aircraft windshield reliability final paper
TRANSCRIPT
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 1/12
Aircraft Windshield Failure Data Analysis
DESE-6070: Statistical Methods for Reliability Engineering
Final Project
Professor Ernesto Gutierrez-Miravete
Erica B. Siegel
December 11, 2008
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 2/12
Introduction: Background/Description of the Problem
The objective of this paper is to look at the data provided and determine the best fit
and distribution for the data. The system being investigated is the failure of aircraft
windshields. An aircraft windshield is comprised of multiple layers which a strong
outer skin and heated layer at the base. All of the layers are laminated under heat
and high pressure. The failures in this case are not actually structural failures but
typically involve related system failures such as the delamination of the outer layer of
the windshield or failure of the heating system. Even though no structural damage
has occurred but the windshield must be replaced. The main reason for analysis of
this data is to predict warranty costs.
The data, given below in Figure 1.1 contains two sets of data from Reliability:
Modeling, Predictions and Optimization1. Eight-eight of the values are considered actual
failures while the other 65 are windshields that have not yet failed but have been
serviced. The data given is incomplete as in that not all failure times have been
observed.
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 3/12
Figure 1. Aircraft Windshield Failure Data
Only the 88 values for the failure times will be used for the following analysis since
the other 65 windshields have not actually failed and it does not state if the work wasdone under warranty or at the owner’s expense.
Methodologies: Summary Description/Application
Three methods will be used to fully analyze the given data. Minitab, Maple and
Excel will be used to analyze the reliability of aircraft windshields. In order to
analyze the data, the assumption was made that the distribution is right-censored
data given there is only a beginning data with no end data in the data set.
Results: Analysis/Discussion
First, Minitab was used to determine the distribution. As can be seen in Figure 2, the
3-parameter weibull distribution, with a correlation coefficient of .922, is the best fit
for this data set. However, after some initial attempts to analyze the data in Maple
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 4/12
using a three parameter Weibull, it was found to be too difficult so a standard
Weibull distribution, with a correlation coefficient of .934, was used instead, see
Figure 3. The correlation coefficient of the standard weibull could have been
improved if the outlier had been removed. However, since no data was provided as
to why the windshield failed so early the data point was included in the analysis.
Without a valid reason for the failure, it seems inappropriate to remove this data
point from the analysis.
101
99.9
90
50
10
1
0.1
Failure Times - Threshold
P e r c e n t
20181614
99.9
99
90
50
10
1
0.1
Failure Times - Threshold
P e r c e n t
10.00001.00000.10000.01000.00100.0001
99.9
90
50
10
1
0.1
Failure Times - Threshold
P e r c e n t
151296
99.9
99
90
50
10
1
0.1
Failure Times - Threshold
P e r c e n t
3-Parameter Weibull
0.992
3-Parameter Lognormal
0.9912-Parameter Exponential
*
3-Parameter Loglogistic
0.984
C orrelation C oefficient
Probability Plot for Failure TimesLSXY Estimates-Complete Data
3-Parameter Weibull 3-Parameter Lognormal
2-Parameter Exponential 3-Parameter Loglogisti c
Figure 2. Minitab Distribution Comparison: 3-Parameter Weibull, Lognormal &
Loglogistic
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 5/12
10.01.00.1
99.9
90
50
10
1
0.1
Failure Times
P e
r c e n t
10.01.00.1
99.9
99
90
50
10
1
0.1
Failure Times
P e
r c e n t
10.0001.0000.1000.0100.001
99.9
90
50
10
1
0.1
Failure Times
P e r c e n t
10.01.00.1
99.9
99
90
50
10
1
0.1
Failure Times
P e r c e n t
Weibull
0.934
Lognormal
0.861
Exponential
*Loglogistic
0.875
C orrelation C oefficient
Probability Plot for Failure TimesLSXY Estimates-Complete Data
Weibull Lognormal
Exponential Loglogistic
Figure 3. Minitab Distribution Comparisons: Weibull, Lognormal and Loglogistic.
After determining a Weibull distribution was the best option, a detailed analysis was
run in Minitab to determine the shape and scale parameters for the data. The results
of that analysis can be seen in Figure 4.
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 6/12
6420
0.3
0.2
0.1
0.0
Failure Times
P D F
10.01.00.1
99.9
90
50
10
1
0.1
Failure Times
P e r c e n t
6420
100
50
0
Failure Times
P e r c e n t
6420
1.5
1.0
0.5
0.0
Failure Times
R a t e
Correlation 0.934
S hape 1.94482
S cale 2.99246
M ean 2.65362
S tD ev 1.42251
Median 2.47847
IQ R 1.96280
Failure 88
C ensor 0
A D* 1.373
Table of S tatisticsProbability Density F unction
Surv iv al F unction Hazard Function
Distribution Overview Plot for Failure Times- Weibull
LSXY Estimates-Complete Data
Weibull
Figure 4. Detailed Weibull Analysis
The next step in the analysis was to use the data from Minitab to create a weibull
failure rate function to analyze in Maple. The shape parameter of 1.945 is used as !
in the analysis, while the scale parameter of 2.992 is used as ".
The two variables are
then plugged into the Weibull equation of F = 1-exp(-!t)". Then, Maple is used to
determine the failure rate function (F), the reliability function (R = 1 - F), the failure
probability density function, and the hazard function (z). The results of the analysis
can be seen in Figure 5 below.
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 7/12
Figure 5. Maple analysis
The equations shown in the Figure 5 were then graphed in Maple in order to see the
distribution.
First, the failure rate function (F) and the reliability function (R = 1 - F), also known
as the survival rate, were graphed against each other. The graph is Figure 6 shows
that, as expected the reliability decreases with time as the failure rate increases. At
approximately 4500 hours, the windshield is as likely to still be working as it is to
fail. After 4500 hours the windshields are more likely to fail and before 4500 hours
the windshields are more likely to be working properly. The failure rate function is
shown in green in Figure 6 while the reliability function is shown in red.
As stated above, the mean time to failure (MTTF) is about 4500 hours. In addition
to being evident in Figure 6, the MTTF was calculated in Maple as verification.
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 8/12
Figure 6. Plot of the failure rate function (F) versus the reliability function (R = 1 -
F).
Next, both the failure probability density function (f(t)) and the hazard function (z)
were graphed individually. The failure probability density function is shown in
Figure 7, demonstrates the likelihood that the windshield will fail, independent of the
amount of time it has been in service. For this system, it might not be an accurate
estimate because the likelihood of failure is dependent on the amount of time the
aircraft has been in service.
The hazard rate function (z) demonstrates the probability that a windshield will fail,
given that it has reached a particular age. The graph in Figure 8 shows that the
likelihood of windshield failure increases as it ages.
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 9/12
Figure 7. Plot of failure probability density function, f(t)
Figure 8. Plot of hazard rate function, z
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 10/12
After all calculations were completed in Maple, excel was used to run a Monte Carlo
simulation to verify the reliability function. The two reliability functions can be seen
below, and, although not perfect, they have a similar shape ensuring that Monte
Carlo is a valid way to replicate the data analyzed in Maple.
0.000
0.200
0.400
0.600
0.800
1.000
0 1 2 3 4 5
Figure 9. The reliability function from Maple versus the reliability function from
Monte Carlo simulation
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 11/12
Conclusion
During this analysis it was determine that the Weibull, although not the best fit for
an estimate, was an appropriate substitute for the three-parameter weibull
distribution originally found to be the best fit. Additionally, a Monte Carlo
simulation has been created in Excel to determine the probability of failure at any
given time.
One of the goals of this analysis was to look at its application to warranty costs.
However, the costs of repair are not available and made that analysis impossible at
this time. However, when looking into warranty costs, users can assume a mean
time to failure of about 4500 hours for all calculations.
8/10/2019 Aircraft Windshield Reliability Final Paper
http://slidepdf.com/reader/full/aircraft-windshield-reliability-final-paper 12/12
References
1. Bliske, Wallace R, “Reliability- Modeling, Prediction, and Optimization”, 2000,
John Wiley & Sons, Inc., Chapter 2: Illustrative Cases and Data Sets, pp 36.