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
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2. 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.
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3. 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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4. Tolerance Design: Process Flow Diagram
Customer Tolerances
Customer Costs & Losses
Product Output Response
Process Capabilities
Product Output
Response Tolerance
System and Assembly
Process Capabilities
System and
Assembly Tolerances
Component Part
Process Capability
Component Part
Tolerances
Manufacturing Process
Parameter Tolerances
Manufacturing Process
Capabilities
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5. 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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6. 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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7. Classical Tolerance Design Process
Select Process
Change Process
Collect Statistical Data
Under
Control? N
Work on process Y
Management Decision
Process
Capable?
Live with it
Test 100% Y N
Change Specs
Stop Production
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8. Classical Tolerance Design Process (Cont’d)
Specs
Being Met? N
Recenter Process Y
Continue Gathering Statistics
For continued process improvement,
conduct designed and controlled
experiments to further reduce variability
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9. 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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10. Taguchi Approach Concept of off-line QC On-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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11. 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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12. 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-target
Performance 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
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13. Traditional Tolerance Curve
Factories would accept or reject product
based on a simple on/off model (step function)
Assumption that customers will behave the same
way is WRONG m
m-Do
m+Do
Equally bad product
Equally good product
Equally bad product
target
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14. 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)
“Thermostat” example
70F
75F
80F 50
100
% of people
complaining
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15. 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
L is the economic loss to my customer if my product deviates “y” from its rated value “m”
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16. 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)=Ao
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17. Customer Loss Functions
m+Do
m-Do Ao y
L(y) m
The nominal-the-best case
The smaller-the-better case
The larger-the-better case
Asymmetric cases
L(y) Ao y
L(y) Do Ao y Do
L(y) Ao y
m-Do m
m+Do
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18. 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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19. Taguchi’s Loss FunctionDo Di
Losses Ao Ai
Financial incentive
Since A<Ao yo
Target (m)
yi=m-Di
customer tolerance
manufacturing tolerance
Note:
Do-Di=range of safety
Do/Di=safety factor
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20. Safety Factor Ai ≤ Ao: manufacturing-allowable loss
For a standard quadratic loss function
Deviation from target
Loss associated with deviation
Ai ≤ Ao: manufacturing-allowable loss
should be smaller than the customer loss
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21. 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:
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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22. 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 to
engineering tolerances
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23. 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 ( =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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24. Example
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25. 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
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26. Example (alternative interpretation)
The manufacturing tolerance can be considered as a deviation
away from the nominal value m  Di=y-m
The cost to modify the manufacturing process can be considered as the loss function  $5
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27. 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)
m+Do
m-Do Ao y
L(y) m m 
Deviation of the average value of y from the target m
Mean squared deviation of y value away from the target m
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28. Average Quality Loss
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29. Example
From the previous example, assume the power supplies manufactured have their mean value centered around the target (m=m) so its loss of quality will be dominated by the standard deviation term: Q=k2
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=$112.5a net decrease of loss =$87.5
with an investment of $50, it seems to be a bargain.
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30. 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.
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