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One significant feature of this life cycle approach was its ability to track time and usage. In this
study, flight hours and cycles fields were essential, but sometimes incomplete. The limitation
was resolved by determining if the removals were the first replacements since the initial
installations. In these instances, accumulated flight hours and cycles for the components were
the same as the accumulated flight hours and cycles for the aircraft. Precise flight hours and
cycles to date were not always available, but the quarterly data were available from the Airclaims
database.
3.6 GRAPHICAL PRESENTATIONS OF LIFE DATA.
Graphical presentations of the life data were informative and often produced to facilitate an
understanding of the removal patterns and failure trends. Figure 4 shows a typical illustration of
the units removed as a function of time. The x axis shows the cumulative flight hours of the
parts, as well as the aircraft. Part hours and removal times are shown in solid circles. Aircraft
hours are shown from the baseline in a straight line (grey color). They are aligned to the left to
allow examination and risk estimation of trends over time. An increased number of removals
associated with older parts, or older aircraft, are potential indicators of aging-related
performance. The y axis is the aircraft or the part serial number from the smallest number
(oldest) to the largest number (youngest). Critical failure modes instead of removals can also be
displayed using different symbols.
Figure 4. A Typical Illustration of Units Removed as a Function of Time
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3.7 ANALYSIS OF RISK.
Data analysis was driven by the need to understand when, why, how often, and how many of the
units needed service over time. If removal or failure patterns are understood, inspection intervals
can be recommended. The analysis was carried out by examining these trends and, based on the
trends, identifying any safety issues and proposing appropriate action.
This life cycle approach is quite extensive in the literature. Meeker and Escobar [6],
Lawless [7], and Nelson [8] were among the references. The approach is summarized in section
3.7.1, and additional technical topics are discussed in section 3.7.2. Section 3.7.1 should be
sufficient for most readers.
3.7.1 Overview of the Life Cycle Approach.
During a typical life cycle analysis, life and failure events of the parts are obtained. The life
histories are aligned at baseline (see figure 4) and proportions of failure and removal are
computed. They are computed by dividing the number of removals, di, by the number of
working parts, ni, at each interval, i, which yields the probability of failing and removal, ,at
each interval from 1 to m.
pˆi
ˆ , 1,..., i
i
i
p dn i m
= =
i .
(1)
If n is the total number of parts in the beginning of the study, then subsequent numbers of
working parts are The intervals are flexible and may be small or large as long as
they do not overlap. Using the individual probabilities at each interval, cumulative probabilities
of surviving and no removals over time are computed.
ni=n-di-1-r-1
(2)
1
ˆ ( ) [1 ˆ ], 1,...,
i
i j
j
S t p i m
=
=Π − =
Subsequently, cumulative probabilities of failing and removals are computed.
Fˆ (ti) =1−Sˆ(ti), i =1,...,m (3)
The computational steps described above result in a step function (figure 5, black curve) where
the probability of failing increases over time from 0 to eventually 1. In the literature, an
underlying probability distribution (a smooth mathematical function) is used for additional
advantages (figure 5, blue curve). The advantages include being able to describe the removal
patterns using a smooth curve with a few parameters and, most importantly, to project (in time)
performance of the parts in the future. This approach is called the parametric approach. The
choice of the best probability distribution is always determined by the model that best fits the
data. Examples of probability distributions include normal distribution, log normal distribution,
Weibull distribution, etc. Using a log normal distribution as an example, cumulative probability
function with μ and σ as the parameters is
9
( ; , ) nor log( )
F t t μ
μ σ
σ
= Φ ⎡⎢⎣ − ⎤⎥⎦
(4)
where the log of time t has a normal distribution with mean μ, and variance σ2 and is the
cumulative probability function for a standard normal distribution. The values of μ and σ2 are
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