jason wojack • quantifiable decision criteria docs...© 2010 motorola – all rights reserved j....

© 2010 Motorola – All Rights Reserved J. Wojack, 2011-03-22

Statistics and Probabilityin

Mechanical DesignJason Wojack

Motorola - Mobile Devices


Why Statistical Methods

• Disciplined Approach

• Quantifiable Decision Criteria

• Optimization


Statistics Review


Parameters:!, σ

Population Sampling


Probability Distributions


ƒ (x) = σ(2π)1/2

e-(x-!) / (2σ2)

! = mean" = standard deviation

2

Normal DistributionNormal (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

Prob

abili

ty D

ensi

ty

1σ1σ

68.27%

-3σ 3σ

99.7%

95.45%6σ-6σ

99.9999998%


Process Capability


Allowable VariationActual Variation

Process Capability

Cp = AllowableActual

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

USLLSL

Allowable

Actual


Process Capability


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

USLLSL

Cp =2.0USLLSL


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

2.0 distributionscan fit within thespecification


Process Capability

< 1.0 Poor Capability

1.0 – 1.5 Marginal Capability

> 1.5 Good Capability

> 2.0 Motorola 6σ Capability

BestBad


Process CapabilityNormal (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

USLLSL


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

USLLSLNormal (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

USLLSL

Cp = 2.0

CpIndependent

of the

target.


Process Capability (Cp, Cpk)

Cp = USL - LSL6σ

Cpk = min USL - ! , ! - LSL3σ 3σ

Cp = 2.0Cpk = 1.0

USLLSL !

USLLSL

Cp = 1.0Cpk = 1.0

!


Mechanical DevelopmentProcess


part / processdesignConcept

ToleranceAnalysis Good?

yes

no trial run

MeasurementSystems Analysis

spec / print

Good?ImproveMeasurement

MeasureQuantity ?

no

MeasurementsAnalyzeDataGood?

ProcessControl

ModifySpec.

yesno

yes

Development Flow ( simplified )

ImproveManf. Process

trial run

ReliabilityModeling




yes

no trial run


spec / print


MeasureQuantity ?

no


ProcessControl

ModifySpec.

yesno

yes



trial run

ReliabilityAnalysis


Development Flow (simplified )

part / processdesign

TA Acceptable?

yes

no

Concept ToleranceAnalysis


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


Root Sum Squared (RSS)

"gap = + Te3Cp

( (2 Tpi3Cpi

( (2i = 1

m

#

Variances can be added……

"2 = "2 + "2 + "2 + "2 + "2A B C D Envelope

"gap = 0.035


?? +/- 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)

"gap = 0.0353"gap = 0.1056"gap = 0.210

Envelope = A+B+C+D+6σgap = 4.46


A B C D

Statistical = 4.46

Worst Case = 4.67

A B C D

Worst Case vs. RSS

" = 4.5%




yes

no trial run


spec / print


MeasureQuantity ?

no


ProcessControl

ModifySpec.

yesno

yes



trial run

ReliabilityAnalysis


Measurement SystemsAnalysis

trial run

…..

spec / print(requirement)Good TA



Measurement Systems Analysis

WHY?Measurement Error Bad Decisions


Measurement Systems AnalysisCharacteristics:

Stability Discrimination Accuracy (Bias)

Linearity

Precision


Measurement Systems AnalysisTotal Variation:

Precision:


MSA; %GR&R

% GR&R

<10% Acceptable

10% - 30% Ok; non-criticalmeasurements

>30% Unacceptable




yes

no trial run


spec / print


MeasureQuantity ?

no


ProcessControl

ModifySpec.

yesno

yes



trial run

ReliabilityAnalysis


Perform Measurements

yes

Determine Quantity toMeasure

5 or 20 or 100???

MSA

Improve theMeasurement System

MSAAcceptable?

no





yes

no trial run


spec / print


MeasureQuantity ?

no


ProcessControl

ModifySpec.

yesno

yes



trial run

ReliabilityAnalysis


Process Control


yes

ModifySpecification

3.65

DataAcceptable?

no

Improve Manf.Process

Cp= 2.0Cpk = 2.0

Reliability Analysis

Analyze DataNormal (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

Prob

abili

ty D

ensi

ty

USLLSL Cp = 2.0Cpk =0.0

MeasurementData


Data Analysis




yes

no trial run


spec / print


MeasureQuantity ?

no


ProcessControl

ModifySpec.

yesno

yes


trial run

ReliabilityAnalysis



Comparative Methods• Analytical method to evaluate changes &

differences.

• Examples:• Different vendors• Change to a process


Comparative Methods


Design of Experiments



LevelFactors -> 22 = 4 Runs



2k -> LevelFactors






yes

no trial run


spec / print


MeasureQuantity ?

no


ProcessControl

ModifySpec.

yesno

yes



trial run

ReliabilityAnalysis


Statistical Process Control

SPC Goals: • Predicable process

• Consistent σ (Cp)

• Centered distribution (Cpk)


Control Charts


Control Charts


Control Charts; X-Bar / R




yes

no trial run


spec / print


MeasureQuantity ?

no


ProcessControl

ModifySpec.

yesno

yes



trial run

ReliabilityAnalysis





DecreasingFailure

Rate

bad region

ConstantFailure

Rate

desired region

IncreasingFailure

Rate

bad region

Useful Life

time

Failu

re R

ate

Infant Mortality! < 1

Normal Operation! = 1

Wearout! > 1

This region is not always perfectly flat.


Tools Used


DFSS Tools…

Tolerance Analysis MeasurementSystem Analysis

Process Capability

ComparitiveMethods

Design ofExperiments

Reliability Modeling

Monte CarloSimulation DACE

Etc….




yes

no trial run


spec / print


MeasureQuantity ?

no


ProcessControl

ModifySpec.

yesno

yes



trial run

ReliabilityAnalysis


Q & A


Back up Slides


Six Sigma Methodologies


DFSS – CDOV ProcessDFSS – Design for Six Sigma

CDOVConcept•PrioritizeCustomer Needs

•Select SuperiorConcept

Design•Baseline Design

•Customer needscaptured inDesignRequirements

Optimize•Optimize“Critical to Quality”Parameters

Verify•Ensurelong termperformance


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


Measurement Systems Analysis

jason wojack • quantifiable decision criteria docs...© 2010 motorola – all rights reserved j....

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