Design with uncertainty

1. Design with uncertainty Prof. Dr. Vasilios Spitas

2. What is uncertainty? • The deviation (u) of an anticipated result (μ) within a margin of confidence (p)

3. How familiar are we with uncertainty? • Hesitation • Chance • Luck • Ambiguity • Expectation • Error • Probability • Risk • Reliability • Tolerance QUANTITATIVE QUALITATIVE

4. Quantitative assessment requires … • Knowledge of the real problem • BOUNDARY CONDITIONS • Knowledge of the physical laws / interactions • CONSTITUTIVE EQUATIONS & CONSTANTS • Solvable / treatable formulation • MODEL • Solution • MATHEMATICS

5. Basic mathematical background • Discrete and continuous probability distribution functions • Metrics:

6. Basic mathematical background Normal distribution

7. Basic mathematical background Weibull distribution

8. From data sets to distribution functions • The sample / measurement set • Follows the statistical distribution • If and only if the likelihood function • Satisfies the equation Maximum Likelihood Method

9. Statistical hypothesis testing • State a null hypothesis • And an alternative hypothesis • Such that either Ho or H1 are true. Then verify the null hypothesis using • Z – tests • Student’s tests • F – tests (ANOVA) • Chi – square tests

10. Central limit theorem • A random sample of size n • Coming from a population of unknown distribution function with mean value (μ) and standard deviation (σ), has an average which follows the normal distribution with mean value: • And standard deviation:

11. Linking uncertainty with standard deviation

12. Combined uncertainty • The uncertainty of a function • With arguments xi and uncertainty ui each, is calculated as:

13. Tolerancing in Embodiment Design • Dimensional tolerance The acceptable uncertainty of a dimension

14. Tolerancing in Embodiment Design • Geometrical tolerance The acceptable uncertainty of a feature form - location Orientation Form Orientation Form Orientation Form Position Form Orientation Runout Form Runout Position Position

15. Tolerancing in Embodiment Design • Understanding tolerancing

16. Tolerancing in Embodiment Design • Communicating a function through tolerancing

17. Tolerancing in Embodiment Design • Communicating functions through tolerancing

18. Example of combined tolerance calculation • A 50mm long 50 piezostack is formed by assembling 50 identical PZT disks, each 1mm in thickness and with a parallelism tolerance of 0.02mm. What is the resulting parallelism of the assembled stacks?

19. Example of combined tolerance calculation • Let Δti be the deviation in parallelism of part i (i=1-50) • The piezostack length is the sum of the individual thicknesses of the parts ti • The requested uncertainty would then be:

20. If we are sure that none of the parts exceeds the tolerance … Tolerance zone … then where is the uncertainty ?

21. Methods for reducing uncertainty in engineering design • Analysis break the complex part into two or more simpler parts • Synthesis combine two or more parts into one monolithic part • Inversion female geometries to male geometries compression to tension internal features to external features • Constraint control

22. Thank you for your attentionGood luck with the workshop assignments