DSES-6070HV2 Class Project
Aircraft Windshield Component Failures
By
Robert A. Charbonneau
14-Aug-2008
1
1. Introduction: Background / Description of the Problem
The purpose of this document is to discuss results of data taken from a case study on
aircraft windshields, as illustrated in “Illustrated Cases and Data Sets”, Bliske. [Ref. 5.1]
As discussed in the referenced document, the eighty-eight points of data were from
aircraft windshields, which failed due to de-lamination of the windshield layers, failure
of the heating system, and/or some form of external damage. Failure of the windshields
does not pose a structural concern, or problem to the airlines. There was a concern with
warranty costs. The collected data was documented in x1000 hours of service. [see Fig.
1.1] Also included with the failure data are Service times. Service times will not be
discussed in this paper.
Fig 1.1 – Aircraft Windshield Failure Data
The analysis programs used, prior to the write up of this report, were the following:
Minitab-v15, Maple-v10 & 12, and MS Excel. These programs were used to determine:
the exponential formula best fitting the data points. Once the formula was selected, two
types of data analysis were performed: the use of Maple to execute the formulas, and MS
Excel to produce a Monte Carlo curve of the reliability and confirm the reliability curve
generated by the selected exponential form.
2
The final output of the data is in the form of graphs generated by Maple, and in addition, Monte-Carlo charts from MS Excel.
Also included with the results are
recommendations and conclusions based on the failure results.
Airplane Windshield Failure Histogram
Normal
Mean
StDev
N
20
2.588
1.123
88
Frequency
15
10
5
0
0
1
2
3
Time (x1000 hr)
4
5
R A Charbonneau
Fig. 1.2 – Minitab v15 Histogram of Aircraft Windshield Failure Data
The grouping of the data, as shown in Fig. 1.2 – Normal Histogram, shows failure
data ranging from 46 to almost 5,000 hours of flight time. The calculated Mean shows a
value of 2,588 hours and a Standard Deviation of 1,129 hours. The maximum rate of
failures appears to be around 2,000 hours. See section 3.0 – Analysis / Discussion for
further results.
3
2. Methodologies: Summary description / Application
In order to correctly analyze the situation, here are first a couple assumptions that
need to be made. First, the assumption is made that the life distribution of units are
right-censored data given there is only a beginning data and no end data with the original
data set. Further, to calculate the probability density function, we will assume that the
data follows a Weibull distribution. We will begin by validating this assumption and
then use this to calculate the MTTF and MRL.
4
3. Results: Analysis / Discussion
The data was first loaded into MS Excel, by hand, from the case failure data table
[see Fig. 1.1]. From there, it was loaded into Minitab. The first analysis performed was
the Histogram (Fig. 1.2), and a Probability Plot [see Fig. 3.1] of the full eighty-eight
points of data.
Airplane Windshield Failure Probability Plot
Normal - 95% CI
99.9
Mean
StDev
N
AD
P-Value
99
95
90
2.588
1.123
88
0.581
0.127
Percent
80
70
60
50
40
30
20
10
5
1
0.1
-2
-1
0
1
2
3
4
Time (x1000 hr)
5
6
7
R A Charbonneau
Fig. 3.1 – Failure Probability Plot w/ Probability of 0.127 (p>0.05 – not significant)
Aircraft Windshield Probability Plot for Time (x1000 hr)
LSXY Estimates-Complete Data
C orrelation C oefficient
Weibull
0.933
Lognormal
0.861
E xponential
*
N ormal
0.991
Lognormal
99.9
99.9
90
99
50
90
P er cent
P er cent
Weibull
10
1
50
10
1
0.1
0.1
1.0
T ime ( x1 0 0 0 hr )
0.1
10.0
0.1
10.0
N ormal
99.9
99.9
90
99
50
90
P er cent
P er cent
E xponential
1.0
T ime (x1 0 0 0 hr )
10
1
50
10
1
0.1
0.001
0.010
0.100
1.000
T ime ( x1 0 0 0 hr )
10.000
0.1
0
2
4
T ime (x1 0 0 0 hr )
6
Fig. 3.2 – Aircraft Windshield Distribution ID Plot (Right Censored)
5
The data results from Fig. 3.1 show the p-value is 0.127. If assuming a hypothesis
that the data should agree with 95% Confidence Interval or greater, then the p-value of
0.127 would indicate the data is not significant and does not meet a normal distribution.
The next step was to perform a distribution analysis.
This was performed in
Minitab by opening up Stat > Reliability/Survival > Distribution Analysis (Right Censoring) > Distribution ID Plot. From there, used the default settings and output a fourplot chart with: Weibull, Lognormal, Exponential, and Normal Probability Plots. [See
Fig. 3.2] The result was a Normal Probability of 0.991, a Lognormal of 0.861 and a
Weibull of 0.933. This would indicate the Normal formulas would work the best for
calculating Mid-Time To Failure (MTTF). However, the problem with the Normal
formulas, the formula for Cumulative Failure Probability, F(t) doesn’t convert well to
calculate t[RAND()] in Monte Carlo.
Based on the results from Fig. 3.2, the lowest time point was treated as an outlier
(46 hours), and removed from the data set. This done, the Stat >> Distribution ID Plot
was redone with the revised dataset. The results (see Fig. 3.3) then became: a Normal
Probability of 0.989, a Lognormal Probability of 0.941, and a Weibull Probability of
0.982.
Probability Plot for T (Rev)
LSXY Estimates-Complete Data
C orrelation C oefficient
Weibull
0.982
Lognormal
0.941
E xponential
*
N ormal
0.989
Lognormal
99.9
99.9
90
99
50
90
P er cent
P er cent
Weibull
10
1
50
10
1
0.1
0.1
1.0
T (Rev)
0.1
0.1
10.0
1.0
T (Rev)
N ormal
99.9
99.9
90
99
50
90
P er cent
P er cent
E xponential
10.0
10
1
50
10
1
0.1
0.001
0.010
0.100
1.000
T (Rev)
10.000
0.1
0
2
T (Rev)
4
6
Fig. 3.3 – Revised Aircraft Windshield Distr. ID Plot (Right Censored)
6
By removing one outlier, the probability factor for Weibull was improved. With the
Weibull and the Normal Probabilities both in the relative 0.98+ range, it was decided to
use the Weibull analysis to best determine MTTF.
To find the Shape and Scale of the data set, a Distribution Overview Plot was generated with Weibull selected for the distribution plots for: probability plot, Probability
Density Function - f, Survival Function - R and Hazard Function – z. [Ref. Fig. 3.4]
Distribution Overview Plot for T (Rev)
LSXY Estimates-Complete Data
P robability D ensity F unction
Table of S tatistics
S hape
2.38447
S cale
2.96874
M ean
2.63143
S tDev
1.17473
M edian
2.54576
IQ R
1.64402
F ailure
87
C ensor
0
A D*
0.543
C orrelation
0.982
Weibull
99.9
90
0.3
P er cent
P DF
50
0.2
0.1
0.0
10
1
0
2
4
0.1
0.1
6
1.0
T (Rev)
T (Rev)
S urv iv al F unction
H azard F unction
2
Rate
100
P er cent
10.0
50
0
1
0
0
2
4
6
0
2
T (Rev)
4
6
T (Rev)
Fig. 3.4 – Aircraft Windshield Weibull Distr. Overview Plot (w/o Outlier)
The resultant Shape (  ) and the Scale (1/  ) was [2.38447, 1/2.96874], respectively. Plugging these results into the Weibull formulas produced the following:

R(t )  e (t )  e (0.33684t )
2.38447
F (t )  1  R(t )  1  e (0.33684t )
2.38447
0.33684  t 2.38447  e 0.33684t 
f (t )  diff ( R(t ), t ) 
2.38447
t
z (t )  f (t )
2.38447  0.33684  t 
t
2.38447
R(t )

By recalculating the formula F(t), solving for t, and replacing RAND() in place of
F(t), the resultant formula became:
1
1
t ( RAND ())     ( LN (1  RAND ())) 

7
This formula was then used to calculate for “T” with specific time steps from 50
hours to 6,000 hours, and then comparing the randomized ratio steps to Reliability at
times, t. See Reference 5.2 for the full results. The resultant Reliability/Survival Monte
Carlo chart is illustrated in (Figure 3.5).
Reliability (ratio)
Airplane Windshield
Reliability/Survival Monte Carlo - Failure
1.0000
0.9000
0.8000
0.7000
0.6000
0.5000
0.4000
0.3000
0.2000
0.1000
0.0000
Reliability
Monte Carlo
0
2
4
6
Time, t (x1000 hrs)
Fig. 3.5 – Aircraft Windshield Monte Carlo Results (Ref. 4.2)
The comparison of the Monte Carlo to the Reliability results at times, t confirmed
the values for  and  was relatively correct.
With the completion of the Minitab and Excel calculations, the remaining work was
performed in Maple v12. In Reference 5.3, the equations for F(t), R(t), f(t), and z(t) were
then plugged in, along with the values for  and  , and numerical results were calculated. [see Fig. 3.6] The resultant formulas were then used to plot Probability Density
Function - f, Survival Function - R and Hazard Function – z. [see Figs. 3.7 – 3.9]
The
typical
way
of
calculating
MTTF
is
with
the
formula,
MTTF : int( Rw, t  inf inity ) . However, this didn’t appear to work and an alternative
 
method was used instead: MTTF  1   1  1 . This resulted in a solution number of

2.631427360 for MTTF.
8
Fig. 3.6 – Maple Weibull Calculation Formula Calculations
To further confirm the Monte Carlo calculations were correct, Reliability was calculated at a specific times, t. The value for t was input for 1,500 hours with a resultant
value of 0.8217, which agrees with the Monte Carlo and R(t) formulas in the Excel
spreadsheet. [Ref. 5.2 & 5.3]
9
Fig 3.7 – Plot of Probability Density Function, f(t) as time varies from 0 to 6,000
hours
10
Fig. 3.8 – Combined Plots of Reliability/Survival – R(t) (green) and Failure Functions – F(t) (red) as time varies from 0 to 6,000 hours
Fig. 3.9 – Plot of Hazard Function as time, t varies from 0 to 6,000 hours
Further calculations to determine a new warranty limit based on cost values were
not available with the original data. It could be assumed where the ratio of failures
become greater than the ratio of survivors would be an appropriate time to place the
warranty limit. This would then be a warranty limit directly connected with Mid Time
To Failure (MTTF) value of 2,630 hours.
11
4. Conclusions
The decision to use Weibull to calculate for f(t), F(t), R(t), z(t), and MTTF were
based on a Hypothesis test where the histogram p-value did not meet test requirements
for a Normal histogram; and with one outlier removed, the Weibull was comparable to
the Normal probability of greater than 0.98. [see Section 3.0] This resulted in a calculated MTTF of 2,630 hours.
From the results of the Minitab, Maple and Monte Carlo results confirmed correlation to the Weibull formulas used, and a Monte Carlo model is now available to predict
the rate of surviving aircraft windshields at any time, t. It was also concluded further
investigation into warranty cost could not proceed based on the lack of information.
It is the recommendation of This Writer to use the MTTF value of 2,630 hours as a
warranty limit for those aircraft windshields now in service.
12
5. References
5.1 Bliske, Wallace R.; “Reliability – Modeling, Prediction, and
Optimization”; 2000, John Wiley & Sons, INC., Chapter 2 –
Illustrative Cases and Data Sets, pp 36.
5.2 Charbonneau, Robert A.; “…/Support Documents/
Monte_Carlo_Weibull.xls”; 14-Aug-08, Excel spreadsheet created to
perform Monte Carlo analysis to confirm Weibull formulas agree –
from time steps 50 hours through 6,000 hours.
5.3 Charbonneau, Robert A.; “…/Support Documents/
Aircraft_Window_Failures.mw” 14-Aug-08, Maple v12 formula
calculations set to determine MTTF.
13