Download - EML 4550: Engineering Design Methods
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.