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MOTOROLA and the Stylized M Logo are trademarks or registered trademarks of Motorola Trademark Holdings, LLC.
All other trademarks are the property of their respective owners. © 2010 Motorola Mobility, Inc. All rights reserved.
Statistics & Probability in
Mechanical Design
Jason Wojack
Monday, March 12th, 2012
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Why Statistical Methods
• Disciplined Approach
•Repeatable Results
• Quantifiable Decision Criteria
• Optimization
© 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary
13-Mar-12
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Parameters:
µ, σ
Population Sampling
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13-Mar-12
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Probability Distributions
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13-Mar-12
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ƒ (x) = σ(2π)1/2
e-(x-µ) / (2σ2)
µ = mean
σ = standard deviation
2
Normal (Gaussian) Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6
Pro
bability D
ensity
Pro
babili
ty D
ensity
1σ 1σ
68.27%
-3σ 3σ
99.7%
95.45%
6σ -6σ
99.9999998%
Normal Distribution
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Process Capability
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Allowable Variation
Actual Variation Cp =
Allowable
Actual
Normal (Gaussian) Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6
Pro
bability D
ensity
USL LSL
Allowable
Actual
Normal Distribution
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13-Mar-12
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Normal (Gaussian) Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6
Pro
bability D
ensity
USL LSL
Cp =2.0
USL LSL
Normal (Gaussian) Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6
Pro
bability D
ensity
2.0 distributions can fit within the specification
Process Capability
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13-Mar-12
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< 1.0 Poor Capability
1.0 – 1.5 Marginal Capability
> 1.5 Good Capability
> 2.0 Motorola 6σ Capability
Best
Bad
Process Capability
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13-Mar-12
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Normal (Gaussian) Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6
Pro
bability D
ensity
USL LSL
Normal (Gaussian) Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6
Pro
bability D
ensity
USL LSL
Normal (Gaussian) Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6
Pro
bability D
ensity
USL LSL
Cp = 2.0
Cp Independent
of the
target.
Process Capability
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13-Mar-12
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Cp = USL - LSL
6σ
Cpk = min USL - µ , µ - LSL
3σ 3σ
Cp = 2.0 Cpk = 1.0
USL LSL µ
USL LSL
Cp = 1.0 Cpk = 1.0
µ
Process Capability; Cp & Cpk
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Cp, Cpk Example
Chip set Stack Up
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Package Height Prediction (post-reflow) Min. Nom. Max. Tol. Min. Nom. Max. Tol.
Memory Mold Cap (2-die) 0.330 0.340 0.350 0.010 0.340 0.370 0.400 0.030
Memory Substrate Thk. 0.110 0.140 0.170 0.030 0.110 0.140 0.170 0.030
Memory Ball Ht. (after reflow) 0.140 0.190 0.240 0.050 0.240 0.290 0.340 0.050
OMAP Mold Cap (Ref.) 0.000 0.000 0.000 0.000 0.220 0.250 0.280 0.030
OMAP / W3G Die Thickness (um) 0.095 0.100 0.105 0.005 0.095 0.100 0.105 0.005
Die-to-Substrate Gap (um) 0.020 0.025 0.030 0.005 0.020 0.025 0.030 0.005
OMAP Top Ball Ht. 0.000 0.000 0.000 0.000 0.160 0.190 0.220 0.030
OMAP/W3G Substrate Thk. (w/o SM) 0.408 0.448 0.488 0.040 0.244 0.284 0.324 0.040
OMAP/W3G Ball Ht. (after reflow) 0.168 0.206 0.244 0.038 0.110 0.160 0.210 0.050
OMAP/W3G Ball Ht. (before reflow) 0.180 0.230 0.280 0.050 0.140 0.190 0.240 0.050
Total 2-Package POP Height (post-reflow): 1.24 1.32 1.40 1.15 1.24 1.34
Max. 2-Package POP Height (post-reflow): 1.40 RSS: 0.081 1.34 RSS: 0.096
Delta to OMAP4 0.06
delta reference
W3G (current) OMAP4430 TMV
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13-Mar-12
Chipset Stack Up
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-6σ +6σ
Etna Targa
0.1154mm
+0.16mm
Cp = 2.8
Cpk = 2.7
Chipset Height Cp, Cpk
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Mechanical Development
Process
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Development Flow (simplified)
part / process
design Concept
Tolerance
Analysis Good?
yes
no trial run
Measurement Systems
Analysis
spec / print
Good? Improve
Measurement
Measure
Quantity ?
no
Measurements Analyze
Data Good?
Process
Control
Modify
Spec.
yes no
yes
Improve
Manf. Process
trial run
Reliability
Modeling
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13-Mar-12
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Tolerance Analysis
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Development Flow (simplified)
part / process
design Concept
Tolerance
Analysis Good?
yes
no trial run
Measurement Systems
Analysis
spec / print
Good? Improve
Measurement
Measure
Quantity ?
no
Measurements Analyze
Data Good?
Process
Control
Modify
Spec.
yes no
yes
Improve
Manf. Process
trial run
Reliability
Modeling
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13-Mar-12
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part / process design
TA Acceptable?
yes
no
Concept Tolerance Analysis
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Development Flow (simplified)
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?? +/- 0.13
A 0.50 +/- 0.05 1.00 +/- 0.07 1.50 +/- 0.10 1.25 +/- 0.10
B
C
D
Components
A B C D
0.55 + 1.07 + 1.60 + 1.35
4.54
Envelope Size
A B C D
4.67
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13-Mar-12
Tolerance Analysis
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σgap =
+ Te
3Cp (
( 2
Tpi
3Cpi (
( 2
i = 1
m
Σ
Variances can be added……
σ2 = σ2 + σ2 + σ2 + σ2 + σ2 A B C D Envelope
σgap = 0.035
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13-Mar-12
Root Sum Squared (RSS)
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?? +/- 0.13
A 0.50 +/- 0.05 1.00 +/- 0.07 1.50 +/- 0.10 1.25 +/- 0.10
B
C
D
Components
Gap Size
A B C D
σgap = 0.035
3σgap = 0.105
6σgap = 0.210
Envelope = A+B+C+D+6σgap = 4.46
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13-Mar-12
Root Sum Squared (RSS)
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A B C D
Statistical = 4.46
Worst Case = 4.67
A B C D ∆ = 4.5%
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13-Mar-12
RSS vs. Worst Case
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TA Example:
uUSB and HDMI
Connectors
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Measurement Systems
Analysis
(MSA)
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Development Flow (simplified)
part / process
design Concept
Tolerance
Analysis Good?
yes
no trial run
Measurement Systems
Analysis
spec / print
Good? Improve
Measurement
Measure
Quantity ?
no
Measurements Analyze
Data Good?
Process
Control
Modify
Spec.
yes no
yes
Improve
Manf. Process
trial run
Reliability
Modeling
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13-Mar-12
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Measurement Systems Analysis
trial run
…..
spec / print (requirement)
Good TA
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Development Flow (simplified)
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WHY?
Measurement Error Bad Decisions
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Measurement Systems Analysis
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Characteristics:
Stability Discrimination Accuracy (Bias)
Linearity
Precision
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Measurement Systems Analysis
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Total Variation:
Precision:
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Measurement Systems Analysis
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% GR&R
<10% Acceptable
10% - 30% Ok; non-critical
measurements
>30% Unacceptable
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13-Mar-12
MSA; %GR & R
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Measurement Systems
Analysis
Example:
Glue Weight
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Development Flow (simplified)
part / process
design Concept
Tolerance
Analysis Good?
yes
no trial run
Measurement Systems
Analysis
spec / print
Good? Improve
Measurement
Measure
Quantity ?
no
Measurements Analyze
Data Good?
Process
Control
Modify
Spec.
yes no
yes
Improve
Manf. Process
trial run
Reliability
Modeling
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13-Mar-12
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© 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary
13-Mar-12
Sample Size
• Dependent on the type of analysis to be performed
• Apply applicable formula
• Example: Sample Mean to a known population
n = (Zα + Zβ) • σ2
δ2
α = level of acceptability of a false positive (0.05 is typical)
β = level of acceptability of a false negative (0.10 is typical)
σ = known standard deviation
m = amount of difference that matters (practical difference)
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Perform Measurements
yes
Determine Quantity to Measure
5 or 20 or 100 ???
MSA
Improve the Measurement System
MSA Acceptable?
no
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13-Mar-12
Development Flow (simplified)
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Development Flow (simplified)
part / process
design Concept
Tolerance
Analysis Good?
yes
no trial run
Measurement Systems
Analysis
spec / print
Good? Improve
Measurement
Measure
Quantity ?
no
Measurements Analyze
Data Good?
Process
Control
Modify
Spec.
yes no
yes
Improve
Manf. Process
trial run
Reliability
Modeling
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13-Mar-12
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13-Mar-12
Data Analysis
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Development Flow (simplified)
part / process
design Concept
Tolerance
Analysis Good?
yes
no trial run
Measurement Systems
Analysis
spec / print
Good? Improve
Measurement
Measure
Quantity ?
no
Measurements Analyze
Data Good?
Process
Control
Modify
Spec.
yes no
yes
trial run
Reliability
Modeling
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13-Mar-12
Improve
Manf. Process
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Comparative Analysis
&
Design of Experiments
(DOE)
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• Analytical method to evaluate changes & differences.
• Examples: • Different vendors • Change to a process
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13-Mar-12
Comparative Methods
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Comparative Methods
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Comparative Analysis
Example:
PCB Bow
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Design of Experiments
• Efficient Experimental Method
• Optimizes Processes and Designs
• Allows for the Analysis of Interactions
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part / process design
Concept Tolerance Analysis
Good? yes
no trial run
Measurement Systems Analysis
spec / print
Good? Improve
Measurement
Measure Quantity ?
no
Measurements Analyze
Data Good?
Process Control
Modify Spec.
yes no
yes
Improve Manf. Process
trial run
Reliability Analysis
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13-Mar-12
Development Flow (simplified)
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SPC Goals: • Predicable process
• Consistent σ (Cp)
• Centered distribution (Cpk)
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Statistical Process Control
Page 52
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Control Charts
Page 53
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Control Charts
Page 54
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Control Charts (X-bar / R)
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part / process design
Concept Tolerance Analysis
Good? yes
no trial run
Measurement Systems Analysis
spec / print
Good? Improve
Measurement
Measure Quantity ?
no
Measurements Analyze
Data Good?
Process Control
Modify Spec.
yes no
yes
Improve Manf. Process
trial run
Reliability Analysis
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13-Mar-12
Development Flow (simplified)
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13-Mar-12
Reliability Analysis
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Decreasing
Failure
Rate
bad region
Constant
Failure
Rate
desired region
Increasing
Failure
Rate
bad region
Useful Life
time
Fa
ilu
re R
ate
Infant Mortality
b < 1
Normal Operation
b = 1
Wearout
b > 1
This region is not always perfectly flat.
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13-Mar-12
Reliability Analysis
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Failure Analysis
Example:
Display Breakage
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Process Control
yes
Modify Specification
3.65
Data Acceptable?
no
Improve Manf. Process
Cp= 2.0 Cpk = 2.0
Reliability Analysis
Analyze Data
Normal (Gaussian) Distribution
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6
Pro
babili
ty D
ensi
ty
USL LSL Cp = 2.0 Cpk =0.0
Measurement Data
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13-Mar-12
Development Flow (simplified)
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Tolerance Analysis
Measurement System Analysis
Process Capability
Comparitive Methods
Design of Experiments
Reliability Modeling
Monte Carlo Simulation
DACE
Etc…. © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary
13-Mar-12
DFSS Tools
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part / process design
Concept Tolerance Analysis
Good? yes
no trial run
Measurement Systems Analysis
spec / print
Good? Improve
Measurement
Measure Quantity ?
no
Measurements Analyze
Data Good?
Process Control
Modify Spec.
yes no
yes
Improve Manf. Process
trial run
Reliability Modeling
© 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary
13-Mar-12
Development Flow (simplified)
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Why Statistical Methods
• Disciplined Approach
•Repeatable Results
• Quantifiable Decision Criteria
• Optimization
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13-Mar-12
Page 66
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13-Mar-12
6 Sigma Methodology
Page 67
DFSS – Design for Six Sigma
CDOV
Concept
•Prioritize Customer Needs
•Select Superior Concept
Design
•Baseline Design
•Customer needs captured in Design Requirements
Optimize
•Optimize “Critical to Quality” Parameters
Verify
•Ensure long term performance
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13-Mar-12
DFSS – CDOV Process
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Concept
Design
Optimize
Verify
MSA Process Capability
Confidence Intervals
Comparative Methods
Monte Carlo
CPM FMEA Control Charts
Regression DOE RSM Robust Design
Tolerance Analysis
DACE
Major Steps
VOC KJ Analysis
Initiate CPM
Pugh Analysis
Reliability Modeling
System Reliability
System Availability
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13-Mar-12
CDOV Process
Page 69
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13-Mar-12
Measurement Systems Analysis
Page 70
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13-Mar-12
Design of Experiments
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LevelFactors -> 22 = 4 Runs
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13-Mar-12
Design of Experiments
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2k -> LevelFactors
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13-Mar-12
Design of Experiments
Page 73
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13-Mar-12
Design of Experiments