Aircraft Windshield Reliability Final Paper

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



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.



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



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


P e r c e n t

10.00001.00000.10000.01000.00100.0001

99.9

90

50

10

1

0.1


P e r c e n t

151296

99.9

99

90

50

10

1

0.1


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



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.



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.



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.



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.



Figure 7. Plot of failure probability density function, f(t)

Figure 8. Plot of hazard rate function, z



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



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.



References

1. Bliske, Wallace R, “Reliability- Modeling, Prediction, and Optimization”, 2000,

John Wiley & Sons, Inc., Chapter 2: Illustrative Cases and Data Sets, pp 36.

Date post:	02-Jun-2018
Category:	Documents
Upload:	anonymous-ffje1rpa
View:	219 times
Download:	2 times

Aircraft Windshield Reliability Final Paper

Documents