javier garcia - verdugo sanchez - six sigma training - w4 statistical tolerance analysis

/4705b BB W4 Tolerancing 05, D. Szemkus/H. Winkler

Statistical Tolerance Analysis

& Scorecards

Week 4

Page 2/4705b BB W4 Tolerancing 05, D. Szemkus/H. Winkler

Six Sigma Tolerance Analysis

Worst case tolerancing

Root sum of squares tolerancing

Statistical tolerancing

- linear applications

- non linear application


1

Y• Dependent• Output• Effect• Symptom• Monitoring

X1 . . . Xn

• Independent• Input• Cause• Problem• Control

Prevention of defects requires more than just an inspection of “Y”. We need to understand the effect from “X” on “Y” and how to

control the “X”. Than we can avoid the inspection of “Y”.

The Six Sigma Focus

Y = f (x1 + x2 + ... + xn)


Product Name:

Part DPU

ProcessDPU

Performance DPU

Software DPU

Assembly Current Opp Current Opp Current Opp Current Opp

Totals 0 0 0 0 0 0 0 0First Time Sigma #NUM! #NUM! #NUM! #NUM!DPU/Opp #DIV/0! #DIV/0! #DIV/0! #DIV/0!Sigma/Opp; LT #DIV/0! #DIV/0! #DIV/0! #DIV/0!Sigma/Opp; ST #DIV/0! #DIV/0! #DIV/0! #DIV/0!

6 σ 6 σ 6 σ 6 σ

Scorecards

Sub System A

Sub System B

Firmware

Test Software

Software Worksheet

Performance RequirementsVariable Test Limits

Performance WorksheetSpecificationRequirements

Performance WorksheetContract Requirements

Part A-1 Process Worksheet

Sub-Assembly A ProcessWorksheet

Fab, Assy & Test Process Worksheet


Co

sts

for

dev

elo

pm

ent

and

man

ufa

ctu

rin

g

SIGMA LEVEL

BEST DESIGN

Optimum

?

...

....

.. ..

Customer Requirements

Detailed Requirements

€ & DPU



Y= 0sy = …

Sub 1 Sub 2

Customer requirements level 1

x1 = s1 =

Y = X1 + X2

Sy2 = S1

2 + S22

Transfer Function

S = ?

x2 = s2 =

Statistical Requirements - „Flow down“


0

0 USL

0


Causes for Product Defects

Deliver

Parts

Process A

Process B

Process C

Process D

Subassembliesor

SystemTest

Buying of subassemblies

Software

Partdefects

Processdefects

Performancedefects

Softwaredefects

Performancedefects

= Test, Inspection


Bu

yin

g o

f m

ater

ial

Store

Production

Distribution

Acceptanceof a

product family

Bu

yin

g o

fS

ervi

ces

IT

Cus

tom

er

Supplierdefects

Logistical defects

Order fulfillmentdefects

Systemdefect

Supplierdefect

Causes for Defects in Business Processes


4.976 ± .003

Part

1.240 ± .003

Specification characteristics

Is it ready for manufacturing?

Part 1 Part 2 Part 3

clearance (Gap)

Part 4

Housing

Housing

What is the “worst case” in terms of tolerance stack ?What is the probability of encountering “worst case”?

„Worst Case Analysis“


… known as Stack-Analysis

Housing Nominal 4.976 Tolerance .003

Part 1 Nominal -1.240 Tolerance .003




Nominal Gap .016 Total Tolerance +/-.015

Minimum Gap .001

Maximum Gap .031

„Worst Case Analysis“

What is the Nominal Gap, the Minimum Gap and the Maximum Gap?


Traditional Analysis Method

Calculation of the Gap dimension

Q < - -

+ -

N e T em

Σi = 1 N P i + T P i

N e T em

Σi = 1 N P i - T P i < R

NPi = nominal design value of ith PartTPi = tolerance assigned of the ithe nominal parts design valuem = total number of partsT = half of each tolerance value

When using worst-case analysis, the designer seeks to minimize the arithmetic certainty that any given combination of assigned dimensions and tolerances will produce a condition wherein the product can not be assembled. The study of such a simultaneous condition is known as worst-case analysis.

Let us say that the minimum and maximum gap constraints are specified as Q and R, respectively. Using worst case methods, one can readily determine whether or not the linear combination of part tolerances is favorable in relation to the gap constraints. This is often called “worst case gap analysis.”.

Ne = nominal design value of the housing (e = Envelope)Te = design tolerance of the housingQ = a specified minimum assembly performance criteriaR = a specified maximum assembly performance criteria

Worst Case Analysis


At this point, we shall consider several other equations related to the assembly gap. Essentially, the listed equations allows us to compute the nominal assembly gap as well as the maximum and minimum.

S m a x = N e + T e -m

Σi = 1 N P i - T P i

S m in = N e - T e -m

Σi = 1 N P i + T P i

S nom = N e -m

Σi = 1N P i

S nom =m

Σi = 1N i V i B i

B is a "diametrical correction" factor for certain dimensions

V is the algebraic vector associated with the ith nominal dimension within the assembly loop

Worst Case Analysis


-1σ +1σ +2σ +3σ +4σ +5σ +6σ-2σ-3σ-4σ-5σ-6σ

68.26 %

95.46 %

99.73 %

99.9937 %

99.999943 %

99.9999998 %

+ -µ

. . . Why do we create such a conservative design?

PWC = YFTm = .00275

= .000000000000143

Because of the uncertainties associated with manufacturing?

Limitations of Worst Case AnalysisLet us suppose that each of the parts has a ±3 σ capability This would translate to a first-time yield expectation of .9973 per part. On the other side, we could say the defect probability per part is .0027. With this information, the likelihood of encountering a simultaneous worst case condition could be calculated.

Worst Case Analysis


Vector Analysis

Gap

Envelope Distribution Part Distribution Part Distribution Part Distribution Part Distributionσ = .001 in.µ = 4.976 in.

σ = .001 in.µ = 1.240 in.

σ = .001 in.µ = 1.240 in.

σ = .001 in.µ = 1.240 in.

σ = .001 in.µ = 1.240 in.

Housing Part 1 Part 2 Part 3 Part 4

Part1

Part2

Part3

Part4

Housing

Assembly gap

How can we include the process capability of the housing and of each component into the gap analysis?



Item 1 Item 2 Item 3 Item 4

F

µ S

PS

Z S = Q - µ S

σS

σ S = Σi = 1

m

= F - µ iΣi = 1

m

Distribution of the assembly gap „S“

µ S σ i2

Q S R S

Note: in this example is , Qs = 0.

QS = lower gap limit

Rs = upper gap limit

Fs represents Qs or Rs



Point of 0 gap ”Q”Point of 0 gap ”Q” µ Gap

Z Q

+ ∞- ∞

σ Gap = σ E2 + σ P1

2 + σ P 22 + σ P 3

2 + σ P42

µ Gap = µ E – (µ P 1 + µ P 2 + µ P 3 + µ P 4

Z Q =

Distribution of the assembly gap

If we know the “typical” mean off-set for the envelope and each of the parts, how could this knowledge be used to make the design robust against such shifts and drifts?

Assignment: Define the probability of a interference fit

)



µ Gap

Z Q

σ Gap = 5 x 0,0012 = 0,00223

µ Gap = 4,976 – 1,24 – 1,24 – 1,24 – 1,24 = 0,016

Z Q =


Distribution of the assembly gap

Point of 0 gap ”Q”

Assignment: Define the probability of a interference fit

If we know the “typical” mean off-set for the envelope and each of the parts, how could this knowledge be used to make the design robust against such shifts and drifts?

+ ∞- ∞


Calculation examples

0,245860,245600,249960.249960.245550.254480.250690.252020.254830.245900,253470,250050.248970.247390.25157

Measurements

Stack 5 = ?

X +/- Y

Minimum rail spacing = ?

Assumption: long term behavior

0.250260.249940.245180.245910.248070.254070.253710.250810.248530.249160.247350.249520.253060.254950.25432



Calculate the mean

Mean of 5 in stack

sigma of 1 part

Sigma of 5 stacked

Six Sigma Design

Minimum Slot

Part Dimension

Stack 5 = ?

X +/- Y

Minimum rail spacing = ?

The calculation

µ = 0.250

5 ∗ µ = 1.250

σ = 0.00312

sqrt (5 ∗ σ^2) = 0.007

1.250 +/- 0.042

1.292

0.250 +/- 0.01872



Example 1: Statistical Tolerance Analysis

A practical application


Assembly Housing Pads

Cylinder

Piston

Rotor group

LC of the spindle



Pads

.750 ± .015

.062 ±.005

Cylinder

PistonHousing

Seals

Cylinder cover

3.700±.02

1.55±.01

Rotor

.750±.035

CylinderPiston

Caliper Assembly Pads

.95±.005



3.700.950 .950

1.551.55

.062 .062

.750 .750.750

Gap

CoverHousing

piston

Cover

Rotor

Loop Diagram

PositiveDesignVector

NegativeDesignVector

piston

Tolerance Vector Analysis

PadsBacking plate



Statistical Design Analysis Spreadsheet

Response Description: Order to Ship Cycle for MTO; Replineshment Cycle for MTI_________ Analyst: _____________________ Date: 22. Feb 05

Analysis Table

Variable Information Tolerance Dist. Type

Factor Short or % or Normal or Sensitivity

Description Mean Long Term Std Dev Lower Upper Actual Uniform Coef. DPU ZST

L A N

Mean Response→Response Upper Spec Limit

Response Lower Spec Limit

Summary TableResponse Components

Worst Case LimitsMean Std Dev DPU ZST Lower Upper DPU ZST

Rev 2.0a

© Dr. Maurice L. Berryman, 1996. All rights reserved.

Calc Sensitivities

Clear Sensitivities

Hide Rows

Unhide Rows




Response Description: Automotiv Break Disk________________________________________ Analyst: _____________________ Date: 22. Feb 05

Analysis TableVariable Information Tolerance Dist. Type



cover 0,95 L 0,0017 0,005 0,005 A N 1 2,102E-02 3,53housing 3,7 L 0,0667 0,2 0,2 A N 1 2,700E-03 4,28cover 0,95 L 0,0017 0,005 0,005 A N 1 2,700E-03 4,28piston 1,55 L 0,0333 0,1 0,1 A N -1 2,700E-03 4,28bp 0,062 L 0,0017 0,005 0,005 A N -1 2,700E-03 4,28pad 0,75 L 0,0050 0,015 0,015 A N -1 2,700E-03 4,28rotor 0,75 L 0,0117 0,035 0,035 A N -1 2,700E-03 4,28pad 0,75 L 0,0050 0,015 0,015 A N -1 2,700E-03 4,28pb 0,062 L 0,0017 0,005 0,005 A N -1 2,700E-03 4,28piston 1,55 L 0,0333 0,1 0,1 A N -1 2,700E-03 4,28

Mean Response→ 0,126Response Upper Spec Limit Response Lower Spec Limit 0



0,126 0,08286 6,418E-02 3,02 -0,359 0,611 4,531E-02 4,11 Rev 2.0a




Gap

Outer leg (72.8 +/- .20)Seal (3.60 +/- .10)

piston (53.2 +/- .90)

Rotor (25 +/- .12)

Inner shoe (6.4 +/- .30)Outer shoe (6.4 +/- .30)

-A-

-A- to seal (11.9 +/- ..25)



-A-

+

-

53.2

6.4

25

Nom. Gap

6.4

72.8

11.9

3.6

Note: The gap should always be positive unless you are working with interference fits!

Loop Diagram




Response Description: Break Assembly, example 2___________________________________ Analyst: _____________________ Date: 22. Feb 05




Outer leg 72,8 S 0,0667 0,2 0,2 A N -1 2,102E-02 3,53A to seal 11,9 S 0,0833 0,25 0,25 A N -1 2,102E-02 3,53Seal 3,6 S 0,0333 0,1 0,1 A N -1 2,102E-02 3,53Outer shoe 6,4 S 0,1000 0,3 0,3 A N 1 2,102E-02 3,53Rotor 25 S 0,0400 0,12 0,12 A N 1 2,102E-02 3,53Inner shoe 6,4 S 0,1000 0,3 0,3 A N 1 2,102E-02 3,53Piston 53,2 S 0,3000 0,9 0,9 A N 1 2,102E-02 3,53

Mean Response→ 2,7Response Upper Spec Limit

Response Lower Spec Limit 0



2,7 0,457962 2,286E-09 7,36 0,53 4,87 1,471E-01 3,53 Rev 2.0a




Example: Order Cycle Time

How good is the fulfilment for 2,7 days (65h)?

Order entry2 +/- 1h

Raw supply28 +/-16 h

Process2 +/- 1 h

Waiting10 +/- 1h

Turn around4+/- 0,5 h

Turn around4+/- 0,5 h

QC + Packaging1 +/- 0,5 h

Order to Delivery = Order entry + Raw supply + Turn around + Waiting + Processing + Waiting + Turn around + QC + Packaging

Waiting10 +/- 1h


Hypothetical Data Baseline



Response Description: Auftragslaufzeit für Prozess XY__________________________________ Analyst: _____________________ Date: 08. Sep 05




Order entry + scheduling 2 L 0,3333 1 1 A N 1 2,700E-03 4,28Raw supply 28 L 5,3333 16 16 A N 1 2,700E-03 4,28Waiting 10 L 0,3333 1 1 A N 2 2,700E-03 4,28Turn around 4 L 0,1667 0,5 0,5 A N 1 2,700E-03 4,28Process 2 L 0,3333 1 1 A N 1 2,700E-03 4,28Turn around 4 L 0,1667 0,5 0,5 A N 1 2,700E-03 4,28Waiting 10 L 0,3333 1 1 A N 1 2,700E-03 4,28QC + packaging 1 L 0,1667 0,5 0,5 A N 1 2,700E-03 4,28

Mean Response→ 61Response Upper Spec Limit 65Response Lower Spec Limit 0



61 5,4134606 2,300E-01 2,24 38,5 83,5 2,160E-02 4,28 Rev 2.0a




Response Description: Auftragslaufzeit für Prozess XY__________________________________ Analyst: _____________________ Date: 08. Sep 05




Order entry + scheduling 2 L 0,3333 1 1 A N 1 2,700E-03 4,28Raw supply 25 L 3,3333 10 10 A N 1 2,700E-03 4,28Waiting 5 L 0,3333 1 1 A N 1 2,700E-03 4,28Turn around 4 L 0,1667 0,5 0,5 A N 1 2,700E-03 4,28Process 2 L 0,3333 1 1 A N 1 2,700E-03 4,28Turn around 4 L 0,1667 0,5 0,5 A N 1 2,700E-03 4,28Waiting 5 L 0,3333 1 1 A N 1 2,700E-03 4,28QC + packaging 1 L 0,1667 0,5 0,5 A N 1 2,700E-03 4,28

Mean Response→ 48Response Upper Spec Limit 65Response Lower Spec Limit 0



48 3,4115816 3,273E-07 6,47 32,5 63,5 2,160E-02 4,28 Rev 2.0a


Hypothetical Data Future alternative



1. Enter the information and data for each variable

2. Enter the Std Dev. optional input; if a value is entered here, the tolerance field is not used. If no value is entered, this is a calculated field based on the tolerance field entries described below.

3. Enter the tolerance information for each variable

4. Selection of the probability distribution based of the toleranceassumption

5. Entry of the equation in the equation field (mean response)

6. Use the “ calc sensitivities” button for calculation

7. Analysis of the area results and components

8. If a new equation has to be used push „clear sensitivities“ and enter new equation for new analysis with the use of the „calculate sensitivities“ button.

Statistical Analysis Spreadsheet, procedure - short cut



1. Designed to analyze tolerance stack-ups on linear and non-linear transfer functions (1-dimensional or multi-dimensional tolerance analysis)

2. Can be used for limited optimization based on sensitivity analysis and tolerancing

3. User Entries are highlighted in color (Both required and optional inputs)

4. Calculated Cells are write protected (Formulas hidden)

5. The analysis spreadsheet can handle up to 20 different variables in a statistical analysis.

6. If more than 20, terms can be combined on lower level analysis spreadsheets

7. Top Section is all user inputs

8. Bottom Section (Summary Box) is response results section

Statistical Analysis Spreadsheet, general information



1 Define the Problem

2


Identify Important Terms for Analysis: (Dimensions and properties of material)

Dimensions of the beam

tensile stress in beam (s)

max bending moment in beam

moment of inertia

distance to where beam is analyzed

yield strength of material (S)

load

performance requirement > 0: P = (S - s)


t

h

w

P

L1

LT

WP

F


Structural Design A Sensitivity Study


σ t

FA

Fht wt t

= =+ −2 2 4 2

( )( )( )

σ

σ

b

bP T P

McI

ch

hPW L L W

wh w t h t

=

=

=− −

− − −

23 2 2

2 21

3 3


Tensile Stress:

Bending Stress:

Maximum Stress:It will be a combination of tensile stress and bending stress


( )( )( )33

P1TP

2s

bts

t2ht2wwhWL2L2hPW3

t4wt2ht2F

−−−−−

+−+

=σ

σ+σ=σ


Total stress will be the sum of bending and tension stress

If the yield strength, Sy of the material is known, the design margin is equal to the difference between the maximum stress and the yield strength, or Sy-ss


( )( )( )33

P1TP

2y t2ht2wwhWL2L2hPW3

t4wt2ht2F

S.Diff−−−−−

+−+

−=


The equation above will be programmed into

the Statistical Design Analysis Spreadsheet

for analysis and optimization

All specifications are entered in the spreadsheet Struc.xls.

Assume all dimensions are 3 sigma long term.

Adjust specifications to achieve 6 sigma design margin.


Appendix:

Procedure for the Spreadsheet

Acronyms


• User Inputs

1) Variable Information:

– Description: Description of the variables in the stack-up

– Name: Name of the variables

– Mean: Mean value of the variable

– Std Dev: This is an optional input; if a value is entered here, the tolerance field is not used. If no value is entered, this is a calculated field based on the tolerance field entries described below. Once a value is typed into this field, it replaces the formula which is used when data is entered into the tolerance fields.

Spreadsheet Field Descriptions



• User Inputs

2) Tolerance Field inputs:

– Upper and Lower Tolerance: The user can enter the upper and lower tolerances of the variable.

– % or actual: When a % sign is typed into it recognizes the lowerand upper tolerance as a percent of Mean value. When an “A” is typed in for “Actual” the lower and upper are recognized as the actual value of the upper and lower limit.

– Tolerance Field Inputs (cont):

– Distribution Type: Enter “N” for normal and “U” for Uniform distributions. If N, then the std dev is calculated as the (tolerance range) / 6. If U, then the std dev is calculated as the (tolerance range) / SQRT(12).





• User Inputs

3) Enter Equation:

– The equation for the relationship between the response and variables is entered here. The entry must be in terms of the cell locations of the means of the variables (e.g... = (C9 +C10); this says the response is the sum of the variables whose mean values show up in C9 and C10).

– Upper and Lower Response Spec Limits: the upper and lower specs for the response is entered here. This can be a one sidedspec if necessary.



• Calculations for Top Section:– Sens: This is the sensitivity of that variable based on the partial

derivative of the transfer function loaded into the equation box.

– DPU: Probability of bad parts based on their tolerance range and mean and standard deviations. DPU is the probability that parts will be produced outside the tolerance range.

– On parts with a uniform distribution this probability is shown to be zero, with truncation outside the tolerance range.

– Sigma: “Sigma” of the individual variables based on their capability relative to the specification. This is a long term Sigma where 3.4 defects/mil = 6 Sigma.

– The calculations for the top section are completed as the data is entered into the variable and tolerance fields.

– These calculations do not change when calculate sensitivity is pushed.






• Calculations of the Summary Section:1) Response Section:

– Mean: Mean value of the response based on the individual variable means and transfer function

– Std Dev: Standard deviation of the response based on individual variable standard deviations and sensitivities from the differentiation of the transfer function

– DPU & Sigma: These are based on the responses long term capability. The limits of the Sigma number are -1.5 to 28. Outside of these values will still produce the same numbers.

– Worst Case Upper and Lower Limit: These are for the response based on the worst combination of the individual variables at their worst case values and their effect on the response.

2) Components Section:

– DPU: The sum of all DPUs for the individual components

– Sigma: Based on the total Component DPU with the number of opportunities being the total number of components


• BOM - Bill of Materials

• DFA - Design for Assembly

• DFM - Design for Manufacturability

• DOE - Design of Experiments

• DPMO - Defects per million opportunities

• DPU - Defects Per Unit

• DPU sub - DPU submitted

• DPU obs - DPU observed

• DPU esc - DPU escaping

• DV - Dependent Variable


Acronyms


• EE - Electrical Engineering

• FEA - Finite Element Analysis

• FMEA - Failure Modes and effects analysis

• GR&R - gauge repeatability & reproducibility

• H/W - Hardware

• I & T - Integration & Test Phase of a Program

• IV - Independent Variable

• LSL - Lower Specification Limit

• ME - Mechanical Engineering

• mil. - million

• oppor - opportunity


Acronyms


• PCB - Printed Circuit Board• PCD - Process Capability Database• PCM - Process Capability Models• QFD - Quality Function Deployment • RSS - Root Sum Squares• SDM - Statistical Design Methods• SE - Systems Engineering• S/W - Software• USL - Upper Specification Limit• Tol - Tolerance• WC - Worst Case• 1-d - one dimensional linear stack-up


Acronyms