EML4550 2007 1

EML 4550: Engineering Design Methods

Tolerance Design

From “Tolerance Design: A Handbook for developing optimal specifications,” by C.M. Creveling, Addison-Wesley, Chapter 11

Also

“Engineering Design,” by G.E. Dieter Chapter 12

EML4550 -- 2007

Definitions

Tolerance

Geometric tolerance - range for a particular dimension General tolerance - acceptable range for a design variable

(dimension, roughness, viscosity, refractive index, etc.)

Most techniques developed for tolerance design apply to dimensions, but many can be generalized to any design tolerance problem

Tolerance design appeared with the Industrial Revolution as the need for interchangeability arose.

EML4550 -- 2007

Definitions

Geometric Dimensioning and Tolerancing (GD&T) Tolerance design geared towards ‘variance

reduction’ as the key to repeatable, low-cost manufacturing

Converging views from East and West Taguchi method

Application of sound statistical and mathematical methods in the design process to reduce variance (design for quality)

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Tolerance Design: Process Flow Diagram

Customer Tolerances Customer Costs & Losses

Product OutputResponse Tolerance

Product Output ResponseProcess Capabilities

System andAssembly Tolerances

System and AssemblyProcess Capabilities

Component PartTolerances

Component PartProcess Capability

Manufacturing ProcessParameter Tolerances

Manufacturing ProcessCapabilities

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Tolerances

Tolerances need to be defined because we live in a probabilistic world and 100% reproducibility in manufacturing is not physically possible

Tolerances are defined in a standard: ANSI Y14-5M-1982 (R1988) (American National Standards Institute-ANSI)

“The total amount by which a given dimension may vary, or the difference between the limits”

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Different Approaches to Tolerancing

Traditional methods in tolerance design Semi-empirical

Experience Manufacturing process capabilities

Computer-aided tolerance design Plug-in packages for CAD software (propagation of tolerance

techniques – “error analysis”) Statistical methods

Monte Carlo simulation Sensitivity analysis Cost-based tolerance design

Modern methods in tolerance design Taguchi approach

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Classical Tolerance Design Process

Select Process

Collect Statistical Data

UnderControl? Work on process

ProcessCapable?

Management Decision

Change Process

Change SpecsLive with itTest 100%

Stop Production

NY

N

Y

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Classical Tolerance Design Process (Cont’d)

SpecsBeing Met? Recenter Process

Continue Gathering Statistics

For continued process improvement,conduct designed and controlled

experiments to further reduce variability

Y

N

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Tolerances and Quality Engineering

Taguchi:

“Tolerances are economically established operating windows of functional variability for optimized control factor set points to limit customer loss”

More general, not just dimensions Economically-driven (trade off) Control factors that are pre-defined (not any variable) Limit, but not eliminate, customer losses

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Taguchi Approach

Concept of off-line QC

Incorporate QC and tolerancing before releasing the design to production

Iterative process as a final step prior to drawing release

On-line QC

Traditional approach of in-plant QC, ‘fix it’ after the fact or scrap

Use on-line QC to maintain or improve quality of the designed product (little or no improvement needed if ‘off-line’ QC was properly implemented)

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The three phases in Tolerance Analysis

Basically the standard approach for the design process

Concept design: selection of technology platform, metrics to assess relative merits, concept robustness (safety, environment, commercial, reliability, etc.)

Parameter design: optimization of concept, parameters to reduce sensitivity to ‘noise’ (uncontrollable parameters)

Tolerance design: Balancing of customer loss function with production cost, ability to determine and limit the variability around the ‘target’ set points (as defined in parameter design).

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Taguchi’s Approach to Tolerancing

Input from the ‘voice of the customer’

Select proper quality-loss function for the design evaluation

Select the customer tolerance values for the Quality Loss Function:Ao ($ lost due to off-target value) and Do (measurement of

Off-target performance in engineering terms)

Determine the cost to the business to adjust the off-targetPerformance back to acceptable range during manufacturing: A

Calculate the manufacturing tolerance: D based on Taguchi’s Equation:

“My” acceptable variability = “Their (customer’s)” acceptable variability x square root of the ratio between “My” cost to stay within production tolerance / “Their” loss if my

product is out of tolerance

oo AADD /

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Traditional Tolerance Curve

mm-Do m+Do

Equally bad product Equally good product Equally bad product

Factories would accept or reject productbased on a simple on/off model (step function)

Assumption that customers will behave the sameway is WRONG

target

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Customer Tolerance

Customer tolerance is not a simple step function Customer tolerance Do corresponds to the point in

which a significant fraction of customers will take some type of action (e.g., 50% of customers would complain)

70F 75F 80F

50

0

100

% of peoplecomplaining

“Thermostat” example

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Customer Loss Function

Quadratic approximation to the customer loss function

L is the loss function k is the quality-loss coefficient y is the performance variable m is the target performance

2)my(k)y(L

L is the economic loss to my customer if my product deviates “y” from its rated value “m”

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Quality Loss Coefficient

The functional limits (m + Do) and (m - Do) represent the deviations from the target in which about 50% of the customers would complain (significant economic loss)

This is essentially a definition of product ‘failure’. The economic loss to the customer associated with product failure is Ao (e.g., losses due to lack of access to product plus cost to repair, generally in terms of $)

Therefore L(y-=m-Do) = L(y+=m+Do)=Ao2o

o

DAk

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Customer Loss Functions

The nominal-the-best case

The smaller-the-better case

The larger-the-better case

Asymmetric cases

22o

o myDA)y(L

22o

o yDA)y(L

2

2o

oyDA)y(L

myifmykL

myifmykL

2

2

Do

Ao

y

L(y)

Do

Aoy

L(y)

m+Dom-Do

Ao

y

L(y)

m

m+Dom-Do

Ao

y

L(y)

m

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Taguchi Tolerancing Equations

Concept of Taguchi ‘safety factor’ in tolerancing What are the maladies for which we need to build a

safety factor? Customer dissatisfaction due to quality problems and

customer financial losses (long-term impact to reputation) Higher manufacturing costs due to re-work and scrap

Define a tolerance level as seen by the customer (losses) and a tolerance level as seen by the manufacturing process

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Taguchi’s Loss Function

Di

yo Target (m)

AoLosses

yi=m-Di

Ai

Note:Do-Di=range of safetyDo/Di=safety factor

manufacturing tolerancecustomer tolerance

Financial incentiveSince A<Ao

Do

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Safety Factor

For a standard quadratic loss function

Deviation from target

Loss associated with deviation

222)( mykmy

DAyLo

o

22i myD

2i2

o

oii D

DAA)y(L

Ai ≤ Ao: manufacturing-allowable loss should be smaller than the customer loss

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Safety Factor

At what level is the company willing to ‘act’ to avoid customer losses by ‘fixing’ the product back to the target value before releasing it?

Economic safety factor

In general notation:

i

o2i

2o

AA

DD

i

o

i

o

i

o

AA

DD

DDS 2

2

AAS o

Derived from statistical considerations, sub-o relates to customer (loss function, and maximum deviation), sub-i relates to manufacturer, cost to re-work and maximum manufacturing tolerance

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Safety Factor

S=SQRT[(average loss to (customer) in $ when a product characteristic exceeds customer tolerance limits)/(average loss to (manufacturer) in $ when a product characteristics exceeds manufacturing tolerance limits)]

The Taguchi Approach relates customer tolerances toengineering tolerances

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Example

A company makes a power supply. The nominal (target) value for the supply voltage is 115V. We know the customer incurs a loss of $200 (Ao, due to damaging to instrument, loss of productivity, recall, etc..) when the voltage exceeds 135V (135-115=20=Do, deviation from nominal). The production department has determined that it costs $5 to re-work (adding current-limiting resistor, etc..) a power supply that is off-target back to the nominal value.

What should the manufacturing tolerance be and what is the economic safety factor?

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Example

22o

o2 myDAmyk)y(L

)V/($5.0Volts20200$

DAk 2

222o

o

V20D200$A

0

o

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Example

The manufacturing tolerance is:

The safety factor is:

If the assembly line detects a power supply with voltage lower than 112V (115-3) or higher than 118V (115+3) it is economical to pull it off and repair it

The difference between the customer loss and the manufacturing cost is relatively large (200/5=40) smaller tolerance is permissible sqrt(Ao/A)=sqrt(40)=6.32~20/3

V3V16.3200520

AADD

oo

32.65

200A

AS o

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Example (alternative interpretation)

2

22

22

2

)(5.0)(

)()20(

200)(

myyL

mymyDAmykyLo

o

The manufacturing tolerance can be considered as a deviationaway from the nominal value m Di=y-m

The cost to modify the manufacturing process can be considered as the loss function $5

316.3)(5.05 2 mymy

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Average Quality Loss

The average quality loss, Q, from a total of n units from a specific process can be given by (derived in the next slide)

large. isn when )(

)(1

1 and1 where

1)(

)()()()()()(1

22

1

2

1

2

22

222

2121

mkQ

yn

yn

nnmk

mymymynkyLyLyL

nQ

n

ii

n

ii

nn

Deviation of the average value of y from the target

Mean squared deviation of y value away from the target

m+Dom-Do

Ao

y

L(y)

m

Average Quality Loss

EML4550 2007

large. is n when)(

)(1

1 and1 where

1)(

1)(21)(

1)2(11)(

21)(1)(

2121

)(22

)2()2()2(

)()()()()()(1

22

1

2

1

2

22

1

22

1

222

1

2

11

22

22

1

222

1

22

2

1

2222

1

2

2

1

22

11

2

2222

22

21

21

222

2121

mkQ

yn

yn

nnmk

yn

mkyyn

mk

ny

ny

nmk

yn

mkyn

mk

mmyn

kmmyn

k

nmnmynknmymy

nk

mmyymmyymmyynk

mymymynkyLyLyL

nQ

n

ii

n

ii

n

ii

n

iii

n

i

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

n

ii

nn

nn

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Example

From the previous example, assume the power supplies manufactured have their mean value centered around the target (=m) so its loss of quality will be dominated by the standard deviation term: Q=k2

If the variance of the power supplies =20 volts, determine the quality loss due to the manufacturing deviation: Q=(0.5)(20)2=$200

If a resistor is added to the unit, it has been demonstrated that it can reduced the variance to 15 volts. The cost of the additional process is $50. Show that whether it is worthwhile?

Q=(0.5)(15)2=$112.5a net decrease of loss 200-112.5=$87.5with an investment of $50, it seems to be a bargain.

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Conclusions

The Taguchi Approach can be used at the system level to interact with outside customers, but it can also be implemented within a company

Each successive step in the manufacturing process can be seen as a ‘customer’ of the previous step (manufacturing, purchased part, service, etc.)

When implemented on a company-wide basis the Taguchi Approach can lead to a quasi-optimal distribution of tolerances among the different components that go into a final product.