Statistics & Probability in Mechanical Design...Mechanical Development Process Development Flow...

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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

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

Normal (Gaussian) Distribution

-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6

bability D

ensity

babili

ensity

1σ 1σ

68.27%

-3σ 3σ

95.45%

6σ -6σ

99.9999998%

Normal Distribution

Process Capability

Allowable Variation

Actual Variation Cp =

Allowable

Actual

-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6

bability D

ensity

USL LSL

Allowable

Actual

Normal Distribution

13-Mar-12

-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6

bability D

ensity

USL LSL

Cp =2.0

USL LSL

-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6

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

Process Capability

13-Mar-12

-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6

bability D

ensity

USL LSL

-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6

bability D

ensity

USL LSL

-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6

bability D

ensity

USL LSL

Cp = 2.0

Cp Independent

of the

target.

Process Capability

13-Mar-12

Cp = USL - LSL

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

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?

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

Measurements Analyze

Data Good?

Process

Control

Modify

yes no

Improve

Manf. Process

trial run

Reliability

Modeling

13-Mar-12

Tolerance Analysis

part / process

design Concept

Tolerance

Analysis Good?

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

Data Good?

Process

Control

Modify

yes no

Improve

Manf. Process

trial run

Reliability

Modeling

13-Mar-12

part / process design

TA Acceptable?

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

Components

A B C D

0.55 + 1.07 + 1.60 + 1.35

Envelope Size

A B C D

13-Mar-12

Tolerance Analysis

σgap =

3Cpi (

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

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

part / process

design Concept

Tolerance

Analysis Good?

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

Data Good?

Process

Control

Modify

yes no

Improve

Manf. Process

trial run

Reliability

Modeling

13-Mar-12

Measurement Systems Analysis

trial run

spec / print (requirement)

Good TA

13-Mar-12

Measurement Error Bad Decisions

13-Mar-12

Characteristics:

Stability Discrimination Accuracy (Bias)

Linearity

Precision

13-Mar-12

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?

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

Data Good?

Process

Control

Modify

yes no

Improve

Manf. Process

trial run

Reliability

Modeling

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

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

Determine Quantity to Measure

5 or 20 or 100 ???

Improve the Measurement System

MSA Acceptable?

13-Mar-12

Data Analysis

part / process

design Concept

Tolerance

Analysis Good?

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

Data Good?

Process

Control

Modify

yes no

Improve

Manf. Process

trial run

Reliability

Modeling

13-Mar-12

Data Analysis

part / process

design Concept

Tolerance

Analysis Good?

no trial run

Measurement Systems

Analysis

spec / print

Good? Improve

Measurement

Measure

Quantity ?

Data Good?

Process

Control

Modify

yes no

trial run

Reliability

Modeling

13-Mar-12

Improve

Manf. Process

Comparative Analysis

Design of Experiments

• 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 ?

Data Good?

Process Control

Modify Spec.

yes no

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 ?

Data Good?

Process Control

Modify Spec.

yes no

trial run

13-Mar-12

Decreasing

Failure

bad region

Constant

Failure

desired region

Increasing

Failure

bad region

Useful Life

Infant Mortality

Normal Operation

Wearout

This region is not always perfectly flat.

13-Mar-12

Failure Analysis

Example:

Display Breakage

Process Control

Modify Specification

Data Acceptable?

Cp= 2.0 Cpk = 2.0

Analyze Data

-6 -4.8 -3.6 -2.4 -1.2 -0 1.2 2.4 3.6 4.8 6

babili

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

13-Mar-12

DFSS Tools

Good? yes

no trial run

spec / print

Good? Improve

Measurement

Measure Quantity ?

Data Good?

Process Control

Modify Spec.

yes no

trial run

13-Mar-12

Why Statistical Methods

• Disciplined Approach

•Repeatable Results

• Quantifiable Decision Criteria

• Optimization

13-Mar-12

Back Up Slides

13-Mar-12

6 Sigma Methodology

DFSS – Design for Six Sigma

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

Major Steps

VOC KJ Analysis

Initiate CPM

Pugh Analysis

System Reliability

System Availability

13-Mar-12

CDOV Process

13-Mar-12

LevelFactors -> 22 = 4 Runs

13-Mar-12

2k -> LevelFactors

13-Mar-12

Statistics & Probability in Mechanical Design...Mechanical Development Process Development Flow...

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