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Chanyoung Park Raphael T. Haftka

Paper Helicopter Project. Chanyoung Park Raphael T. Haftka. Problem1: Conservative estimate of the fall time. Estimating the 5 th percentile of the fall time of one helicopter Estimating the 5 th percentile to compensate the variability in the fall time ( aleatory uncertainty)

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Chanyoung Park Raphael T. Haftka

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  1. Paper Helicopter Project Chanyoung Park Raphael T. Haftka

  2. Problem1: Conservative estimate of the fall time • Estimating the 5th percentile of the fall time of one helicopter • Estimating the 5th percentile to compensate the variability in the fall time(aleatory uncertainty) • The sampling error (epistemic uncertainty) • Estimating the sampling uncertaintyin the mean and the STD • Obtaining a distribution of the 5th percentile • Taking the 5th percentile of the 5thpercentile distribution to compensatethe sampling error mt,P st,P Sampling tP Sampling t0.05,P

  3. Problem1: Conservative estimate of the fall time • Estimating the 5th percentile of the fall time of first helicopter (mean 3.78, std 0.37) • 100,000 5th percentiles of fall time • Helicopter 1 of the dataset 3 • Height: 148.5 in • 2.88 (sec) is the 5th percentile of the histogram(a conservative estimate ofthe 5th percentile of the fall time for 95% confidence)

  4. Problem2: Predicted variability using prior • Calculating predicted variability in the fall time • We assume that the variability in the fall time is caused by the variability in the CD • The variability in the fall time is predicted using the computational model (quadratic model) and the distribution of the CD • The prior distribution represents our initial guess for the distribution of the CD Height at time t where Steady state speed

  5. Problem2: Comparing predicted variability and observed variability using prior • Area metric with the prior • Data set 3

  6. Problem3: Calibration: Posterior distribution of mean and standard deviation • Estimating parameters of the CD distribution • We assume that CD of each helicopter follows the normal distribution • The parameters, CD and σtest are estimated using 10 data • Posterior distribution is obtained based on 10 fall time data • Non informative distribution is used for the standard deviation After 1 update After 5 updates After 10 updates

  7. Problem3: Comparing predicted variability using posterior and observed variability • Comparing MLE and sampling statistics • MCMC sampling • 10,000 pairs of the CD and the STD of CD are generated using Metropolis-Hastings algorithm • An independent bivariate normal distribution is used as a proposal distribution • MLE of the posterior distribution is used as a starting point

  8. Problem3: Comparing predicted variability using posterior and observed variability • Handling the epistemic uncertainty due to finite sample • How to handle epistemic uncertainty in the CD and the test standard deviation estimates? • Comparing the posterior predictive distribution of the fall time and the empirical CDF of tests (combining epistemic and aleatory uncertainties) • Using p-box with 95% confidence interval of epistemic uncertainty (separating epistemic and alreatory uncertainties)

  9. Problem3: Comparing predicted variability using posterior and observed variability • Area metric of the posterior predictive distribution of CD

  10. Problem3: Comparing predicted variability using posterior and observed variability • Area metric of the posterior predictive distribution of CD

  11. Problem4: Predictive validation for the same height and different weight • Area metric of the posterior predictive distribution of CD

  12. Problem4: Predictive validation for the same height and different weight • Area metric of the distribution of CD with p-box

  13. Problem6: Predictive validation for different height and the same weight • Area metric of the posterior predictive distribution of CD

  14. Problem6: Predictive validation for different height and the same weight • Area metric of the distribution of CD with p-box

  15. Problem5: Linear model • Area metric with the prior • Data set 3

  16. Comparison to predictive validation • Area metric of the posterior predictive distribution of CD • The predictive validation with the linear model is not as successful as that with the quadratic model • Area metric with p-box • Area metric with p-box tries to capture the extreme discrepancy between the predicted variability and the observed variability

  17. Problem5: Linear model • Area metric of the posterior predictive distribution of CD

  18. Problem5: Linear model • Area metric of the distribution of CD with p-box

  19. Problem5: Linear model with one clip • Area metric of the posterior predictive distribution of CD

  20. Problem5: Linear model with one clip • Area metric of the distribution of CD with p-box

  21. Comparison between quadratic and linear models • Area metric of the posterior predictive distribution of CD • Area metric of the distribution of CD with p-box

  22. Concluding remarks • Predictive validation for both quadratic and linear models • The predictive validation for different mass is a partially success • The predictive validation for different height is a success but the assumption of constant CD is not clearly proven • Comparison between models • Cannot conclude • Overall • Reason for the differences in the area metric is not clear • The effect of the manufacturing uncertainty is significant (i.e. very different area metrics for the same test condition)

  23. Kaitlin Harris, VVUQ Fall 2013 • Comments: • Chose to maintain uniform distribution for calibration parameter based on histogram results (vs normal) • Conclusions: • Best models: calibrated quadratic at both heights and calibrated linear with 2 paper clips • Worst models: un-calibrated linear with 2 paper clips and un-calibrated linear with 1 clip for data set 1

  24. Validation of analytical model used to predict fall time for Paper HelicopterBy Nikhil Londhe *Calibrated Analytical Model is validated to represent experimental data *Quadratic dependence is valid assumption between drag force and speed *For given difference in fall height, Cd can be assumed to be constant

  25. Course Project: Validation of Drag Coefficient -Yiming Zhang Validation based on 1 set of data: Validation based on 3 set of data: Quadratic Dependence 2 clips Prior VS. Posterior Dist. Prior Area Metric:0.4920 Prior Area Metric:0.4042 Post Area Metric:0.2 Post Area Metric:0.1695 95% Confidence 0.77 95% Confidence 0.83 Quadratic Dependence 1 clip Predictive Validation. Validation Area Metric:0.1267 Validation Area Metric:0.1923 If just use this set to calculate posterior.95% Confidence 0.86 If just use this set to calculate posterior.95% Confidence 0.82 Linear Dependence 2 clips & 1clip Prior VS. Posterior Dist. Summary: 2 clips: 1 clip: 2 clips: Summary: 1 clip: Posterior Area Metric:0.1987 Validation Area Metric:0.1960 Validation Area Metric: 0.1253 Posterior Area Metric:0.1729 Seems reasonable, but not accurate Two confi. interval of Cd don’t coincide. Linear dependence seems inaccurate. 95% Confidence 0.8 95% Confidence 0.88 95% Confidence 0.8 95% Confidence 0.91 Quadratic Dependence 2 clips Different height. Validation Area Metric:0.1394 Validation Area Metric:0.0970 If just use this set to calculate posterior.95% Confidence 0.76 If just use this set to calculate posterior.95% Confidence 0.79 Summary: (1) Quadratic dependence seems accurate using one set; Quadratic and linear dependence both don’t match well using 3 sets; (2) SRQ is required to be fall time. If use Cd as SRQ, comparison could be more consistent and clear; (3) 0.8 seems a reasonable estimation of Cd. This estimation would be more accurate while introducing more sets of data.

  26. Backup Slides

  27. Problems • Problem1: Conservative estimate of the fall time • Problem2: Comparing predicted variability and observed variability using prior • Problem3: Comparing predicted variability and observed variability using posterior • Problem4: Predictive validation for the same height and different weight • Problem5: Comparing the quadratic and linear models • Problem6: Predictive validation for different height and the same weight (proving the assumption of constant CD)

  28. Problem1: Conservative estimate of the fall time • Estimating the 5th percentile of the fall time of one helicopter • Since fall time follows a normal distribution, estimating the 5th percentile is based on estimating the mean and standard deviation (STD) of the fall time distribution • The mean and STD are estimated based on 10 samples • There is epistemic uncertainty in the estimated mean and STD due to a finite number of samples • To compensate the epistemic uncertainty, a conservative measure to compensate the epistemic uncertainty is required • Estimating the 5th percentile with 95% confidence level

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