Statistics & Probability in Mechanical Design...Mechanical Development Process Development Flow (simplified) part / process design Concept Tolerance Analysis Good? yes no trial run

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Statistics & Probability in

Mechanical Design

Jason Wojack

Monday, March 12th, 2012

Why Statistical Methods

• Disciplined Approach

•Repeatable Results

• Quantifiable Decision Criteria

• Optimization

© 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

13-Mar-12

Statistics Review

Parameters:

µ, σ

Population Sampling


13-Mar-12

Probability Distributions


13-Mar-12

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

Process Capability

Allowable Variation

Actual Variation Cp =

Allowable

Actual


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


13-Mar-12


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


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


13-Mar-12

< 1.0 Poor Capability

1.0 – 1.5 Marginal Capability

> 1.5 Good Capability

> 2.0 Motorola 6σ Capability

Best

Bad

Process Capability


13-Mar-12


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


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


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


13-Mar-12

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


13-Mar-12

Cp, Cpk Example

Chip set Stack Up

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


13-Mar-12

Chipset Stack Up

-6σ +6σ

Etna Targa

0.1154mm

+0.16mm

Cp = 2.8

Cpk = 2.7

Chipset Height Cp, Cpk

Mechanical Development

Process

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


13-Mar-12

Tolerance Analysis


part / process

design Concept

Tolerance

Analysis Good?

yes

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

no


Data Good?

Process

Control

Modify

Spec.

yes no

yes

Improve

Manf. Process

trial run

Reliability

Modeling


13-Mar-12

part / process design

TA Acceptable?

yes

no

Concept Tolerance Analysis


13-Mar-12


?? +/- 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


13-Mar-12

Tolerance Analysis

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


13-Mar-12

Root Sum Squared (RSS)

?? +/- 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


13-Mar-12

Root Sum Squared (RSS)

A B C D

Statistical = 4.46

Worst Case = 4.67

A B C D ∆ = 4.5%


13-Mar-12

RSS vs. Worst Case

TA Example:

uUSB and HDMI

Connectors

Measurement Systems

Analysis

(MSA)


part / process

design Concept

Tolerance

Analysis Good?

yes

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

no


Data Good?

Process

Control

Modify

Spec.

yes no

yes

Improve

Manf. Process

trial run

Reliability

Modeling


13-Mar-12

Measurement Systems Analysis

trial run

…..

spec / print (requirement)

Good TA


13-Mar-12


WHY?

Measurement Error Bad Decisions


13-Mar-12


Characteristics:

Stability Discrimination Accuracy (Bias)

Linearity

Precision


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Total Variation:

Precision:


13-Mar-12


% GR&R

<10% Acceptable

10% - 30% Ok; non-critical

measurements

>30% Unacceptable


13-Mar-12

MSA; %GR & R

Measurement Systems

Analysis

Example:

Glue Weight

Measurement Size


part / process

design Concept

Tolerance

Analysis Good?

yes

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

no


Data Good?

Process

Control

Modify

Spec.

yes no

yes

Improve

Manf. Process

trial run

Reliability

Modeling


13-Mar-12


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)

Perform Measurements

yes

Determine Quantity to Measure

5 or 20 or 100 ???

MSA

Improve the Measurement System

MSA Acceptable?

no


13-Mar-12


Data Analysis


part / process

design Concept

Tolerance

Analysis Good?

yes

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

no


Data Good?

Process

Control

Modify

Spec.

yes no

yes

Improve

Manf. Process

trial run

Reliability

Modeling


13-Mar-12


13-Mar-12

Data Analysis


part / process

design Concept

Tolerance

Analysis Good?

yes

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

no


Data Good?

Process

Control

Modify

Spec.

yes no

yes

trial run

Reliability

Modeling


13-Mar-12

Improve

Manf. Process

Comparative Analysis

&

Design of Experiments

(DOE)

• Analytical method to evaluate changes & differences.

• Examples: • Different vendors • Change to a process


13-Mar-12

Comparative Methods


13-Mar-12

Comparative Methods

Comparative Analysis

Example:

PCB Bow


13-Mar-12


• Efficient Experimental Method

• Optimizes Processes and Designs

• Allows for the Analysis of Interactions

Process Control



Good? yes

no trial run


spec / print

Good? Improve

Measurement

Measure Quantity ?

no


Data Good?

Process Control

Modify Spec.

yes no

yes

Improve Manf. Process

trial run

Reliability Analysis


13-Mar-12


Process Control

SPC Goals: • Predicable process

• Consistent σ (Cp)

• Centered distribution (Cpk)


13-Mar-12

Statistical Process Control


13-Mar-12

Control Charts


13-Mar-12

Control Charts


13-Mar-12

Control Charts (X-bar / R)



Good? yes

no trial run


spec / print

Good? Improve

Measurement

Measure Quantity ?

no


Data Good?

Process Control

Modify Spec.

yes no

yes


trial run



13-Mar-12



13-Mar-12


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.


13-Mar-12


Failure Analysis

Example:

Display Breakage

Process Control

yes

Modify Specification

3.65

Data Acceptable?

no


Cp= 2.0 Cpk = 2.0


Analyze Data


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


13-Mar-12


Tools Used

Tolerance Analysis

Measurement System Analysis

Process Capability

Comparitive Methods


Reliability Modeling

Monte Carlo Simulation

DACE

Etc…. © 2010 Motorola Mobility, Inc. - Motorola Confidential Proprietary

13-Mar-12

DFSS Tools



Good? yes

no trial run


spec / print

Good? Improve

Measurement

Measure Quantity ?

no


Data Good?

Process Control

Modify Spec.

yes no

yes


trial run



13-Mar-12


Why Statistical Methods

• Disciplined Approach

•Repeatable Results

• Quantifiable Decision Criteria

• Optimization


13-Mar-12

Q & A

Back Up Slides


13-Mar-12

6 Sigma Methodology

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


13-Mar-12

DFSS – CDOV Process

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


System Reliability

System Availability


13-Mar-12

CDOV Process


13-Mar-12



13-Mar-12


LevelFactors -> 22 = 4 Runs


13-Mar-12


2k -> LevelFactors


13-Mar-12



13-Mar-12


Documents

Statistics & Probability in Mechanical Design...Mechanical Development Process Development Flow (simplified) part / process design Concept Tolerance Analysis Good? yes no trial run