1. Probabilistic Mechanism Analysis

2. Outline Uncertainty in mechanisms Why consider uncertainty Basics of uncertainty Probabilistic mechanism analysis Examples Probabilistic mechanism synthesis Conclusions 2

3. Uncertainty in Mechanisms 3

4. The joint clearances at A, B, C, and D are also random due to manufacture imprecision and installation errors. 4

5. Uncertainty in Mechanisms Loads and material properties are also random. 5

6. Impact of Uncertainty 6

7. Why Consider Uncertainty? We know the true solution. We know the effect of uncertainty. We can design mechanisms whose performance is not sensitive to uncertainty We can make more reliable decisions. 7 In this presentation, we focus on uncertainty only in R.

8. We use probability distributions to model parameters with uncertainty. How Do We Model Uncertainty? 8

9. Probability Distribution 9

10. Normal Distribution 10

11. It indicates how data spread around the mean. It is always non-negative. High std means – High dispersion – High uncertainty – High risk 11

12. More Than One Random Variables 12

13. Mechanism Reliability 13

14. First Order Second Moment Method (FOSM) 14

15. Monte Carlo Simulation (MCS)* A sampling-based simulation method 15 *This topic is optional. Step 3: Statistic Analysis on model output Extracting probabilistic information Step 3: Statistic Analysis on model output Extracting probabilistic information Step 2: Numerical Experimentation Evaluating performance function Step 2: Numerical Experimentation Evaluating performance function Step 1: Sampling of random variables Generating samples of random variables Step 1: Sampling of random variables Generating samples of random variables Analysis Model Samples of input variables Samples of output variables Probabilistic characteristics of output variables Distributions of input variables

16. Step 1: Sampling on random variables

17. Step 2: Obtain Samples of Output

18. Step 3: Statistic Analysis on output

19. FORM vs MCS FORM is more efficient FORM may not be accurate when a limit-state function is highly nonlinear MCS is very accurate if the sample size is sufficiently large MCS is not efficient 19

20. Example - FOSM 20 A C B

21. Example - FOSM 21

22. Example - FOSM 22

23. Example - MCS 23 A C B

24. 1e6 Simulations

25. Reliability–Based Mechanism Synthesis 25

26. Conclusions For important mechanisms in important applications, it is imperative to consider reliability. Uncertainty can be modeled probabilistically. Reliability can be estimated by FOSM and MCS. Same methodologies can also be used for cams, gears, and other mechanisms. 26