ME317 dfM at Stanford
Design for Manufacturability
ME317 dfM
Robust Parameter Design using
the Design of Experiments
“Robust design involves making the product’s function
least sensitive to various sources of noise”
Phadke, 1985
Kos Ishii, Professor
Department of Mechanical Engineering
Stanford University
ishii@stanford.edu
http://me317.stanford.edu
©2006 K. Ishii
ME317 dfM at Stanford
Robustness and DOE...
“What do you do when you don’t
have a good analytical model that
relates the variables and output?”
©2006 K. Ishii
ME317 dfM at Stanford
Agenda
 See ‘N Say
Too difficult to develop a numerical model
Must resort to experiments
 Robust Design: Design of Experiments
DoE: Basics
Orthogonal Arrays
Fractional Factorials
Inner / Outer Array Experiments
 Next Week (Please note sequence change)
Conceptual Robust Design
Robust Design Case Study / Confounding
©2006 K. Ishii
ME317 dfM at Stanford
DoE Robustness Process
1. Establish the concept configuration
2. Define performance goals
3. Identify factors which influence performance
4. Set ranges of factors to study
5. Design a set of experiments
Use orthogonal arrays to determine the effect
of each factor on mean and variance
Inner--Outer array to analyze environmental
effects (often called Blocking)
6. Build Models required by plan, run tests
7. Analyze results by analysis of variance
©2006 K. Ishii
ME317 dfM at Stanford
Robust Design by Experiments: Goal
 Use a limited set of experiments to determine the
design sensitivities
 Design the product and process to minimize the
sensitivity of quality measures to environment
©2006 K. Ishii
ME317 dfM at Stanford
Design of Experiments
 Factorial Experiments
Analyze the effects of variables simultaneously
Two-level factorial: monotonic and mostly linear
Three-level factorial: non-monotonic or non-linear
 General Linear Factorial Model (3 Factors)
Response = mean + A + B + C + AB + BC + AC + ABC
©2006 K. Ishii
ME317 dfM at Stanford
More Precisely...
 Taylor Series Expansion of Y around yo
Y = f(A, B, C)
Y  y0 
f 
f 
f 
A +
B +
C
A 
B
C 
Main Effects
+
 f f 
 f f 
f f 
AB 
AC 
BC
A B 
A C 
B C 
+
 f f f 
ABC
A B C 
Interaction
Effects
©2006 K. Ishii
ME317 dfM at Stanford
Full Factorial Experiments
 Full Factorial Experiments
Estimate all the main and interaction effects
Number of experiments multiply
Three factors at two levels: 23 = 8
Seven factors at two levels: 27 = 128
(-1,-1,+1)
C
(+1,-1,+1)
(-1,+1,+1)
(+1,+1,+1)
B
A
(-1,-1,-1)
(+1,-1,-1)
( -1,+1,-1)
(+1,+1,-1)
©2006 K. Ishii
ME317 dfM at Stanford
Fractional Factorial Experiments
 Fractional Factorial Experiments
Neglect higher order interactions
mean + A + B + C + AB + BC + AC + ABC
Interactions confounded with main effects can be dangerous
Smaller number of experiments
Three factors at two levels: 23-1 = 4 runs
Seven factors at two levels: 27-4 = 8 runs
(-1,-1,+1)
C
(+1,+1,+1)
B
A
( -1,+1,-1)
(+1,-1,-1)
©2006 K. Ishii
ME317 dfM at Stanford
Orthogonal Arrays
 For any pair of columns, all
combinations of factor levels occur, and
occur an equal number of times.
(-1,-1,+1)
(+1,+1,+1)
( -1,+1,-1)
 Fractional Factorial Arrays: Special Case(+1,-1,-1)
Orthogonal Arrays: up to 3 factors (23-1 = 4 runs)
Trial
1
2
3
4
A
+
+
Factors
B
+
+
C
+
+
-
NOTE: Also called Taguchi L4 Array
©2006 K. Ishii
ME317 dfM at Stanford
Derive the Sensitivities
using Orthogonality
Y1  y 0 
Y 
Y 
Y 
A 
B 
C
A 
B 
C 
Y2  y 0 
Y
Y
Y 
A 
B 
C
A
B 
C 
Y3  y 0 
Y 
Y 
Y 
A 
B 
C
A 
B 
C 
Y4  y0 
Y
Y 
Y 
A 
B 
C
A
B 
C 
Y
 Y 
 Y1  Y2  2y 0  2
A
A
 A 
 Add all 4 equations
y0 is mean of Yi
Y
 Y 
 Y2  Y3  2y 0  2
C
C
 C 
©2006 K. Ishii
ME317 dfM at Stanford
Taguchi’s Orthogonal Array
 Eight Run Orthogonal Array: up to 7 factors
at two levels (27-4 runs)
Trial No
1
2
3
4
5
6
7
8
1
1
1
1
1
2
2
2
2
2
1
1
2
2
1
1
2
2
3
1
1
2
2
2
2
1
1
Columns
4
1
2
1
2
1
2
1
2
5
1
2
1
2
2
1
2
1
6
1
2
2
1
1
2
2
1
7
1
2
2
1
2
1
1
2
NOTE: Also called Taguchi L8
©2006 K. Ishii
ME317 dfM at Stanford
Three Level Arrays
 Nine Run Orthogonal Array: up to 4 factors at
3 levels (34-2 runs)
T rial
1
2
3
4
5
6
7
8
9
1
1
1
1
2
2
2
3
3
3
Columns
2
1
2
3
1
2
3
1
2
3
3
1
2
3
2
3
1
3
1
2
4
1
2
3
3
1
2
2
3
1
NOTE: Also called Taguchi L9
©2006 K. Ishii
ME317 dfM at Stanford
Interaction Effects
 If value of A influences sensitivity of B
Interaction!
Need to consider factor AB
50
40
30
20
60
B2
B1
B2
B1
Response
Response
60
50
40
B1
B2
30
20
B1
B2
10
10
A1
A2
Factor A
A1
A2
Factor A
©2006 K. Ishii
ME317 dfM at Stanford
A Numerical Example
 Three factors at two levels
A = Ao + a
B = Bo + b
C = Co + c
 Identify range
90 < Ao < 110
8 < Bo < 12
0.8 < Co <1.2
 Environment Variables (Blocks)
a = + 1.0
b = + 0.1
c = + 0.01
©2006 K. Ishii
ME317 dfM at Stanford
Plan the Experiment
 Assign columns
Trial
1
2
3
4
B
+
+
Factors
C
+
+
A
+
+
-
B
8
8
12
12
Factors
C
0.8
1.2
0.8
1.2
A
90
110
90
110
 Enter Values
Trial
1
2
3
4
©2006 K. Ishii
ME317 dfM at Stanford
Let’s say we collected 4 data per trial
(Production Setting)
 A = 90; B = 8.0; C = 0.8
 Data 1: Y = 13.8
 Data 2: Y = 13.9
 Data 3: Y = 13.25
 Data 4: Y = 15.26
 Analysis of Mean and Variance
1
 Ymean
Ym   Yi  14.06
N
Variance
V
1
2

Y

Y

 0.547

i
m
N 1

Ym 2 

S/N = 10 log1 0
 25.58
 V 
©2006 K. Ishii
ME317 dfM at Stanford
Systematic Study of Noise Effects
 Inner Array
Factorial for Control Factors
 Outer Array
Factorial for Environmental Factors
 Simulation of how environment affects each design
 Lab experiments with much tighter control
OUTER ARRAY
x
y
z
INNER ARRAY
Trial
1
2
3
4
A
L
L
H
H
B
L
H
L
H
C
L
H
H
L
L
L
L
1
*
*
*
*
H
H
L
2
*
*
*
*
H
L
H
3
*
*
*
*
L
H
H
4
*
*
*
*
Mean S/N
*
*
*
*
*
*
*
*
©2006 K. Ishii
ME317 dfM at Stanford
L4-L4 Array
Spreadsheet
Inner Array
B
1
8
2
8
3
12
4
12
Outer Array
C
0.8
1.2
0.8
1.2
A
90
110
110
90
b
1
2
3
4
-0.1
-0.1
0.1
0.1
c
-0.01
0.01
-0.01
0.01
a
-1
1
1
-1
Raw Data Array for L4 Inner and L4 Outer
Trial
 Template
available
on ME317
website
B
C
A
Y
Y Mean
Y Variance
Y SN
11
12
13
14
7.9
7.9
8.1
8.1
0.79
0.81
0.79
0.81
89
91
91
89
13.81200
13.92800
13.24900
15.26500
14.06350
0.729542 24.331367
21
22
23
24
7.9
7.9
8.1
8.1
1.19
1.21
1.19
1.21
109
111
111
109
25.67000
25.58300
24.95100
27.70100
25.97625
1.424778 26.754056
31
32
33
34
11.9
11.9
12.1
12.1
0.79
0.81
0.79
0.81
109
111
111
109
11.32600
11.56000
10.90500
12.41300
11.55100
0.403689 25.191926
41
42
43
44
11.9
11.9
12.1
12.1
1.19
1.21
1.19
1.21
89
91
91
89
71.17900
69.93900
67.64600
76.01700
71.19525
12.47496 26.088629
©2006 K. Ishii
ME317 dfM at Stanford
Back to the Example
Compute the Mean (Use Orthogonality)
Y (B1) = (Y1 + Y2) /2 = (14.064 + 25.98) /2 = 20.02
Y (B2) = (Y3 + Y4) /2 = (11.551 + 71.179) /2 = 41.365
T rial
1
2
3
4
B
8
8
12
12
Columns
C
0.8
1.2
0.8
1.2
A
90
110
110
90
Y (C1) = (Y1 + Y3) /2 = 12.81
Y (C2) = (Y2 + Y4) /2 = 48.58
Y (A1) = (Y1 + Y4) /2 = 42.621
Y (A2) = (Y2 + Y3) /2 = 18.767
©2006 K. Ishii
ME317 dfM at Stanford
Mean Response
 All factors affect mean
35
Mean
30
25
20
A1
A2
B1
B2
C1
C2
©2006 K. Ishii
ME317 dfM at Stanford
Variance Response
 Which factors affect variance?
800
Variance
750
700
650
600
A1
A2
B1
B2
C1
C2
©2006 K. Ishii
ME317 dfM at Stanford
Four Types of Control Factors
 Classify based on effects to the response
1. Affect Variation and Mean
3. Affect Mean Only
2. Affect Variation Only
4. No effect on Variation or Mean
©2006 K. Ishii
ME317 dfM at Stanford
Strategy of Parameter Design
 Classification of Control Factors
Class I: affect both performance mean and variation
Class II: affect performance variation only
Class III: affect performance mean only
Class IV: affect nothing.
 Strategy
Select levels of class I and II to reduce variations
Select class III to adjust mean to target value
Set class IV at the most economical level
Big assumption: no significant interactions
©2006 K. Ishii
ME317 dfM at Stanford
Strategy for the Example
 All Parameters affect the Mean
 Parameter C
 Affects Variance most (Category I)
 Set at C=C? for least sensitivity
 Parameter A
 Also affects Variance (Category I)
 Set at A=A? for least sensitivity
 Parameter B
 Little effect on Variance (Cat. III)
 Use this to adjust response
800
Variance
750
700
650
600
A1
A2
B1
B2
C1
©2006 K. Ishii
C2
ME317 dfM at Stanford
What about See’N Say?
 Noise Factors
Spring Stiffness
Belt Tension
 Control Factors
Rotor Ball Bearing Size (3)
Friction Pad (3)
Pad Placement (3)
Rotor Pulley Diameter (3)
 Design of Experiments
Full Factorial requires 81 prototypes!
Actually made 9 prototypes.
 Caution: Beware of Confounding!
This lecture ignored interactions
DoE requires careful planning; Will address in next lecture
©2006 K. Ishii
ME317 dfM at Stanford
HW#3 Steps 1 and 2
 Apply Orthogonal Arrays to Force Sensor
Analyze L4 inner and L4 outer
Inspect L8 inner and L4 outer
Use excel template on the web
Strain
Gauge
Material Aluminum
7
E = 1.25x10 psi
h±h
L±L
b±b
©2006 K. Ishii
ME317 dfM at Stanford
HW #3 Robust Design of a Helicopter
 Competition for longest flight time!
Use DOE to optimize
©2006 K. Ishii