Chanyoung Park Raphael T. Haftka

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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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Paper Helicopter Project

Chanyoung Park

Raphael T. Haftka

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

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)
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

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

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
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)
Problem3: Comparing predicted variability using posterior and observed variability
• Area metric of the posterior predictive distribution of CD
Problem3: Comparing predicted variability using posterior and observed variability
• Area metric of the posterior predictive distribution of CD
Problem4: Predictive validation for the same height and different weight
• Area metric of the posterior predictive distribution of CD
Problem4: Predictive validation for the same height and different weight
• Area metric of the distribution of CD with p-box
• Area metric of the posterior predictive distribution of CD
• Area metric of the distribution of CD with p-box
Problem5: Linear model
• Area metric with the prior
• Data set 3
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
Problem5: Linear model
• Area metric of the posterior predictive distribution of CD
Problem5: Linear model
• Area metric of the distribution of CD with p-box
Problem5: Linear model with one clip
• Area metric of the posterior predictive distribution of CD
Problem5: Linear model with one clip
• Area metric of the distribution of CD with p-box
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
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)

Kaitlin Harris, VVUQ Fall 2013

• 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

### 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

Course Project: Validation of Drag Coefficient -Yiming Zhang

Validation based on 1 set of data:

Validation based on 3 set of data:

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

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

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.

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)
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