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

Statistics Review

Population Sampling Parameters:

µ, σ

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13-Mar-12

Probability Distributions

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13-Mar-12

Normal Distribution Normal (Gaussian) Distribution

0.45

Probability Density

0.4

0.35

ƒ (x) =

0.3

e-(x-µ)

2

/ (2σ2)

σ(2π)1/2

0.25

68.27%

0.2

µ = mean σ = standard deviation

0.15

0.1

0.05 0 -6 -6σ

-4.8

-3.6 -3σ-2.4

-1.2 1σ

-0

1.2 1σ

95.45% 99.7% 99.9999998%

2.43σ 3.6

4.8

6 6σ

Process Capability

Normal Distribution

Allowable Variation Cp = Actual Variation Allowable Normal (Gaussian) Distribution 0.45 0.4

Probability Density

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -6

LSL

-4.8

-3.6

-2.4

-1.2

-0

1.2

2.4

3.6

4.8

6

USL

Actual © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

13-Mar-12

Process Capability

Normal (Gaussian) Distribution 0.45 Normal (Gaussian) Distribution 0.4

0.4

0.35

Probability Density

0.45

Probability Density

0.35 0.3 0.25 0.2

0.3 0.25 0.2 0.15

0.15

0.1

0.1

0.05 0

0.05

-6

0 -6

LSL

-4.8

-3.6

-2.4

-1.2

-0

1.2

2.4

3.6

4.8

6

LSL

-4.8

-3.6

-2.4

-1.2

-0

1.2

2.4

3.6

4.8

6

USL

USL

Cp =2.0

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2.0 distributions can fit within the specification

13-Mar-12

Process Capability Poor Capability

Bad

< 1.0

1.0 – 1.5 Marginal

Capability

Good Capability

> 2.0

Motorola 6σ Capability

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Best

> 1.5

13-Mar-12

Process Capability Normal (Gaussian) Distribution 0.45 0.4

Probability Density

0.35 0.3 0.25 0.2 0.15 0.1

Cp

0.05 0 -6

-4.8

-3.6

-2.4

LSL

-1.2

-0

1.2

2.4

3.6

4.8

6

USL

Normal (Gaussian) Distribution 0.45

Independent of the

0.4

Probability Density

Cp = 2.0

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -6

LSL

-4.8

-3.6

-2.4

-1.2

-0

1.2

2.4

3.6

4.8

6

USL

Normal (Gaussian) Distribution 0.45 0.4

target.

Probability Density

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

LSL © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

-6

-4.8

-3.6

-2.4

-1.2

-0

1.2

2.4

3.6

USL

4.8

6

13-Mar-12

Process Capability; Cp & Cpk

USL - LSL Cp = 6σ

LSL

µ

Cp = 2.0 Cpk = 1.0

Cpk = min

USL

USL - µ , µ - LSL 3σ

3σ LSL

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µ

Cp = 1.0 Cpk = 1.0

USL

13-Mar-12

Cp, Cpk Example

Chip set Stack Up

Chipset Stack Up

W3G (current)

OMAP4430

Package Height Prediction (post-reflow) Memory Mold Cap (2-die) Memory Substrate Thk. Memory Ball Ht. (after reflow)

Min. 0.330 0.110 0.140

Nom. 0.340 0.140 0.190

Max. 0.350 0.170 0.240

Tol. 0.010 0.030 0.050

Min. 0.340 0.110 0.240

Nom. 0.370 0.140 0.290

M 0 0 0

OMAP Mold Cap (Ref.) OMAP / W3G Die Thickness (um) Die-to-Substrate Gap (um) OMAP Top Ball Ht. OMAP/W3G Substrate Thk. (w/o SM) OMAP/W3G Ball Ht. (after reflow) OMAP/W3G Ball Ht. (before reflow)

0.000 0.095 0.020 0.000 0.408 0.168 0.180

0.000 0.100 0.025 0.000 0.448 0.206 0.230

0.000 0.105 0.030 0.000 0.488 0.244 0.280

0.000 0.005 0.005 0.000 0.040 0.038 0.050

0.220 0.095 0.020 0.160 0.244 0.110 0.140

0.250 0.100 0.025 0.190 0.284 0.160 0.190

0 0 0 0 0 0 0

Total 2-Package POP Height (post-reflow): Max. 2-Package POP Height (post-reflow): Delta to OMAP4

1.24

1.32 1.40 0.06 delta

1.40 RSS:

1.15

1.24 1.34

1

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0.081

reference 13-Mar-12

Chipset Height Cp, Cpk +0.16mm

-6σ Cp = 2.8 Cpk = 2.7

Etna

Targa

+6σ

0.1154mm

Mechanical Development Process

Development Flow (simplified) Concept

Tolerance Analysis

part / process design

yes Good?

spec / print

no

Process Control

trial run

Improve Measurement

no yes

Reliability Modeling

Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

Good?

trial run

no

Good?

yes Analyze Data

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Measurements

Measure Quantity ?

13-Mar-12

Tolerance Analysis

Development Flow (simplified) Concept

Tolerance Analysis

part / process design

yes Good?

spec / print

no

Process Control

trial run

Improve Measurement

no yes

Reliability Modeling

Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

Good?

trial run

no

Good?

yes Analyze Data

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Measurements

Measure Quantity ?

13-Mar-12

Development Flow (simplified)

Concept

Tolerance Analysis

part / process design

no

TA Acceptable?

yes © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

13-Mar-12

Tolerance Analysis

Components

A B

4.54

0.50 +/- 0.05

1.00 +/- 0.07

A

B

C

0.55 + 1.07 +

C

1.60

D +

1.35

1.50 +/- 0.10

Envelope Size

D

1.25 +/- 0.10

?? +/- 0.13 © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

4.67 A

B

C

D 13-Mar-12

Root Sum Squared (RSS)

Variances can be added…… 2 σ2 = σA2 + σB2 + σC2 + σD2 + σEnvelope

( Σ

Te + 3Cp

(

(

m

Tpi 3Cpi

2

(

σgap =

2

i=1

σgap = 0.035 © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

13-Mar-12

Root Sum Squared (RSS)

Components

A

0.50 +/- 0.05

B

1.00 +/- 0.07

C

1.50 +/- 0.10

D

1.25 +/- 0.10

Gap Size

σgap = 0.035 3σgap = 0.105 6σgap = 0.210 Envelope = A+B+C+D+6σgap = 4.46

?? +/- 0.13 © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

A

B

C

D 13-Mar-12

RSS vs. Worst Case

Worst Case = 4.67

A

B

C

D

∆ = 4.5%

Statistical = 4.46

A

B

C

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D 13-Mar-12

TA Example: uUSB and HDMI Connectors

Measurement Systems Analysis (MSA)

Development Flow (simplified) Concept

Tolerance Analysis

part / process design

yes Good?

spec / print

no

Process Control

trial run

Improve Measurement

no yes

Reliability Modeling

Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

Good?

trial run

no

Good?

yes Analyze Data

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Measurements

Measure Quantity ?

13-Mar-12

Development Flow (simplified)

Good TA

spec / print

trial run

(requirement)

….. Measurement Systems Analysis

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Measurement Systems Analysis

WHY? Measurement Error

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Bad Decisions

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Measurement Systems Analysis

Characteristics: Stability

Discrimination

Accuracy (Bias)

Linearity

Precision

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Measurement Systems Analysis

Total Variation:

Precision:

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MSA; %GR & R

% GR&R 30%

Unacceptable

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13-Mar-12

Measurement Systems Analysis Example: Glue Weight

Measurement Size

Development Flow (simplified) Concept

Tolerance Analysis

part / process design

yes Good?

spec / print

no

Process Control

trial run

Improve Measurement

no yes

Reliability Modeling

Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

Good?

trial run

no

Good?

yes Analyze Data

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Measurements

Measure Quantity ?

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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13-Mar-12

Development Flow (simplified)

MSA

MSA Acceptable?

Determine Quantity to Measure

yes

no Improve the Measurement System

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5 or 20 or 100 ???

Perform Measurements

13-Mar-12

Data Analysis

Development Flow (simplified) Concept

Tolerance Analysis

part / process design

yes Good?

spec / print

no

Process Control

trial run

Improve Measurement

no yes

Reliability Modeling

Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

Good?

trial run

no

Good?

yes Analyze Data

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Measurements

Measure Quantity ?

13-Mar-12

Data Analysis

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13-Mar-12

Development Flow (simplified) Concept

Tolerance Analysis

part / process design

yes Good?

spec / print

no

Process Control

trial run

Improve Measurement

no yes

Reliability Modeling

Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

Good?

trial run

no

Good?

yes Analyze Data

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Measurements

Measure Quantity ?

13-Mar-12

Comparative Analysis & Design of Experiments (DOE)

Comparative Methods

• 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 Analysis Example: PCB Bow

Design of Experiments

• Efficient Experimental Method • Optimizes Processes and Designs • Allows for the Analysis of Interactions

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13-Mar-12

Process Control

Development Flow (simplified)

Concept

Tolerance Analysis

part / process design

Good?

yes

no

Process Control

trial run

no

yes Reliability Analysis

Good?

trial run Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

spec / print

Improve Measurement

no

Good?

yes Analyze Data

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Measurements

Measure Quantity ?

13-Mar-12

Process Control

Statistical Process Control

SPC Goals:

• Predicable process • Consistent σ (Cp) • Centered distribution © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

(Cpk)

13-Mar-12

Control Charts

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13-Mar-12

Control Charts

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13-Mar-12

Control Charts (X-bar / R)

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13-Mar-12

Development Flow (simplified)

Concept

Tolerance Analysis

part / process design

Good?

yes

no

Process Control

trial run

no

yes Reliability Analysis

Good?

trial run Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

spec / print

Improve Measurement

no

Good?

yes Analyze Data

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Measurements

Measure Quantity ?

13-Mar-12

Reliability Analysis

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13-Mar-12

Failure Rate

Reliability Analysis

Decreasing Failure Rate

Constant Failure Rate

Increasing Failure Rate

bad region

desired region

bad region

Normal Operation b=1 Infant Mortality b1 This region is not always perfectly flat.

13-Mar-12

Failure Analysis Example: Display Breakage

Development Flow (simplified) Analyze Data

Measurement Data

Normal (Gaussian) Distribution 0.45 0.4

Probability Density

0.35 0.3 0.25 0.2

Process Control

0.15 0.1 0.05 0 -6

-4.8

-3.6

-2.4

-1.2

-0

1.2

LSL

2.4

3.6

4.8

6

Cp = 2.0 Cpk =0.0

USL

Modify Specification 3.65

no

Improve Manf. Process

Data Acceptable?

yes Reliability Analysis

Cp= 2.0 Cpk = 2.0 © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

13-Mar-12

Tools Used

DFSS Tools

Tolerance Analysis

Measurement System Analysis

Process Capability

Comparitive Methods

Design of Experiments

Reliability Modeling

Monte Carlo Simulation

DACE

Etc….

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13-Mar-12

Development Flow (simplified)

Concept

Tolerance Analysis

part / process design

Good?

yes

no

Process Control

trial run

no

yes Reliability Modeling

Good?

trial run Measurement Systems Analysis

Improve Manf. Process

Modify Spec.

spec / print

Improve Measurement

no

Good?

yes Analyze Data

© 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

Measurements

Measure Quantity ?

13-Mar-12

Why Statistical Methods

• Disciplined Approach •Repeatable Results • Quantifiable Decision Criteria • Optimization © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

13-Mar-12

Q&A

Back Up Slides

6 Sigma Methodology

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13-Mar-12

DFSS – CDOV Process

DFSS – Design for Six Sigma

CDOV Concept •Prioritize Customer Needs •Select Superior Concept

Design •Baseline Design •Customer needs captured in Design Requirements

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Optimize

Verify

•Optimize “Critical to Quality” Parameters

•Ensure long term performance

13-Mar-12

CDOV Process

Concept Major Steps

VOC

KJ Analysis

Initiate CPM

Pugh Analysis

Design MSA

Process Capability

Confidence Comparative Intervals Methods

Monte Carlo

CPM

FMEA

Control Charts

Optimize Regression

DOE

RSM

Robust Design

Tolerance Analysis

DACE

Verify Reliability Modeling

System Reliability

System Availability

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13-Mar-12

Measurement Systems Analysis

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13-Mar-12

Design of Experiments

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13-Mar-12

Design of Experiments

LevelFactors -> 22 = 4 Runs © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

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Design of Experiments

2k -> LevelFactors

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13-Mar-12

Design of Experiments

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13-Mar-12