1
Chanyoung ParkRaphael T. HaftkaPaper Helicopter Project
1Problem1: Conservative estimate of the fall timeEstimating the
5th percentile of the fall time of one helicopterEstimating 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 STDObtaining a
distribution of the 5th percentileTaking the 5th percentile of the
5th percentile distribution to compensatethe sampling error
SamplingSamplingmt,Pst,PtPt0.05,P
Structural & Multidisciplinary Optimization
Group#/252Estimating the 5th percentile of the fall time of first
helicopter (mean 3.78, std 0.37)100,000 5th percentiles of fall
timeHelicopter 1 of the dataset 3Height: 148.5 in2.88 (sec) is the
5th percentile of the histogram(a conservative estimate ofthe 5th
percentile of the fall time for 95% confidence)
Problem1: Conservative estimate of the fall time
Structural & Multidisciplinary Optimization
Group#/253Problem2: Predicted variability using priorCalculating
predicted variability in the fall timeWe assume that the
variability in the fall time is caused by the variability in the
CDThe 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 tSteady state speedwhere
Structural & Multidisciplinary Optimization
Group#/25Problem2: Comparing predicted variability and observed
variability using priorArea metric with the priorData set 3
0.290.340.49CD from the fall time
dataHelicopter1Helicopter2Helicopter3Sample mean of
CDs0.8960.8420.783Sample STD of CDs0.1900.1540.084PriorMean of CD =
1 / STD of CD = 0.28
Structural & Multidisciplinary Optimization
Group#/25Problem3: Calibration: Posterior distribution of mean and
standard deviationEstimating parameters of the CD distributionWe
assume that CD of each helicopter follows the normal
distribution
The parameters, CD and test are estimated using 10 dataPosterior
distribution is obtained based on 10 fall time data
Non informative distribution is used for the standard
deviation
After 1 updateAfter 5 updatesAfter 10 updates
Structural & Multidisciplinary Optimization
Group#/25Problem3: Comparing predicted variability using posterior
and observed variabilityComparing MLE and sampling statistics
MCMC sampling10,000 pairs of the CD and the STD of CD are
generated using Metropolis-Hastings algorithmAn independent
bivariate normal distribution is used as a proposal distributionMLE
of the posterior distribution is used as a starting point
CD from the fall time
dataHelicopter1Helicopter2Helicopter3Sample mean of
CDs0.8960.8420.783Sample STD of CDs0.1900.1540.084MLE of the CD
mean0.8960.8430.783MLE of the CD STD0.1820.1460.081
Structural & Multidisciplinary Optimization Group#/25MLE
individual or both. MLE and mode?Problem3: Comparing predicted
variability using posterior and observed variabilityHandling the
epistemic uncertainty due to finite sampleHow 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)
Structural & Multidisciplinary Optimization
Group#/25Problem3: Comparing predicted variability using posterior
and observed variabilityArea metric of the posterior predictive
distribution of CD
0.150.110.07CD from the fall time
dataHelicopter1Helicopter2Helicopter3148.5 in2 clips (ref)Sample
mean of CDs0.8960.8420.783Sample STD of CDs0.1900.1540.084
Structural & Multidisciplinary Optimization
Group#/25Problem3: Comparing predicted variability using posterior
and observed variabilityArea metric of the posterior predictive
distribution of CD
0.010.000.00
CD from the fall time dataHelicopter1Helicopter2Helicopter3148.5
in2 clips (ref)Sample mean of CDs0.8960.8420.783Sample STD of
CDs0.1900.1540.084
Structural & Multidisciplinary Optimization
Group#/25Problem4: Predictive validation for the same height and
different weightArea metric of the posterior predictive
distribution of CD
0.190.340.38
CD from the fall time dataHelicopter1Helicopter2Helicopter3148.5
in1 clipsSample mean of CDs0.9160.9920.968Sample STD of
CDs0.1470.0690.049148.5 in2 clips (ref)Sample mean of
CDs0.8960.8420.783Sample STD of CDs0.1900.1540.084
Structural & Multidisciplinary Optimization
Group#/25Problem4: Predictive validation for the same height and
different weightArea metric of the distribution of CD with
p-box
0.010.110.22
CD from the fall time dataHelicopter1Helicopter2Helicopter3148.5
in1 clipsSample mean of CDs0.9160.9920.968Sample STD of
CDs0.1470.0690.049148.5 in2 clips (ref)Sample mean of
CDs0.8960.8420.783Sample STD of CDs0.1900.1540.084
Structural & Multidisciplinary Optimization
Group#/25Problem6: Predictive validation for different height and
the same weightArea metric of the posterior predictive distribution
of CD
0.340.190.09
CD from the fall time
dataHelicopter1Helicopter2Helicopter3181.25 in2 clipsSample mean of
CDs0.8860.8660.816Sample STD of CDs0.0790.1190.066148.5 in2 clips
(ref)Sample mean of CDs0.8960.8420.783Sample STD of
CDs0.1900.1540.084
Structural & Multidisciplinary Optimization
Group#/25Problem6: Predictive validation for different height and
the same weightArea metric of the distribution of CD with p-box
0.050.000.01
CD from the fall time
dataHelicopter1Helicopter2Helicopter3181.25 in2 clipsSample mean of
CDs0.8860.8660.816Sample STD of CDs0.0790.1190.066148.5 in2 clips
(ref)Sample mean of CDs0.8960.8420.783Sample STD of
CDs0.1900.1540.084
Structural & Multidisciplinary Optimization
Group#/25Problem5: Linear modelArea metric with the priorData set
30.650.680.81CD from the fall time
dataHelicopter1Helicopter2Helicopter3Sample mean of
CDs0.9780.9490.916Sample STD of CDs0.1020.0910.050PriorMean of CD =
1 / STD of CD = 0.28
Structural & Multidisciplinary Optimization
Group#/25Comparison to predictive validationArea metric of the
posterior predictive distribution of CDThe predictive validation
with the linear model is not as successful as that with the
quadratic model
Area metric with p-boxArea metric with p-box tries to capture
the extreme discrepancy between the predicted variability and the
observed variabilityHelicopter1Helicopter2Helicopter3148.5 in / 1
clips0.130.060.02148.5 in / 2 clips
(ref)0.010.000.00Helicopter1Helicopter2Helicopter3148.5 in / 1
clips0.440.300.14148.5 in / 2 clips (ref)0.130.110.07
Structural & Multidisciplinary Optimization
Group#/25Problem5: Linear modelArea metric of the posterior
predictive distribution of CD
0.130.110.07CD from the fall time
dataHelicopter1Helicopter2Helicopter3148.5 in2 clips (ref)Sample
mean of CDs0.9780.9490.916Sample STD of CDs0.1020.0910.050
Structural & Multidisciplinary Optimization
Group#/25Problem5: Linear modelArea metric of the distribution of
CD with p-box
0.010.000.00CD from the fall time
dataHelicopter1Helicopter2Helicopter3148.5 in2 clips (ref)Sample
mean of CDs0.9780.9490.916Sample STD of CDs0.1020.0910.050
Structural & Multidisciplinary Optimization
Group#/25Problem5: Linear model with one clipArea metric of the
posterior predictive distribution of CD
0.440.300.14CD from the fall time
dataHelicopter1Helicopter2Helicopter3148.5 in1 clipsSample mean of
CDs0.8820.9230.911Sample STD of CDs0.0780.0330.025148.5 in2 clips
(ref)Sample mean of CDs0.9780.9490.916Sample STD of
CDs0.1020.0910.050
Structural & Multidisciplinary Optimization
Group#/25Problem5: Linear model with one clipArea metric of the
distribution of CD with p-box
0.130.060.02CD from the fall time
dataHelicopter1Helicopter2Helicopter3148.5 in1 clipsSample mean of
CDs0.8820.9230.911Sample STD of CDs0.0780.0330.025148.5 in2 clips
(ref)Sample mean of CDs0.9780.9490.916Sample STD of
CDs0.1020.0910.050
Structural & Multidisciplinary Optimization
Group#/25Comparison between quadratic and linear modelsArea metric
of the posterior predictive distribution of CD
Area metric of the distribution of CD with p-box
Helicopter1Helicopter2Helicopter3148.5 in / 1
clipLinear0.190.340.38Quadratic0.440.300.14148.5 in / 2 clips
(ref)Linear0.130.110.07Quadratic0.150.110.07Helicopter1Helicopter2Helicopter3148.5
in / 1 clipLinear0.130.060.02Quadratic0.010.110.22148.5 in / 2
clips (ref)Linear0.010.000.00Quadratic0.010.000.00
Structural & Multidisciplinary Optimization
Group#/25Concluding remarks Predictive validation for both
quadratic and linear modelsThe predictive validation for different
mass is a partially successThe predictive validation for different
height is a success but the assumption of constant CD is not
clearly proven Comparison between modelsCannot conclude
OverallReason for the differences in the area metric is not
clearThe effect of the manufacturing uncertainty is significant
(i.e. very different area metrics for the same test condition)
Structural & Multidisciplinary Optimization Group#/25Many
designs have multiple failure modes. Probabilistic design approach
has clear advantages over deterministic design in providing
tradeoff between improving performance and protecting failure
modes. Dr. Acar studied the effect of allocating risks for aircraft
design with wings and a horizontal tail. Instead of applying the
same design margins, allocating risks by moving a small amount of
material from the wing to horizontal tail provides safer and
lighter design. Dr. Qu investigated probabilistic design approach
for cryogenic tank design, which has two failure modes, thermal and
mechanical failure modes. To apply probabilistic design, system
reliability has be to determined. Applying independence assumption
for failure modes can simplify evaluating system reliability.
However, in many cases, failure modes are correlated, ignoring
dependence results error in system reliability. The effect of
ignoring dependence with respect to important factors will be
presented. 22First overall comment is not clear
Paper ClipsHeightCalibrated?ModelArea MetricSet 1 Model 1Set 1
Model 2Set 1 Model 3Set 3 Model 1Set 3 Model 2Set 3 Model
321noquadratic0.53760.83970.57170.30040.37060.529921yesquadratic0.1430.4420.190.3050.2670.094612yesquadratic0.62720.8330.65030.39270.5930.544221nolinear0.96391.02021.00620.63850.57370.738821yeslinear0.10730.50030.25870.35710.31040.139811yeslinear1.32161.53981.34570.2130.2070.16822yesquadratic0.1690.2140.2840.4610.4240.269
Kaitlin Harris, VVUQ Fall 2013Comments: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 Comparison of
Analytical Cdf and Empirical CdfData Set 5, H=18.832, No. of
Pins=2Helicopter 1Helicopter 2Helicopter 3Validation Area
MetricBefore Calibration0.73090.69320.6169After
Calibration0.28090.09060.142Calibration ResultsHelicopter
1Helicopter 2Helicopter 3Maximum Likelihood Estimate of
Cd0.78890.89650.9111Standard deviation in Posterior pdf of
Cd0.0460.01910.0238Predictive Validation for Data Set 5, No. of
Pins =1H=18.832ft.Helicopter 1Helicopter 2Helicopter 3Validation
Area Metric0.18690.24530.192Validation Area Metric for Linear
Dependence ModelData Set 5Helicopter 1Helicopter 2Helicopter
3Validation Area Metric0.44350.26060.3506
Validation of Cd is constant at different height, h=11.482ftData
Set 5, Pins = 2Helicopter 1Helicopter 2Helicopter 3Validation Area
Metric0.41860.15920.1513Quadratic Dependence 2 clipsDifferent
height.Quadratic Dependence 1 clipPredictive Validation.Linear
Dependence 2 clips & 1clipPrior VS. Posterior Dist.Quadratic
Dependence 2 clipsPrior VS. Posterior Dist.Course Project:
Validation of Drag Coefficient -Yiming ZhangValidation based on 1
set of data:Validation based on 3 set of data:Summary: (1)
Quadratic dependence seems accurate using one set; Quadratic and
linear dependence both dont match well using 3 sets;
Prior Area Metric:0.4042
Post Area Metric:0.1695
Prior Area Metric:0.4920
Post Area Metric:0.2
Validation Area Metric:0.1923
Validation Area Metric:0.1267
2 clips:
1 clip:
Posterior Area Metric:0.1729
Validation Area Metric:0.1960 Summary:
Two confi. interval of Cd dont coincide. Linear dependence seems
inaccurate.2 clips:
Posterior Area Metric:0.19871 clip:
Validation Area Metric: 0.1253 Summary:
Seems reasonable, but not accurate
Validation Area Metric:0.0970
Validation Area Metric:0.1394
(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. Backup SlidesProblemsProblem1:
Conservative estimate of the fall timeProblem2: Comparing predicted
variability and observed variability using priorProblem3: Comparing
predicted variability and observed variability using
posteriorProblem4: Predictive validation for the same height and
different weightProblem5: Comparing the quadratic and linear
modelsProblem6: Predictive validation for different height and the
same weight (proving the assumption of constant CD)
Structural & Multidisciplinary Optimization
Group#/2527Problem1: Conservative estimate of the fall
timeEstimating the 5th percentile of the fall time of one
helicopterSince fall time follows a normal distribution, estimating
the 5th percentile is based on estimating the mean and standard
deviation (STD) of the fall time distributionThe 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 samplesTo
compensate the epistemic uncertainty, a conservative measure to
compensate the epistemic uncertainty is requiredEstimating the 5th
percentile with 95% confidence levelStructural &
Multidisciplinary Optimization Group#/2528
