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Robust Design,Propagation of Error and Tolerance Analysisp g y

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Robust Design 1

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Robust Design,Propagation of Error and Tolerance Analysisp g y

Robust Design Concepts Robust Design Concepts

Propagation of Error (POE)

RSM Analysis with TA Lathe Machined Parts HDTV signal (as time allows)

1. Raymond H. Myers, Douglas C. Montgomery and Christine M. Anderson-Cook (2009), 3rd edition, Response Surface Methodology, John Wiley and Sons, Inc, Sections 2.7-2.8 and 6.6, Chapters 1, 7, 8 and 10.

2. George E.P. Box, William G. Hunter and J. Stuart Hunter (2005), Statistics for Experimenters, 2nd

diti J h Wil Ch t 12edition John Wiley, Chapter 12.

Robust Design 2

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Agenda Transition

Robust Design Concepts

Propagation of Error (POE) Propagation of Error (POE)

RSM Analysis with TA Lathe Machined Parts Lathe Machined Parts HDTV signal (as time allows)

Robust Design 3

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Robust Design

The goal of robust design is to increase profits by consistently ti f i t dsatisfying customer needs.

This is accomplished by generating robust designs toMi i i d i k ft d i f Minimize design re-work after design freeze Minimize surprises during design verification Accelerate scale up and commercializationcce e a e sca e up a d co e c a a o

Robust designs depends heavily on DOE to translate the Voice-of-Customers (VOCs) and product specifications into process specifications

Robust Design 4

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Robust Design

Functional Design: Strategy of ExperimentationStrategy of Experimentation

Use DOE to model response mean as a function of controllable factor levels

Phase: ScreeningKnownFactors

UnknownFactors

Screening


UnknownFactors

Screening

controllable factor levels.

Choose levels of controllable factors to achieve targeted

Phase: Characterization

Factor effectsand interactions

Curvature?no



Curvature?no

values of the responses. Phase: Optimization

Ph V ifi i

yes

ResponseSurfaceMethods

Phase: Optimization

Ph V ifi i

yes


Phase: Verification Confirm? Backup

Celebrate!

no

yes


Celebrate!

no

yes

Robust Design 5

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Robust Design

Robust Design: Strategy of ExperimentationStrategy of Experimentation

Use DOE to model response variability as a function of control and uncontrolled


UnknownFactors

Screening


UnknownFactors

Screening

control and uncontrolled factor levels.

Choose levels of control



Curvature?no



Curvature?no

factors to reduce variation caused by:• Lack of control of the

Phase: Optimization

Ph V ifi i

yes


Phase: Optimization

Ph V ifi i

yes


control factors.• Variation of the

uncontrolled factors


Celebrate!

no

yes


Celebrate!

no

yes

uncontrolled factors.

Robust Design 6

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Concept: Choose levels of the control1 factors in a way that d t t i ti I th d k th d treduces output variation. In other words, make the product,

process or system robust to variation in the inputs; both control and uncontrolled2 factors. Quality is then improved without

i th f i tiremoving the cause of variation.1 Control factors (x) are parameters whose nominal values can

be cost-effectively adjusted by the engineerbe cost effectively adjusted by the engineer. Example: oven temperature.

2 Uncontrolled factors (z) are parameters that are difficult, expensive, or impossible to control.Example: ambient temperature.

Robust Design 7

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Control vs Uncontrolled Factors

Determining whether a factor is an uncontrolled or a controlled one ft d d th t ’ bj ti th f th j toften depends on the team’s objective or the scope of the project.

A factor considered controlled in some cases might be considered uncontrolled in others.

For example material durometer (hardness):

is controllable to a design engineer, who gets to chose the s co o ab e o a des g e g ee , o ge s o c ose ematerial.

but may be uncontrolled to a process engineer who only th i ti ithi th h t i lsees the variation within the chosen material.

Robust Design 8

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Agenda Transition


Propagation of Error (POE) Propagation of Error (POE)

RSM Analysis with TA Lathe Machined Parts Lathe Machined Parts HDTV signal (as time allows)

Robust Design 9

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Propagation of Error (POE)Transmitted Variation

Objective: Reduce the variation transmitted to the response from variation in control factorsresponse from variation in control factors.

Robust Design 10

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Propagation of ErrorHow it works

Once a relationship has been established between a factor and a th i ti i th t t bresponse, the variation in the output can be:

1. Dependent on the level of the control factor

2. Independent of the level of the control factor

See pictures on next two pagesSee pictures on next two pages

Robust Design 11

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Propagation of errorDependentp

The transmitted variation is d d t th l l f

Effect of Inputdependent on the level of the control factor.

Therefore, set the level of the

on Response

Therefore, set the level of the control factor to reduce variation transmitted to the response from variationresponse from variation (lack-of-control) of the control factor.

Control Factor

A B

Robust Design 12

Control Factor©Stat-E

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Propagation of errorIndependentp

The transmitted variation is i d d t f th l lindependent of the level of the control factor.

Therefore, set the level of theTherefore, set the level of the control factor to center the process mean on target.

Res

pons

e

Effect of Inputon Response

R

Control FactorA B

Robust Design 13

Control Factor©Stat-E

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Power Circuit Design Example

Consider two control factors:1. Transistor Gain – nonlinear relationship to output voltage 2. Resistance – linear relationship to output voltage

The variation in gain and resistance about their nominal values isThe variation in gain and resistance about their nominal values is known. Both variances are constant over the range of nominal values being considered.

Robust Design 14

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Power Circuit Design Example(reduce variation)( )

Variation is reduced by using a nominal gain of 350.

That shifts the output off target to 125 voltsThat shifts the output off-target to 125 volts.

Robust Design 15

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Power Circuit Design Example(return to target)( g )

Decrease the nominal resistance from 500 to 250.

Thi t th t t t th t t d 115 ltThis corrects the output to the targeted 115 volts.

Robust Design 16

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Power Circuit Design Exampleon target with reduced variationg

To illustrate the theory, the control f d ifactors were used in two steps: first to decrease variation and second to move back on target.g

In practice, numerical optimization can be used to simultaneouslycan be used to simultaneously obtain all the goals.

Robust Design 17

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Propagation of errorJust a little mathematical explanationp

Find regions where variation in the control factors transmits the least variation to the responsevariation to the response.

2Y x x 0 1 1 11 1

21 1

Y x x

Y 15 25x 0.7x

The goal is to minimize the slope, which is th 1st d i ti f th di ti tithe 1st derivative of the prediction equation.

Robust Design 18

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Propagation of errorJust a brief mathematical explanationp

2ˆ Y x x Assume σx = 1 and σresid = 0

0 1 1 11 1

21 1

ˆ 15 25 0.7

Y x x

Y x x

22 2 2Y

x resid

Yx

2 2 2ˆ 125 1.4

x residYx

As the slope of the relationship between X and Y decreases the variation transmitted to Y decreases

Robust Design 19

decreases, the variation transmitted to Y decreases.©Stat-E

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Propagation of errorGoal: Minimize propagated error (POE)p p g ( )

What is POE?222

2 2 2 2 2ˆ ˆ

i jx z residY Y

i ji j

f f POEx z

The amount of variation transmitted to the response(using the transfer function): from the lack of control of the control factors and variability from the lack of control of the control factors and variability

from uncontrolled factors(you enter these standard deviations),l th l i ti plus the normal process variation

(obtained from the ANOVA).

It is expressed as a standard deviation.

Robust Design 20

It is expressed as a standard deviation. ©Stat-E

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Propagation of errorSimple One-Factor Illustration (page 1 of 2)p (p g )

1. Build a One FactorRSM d iRSM design.

2. Factor A: low level = 0and high level = 15and high level 15

3. Design for a cubicmodel.

4. Sort by Factor A andenter this data:

(continued on next page)

Robust Design 21

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Propagation of errorSimple One-Factor Illustration (page 2 of 2)p (p g )

5. Compute effects and select appropriate model.

Fitted equation (in terms of actual factor values) is: . . .Y A A 14 95 25 05 0 71 2

6. Enter the standard deviation for each factor:

From the Design Layout Screen - View, Column Info Sheet –t 1 00 f F t Aenter 1.00 for Factor A.

The response standard deviation is filled in automatically from the ANOVA after the analysis is completedfrom the ANOVA after the analysis is completed.

Robust Design 22

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Propagation of errorSimple One-Factor Illustrationp

Analyze the response (R1) and look at the one factor plot and the propagation of error plot (from the View menu )

233.081

One Factor322 25.0693

One Factor333

and the propagation of error plot (from the View menu.)

123.497

178.289

R1 14.5387

19.804

POE(

R1)

22

13.9131

68.7051

322 4.00821

9.27348

333

0.00 3.75 7.50 11.25 15.00

A: A

0.00 3.75 7.50 11.25 15.00

A: A

y . x . x 21 114 95 25 0 7 2 2 2

ˆ 125 1.4 x residy x

Robust Design 23

x residassume 1 and 0.95

y

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Propagation of error(linear relation between factor and response)( p )

If the response is a linear function of the independent factors, the transmitted variation is a constantthe transmitted variation is a constant.

y x 0 1 1

y x 115 25

Robust Design 24

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Propagation of errorLinear relationship (page 1 of 2)p (p g )

ˆ y x x residassume 1 and 3.17

0 1 1

1ˆ 15 25

y x

y x

22 2 2ˆ

x residy

Yx

2 2 2ˆ 25 Constant x residy

Since the slope of the relationship between X and Y is constant, the variation transmitted to the response also remains constant.

Robust Design 25

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Propagation of errorLinear relationship (page 2 of 2)p (p g )

Point Prediction node:

2 2 2ˆ

2 2 2

25 x residy

2 2 225 1 3.17324

25.206

Robust Design 26

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Agenda Transition



RSM Analysis with TA RSM Analysis with TA Lathe Machined Parts DOE (Design-Expert)DOE (Design Expert) Tolerance analysis (VarTran)

HDTV signal (as time allows) HDTV signal (as time allows)

Robust Design 27

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Robust RSM SimulationPrecise Machined Parts

Acme precision machine company has been having trouble holding nominal values on their new highly automated lathe Your job is tonominal values on their new highly-automated lathe. Your job is to study the process and reduce deviations from nominal. Previous work has determined that three factors are the key influencers on the process:process:

Factor Units Range

Cutting Speed fpm 330 700Cutting Speed fpm 330 - 700

Feed Rate ipr 0.01 – 0.022

Depth of cut inches 0 05 0 10

The factor levels given are the extreme values, do not exceed them.

Depth of cut inches 0.05 – 0.10

Robust Design 28

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Which design is an appropriate h i f th L th DOE?

KnownFactors

UnknownFactors

Screening KnownFactors

UnknownFactors

Screening

choice for the Lathe DOE?

What model should we design f ?

Screening Trivialmany

Vital fewCharacterization

Screening Trivialmany

Vital fewCharacterizationfor?Factor effects

and interactions

Curvature?no

CharacterizationFactor effects

and interactions

Curvature?no

Characterization

yes

ResponseSurfacemethods

Curvature?

Optimizationyes

ResponseSurfacemethods

Curvature?

Optimization

Confirm? Backup

Celebrate!

no

yes

VerificationConfirm? Backup

Celebrate!

no

yes

Verification

Robust Design 29

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The experimenters chose to run a Box-Behnken design. The significant factors are already known, and optimization is

the focus. The region of interest and the region of operability are veryThe region of interest and the region of operability are very

similar (can’t exceed the stated factor levels.) They would like to fit a quadratic model.

The key response is delta, i.e. the deviation of the finished part’s dimension from its nominal value. Delta is measured in mils, 1 mil = 0.001 inches.1 mil 0.001 inches.

Robust Design 30

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1. Build a three factor Box-Behnken response surface design. (Th t l d fi B B h k d i ) S th(The extreme values define a Box-Behnken design.) Save the design as “Lathe.dxp”.

2. Right click on the response column header and run the2. Right click on the response column header and run the simulation: Lathe.sim

3. Fit an appropriate model (reduce as needed) to the response: d ltdelta.

4. Examine the response surface to find factor levels where delta is zero.is zero.

Robust Design 31

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Note the variety of0.022

delta0.3 Note the variety of

speed and feed combinations that can produce a delta of 0

0.019-0.2

0.0

0.1

0.2

produce a delta of 0.

You could also explore the other 0.016

0 2

-0.1

5

B: F

eed

graph combinations of speed vs. depth and feed vs. depth.

0.013

-0.2

0 1

330 423 515 608 700

0.010

-0.3-0.1

Robust Design 32

A: Speed©Stat-E

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Robust RSM SimulationPrecise Machined Parts – Add POE

1. Enter information on the expected variation of the controllable f t b t th i t i t F th D i d tfactors about their set points. From the Design node, go to “Column Info Sheet” and enter:

Variable Standard DeviationVariable Standard DeviationA – Speed 5 fpmB – Feed 0.00175 ipr

2. Use the POE model graphs to explore the transmitted error as a function of the independent factors

C – Depth 0.0125 inches

function of the independent factors.

3. Re-save (“Lathe.dxp”)

4 Save as a VarTran file: “Lathe vta”4. Save as a VarTran file: Lathe.vtaRobust Design 33

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Robust RSM SimulationPrecise Machined Parts – Add POE

0.022POE(delta)

0.130.100

POE(delta)

0.019

d

0.12

0.14

0.088

th 0 160.180.20.220.240.26

0.013

0.016

B: F

eed

0.140.15

0.160.17

0.180.19 55555

0.063

0.075

C: D

ept

0.12

0.14

0.140.16

55555

330 423 515 608 700

0.010

330 423 515 608 700

0.050

0.140.16 0.18

0.20.220.240.26

These are two of the three views of the propagated error - - where is POE minimized?

A: Speed A: Speed

where is POE minimized?

Robust Design 34

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Robust RSM SimulationPrecise Machined Parts – Optimizationp

1. Use numerical optimization to find factor levels near zero delta th t l b tthat are also robust:

Response Goal Low HighDelta Target = 0 -0 4 0 4

2 Choose settings to operate the lathe:

Delta Target 0 0.4 0.4

POE(Delta) Minimize 0.0 0.3

2. Choose settings to operate the lathe:

Speed fpmFeed iprFeed iprDepth inchesDelta mils

Robust Design 35

POE(Delta) mils©Stat-E

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Robust RSM SimulationPrecise Machined Parts – Point Prediction

The tolerance interval using the POE standard deviation is from -0.4462 to +0.4462.

Since the specifications are -0.4400 to 0.4400 VarTran should find the process marginally capable at best.

Robust Design 36

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Agenda Transition



RSM Analysis with TA RSM Analysis with TA Lathe Machined Parts DOE (Design-Expert)DOE (Design Expert) Tolerance analysis (VarTran)

HDTV signal (as time allows) HDTV signal (as time allows)

Robust Design 37

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How Do We Model Tolerancesfor Complex Systems?p y

VarTran® is a software package developed by Dr. Wayne Taylor.

VarTran can be used to establish and assess targets and tolerances for product and process inputs (x’s)

The targets and tolerances selected are those which: minimize the variation in the output (y)

t th t t ( ) it t t l center the output (y) on its target value

RDTA section 1 38

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Requirements for VarTran

Vartran requires that a y = f(x) model has been established

The y = f(x) model can come from the following sources:

Pre-existing mathematical relationship

Design of experiments (factorial or response surface)

A combination of the above

RDTA section 1 39

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Requirements for VarTran

In order to fully optimize a product or process, the y p p pfollowing elements are required for input to VarTran:

1. Inputs (target and range)

2. Capabilities of inputs (variation about their set points)

3 Outputs (responses)3. Outputs (responses)

4. Model (y = f(x))

5 Specifications for outputs5. Specifications for outputs

RDTA section 1 40

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Precise Machined PartsTolerance Analysisy

Open VarTran file “Lathe.vta”:

Lathe.dxpA : Speed R1 : delta

I/O

SYSTEM

B : Feed

C : Depth

s_R1 : delta residual std dev

Robust Design 41

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Tolerance AnalysisPCA Requires Five Itemsq

3) Outputs1) Inputs

2) Capabilities) pof Inputs

4) Model: Y=f(x)

Robust Design 425) Specs for Outputs©Stat

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Need to enter specifications to conduct tolerance analysis:

Robust Design 43

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Use Intervaloptimization.

Robust Design 44

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Cpk of 1.11 < 1.3Marginal at bestMarginal at best.

Robust Design 45

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Tolerance AnalysisImproving Cpk – It’s a Business Decisionp g pk

Change Specifications( i i i t d VOC)(engineering requirements and VOC)

Improve the Controls(more cost)

Improve Measurement System(more cost and/or R&D)

Change the Design Change the Design(more R&D)

Refuse the Business(cost of lost opportunity)

Accept 2-3% Failure Rates(depends on criticality – back to VOC)( p y )

Robust Design 46

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Precise Machined PartsImproving Cpkp g pk

Better control of factor C (depth)factor C (depth) does the most to improve capability.

Robust Design 47

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Precise Machined PartsImproving Cpkp g pk

Reduce std dev by ½:Reduce std dev by ½:0.0125 → 0.00625

Robust Design 48

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Re-optimize, then find newthen find new process capability.

Cpk of 1.53is goodis good

improvement!

Robust Design 49

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How Does VarTran Work?

VarTran relies on having a good y=f(x) model.

The variability in outputs (y’s) depends on the variability of the inputs (x’s).

We input the expected variation in the x’s and VarTran uses statistical tolerancing techniques to calculate the resultant distribution of y(s)resultant distribution of y(s).

VarTran takes advantage of non-linear effects and interactions to find the values of x which minimize the variation in y and center y on the target value.

RDTA section 1 50

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Empirical Tolerancing

Use DOE and Empirical Tolerancing:

1. When we cannot write the transfer function due to: Predictive equation not being known from first principles.

E i ti lt diff i f d l ! Existing results differing from our models!

2. When the Y(‘s) are too risky to be left untested and tolerance modeling is insufficient to guarantee resultstolerance modeling is insufficient to guarantee results.

RDTA section 1 51

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Agenda Transition



RSM Analysis with TA RSM Analysis with TA Lathe Machined Parts HDTV signal (as time allows) HDTV signal (as time allows)

Incorporates uncontrolled (noise) DOE (Design-Expert) Tolerance analysis (VarTran)

Robust Design 52

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Uncontrolled Factors in RSMHigh Definition TVg

The goal for a robust product design is to achieve high signal with l ilow variance.

There are four control factors and three uncontrolled factors.

C t l f t d Control factors: x1, x2, x3 and x4.

Uncontrolled factors: z1, z2 and z3.

We are interested in a quadratic model, but the experiments are expensive and time is short. Let’s try to minimize the runs in the DOE.

Robust Design 53

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Uncontrolled Factors in RSMBuilding a Design (page 1 of 2)g g (p g )

1. Start with a 7 factor “Min Run Res V” (MR5) CCD.

Robust Design 54

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Uncontrolled Factors in RSMBuilding a Design (page 2 of 2)g g (p g )

2. Enter factor names and levels:

3. There is one response: “signal”.4. Save your design as “HDTV.dxp”.

Robust Design 55

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Uncontrolled Factors in RSMSimulate and Analyze Responsey p

1. Run the simulation by:

right-clicking on the response column heading

choosing “Simulate Response”

choosing the “HDTV.sim” file and clicking OK.

2 Analyze the response (signal)2. Analyze the response (signal).

Robust Design 56

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Uncontrolled Factors in RSMAdd POE and Optimize (page 1 of 2)p (p g )

1. Click on the “Design” node and choose “Column Info Sheet” d t th t d d d i ti f h f tand enter the standard deviation for each factor:

Robust Design 57

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Uncontrolled Factors in RSMAdd POE and Optimize (page 2 of 2)p (p g )

2. Find levels of the controllable factors to: maximize signal (LL = 70, UL = 100) minimizing variation (LL = 4, UL = 20)

3 R “HDTV d ” d S “HDTV t ”3. Re-save “HDTV.dxp” and Save as “HDTV.vta”

Note: We can’t select (optimize) levels for the uncontrolledNote: We can t select (optimize) levels for the uncontrolled factors. Therefore set them to their midpoint using the goal of “equal to →”.

Robust Design 58

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Uncontrolled Factors in RSMPoint Prediction (Tolerance Interval)( )

The tolerance interval using the POE standard deviation starts at 75.16.Si h l ifi i li i i 70 V T h ld fi d h Since the lower specification limit is 70 VarTran should find the process capable.

Robust Design 59

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Agenda Transition



RSM Analysis with TA RSM Analysis with TA Lathe Machined Parts HDTV signal (as time allows) HDTV signal (as time allows)

Incorporates uncontrolled (noise) DOE (Design-Expert) Tolerance analysis (VarTran)

Robust Design 60

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HDTV with Uncontrolled Factors Tolerance Analysisy

Open VarTran file “HDTV.vta”:HDTV dxpHDTV.dxp

A : x1 band width

B : x2 freq

R1 : signal

I/O

B : x2 freq

C : x3 power

D : x4 colorSYSTEM

D : x4 color

E : z1 voltage

F : z2 compressionF : z2 compression

G : z3 bits

s R1 : signal residual std dev

Robust Design 61

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Uncontrolled Factors Tolerance Analysisy

Handling uncontrolled factors during tolerance analysis: Long-term this factor will fluctuate within some range. The best guess for the current setting of this factor is it’s

midpointmidpoint. Set uncontrolled factors to their nominal values.

Robust Design 62

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Uncontrolled Factor z1 Tolerance Analysisy

Min = NominalMax = Nominaldo Not include in optimizationin optimization

Robust Design 63

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Robust Design 64

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Robust Design 65

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Need to enter specifications to conduct tolerance analysis:

Robust Design 66

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Statistical ToleranceLSLBefore optimizing let’s check

the Cpk you get by simply ran th i l tti th

67.378 109.07terrestrial

Characteristic Value

the nominal settings on the input factors.

Average: 88.224Standard Deviation: 6.9486Cp: ----Cc: ----Cpk: 0.87Def. Rate (normal): 0.436 %Z-Score (short-term): 2.62Sigma Level: 2.62

Robust Design 67 Interval for Values = (67.378, 109.07) +/-3SD©Stat-E

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HDTV with Uncontrolled Factors Tolerance Analysis (before optimization)y ( p )

Robust Design 68

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Use Simplex Optimization

Robust Design 69

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Statistical ToleranceLSL

70 105.3terrestrial

Characteristic ValueA 89 999Average: 89.999Standard Deviation: 5.1002Cp: ----Cc: ----C k 1 31 Cpk of 1.31Cpk: 1.31Def. Rate (normal): 44 dpmZ-Score (short-term): 3.92Sigma Level: 3.92

pkis acceptable!

Robust Design 70 Interval for Values = (74.699, 105.3) +/-3SD©Stat

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HDTV with Uncontrolled Factors Tolerance Analysis (after optimization)y ( p )

Robust Design 71

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Targets and Tolerances of Inputs

Input Type Tolerance / Category LSL USL Average and Variation Requirements

A Statistical 127.24 + ---- - ---- ---- ---- Average = 127.24, Std. Dev. <= 5

B Statistical 9.6518 + ---- - ---- ---- ---- Average = 9.6518, Std. Dev. <= 1

C Statistical 53525 + ---- - ---- ---- ---- Average = 53525, Std. Dev. <= 2500

D Statistical 588.88 + ---- - ---- ---- ---- Average = 588.88, Std. Dev. <= 60

E Statistical 150 + ---- - ---- ---- ---- Average = 150, Std. Dev. <= 35

F Statistical 6 + ---- - ---- ---- ---- Average = 6, Std. Dev. <= 2

G Statistical 7.5 + ---- - ---- ---- ---- Average = 7.5, Std. Dev. <= 3

s_R1 Statistical 0 + ---- - ---- ---- ---- Average = 0, Std. Dev. <= 1.9194

Tolerance / Value Targets and Tolerances of Outputs

Output Type At Target LSL USL Average and Variation Requirements

Robust Design 72

R1 Statistical ---- + ----- - ------ 70 �� Ave = 89.999, Std. Dev. <= 5.1002©Stat-E

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E-mail [email protected] for answers from Stat-Ease’s staff of statistical consultants

Call 612.378.9449 and ask for “statistical help”

Thank YOU for attending!Thank YOU for attending!Robust Design 73

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