EML 4550: Engineering Design Methods

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

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.

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)

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

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”

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

7. Under Control? Process Capable? Classical Tolerance Design Process Select Process Change Process Collect Statistical Data N Work on process Y Management Decision Live with it Test 100% Y N Change Specs Stop Production

8. Specs Being Met? Classical Tolerance Design Process (Cont’d) N Recenter Process Y Continue Gathering Statistics For continued process improvement, conduct designed and controlled experiments to further reduce variability

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

10. 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)

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).

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

13. m m-Do m+Do 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 Equally bad product Equally good product Equally bad product target

14. 100 % of people complaining 50 0 70F 75F 80F 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

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”

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

17. L(y) Ao y m-Do m m+Do Customer Loss Functions • 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

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

19. Taguchi’s Loss Function Do 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

20. Safety Factor • 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

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

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

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 (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?

24. Example

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

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

27. L(y) Ao y m-Do m m+Do 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  Deviation of the average value of y from the target m Mean squared deviation of y value away from the target m

28. Average Quality Loss

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 200-112.5=$87.5 with an investment of $50, it seems to be a bargain.

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.