statistics and probability docs/fa10so… · © 2010 motorola – all rights reserved j. wojack,...

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

Statistics and Probabilityin

Mechanical DesignJason Wojack

Motorola -

Mobile Devices


Why Stastical

Methodologies

• Disciplined Approach

• Quantifiable Decision Criteria

• Optimization


Six Sigma Methodologies


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


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


DFSS

Tools…

Tolerance Analysis Measurement System Analysis

Process Capability

Comparitive

Methods

Design of Experiments

Reliability Modeling

Monte Carlo Simulation DACE

Etc….


Parameters:µ, σ

Population Sampling


Probability Distributions


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%


Allowable VariationActual Variation

Process Capability

Cp = AllowableActual

( )

USLLSL

Cp =2.0USLLSL

( )

2.0 distributions can fit within the specification


Process Capability

< 1.0

Poor Capability

1.0 –

1.5

MarginalCapability

> 1.5

Good Capability

> 2.0

Motorola 6σCapability

BestBad


Process Capability( )

USLLSL

Cp = 2.0

( )

USLLSL( )

USLLSL

CpIndependent

of the

target.


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


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




yes

no trial run


spec / print


Measure Quantity ?

no


Process Control

Modify Spec.

yesnoyes



trial run

ReliabilityAnalysis


Development Flow (simplified )

part / processdesign

TA Acceptable?

yes

no

ConceptTolerance Analysis


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)

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


??

+/-

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




yes

no trial run


spec / print


Measure Quantity ?

no


Process Control

Modify Spec.

yesnoyes



trial run

ReliabilityAnalysis



trial run

…..

spec / print(requirement)Good TA




WHY?Measurement Error Bad Decisions


Measurement Systems AnalysisCharacteristics:

Stability Discrimination Accuracy (Bias)

Linearity

Precision


Measurement Systems AnalysisTotal Variation:

Precision:




yes

no trial run


spec / print


Measure Quantity ?

no


Process Control

Modify Spec.

yesnoyes



trial run

ReliabilityAnalysis


Perform Measurements

yes

Determine Quantity to Measure

5 or 20 or 100???

MSA

Improve the Measurement System

MSAAcceptable?

no





yes

no trial run


spec / print


Measure Quantity ?

no


Process Control

Modify Spec.

yesnoyes



trial run

ReliabilityAnalysis


Process Control


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


Data Analysis




yes

no trial run


spec / print


Measure Quantity ?

no


Process Control

Modify Spec.

yesnoyes


trial run

ReliabilityAnalysis



Comparative Methods



LevelFactors

-> 22 = 4 Runs



2k

-> LevelFactors




yes

no trial run


spec / print


Measure Quantity ?

no


Process Control

Modify Spec.

yesnoyes



trial run

ReliabilityAnalysis


Statistical Process Control

SPC Goals: • Predicable process

• Consistent σ

(Cp)

• Centered distribution (Cpk)


Control Charts


Control Charts; X-Bar / R




yes

no trial run


spec / print


Measure Quantity ?

no


Process Control

Modify Spec.

yesnoyes



trial run

ReliabilityAnalysis



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.




yes

no trial run


spec / print


Measure Quantity ?

no


Process Control

Modify Spec.

yesnoyes



trial run

ReliabilityAnalysis

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