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© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Statistics and Probabilityin
Mechanical DesignJason Wojack
Motorola -
Mobile Devices
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Why Stastical
Methodologies
• Disciplined Approach
• Quantifiable Decision Criteria
• Optimization
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Six Sigma Methodologies
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
DFSS
–
CDOV
ProcessDFSS
–
Design for Six Sigma
CDOVConcept•Prioritize Customer Needs
•Select Superior Concept
Design•Baseline Design
•Customer needs captured in Design Requirements
Optimize•Optimize “Critical to Quality”
Parameters
Verify•Ensurelong term performance
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
CDOV
Process
Concept
Design
Optimize
Verify
MSAProcessCapability
ConfidenceIntervals
ComparativeMethods
MonteCarlo CPM FMEA Control
Charts
Regression DOE RSM RobustDesign
ToleranceAnalysis DACE
MajorSteps VOC KJ
AnalysisInitiateCPM
PughAnalysis
ReliabilityModeling
SystemReliability
SystemAvailability
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
DFSS
Tools…
Tolerance Analysis Measurement System Analysis
Process Capability
Comparitive
Methods
Design of Experiments
Reliability Modeling
Monte Carlo Simulation DACE
Etc….
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Parameters:µ, σ
Population Sampling
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Probability Distributions
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
ƒ (x) =σ(2π)1/2
e-(x-µ)
/ (2σ2)
µ
= meanσ
= standard deviation
2
Normal Distribution( )
Prob
abili
ty D
ensi
ty
1σ1σ
68.27%
-3σ 3σ
99.7%
95.45%6σ-6σ
99.9999998%
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Allowable VariationActual Variation
Process Capability
Cp = AllowableActual
( )
USLLSL
Cp =2.0USLLSL
( )
2.0 distributions can fit within the specification
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Process Capability
< 1.0
Poor Capability
1.0 –
1.5
MarginalCapability
> 1.5
Good Capability
> 2.0
Motorola 6σCapability
BestBad
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Process Capability( )
USLLSL
Cp = 2.0
( )
USLLSL( )
USLLSL
CpIndependent
of the
target.
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Process Capability (Cp, Cpk)
Cp = USL -
LSL6σ
Cpk = min USL -
µ
, µ
-
LSL3σ 3σ
Cp = 2.0Cpk = 1.0
USLLSL µ
USLLSL
Cp = 1.0Cpk = 1.0
µ
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
ImproveManf. Process
trial run
ReliabilityModeling
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
ImproveManf. Process
trial run
ReliabilityAnalysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Development Flow (simplified )
part / processdesign
TA Acceptable?
yes
no
ConceptTolerance Analysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
TA: Worst Case Analysis
??
+/-
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 D0.55 + 1.07 + 1.60 + 1.35
4.54
Envelope Size
A B C D
4.67
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Root Sum Squared (RSS)
σgap
= +Te
3Cp( (2 Tpi
3Cpi( (2
i = 1
m
Σ
Variances can be added……
σ2
= σ2 + σ2
+ σ2
+ σ2
+ σ2A B C D Envelope
σgap
= 0.035
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
??
+/-
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
Root Sum Squared (RSS)
σgap
= 0.035 3σgap
= 0.105 6σgap
= 0.210
Envelope = A+B+C+D+6σgap = 4.46
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
ImproveManf. Process
trial run
ReliabilityAnalysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Measurement Systems Analysis
trial run
…..
spec / print(requirement)Good TA
Development Flow (simplified )
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Measurement Systems Analysis
WHY?Measurement Error Bad Decisions
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Measurement Systems AnalysisCharacteristics:
Stability Discrimination Accuracy (Bias)
Linearity
Precision
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Measurement Systems AnalysisTotal Variation:
Precision:
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Measurement Systems Analysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Measurement Systems Analysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
ImproveManf. Process
trial run
ReliabilityAnalysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Perform Measurements
yes
Determine Quantity to Measure
5 or 20 or 100???
MSA
Improve the Measurement System
MSAAcceptable?
no
Development Flow (simplified )
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
ImproveManf. Process
trial run
ReliabilityAnalysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Process Control
Development Flow (simplified )
yes
ModifySpecification
3.65
DataAcceptable?
no
Improve Manf. Process
Cp= 2.0Cpk = 2.0
Reliability Analysis
Analyze Data
USLLSL Cp = 2.0Cpk =0.0
MeasurementData
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Data Analysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
trial run
ReliabilityAnalysis
ImproveManf. Process
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Comparative Methods
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Comparative Methods
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Design of Experiments
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Design of Experiments
LevelFactors
-> 22 = 4 Runs
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Design of Experiments
2k
-> LevelFactors
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Design of Experiments
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
ImproveManf. Process
trial run
ReliabilityAnalysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Statistical Process Control
SPC Goals: • Predicable process
• Consistent σ
(Cp)
• Centered distribution (Cpk)
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Control Charts
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Control Charts
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Control Charts; X-Bar / R
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
ImproveManf. Process
trial run
ReliabilityAnalysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Reliability Analysis
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
Reliability Analysis
DecreasingFailureRate
bad region
ConstantFailureRate
desired region
IncreasingFailureRate
bad region
Useful Life
time
Failu
re R
ate
Infant Mortalityβ < 1
Normal Operationβ = 1
Wearoutβ > 1
This region is not always perfectly flat.
© 2010 Motorola – All Rights Reserved J. Wojack, 2010-03-22
part / processdesignConcept
Tolerance Analysis Good?
yes
no trial run
Measurement Systems Analysis
spec / print
Good?Improve Measurement
Measure Quantity ?
no
MeasurementsAnalyzeDataGood?
Process Control
Modify Spec.
yesnoyes
Development Flow ( simplified )
ImproveManf. Process
trial run
ReliabilityAnalysis
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