Oven Dwell Time
Press Size
Press Tonnage
Press Closing Speed
Tool Temperature
Process Dwell Time
IR
210C (upper), 210C (lower)
210C (upper), 210C (lower)
190C (upper), 190C (lower)
190C (upper), 190C (lower)
114 sec/zone
907 t
454 t to start
18 cm/sec
55C
45 sec
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Improving Performance of GMT
Bumper Beams
DOE Study
• Pre-preg Materials Placed & Indexed in Oven
to exit at the Same Time as BMC log
• Material Temperature 210-240C @ Oven Exit
• Prepreg Heated at same Rate due to Similar
Thermal Properties & Thickness
• Materials Placed in Tool (Transfer Time 20-30
sec) and then compression molded
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Experimental Layout for the Taguchi L-9
EXPERIMENT
NUMBER
TP-BMC
GLASS %
(wt)
PREPREG
% BY WT.
TONNAGE
(t)
TYPE OF PREPREG
NUMBER OF PARTS
1
2
3
4
5
6
7
8
9
Control 20%
Control 30%
Control 40%
20%
20%
20%
30%
30%
30%
40%
40%
40%
20%
30%
40%
25%
50%
75%
25%
50%
75%
25%
50%
75%
0%
0%
0%
450
650
900
650
900
450
900
450
650
450
450
450
Satin
Uni
Twill
Twill
Satin
Uni
Uni
Twill
Satin
none
none
none
3
3
3
3
3
3
3
3
3
5
5
5
Note: Equivalent Full Factorial Design would require 81 experiments
: Number of Experiments = (levels)Factors = 34 = 81 experiments
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Improving Performance of BMC
Bumper Beams
DOE Study
• 27 Beams, 3 each of 9 Variants, plus 15 BMCOnly
• Control Beams were BMC 20%, BMC 30%,
and BMC 40%,
• Experiment was not Randomized to Minimize
Duration
Taguchi Study Permits Interpolation of Results
to Other Variants not Physically
Manufactured or Tested
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Static Load for BMC and Prepreg
Sta tic Loa d for BMC/ Prepreg Experiment
35
Ave ra g e
30
25
20
15
BMC40
BMC30
BMC20
9
8
7
6
5
4
3
2
10
1
Ave ra ge S ta tic Loa d, kN
Sta nd a rd De via tio n
Expe rime nt Numbe r
Experiment Number
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Mass of Beams for GMT/Prepreg
Ma ss for BMC/ Prepreg Experiment
Sta nd a rd De via tio n
Ave ra g e
4.5
4
M a ss, kg
3.5
3
2.5
2
1.5
1
0.5
Taguchi
BMC40
BMC30
BMC20
C-GMT40+
C-GMT30+
9
8
Experiment Number
Expe rime nt Numbe r
GMT+
GMT
7
9
6
8
7
65
5
4
4
33
2
2
1
1
0
30
Mean Static Load vs. Beam Mass
for BMC/Prepreg
Static Load versus Beam Mass for GMT/Prepreg Experiment
Sta tic Loa d versus Bea m Ma ss for BMC/ Prepreg Experiment
Ave ra g e
S ta tic Loa d, kN
35
Sta nd a rd De via tio n
30
25
20
15
10
5
0
3
3.2
3.4
3.6
3.8
4
4.2
Be a m M a ss, kg
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Mean Specific Static Load for
BMC/Prepreg
Sta tic Loa d versus Gla ss Volume %
Ave ra g e
35
Sta nd a rd De via tio n
25
20
15
0
5
10 Experiment
15
20
Number
25
C-GMT40+
0
C-GMT30+
5
GMT+
10
GMT
S ta tic Loa d,kN
30
30
35
40
Gla ss Volume %
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Level Averages for
GMT/Prepreg
Dimensionless Load** Level Averages
850
750
700
650
600
550
Uni
Twill
Satin
TonHigh
TonMed
TonLow
Prepreg75
Prepreg50
Prepreg25
BMC40
BMC30
500
BMC20
Dimensionless Load
800
Variable Level
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Analysis of Variance(ANOVA) Results
Significance Of Each Variable
Source
TP-BMC Glass
Weight Percentage
Weight Percentage
of Prepreg
Press Tonnage
GMT Type
(metric)
Prepreg Type
e1
e2
Lay-Up
Total
Df
Sum of Squares
Variance
2
6604
3302
2
22065
2
8121
4210
5710
2855
2
2
0
18
26
2
82651
0
4645
4622
47444
F
S'
10.66
rho %
5984
11032 1498535.63 19.682144528446
310
2063
13.60
41325
9.20
12.52
2311
3.04
0
18
0
13704
761
Total
26
150007
5770
81128
18.96 45.20
54.08
5091
9531
e1
e2
Taguchi
54.28
7802
12.61
17539
7122
16.44
10.73
11.69
3100
2.07
19795
13.20
15.01
100
100
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Optimum Levels of Each Variable
as Determined from Level
Averages Graph
Variable
Optimum Level
TP-BMC Glass Weight Percentage
Weight fraction of prepreg
Press Tonnage (metric)
Prepreg type
TP-BMC with 30 % glass fibers
75% percentage prepreg
900 tons
Satin (1:1)
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Improving Performance of GMT
Bumper Beams
Confirmation Run
• Confirmation Run used same Process
Settings
• Used the GMT product with 30% Chopped
Fiber & Each of the Prepreg Materials
• 5 Beams with Each Prepreg Material were
made, plus 5 Control Beams
• Test Results confirmed C-GMT 30+
Product was Improved by Adding Prepreg
Material
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Conclusions
• Comingled thermoplastic prepregs improve the stiffness
properties of TP-BMC composites by 15% to 20%.
• The static load of composite bumper beams increases with up to
a maximum of 22% to 25% glass (by volume).
• Comingled thermoplastic prepregs improve the static bumper
performance of TP-BMC composites to a level superior to
published results for standard GMT materials.
• The significant material and processing parameters in this
experiment are TP-BMC glass weight percentage, weight
percentage prepreg, press tonnage, and prepreg type.
• The optimum levels for maximum dimensionless static load
are:




TP-BMC glass weight percentage = 30%
Weight percentage prepreg = 75%
Press tonnage (metric) = 900 t
Prepreg type = Satin
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