Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 1, pp. 981-990, Warsaw 2016
DOI: 10.15632/jtam-pl.54.1.981

THE METHOD OF ESTIMATING LIFETIMES OF AIRCRAFT DEVICES
OPERATING UNDER AGEING-ATTRIBUTABLE WEARING CONDITIONS

Mariusz Ważny
Military University of Technology, Warsaw, Poland

e-mail: mariusz.wazny@wat.edu.pl

The paper presents a probabilistic method of assessing lifetimes of selected structural com-
ponents or assemblies of devices/systems affected by destructive processes that occur during
aircraft operation. Reliability status of the device is evaluated by means of diagnostic or
operational parameters. It is assumed that these devices (systems, assemblies) operate re-
liably if effects of wear and tear processes described by diagnostic parameters do not exceed
boundary conditions/regimes. From the mathematical aspect, the method has been based
on difference equations from which, when rearranged, a partial differential equation of the
Fokker-Planck type is derived. A density function of the component wearing is a particular
solution to this equation.With the density function of the componentwearing applied, after
suitable rearrangements, one can determine a density function of time for the exceeding
the boundary condition. Now, with the density function of time of reaching the boundary
condition found, and after rearrangement of this function, one receives dependences that
can be applied to determine lifetime of the device given consideration. An example at the
end of the paper illustrates how this method can be applied to analyse an airborne sighting
system.

Keywords: reliability, lifetime, destructive processes, permissible condition

1. Introduction

Any manifestations of aeronautical engineering, i.e. aeronautical devices, systems, etc., make
design engineers, manufacturers and users meet requirements connected with maintaining high
values of safety and reliability parameters. Examining the safety and reliability of an aircraft
throughout the operational process involves predictions about health/maintenance status of
particular devices and systems of an aircraft and the aircraft itself as a platform combining
all the above-mentioned elements. Analyzing an aircraft as an object intended to provide, e.g.
transportation of passengers and cargo, we can assume that the operating conditions are of
special importance comparedwith other popularmeans of transport (Pamuła, 2011). A series of
factorsmake values of parameters that describe health/maintenance status of an aircraft change
over time. Destructive processes resulting from overloads, friction, vibration, ageing processes,
etc. prove to have the crucial effect on that change.
An aircraft is a platform upon which systems like the following ones are integrated:

• automatic control systems,
• communication systems,
• avionic systems,
• air armament systems,
• sighting systems,
• systems to abandon the aircraft in emergency, etc.



982 M.Ważny

The health/maintenance status of aircraft devices is mainly evaluated with a set of diagno-
stic parameters. The effect of destructive processes is visible in changes of values of diagnostic
parameters, which cause increments in deviations from nominal values of these parameters. De-
viations from nominal values of diagnostic parameters are used to estimate reliability of a given
device. This question is addressed in (Niu et al., 2011; Tomaszek andWróblewski, 2001; Toma-
szek et al., 2004, 2011, 2013; Tomaszek and Szczepanik, 2005; Ye et al., 2011; Zhanshan and
Krings, 2008). Among the above mentioned works, (Niu et al., 2011) is an interesting publica-
tion where health/maintenance status of a piece of equipment is analysedwith theMahalanobis
distance indicator and theWeibull distribution applied.
This paper is an attempt to analyse and describe degradation of health/maintenance status

of selected devices as a result of destructive processes affecting them.
Classification of correlations between effects of destructive processes and changes in values

of diagnostic parameters is presented by Ważny (2011a). The density function of changes in
deviations of the diagnostic parameter has been determined in this article, with the following
assumptions applied:
• the device health is determined by one dominant diagnostic parameter (its current value
is denoted by x),

• the change in value of the diagnostic parameter due to the destructive effect of ageing
processes occurs as the calendar time passes by,

• the deviation of the diagnostic parameter from the nominal value is

z = |xp −xn|

where xp is measured value of the diagnostic parameter, xn – nominal value of the diagno-
stic parameter,

• the value of the diagnostic parameter deviation determines the level of reliability of a given
structural component. If it remains within the interval z ∈ [0,Zd], the component will be
recognised as serviceable (fit for use). Otherwise, it will be recognised unserviceable (unfit
for use),

• the increase in the diagnostic parameter deviation against the calendar time satisfies the
relationship

dz

dt
= c

where c is a random variable that depends on ageing processes, t – the calendar-based
time.

2. Determining the density function of changes in values of deviations of the
diagnostic parameter

It is assumed that the intensity of growth of the component wear and tear takes the same form
as the failure rate for theWeibull distribution

λ(t)=
α

θ
tα−1 (2.1)

where α and θ are constants in theWeibull distribution with the following denotations: α – the
shape factor, θ – the scale factor.
The stochastically approached dynamics of changes in values of diagnostic parameters, inclu-

ding the parameter deviation, is describedbydifference equation. Let Uz,t denote the probability
that at the time instance t the diagnostic parameter deviation takes value z.



The method of estimating lifetimes of aircraft devices... 983

A difference equation takes the following form for the assumed conditions

Uz,t+∆t =
(
1−

α

θ
tα−1∆t

)
Uz,t +

α

θ
tα−1∆tUz−∆z,t (2.2)

where ∆z is the increment in the diagnostic parameter deviation in the time interval ∆t.
Equation (2.2) written down in functional notation takes the following form

u(z,t+∆t)=
(
1− α

θ
tα−1∆t

)
u(z,t)+

α

θ
tα−1∆tu(z−∆z,t) (2.3)

where u(z,t) is the density function of the diagnostic parameter deviation, [1− (α/θ)tα−1∆t] –
probability that in the time interval ∆t there will be no increment in the diagnostic parameter
deviation, (α/θ)tα−1∆t – probability that in the time interval ∆t therewill be the ∆z increment
in the parameter deviation, and the following condition is satisfied

α

θ
tα−1∆t ¬ 1

Rearrangement of equation (2.3) into a partial differential equation results in the following
approximation

u(z,t+∆t)=u(z,t)+
∂u(z,t)
∂t

∆t

u(z −∆z,t)= u(z,t)− ∂u(z,t)
∂z

∆z+
1
2
∂2u(z,t)
∂z2

(∆z)2
(2.4)

Having substituted relationships (2.4) into equation (2.3) and with some rearrangements
done, the following is arrived at

∂u(z,t)
∂z

=−α
θ
tα−1∆z

∂u(z,t)
∂z

+
1
2
α

θ
tα−1(∆z)2

∂2u(z,t)
∂z2

(2.5)

Consideration is given to the increment in the diagnostic parameter deviation per time unit
(when ∆t=1), hence

∆z

∆t
= c ⇒ ∆z = c∆t =⇒

∆t=1
c

where c denotes increment in the diagnostic parameter deviation per one time unit.
The final form of equation (2.5) is as follows

∂u(z,t)
∂z

=−
αc

θ
tα−1

︸ ︷︷ ︸
γ(t)

∂u(z,t)
∂z

+
1
2
αc2

θ
tα−1

︸ ︷︷ ︸
β(t)

∂2u(z,t)
∂z2

(2.6)

The solution to equation (2.6) takes the form

u(z,t) =
1

√
2πA(t)

exp
(
−(z −B(t))

2

2A(t)

)
(2.7)

where B(t) is an average value of the diagnostic parameter deviation for the operating time t

B(t)=
t∫

0

γ(t) dt (2.8)



984 M.Ważny

A(t) – a variance of the diagnostic parameter deviation for the operating time t

A(t)=
t∫

0

β(t) dt (2.9)

With integrals (2.8) and (2.9) calculated, the following expressions are arrived at

B(t)=
t∫

0

αc

θ
tα−1 dt =

c

θ
tα A(t)=

t∫

0

αc2

θ
tα−1 =

c2

θ
tα (2.10)

Hence, relationship (2.7) takes the following form

u(z,t)=
1

√
2πc

2

θ
tα
exp


−

(
z − c

θ
tα
)2

2c
2

θ
tα


 (2.11)

Relationship (2.11) presents the density function of the diagnostic parameter deviation from
the nominal value.
With the following substitutions

c

θ
= b

c2

θ
= a

relationship (2.11) takes the following form

u(z,t)=
1√
2πatα

exp
(
−
(z − btα)2
2atα

)
(2.12)

With thedensity function found, one canwritedowna relationship for reliabilitywith respect
to time of the diagnostic parameter deviation increasing up to the boundary value. The formula
takes the form

R(t)=

zd∫

−∞
u(z,t) dz (2.13)

where zd is a permissible value of the diagnostic parameter deviation.

3. Determining distribution of time of exceeding the permissible condition by
the diagnostic parameter deviation

The probability of exceeding the boundary value by the diagnostic parameter with the density
function of changes in the diagnostic parameter deviation can be written down in the following
form

Q(t,zd)=
∞∫

zd

1√
2πatα

exp
(
−(z− bt

α)2

2atα
)
dz (3.1)

The density function of the distribution of time of exceeding the permissible value of the
diagnostic parameter zd equals

f(t)=
∂

∂t
Q(t,zd) (3.2)



The method of estimating lifetimes of aircraft devices... 985

With (3.1) taken into account, this equation takes the form

f(t)=
∂

∂t

∞∫

zd

1√
2πatα

exp
(
−(z− bt

α)2

2atα
)
dz (3.3)

Now, the time derivative of the integrand for relationship (3.3) can be found

f(t)=
∞∫

zd

[2(z − btα)bαtα +(z − btα)2α
2atα+1

− α
2t

]
u(z,t) dz (3.4)

In order to calculate integral (3.4), we need to determine an antiderivative. We assume the
following form of the antiderivative of the integrand in relationship (3.4)

w(z,t) = u(z,t)θ(z,t) (3.5)

Thederivative of the indefinite integralwith respect to thevariable z is equal to the integrand
of relationship (3.4). Hence

∂u(z,t)
∂z

θ(z,t)+u(z,t)
∂θ(z,t)
∂z

=
[2(z − btα)bαtα +(z − btα)2α

2atα+1
− α
2t

]
u(z,t) (3.6)

Now, the derivative ∂u(z,t)/∂z is calculated

∂u(z,t)
∂z

=
1√
2πatα

exp
(
−
(z −btα)2
2atα

)[
−
2(z − btα)
2atα

]
= u(z,t)

[
−
(z− btα)
atα

]
(3.7)

After substitution of (3.7) into (3.6), the following relationship can be written down

u(z,t)
[
−
(z− btα)
atα

θ(z,t)
︸ ︷︷ ︸

I−L

+
∂θ(z,t)
∂z︸ ︷︷ ︸
II−L

]
= u(z,t)

[2(z − btα)bαtα +(z − btα)2α
2atα+1︸ ︷︷ ︸
I−P

−
α

2t︸︷︷︸
II−P

]
(3.8)

Using relationship (3.8), the function θ(z,t) is determined in such away that the left side of
relationship (3.8) equals its right side. Therefore

I −L = I −P ⇒ θ(z,t)=−
2bαtα +α(z − btα)

2t

II −L = II −P ⇒
∂θ(z,t)
∂z

=−
α

2t

(3.9)

After reduction

θ(z,t)=−
α(z + btα)
2t

(3.10)

The conclusion is that the primitive of the integrand takes the following form

w(z,t) = u(z,t)
[
−
α(z +btα)
2t

]
(3.11)

where

u(z,t) =
1√
2πatα

exp
(
−(z− bt

α)2

2atα
)



986 M.Ważny

We calculate integral (3.3)

f(t)zd = u(zd, t)
α(zd + bt

α)
2t

(3.12)

where

u(zd, t)=
1√
2πatα

exp
(
−
(zd − btα)2
2atα

)

Thus, relationship (3.12) determines the density function of time of exceeding the boundary
(permissible) condition by the diagnostic parameter zd deviation

f(t)zd =
α(zd +bt

α)
2t

1√
2πatα

exp
(
−
(zd − btα)2
2atα

)
(3.13)

4. A method to assess the lifetime of a device with respect to the diagnostic
parameter being analyzed

The formula for the reliability of a device with respect to the diagnostic parameter deviation
from the nominal value can be expressed as

R(τ)= 1−
τ∫

0

f(t,zd) dt (4.1)

where

f(t,zd)=
α(zd +bt

α)
2t

1√
2πatα

exp
(
−
(zd − btα)2
2atα

)
(4.2)

Thus, the unreliability of a device can be expressed by the following formula

Q(τ)=
τ∫

0

α(zd + bt
α)

2t
1√
2πatα

exp
(
−
(zd − btα)2
2atα

)
dt (4.3)

One can simplify integral (4.3) bymaking the following substitution u = tα

Q(u)=

α
√
u∫

0

zd + bu
2u

1√
2πau

exp
(
−(zd − bu)

2

2au

)
du (4.4)

Integral (4.4) shouldbe converted into a simpler formand theproblemcomes down to solving
the indefinite integral

∫
f(u,zd) du (4.5)

where f(u,zd) is the integrand of equation (4.5).
For the integrand of integral (4.5), the following change (zd − bu)2 =(bu−zd)2 and substi-

tution are performed ω =(bu−zd)2/(2au)

1
2
√
π

∫
1√
ω
e−ω dω (4.6)



The method of estimating lifetimes of aircraft devices... 987

The following substitution into integral (4.6),
√
ω = w results in what follows

1
2
√
π

∫
1√
ω
e−ω dω =

1√
π

∫
e−w

2
dw (4.7)

With onemore substitution into the integral (4.7)

w2 =
y2

2
2w dw = y dy (4.8)

formula (4.7) can be written down as

1√
π

∫
e−w

2
dw =

1√
2π

∫
exp

(
−y
2

2

)
dy (4.9)

Considering the above substitution y takes the form expressed by

y =

√
2(btα −zd)√
2atα

(4.10)

Hence, remembering about the appropriate notation for the limits of integration, the unre-
liability of a device with respect to the growth of the diagnostic parameter deviation will be
expressed by the following equation

Q(τ)=
1√
2π

√
2(btα−zd)√
2atα∫

−∞
exp

(
−y
2

2

)
dy (4.11)

Assuming some specific level of risk of failure, i.e. the level of probabilities of exceeding the
permissible value of the parameter deviation, the following equation can be written down

Q(τ)= Q∗ (4.12)

Hence

Q∗ =
1√
2π

γ∫

−∞
exp

(−y2
2

)
dy (4.13)

For the assumed value of Q∗, the value of the upper limit of integral (4.13) is to be found in
the Standard Normal Distribution Table.
In this way, we obtain the value of γ. Thus, the equation to determine lifetime of an element

takes the following form

γ =

√
2(btα −zd)√
2atα

(4.14)

Having solved equation (4.14), one can find the lifetime of a device with respect to the
assumed diagnostic parameter.
Therefore, having the value of the parameter γ and using relation (4.14), one can determine

the lifetime of the device with respect to the diagnostic parameter given consideration. The
following substitution has beenmade to do this

s = tα (4.15)

Using formula (4.15), one candetermine the lifetimeof thedevice on thebasis of the following
relationship

t∗ =
1
a
ln
2bzd +γ

2a+γ
√
4bzda+γ2a2

2b2
(4.16)



988 M.Ważny

5. Example of calculation

Tostart verification of themethod intended todetermine the lifetimeof adevicewith theWeibull
distribution parameters applied, the coefficients of this distribution should be determined first.
Both the parameter θ reflecting the scale factor and the parameter α representing the ratio of
the shape have been pre-assumed to equal unity (determination thereof will provide the basis
for further work).
Data recorded in the course of operating one of the sighting system units, i.e. the sighting

head, has been assumed a good example, on the basis of which the above-presented model has
been verified. The construction of this head enables visualisation of the sighting data in the
form of the sightingmarker that supports the sighting process whilemaking use of air weapons.
Undermaintenance procedures/works performed every 100 hours’ flight time recorded are values
of diagnostic parameters of the sighting head in form of two co-ordinates ε and β that define the
co-ordinates of the check position of the sighting marker. Nominal values of these co-ordinates
(with permissible error range included) are determined by the aircraftmanufacturer responsible
for adjusting the sighting head with the sighting system at the stage the aircraft is introduced
in service. If in the course of maintenance, the measured values of co-ordinates of the sighting
marker position are within the limits of permissible error, no maintenance efforts are taken to
correct the sighting marker position. On the other hand, if the value of at least one of the two
diagnostic parameters exceeds the permissible error value, the sighting system is subjected to
maintenance to remove the error. This is done by means of introducing into the system the
co-ordinates that remove the deviation of the sighting marker position from the nominal one.
With data recorded in the course of the operational process (ε1, t1);(ε2, t2); . . . ;(εn, tn) and

related to changes in values of one of the parameters that define health/maintenance status of
the sighting system, the rate of changes (of the increase) in deviation of the diagnostic parameter
can be written down as

ct1 = ε1 c =
ε1
t1

(5.1)

For thedatapresented inFig. 1, recorded in the courseofmaintenanceworks every 100hours’
flight time of the aircraft, the following parameters have been found that enable determination
of lifetime described with relationship (4.14)

b =0.00123 a =1.524 ·10−6 c =0.00123 (5.2)

Fig. 1. Changes in the value of the diagnostic parameter that describes the position of the sighting
marker and the value of the lifetime found

It is important for aeronautical systems that the level of reliability of the device in question
nears unity. That is why the reliability has been assumed close to unity to determine the lifetime



The method of estimating lifetimes of aircraft devices... 989

of the device. Assuming the level of reliability R∗(t)= 0.98, the value of the parameter γ∗ =2.32
could be found from theStandardNormalDistributionTables. Theparameter εd could be found
in the system technical documentation (Maintenance Manual), where information is given on
the value of permissible deviation of the diagnostic parameter that describes co-ordinates of the
sightingmarker position. Using operational data (Fig. 1) and the above-determined parameters,
the lifetime has been calculated from Eq. (4.14). In the analyzed case, it is

t∗ =124months (5.3)

6. Summary

The process of operating technical devices installed on an aircraft involves influences of change-
able weather andmechanical conditions resulting from, among other things, in-flight overloads.
These factors lead to the accumulation of destructive factors affecting the system(s) and causing
the nominal performance characteristics of selected system components decline. The material
presented in this article is a continuation of analyses to developmethods of finding density func-
tion of changes in deviations of parameters, and the time of exceeding the permissible condition
by the diagnostic parameter under analysis, the parameter being affected by particular destruc-
tive factors (Ważny, 2011a,b). The presentedmethod enables determination of residual lifetime
of a device and can be used to modify the process of operating the device in question. In this
way, the number of checks of health/maintenance status can be limited, which consequently can
reduce the time the device remains beyond the operational-use system.
The relationship for the failure rate described by parameters typical of theWeibull distribu-

tion, as applied to determine the rate of wear and tear of the component/device, is an important
part of this study. Further work should be focused on verification of the method in question as
referred to larger population of objects under analysis. Parameters that describe the rate of
the ever increasing intensity of the component/device wear and tear should be found. Both
objectives determine future lines of activity for the Author.
The above-presented considerations have been based on analyses of the process of opera-

ting aeronautical devices/systems. However, they can be successfully applied for other technical
objects, the health/maintenance status of which is usually determined by means of diagnostic
parameters. Implementation of this method is possible if we, firstly, have values of diagnostic
parameters located along the axis of timeof thedevice service/operation and, secondly, know the
value of the boundary/limiting error that –when exceeded by the diagnostic parameter –makes
operation of the device less effective (i.e. the device is at the medium level of serviceability).

References

1. Abezgauz G., 1973,Calculus Probabilistic. Handbook (in Polish),WydawnictwoMON,Warszawa

2. Gniedenko B.W., Bielajew J.K., Sołowiew A.D., 1968,Mathematical Methods in Reliability
Theory (in Polish),WNT,Warszawa

3. Gradsztejn I.S., Ryżyk I.M., 1964,Tables of Integrals, Sums, Rows and Products (in Polish),
PWN,Warszawa

4. NiuG., Singh S.,HollandS., PechtM., 2011,Healthmonitoring of electronic products based
onMahalanobis distance andWeibull decisionmetrics,Microelectronics Relibility, 51, 279-284

5. Pamuła W., 2011,The Reliability and the Safety (in Polish),Wydawnictwo Politechniki Śląskiej,
Gliwice



990 M.Ważny

6. TomaszekH., JasztalM., ZiejaM., 2011,A simplifiedmethod to assess fatigue life of selected
structural components of an aircraft for a variable load spectrum, Eksploatacja i Niezawodność –
Maintenance and Reliability, 4, 29-34

7. Tomaszek H., Jasztal M., Zieja M., 2013, Application of the Paris formula with m = 2 and
the variable load spectrum to a simplified method for evaluation of reliability and fatigue life
demonstrated by aircraft components,Eksploatacja i Niezawodność –Maintenance and Reliability,
4, 297-304

8. Tomaszek H., Szczepanik R., 2005, Method of evaluation distribution of time of exceed a
limiting state (in Polish), Zagadnienia Eksploatacji Maszyn, 40, 4, 35-44

9. Tomaszek H., Wróblewski M., 2001, Bases of Assessment of the Effectiveness of the Use of
Weapon Systems Air (in Polish), DomWydawniczy Bellona,Warszawa

10. Tomaszek H., Żurek J., Stępień S., 2008, The airship maintenance with its renovation and
the risk of its loss, Scientific Problems of Machines Operation and Maintenance, 43, 4, 39-49

11. Tomaszek H., Żurek J., Loroch L., 2004, The outline of a method of estimation reliability
and durability of aircraft’s structure elements on the basis of destruction process description (in
Polish), Zagadnienia Eksploatacji Maszyn, 39, 3, 73-85

12. Ważny M., 2011a, An outline of the method for determining the density function of changes in
diagnostic parameter deviations with the use of the Weibull distribution, Scientific Problems of
Machines Operation and Maintenance, 46, 2, 105-144

13. Ważny M., 2011b, An outline of a method for determining the density function of the time of
exceeding the limit state with the use of theWeibull distribution, Scientific Problems of Machines
Operation and Maintenance, 46, 3, 77-91

14. Ważny M., 2013,Effectiveness Analysis of the Air Sighting System in the Aspect of the Effect of
Destructive Processes (in Polish),WAT

15. YeZ., TangL., XieM., 2011,Aburn-in schemebased on precentiles of the residual life, Journal
of Quality Technology, 43, 4, 334-345

16. Zhanshan M., Krings, A.W., 2008, Survival analysis approach to reliability, survivability and
prognostics and healthmanagement (PHM),Aerospace Conference, IEEE, 1-20

Manuscript received July 21, 2014; accepted for print May 24, 2015