3. Types of damage causing threats and models for assessing the probability of their occurrence

#### 3.1. Classification of construction systems and wear processes

By assumption, the aviation technology has high reliability requirements, which, in practice, are implemented through special inspection procedures and appropriate design solutions involving the introduction of excesses of structure, strength, power, information, etc. The structural excess is characterised by elements or functional systems, basic and reserveprotective ones. After damage to the basic system, the protection systems start functioning. It ensures a high safety level of aircraft flights, which is one of the most important issues in the air transport. Despite these protections and great efforts of technical services, the failures that cause accidents occur.

increase in the number of experiments. Based on operational tests, it can be concluded that the normal distribution provides an approximate (asymptotic) description of the random variable of T time of proper operation of the device's element to damage, and it can be used when the element's wear and ageing parameters create a continuous random process to achieve the limit state.

The random variable of T life time of tested objects has normal distribution, if its probability

The shape of f(t) density function curve of normal distribution characterises the population of objects in terms of homogeneity. The homogeneity of the population of the same elements of devices in terms of their durability in operation is represented by the coefficient of variation

For v small values, it is possible to accurately predict the moment of time for achieving the

þ ð∞

t

In order to simplify the calculations in practice, the so-called standardised variable is adopted:

<sup>u</sup> <sup>¼</sup> <sup>t</sup> � <sup>m</sup>

it indicates a number of average (standard) deviation in terms of which the random variable Tt being the implementation of life time of the particular i element differs from its expected value

f tð Þdt <sup>¼</sup> <sup>1</sup>

σ ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> þ ð∞

t

<sup>2</sup><sup>π</sup> <sup>p</sup> � exp � ð Þ <sup>t</sup> � <sup>m</sup>

2

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ð Þ T —variance of the random variable are the

exp � ð Þ <sup>t</sup> � <sup>m</sup>

<sup>σ</sup> (3)

� � (4)

� �du (5)

2

dt (2)

2σ<sup>2</sup> " # (1)

41

2σ<sup>2</sup> " #

f tðÞ¼ <sup>1</sup>

σ ffiffiffiffiffiffi

density is given by the following formula:

where <sup>m</sup> <sup>=</sup> <sup>E</sup>(T)—expected value and <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup>

limit state in the operating time interval (0, t),

R tðÞ¼ 1 � F tðÞ¼ 1 �

The reliability function value is calculated as follows:

ðt

f tð Þdt ¼

With t=m+ σu, taking into account that dt = σdu, the above formula is as follows:

f tðÞ¼ <sup>1</sup>

R tðÞ¼ <sup>1</sup>

ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup>

However, the formula for the reliability function is as follows:

ffiffiffiffiffiffi

þ ð∞

mþσ�u

<sup>2</sup><sup>π</sup> <sup>p</sup> � exp � <sup>u</sup><sup>2</sup>

2

exp � <sup>u</sup><sup>2</sup> 2

�∞

distribution parameters.

v = σ/m (Figure 2).

m = E(T).

Due to the fact that the integral

The protecting systems constituting the reserve of basic systems significantly increase the production costs and reduce the overall performance, such as capacity, range, fuel consumption, etc. They also require special treatment in the operation of aircraft, so that they have very high probability of correct functioning at the very low probability of use.

The accuracy of continuous or periodic identification of the state of usability is an important issue. The person stating the state of usability of basic and reserve technical systems can make two types of errors:


The result of the erroneous statement on the system activating the emergency release of the landing gear was the emergency landing of PLL LOT plane, Boeing 767-300ER, on November 1, 2011 at Warsaw Chopin Airport, which will be discussed in the further part of the chapter.

The wear and ageing processes of various elements are correlated with time or the functioning duration, or with calendar time in a varying degree. Generally, the construction elements and functional systems may be classified into three types:


#### 3.2. Elements strongly correlated with functioning time

In case of elements of the first group, it is possible to create the technical condition trajectory and to expect a moment of time, in which the limit state will occur. It is also possible to predict a moment of the element or unit secure taking out of service. In this case, a process of damage can be described with a suitably selected model for normal distribution, even with a small variance [2]. The suitable quantile of the random variable of functioning duration between damage can be a basis for developing a programme of diagnosing, maintenance and repairs. This group of elements can include slide bearings, gears, tyre treads of gear wheels, etc. A good model describing the time of the correct operation is normal distribution.

#### 3.2.1. Normal distribution

The normal distribution sometimes constitutes limit distribution, to which many other types of distribution asymptotically approach in the operational processes of devices, together with an

increase in the number of experiments. Based on operational tests, it can be concluded that the normal distribution provides an approximate (asymptotic) description of the random variable of T time of proper operation of the device's element to damage, and it can be used when the element's wear and ageing parameters create a continuous random process to achieve the limit state.

The random variable of T life time of tested objects has normal distribution, if its probability density is given by the following formula:

$$f(t) = \frac{1}{\sigma\sqrt{2\pi}} \cdot \exp\left[-\frac{(t-m)^2}{2\sigma^2}\right] \tag{1}$$

where <sup>m</sup> <sup>=</sup> <sup>E</sup>(T)—expected value and <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup> ð Þ T —variance of the random variable are the distribution parameters.

The shape of f(t) density function curve of normal distribution characterises the population of objects in terms of homogeneity. The homogeneity of the population of the same elements of devices in terms of their durability in operation is represented by the coefficient of variation v = σ/m (Figure 2).

For v small values, it is possible to accurately predict the moment of time for achieving the limit state in the operating time interval (0, t),

The reliability function value is calculated as follows:

air transport. Despite these protections and great efforts of technical services, the failures that

The protecting systems constituting the reserve of basic systems significantly increase the production costs and reduce the overall performance, such as capacity, range, fuel consumption, etc. They also require special treatment in the operation of aircraft, so that they have very

The accuracy of continuous or periodic identification of the state of usability is an important issue. The person stating the state of usability of basic and reserve technical systems can make

The result of the erroneous statement on the system activating the emergency release of the landing gear was the emergency landing of PLL LOT plane, Boeing 767-300ER, on November 1, 2011 at Warsaw Chopin Airport, which will be discussed in the further part of the chapter. The wear and ageing processes of various elements are correlated with time or the functioning duration, or with calendar time in a varying degree. Generally, the construction elements and

• Elements having strongly correlated parameters determining the state of usability with the functioning duration or time, which can be identified with the existence of the mem-

• Elements having poorly correlated parameters of the state of usability with the functioning duration or time, which imply weak relationships of operating time with the technical

• Elements without correlation with the operating time, number of activation, or other

In case of elements of the first group, it is possible to create the technical condition trajectory and to expect a moment of time, in which the limit state will occur. It is also possible to predict a moment of the element or unit secure taking out of service. In this case, a process of damage can be described with a suitably selected model for normal distribution, even with a small variance [2]. The suitable quantile of the random variable of functioning duration between damage can be a basis for developing a programme of diagnosing, maintenance and repairs. This group of elements can include slide bearings, gears, tyre treads of gear wheels, etc. A

The normal distribution sometimes constitutes limit distribution, to which many other types of distribution asymptotically approach in the operational processes of devices, together with an

measure of the functioning duration, with randomly occurring damage.

good model describing the time of the correct operation is normal distribution.

high probability of correct functioning at the very low probability of use.

• an error of the first type consists of qualifying the usable device as unfit;

functional systems may be classified into three types:

condition change, wear and damage.

3.2. Elements strongly correlated with functioning time

ory related to the past.

3.2.1. Normal distribution

• an error of the second type consists of qualifying the unfit device as usable;

cause accidents occur.

40 System of System Failures

two types of errors:

$$R(t) = 1 - F(t) = 1 - \int\_{-\infty}^{t} f(t)dt = \int\_{t}^{+\infty} f(t)dt = \frac{1}{\sigma\sqrt{2\pi}} \int\_{t}^{+\infty} \exp\left[-\frac{(t-m)^2}{2\sigma^2}\right]dt\tag{2}$$

In order to simplify the calculations in practice, the so-called standardised variable is adopted:

$$
\mu = \frac{t - m}{\sigma} \tag{3}
$$

it indicates a number of average (standard) deviation in terms of which the random variable Tt being the implementation of life time of the particular i element differs from its expected value m = E(T).

With t=m+ σu, taking into account that dt = σdu, the above formula is as follows:

$$f(t) = \frac{1}{\sqrt{2\pi}} \cdot \exp\left(-\frac{\mu^2}{2}\right) \tag{4}$$

However, the formula for the reliability function is as follows:

$$R(t) = \frac{1}{\sqrt{2\pi}} \int\_{u + \sigma \cdot u}^{+\infty} \exp\left(-\frac{u^2}{2}\right) du\tag{5}$$

Due to the fact that the integral

Figure 2. The probability density of normal distribution for different values of v coefficient of variation.

$$\frac{1}{\sqrt{2\pi}} \cdot \int\_{-\infty}^{0} \exp\left(-\frac{u^2}{2}\right) du = 0.5\tag{6}$$

The use of the truncated normal distribution in the reliability tests of technical objects has the following practical sense. The equation for the density function of the normal distribution applies for all t values, from �∞ to þ∞. In the operational reliability tests of cars and their elements, there is always the relationship that t > 0, for which the density function is given by

<sup>2</sup><sup>π</sup> <sup>p</sup> � exp � ð Þ <sup>t</sup> � <sup>m</sup>

2

2

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<sup>σ</sup> <sup>¼</sup> <sup>u</sup> (13)

dt ¼ σdu (14)

� �du (15)

� � (16)

2σ<sup>2</sup> " # (11)

43

dt (12)

2σ<sup>2</sup> " #

exp � ð Þ <sup>t</sup> � <sup>m</sup>

f tðÞ¼ <sup>1</sup>

however, R(t) reliability function is provided by the following formula:

R tðÞ¼ <sup>1</sup>

The solution of the above integral includes the expression:

3.3. Elements poorly correlated with functioning time

damage threatening the safety or the most common ones.

aσ ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup>

R tðÞ¼ <sup>ð</sup> ∞

substituting these figures, it is possible to obtain:

t

aσ ffiffiffiffiffiffi

f tð Þdt <sup>¼</sup> <sup>1</sup>

aσ ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> ð ∞

t � m

ð ∞

exp � <sup>1</sup> 2 u2

mþσu

R tðÞ¼ <sup>F</sup>0ð Þ <sup>u</sup> F<sup>0</sup> <sup>m</sup> σ

The second group includes elements and structures operating in the variable conditions that are subject to the material fatigue, vibration, corrosion, etc. The process of damage to the other group's elements can be described by the models with variable parameters and high dispersion, such as: gamma, log-normal, Weibull and others [2]. The selection of operating programmes is very difficult, especially in cases of aviation technology, where the failure of a function of the object's construction system threatens the safety of people, the environment or causes significant material losses. In this case, it is important to apply the density of services, matching them to the

t

the above formula

Because

where

and the function

$$\frac{1}{\sqrt{2\pi}}\int\_0^{u+\alpha u} \exp\left(-\frac{u^2}{2}\right) du = \Phi(u) \tag{7}$$

are called the Laplace function (integral), the final form of the equation of the reliability function will be as follows:

$$R(\mu) = 0.5 - \Phi(\mu) \quad \text{and} \quad F(\mu) = 0.5 + \Phi(\mu) \tag{8}$$

The above presented formulas for normal distribution of life time of the aircraft elements provide the right accuracy of calculations at a high degree of homogeneity of a feature and tested objects, which is characterised by small deviation values and the standard one, that is, when the expected value E(T) = m, where m >> σ, m > ð Þ 2÷3 σ is practically accepted.

In these cases, for which <sup>m</sup> <sup>σ</sup> < 2, it is recommended to use truncated normal distribution with the parameters of m, σ, for which the probability density function is as follows:

$$f(t) = \frac{1}{a\sigma\sqrt{2\pi}} \cdot \exp\left[-\frac{\left(t - m\right)^2}{2\sigma^2}\right] \tag{9}$$

where m means the average life time of the object to damage, t means a current variable, <sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup> ð Þ T means a variance, while σ > 0 and t > 0, and a constant is determined on the basis of the following formula:

$$a = \frac{1}{F\_0 \frac{m}{\sigma}} \tag{10}$$

The use of the truncated normal distribution in the reliability tests of technical objects has the following practical sense. The equation for the density function of the normal distribution applies for all t values, from �∞ to þ∞. In the operational reliability tests of cars and their elements, there is always the relationship that t > 0, for which the density function is given by the above formula

$$f(t) = \frac{1}{a\sigma\sqrt{2\pi}} \cdot \exp\left[-\frac{(t-m)^2}{2\sigma^2}\right] \tag{11}$$

however, R(t) reliability function is provided by the following formula:

$$R(t) = \bigcap\_{t}^{\alpha} f(t)dt = \frac{1}{a\sigma\sqrt{2\pi}} \left[ \exp\left[ -\frac{(t-m)^2}{2\sigma^2} \right] dt \tag{12}$$

Because

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> �

1 ffiffiffiffiffiffi <sup>2</sup><sup>π</sup> <sup>p</sup> mþ ðσu

0

and the function

42 System of System Failures

function will be as follows:

In these cases, for which <sup>m</sup>

of the following formula:

<sup>σ</sup><sup>2</sup> <sup>¼</sup> <sup>D</sup><sup>2</sup>

ð 0

Figure 2. The probability density of normal distribution for different values of v coefficient of variation.

exp � <sup>u</sup><sup>2</sup> 2 � �

exp � <sup>u</sup><sup>2</sup> 2 � �

are called the Laplace function (integral), the final form of the equation of the reliability

The above presented formulas for normal distribution of life time of the aircraft elements provide the right accuracy of calculations at a high degree of homogeneity of a feature and tested objects, which is characterised by small deviation values and the standard one, that is,

<sup>2</sup><sup>π</sup> <sup>p</sup> � exp � ð Þ <sup>t</sup> � <sup>m</sup>

ð Þ T means a variance, while σ > 0 and t > 0, and a constant is determined on the basis

where m means the average life time of the object to damage, t means a current variable,

<sup>a</sup> <sup>¼</sup> <sup>1</sup> F<sup>0</sup> <sup>m</sup> σ

when the expected value E(T) = m, where m >> σ, m > ð Þ 2÷3 σ is practically accepted.

the parameters of m, σ, for which the probability density function is as follows:

aσ ffiffiffiffiffiffi

f tðÞ¼ <sup>1</sup>

R uð Þ¼ 0:5 � Φð Þ u and F uð Þ¼ 0:5 þ Φð Þ u (8)

<sup>σ</sup> < 2, it is recommended to use truncated normal distribution with

2

2σ<sup>2</sup> " #

du ¼ 0:5 (6)

du ¼ Φð Þ u (7)

(9)

(10)

�∞

$$\frac{t-m}{\sigma} = \mu \tag{13}$$

where

$$dt = \sigma du\tag{14}$$

substituting these figures, it is possible to obtain:

$$R(t) = \frac{1}{a\sigma\sqrt{2\pi}} \int\_{m+\sigma u}^{\infty} \exp\left(-\frac{1}{2}u^2\right) du\tag{15}$$

The solution of the above integral includes the expression:

$$R(t) = \frac{F\_0(u)}{F\_0\left(\frac{m}{\sigma}\right)}\tag{16}$$

#### 3.3. Elements poorly correlated with functioning time

The second group includes elements and structures operating in the variable conditions that are subject to the material fatigue, vibration, corrosion, etc. The process of damage to the other group's elements can be described by the models with variable parameters and high dispersion, such as: gamma, log-normal, Weibull and others [2]. The selection of operating programmes is very difficult, especially in cases of aviation technology, where the failure of a function of the object's construction system threatens the safety of people, the environment or causes significant material losses. In this case, it is important to apply the density of services, matching them to the damage threatening the safety or the most common ones.

With the development of the construction, it is important to mount the diagnosing systems for tracing (monitoring) the technical condition and signalling the pre-failure states in the units and functional systems. A certain way out of the situation involves monitoring of the course of induced forces with the use of a system of recorders adapted to record all relevant operational events, especially those threatening the safety of use. With the diagnosing and IT system for monitoring the state and the process of damage, it is possible to determine the area, in which the technical condition trajectory is placed, or to identify the durability resource.

#### 3.3.1. Gamma distribution

In this distribution, it is assumed that for randomly selected moments of t time in the object, the energy with the same value of individually operating induced forces (external loads) is cumulated, and that after putting k number of induced forces, the object is damaged.

The density function of this probability is as follows:

$$f(t) = \begin{cases} \frac{1}{\Gamma(k)}^{\lambda^k t^{k-1} e^{-\lambda t}} for & t \ge 0\\ 0 & for \quad t < 0 \end{cases} \tag{17}$$

p tðÞ¼ p ¼ λΔt þ 0Δt (21)

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45

N tð Þ ≥ SG (22)

<sup>y</sup> (23)

dt (24)

does not depend on the number of such increases, which occurred in time preceding t moment. In other words, the condition 'without consequences'that is significant for the simple Poisson's stream of damage is assumed. The above assumptions remain valid also for the normal distribution.

In case of the assumption that SG is a maximum permissible level of cumulation of n(t) stimuli, which result in ageing of a tested element of the aircraft and that for the number of stimuli

this object becomes unfit for further correct operation in the system, k number of induced forces, the cumulated energy of which is necessary for causing its damage, is calculated from

<sup>k</sup> <sup>¼</sup> SG

where y means the value, by which the ageing takes place (e.g., wear) of the element in a stepped manner under the impact of a single stimulus. However, λ magnitude is characterised

<sup>y</sup> <sup>¼</sup> dEf g <sup>η</sup>ð Þ<sup>t</sup>

By using the formula for the function of F(t) cumulated density of damage and R(t) = 1 � F(t) relationship, the element's reliability function for the Erlang distribution will be expressed by

> ð Þ <sup>λ</sup> � <sup>t</sup> <sup>n</sup> <sup>n</sup>! <sup>e</sup>

�λ<sup>t</sup> <sup>¼</sup> <sup>e</sup> �λt X k�1

n¼0

ð Þ <sup>λ</sup> � <sup>t</sup> <sup>n</sup>

<sup>n</sup>! (25)

<sup>λ</sup> <sup>¼</sup> <sup>1</sup>

k�1

n¼0

R tðÞ¼ <sup>1</sup> � F tðÞ¼ <sup>X</sup>

the following relationship:

the following formula:

by the average intensity of the aircraft ageing:

Figure 3. Gamma distribution density with different values k and λ.

where

k—number of events enforcing the ageing process, the cumulated effects of which cause the occurrence of damage in the object,

Γð Þk —gamma function is determined by the following formula:

$$
\Gamma(k) = \int\_0^\infty x^{k-1} e^{-x} dx \tag{18}
$$

For total k, there is the relationship:

$$
\Gamma(k) = (k-1)! \tag{19}
$$

and the gamma distribution is called the Erlang distribution.

In this case, F(t) distribution function has the following form:

$$F(t) = 1 - \sum\_{n=0}^{k-1} \frac{(\lambda \cdot t)^n}{n!} e^{-\lambda t} \tag{20}$$

In Figure 3, the gamma distribution density for various values k and λ was presented. At the same time, it is characteristic that the individual induced force (load) action results in the aircraft ageing (or increase of the energy cumulated in it in a stepped manner). The individual increase of effects of such an induced force has a constant value. Furthermore, the probability of the occurrence of the aircraft ageing increases in the time interval (t, t + Δt):

Figure 3. Gamma distribution density with different values k and λ.

With the development of the construction, it is important to mount the diagnosing systems for tracing (monitoring) the technical condition and signalling the pre-failure states in the units and functional systems. A certain way out of the situation involves monitoring of the course of induced forces with the use of a system of recorders adapted to record all relevant operational events, especially those threatening the safety of use. With the diagnosing and IT system for monitoring the state and the process of damage, it is possible to determine the area, in which

In this distribution, it is assumed that for randomly selected moments of t time in the object, the energy with the same value of individually operating induced forces (external loads) is

cumulated, and that after putting k number of induced forces, the object is damaged.

1 Γð Þk 0

Γð Þ¼ k

F tðÞ¼ <sup>1</sup> �<sup>X</sup>

of the occurrence of the aircraft ageing increases in the time interval (t, t + Δt):

k�1

ð Þ <sup>λ</sup> � <sup>t</sup> <sup>n</sup> <sup>n</sup>! <sup>e</sup>

n¼0

In Figure 3, the gamma distribution density for various values k and λ was presented. At the same time, it is characteristic that the individual induced force (load) action results in the aircraft ageing (or increase of the energy cumulated in it in a stepped manner). The individual increase of effects of such an induced force has a constant value. Furthermore, the probability

8 < :

λk t <sup>k</sup>�1e�λ<sup>t</sup> for for

k—number of events enforcing the ageing process, the cumulated effects of which cause the

ð ∞

xk�<sup>1</sup> e �x

0

t ≥ 0 t < 0

dx (18)

�λ<sup>t</sup> (20)

Γð Þ¼ k ð Þ k � 1 ! (19)

(17)

The density function of this probability is as follows:

f tðÞ¼

Γð Þk —gamma function is determined by the following formula:

and the gamma distribution is called the Erlang distribution. In this case, F(t) distribution function has the following form:

the technical condition trajectory is placed, or to identify the durability resource.

3.3.1. Gamma distribution

44 System of System Failures

occurrence of damage in the object,

For total k, there is the relationship:

where

$$p(t) = p = \lambda \Delta t + 0 \Delta t \tag{21}$$

does not depend on the number of such increases, which occurred in time preceding t moment. In other words, the condition 'without consequences'that is significant for the simple Poisson's stream of damage is assumed. The above assumptions remain valid also for the normal distribution.

In case of the assumption that SG is a maximum permissible level of cumulation of n(t) stimuli, which result in ageing of a tested element of the aircraft and that for the number of stimuli

$$N(t) \ge S\_G \tag{22}$$

this object becomes unfit for further correct operation in the system, k number of induced forces, the cumulated energy of which is necessary for causing its damage, is calculated from the following relationship:

$$k = \frac{\mathbf{S}\_G}{y} \tag{23}$$

where y means the value, by which the ageing takes place (e.g., wear) of the element in a stepped manner under the impact of a single stimulus. However, λ magnitude is characterised by the average intensity of the aircraft ageing:

$$
\lambda = \frac{1}{y} = \frac{d\mathbb{E}\{\eta(t)\}}{dt} \tag{24}
$$

By using the formula for the function of F(t) cumulated density of damage and R(t) = 1 � F(t) relationship, the element's reliability function for the Erlang distribution will be expressed by the following formula:

$$R(t) = 1 - F(t) = \sum\_{n=0}^{k-1} \frac{(\lambda \cdot t)^n}{n!} e^{-\lambda t} = e^{-\lambda t} \sum\_{n=0}^{k-1} \frac{(\lambda \cdot t)^n}{n!} \tag{25}$$

The expected value E(T), D<sup>2</sup> (T) variance and v(T) coefficient of variation for this distribution is as follows:

$$E(T) = \frac{k}{\lambda};\ D^2(T) = \frac{k}{\lambda^2};\ \mathbf{v}(T) = \frac{D^2(T)}{E(T)} = \sqrt{\frac{1}{k}}\tag{26}$$

does not decrease with time of operation. The value of λ parameter affects the shape of the

When (Tk) time of proper operation to damage is treated discretely [3] (e.g., by K�1 number of activating the object without damage to the moment of K activation, during which the failure will occur). Then, the geometric distribution is used. The F(t) distribution function of (Tk) time

p—means the probability of damage to the unit at K activation, it can be also calculated from

K

F tðÞ¼ P Tf g¼ <sup>k</sup> <sup>≤</sup><sup>K</sup> <sup>1</sup> � ð Þ <sup>1</sup> � <sup>p</sup> <sup>k</sup>þ<sup>1</sup> (31)

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E Tð Þ (32)

E Tð Þ (33)

R tð Þþ F tðÞ¼ 1 (34)

E Tð Þ <sup>¼</sup> <sup>1</sup> � exp � <sup>K</sup>

exponential distribution density curve presented in Figure 4.

for proper operation is calculated in the following way:

the relationship providing the approximate values:

F tðÞ¼ P Tf g ≤K ¼ 1 � e

Therefore, for the purposes of operation, it is possible to use the following formula:

Figure 4. Example courses of the exponential distribution density for different values of the parameter λ.

R tðÞ¼ exp ½ �¼ �λ<sup>t</sup> exp � <sup>t</sup>

In relation to the fact that F(t) distribution function is the complement to the reliability function

E Tð Þ—expected value of time of proper operation to damage.

When t < 0, the function is f(t) = 0,

where

where

unity:

#### 3.4. Elements without correlation with operating time

The elements of the third group are subject to the exponential reliability law, in which the constant intensity of damage is assumed. The damage have a random nature and most often come from:


The elements of the third group include bodies, glass housings and covers made of plastic, electronics components, etc.

#### 3.4.1. Exponential distribution

If (T) time of correct operation to damage is recorded in a continuous manner and the intensity of damage λð Þt is constant and does not depend on time in the entire interval (0, t), that is,

$$
\lambda(t) = \lambda = \text{const} \tag{27}
$$

the exponential distribution is used.

The F(t) distribution function of this distribution of (T) time of the correct operation in the interval (0,t) is calculated on the basis of the following relationship:

$$F(t) = P\{T \le t\} = 1 - e^{-\lambda t} = 1 - \exp\left[-\lambda t\right] \tag{28}$$

and f(t) function of distribution density for t > 0 is calculated on the basis of the relationship:

$$f(t) = \frac{dF(t)}{dt} = \lambda e^{-\lambda t} = \lambda \exp\left[-\lambda t\right] \tag{29}$$

where

λ > 0—means the distribution parameter (intensity of damage).

Moments of the exponential distribution are given by the following formula:

$$E(T) = \frac{1}{\lambda} \quad \text{and} \quad D^2(T) = \frac{1}{\lambda^2} \tag{30}$$

The equality E Tð Þ¼ <sup>1</sup> <sup>λ</sup> is true only for those elements of the device, for which the intensity of damage in the entire range of operation (0, t) is constant, and therefore, it does not increase or does not decrease with time of operation. The value of λ parameter affects the shape of the exponential distribution density curve presented in Figure 4.

When t < 0, the function is f(t) = 0,

When (Tk) time of proper operation to damage is treated discretely [3] (e.g., by K�1 number of activating the object without damage to the moment of K activation, during which the failure will occur). Then, the geometric distribution is used. The F(t) distribution function of (Tk) time for proper operation is calculated in the following way:

$$F(t) = P\{T\_k \le K\} = 1 - (1 - p)^{k+1} \tag{31}$$

where

The expected value E(T), D<sup>2</sup>

• overloads of a different nature;

electronics components, etc.

3.4.1. Exponential distribution

the exponential distribution is used.

E Tð Þ¼ <sup>k</sup>

3.4. Elements without correlation with operating time

• manufacture errors (material and technological errors);

<sup>λ</sup> ; D<sup>2</sup>

• non-compliance with the instructions for use or operation technology.

interval (0,t) is calculated on the basis of the following relationship:

F tðÞ¼ P Tf g ≤ t ¼ 1 � e

f tðÞ¼ dF tð Þ

Moments of the exponential distribution are given by the following formula:

E Tð Þ¼ <sup>1</sup>

λ > 0—means the distribution parameter (intensity of damage).

ð Þ¼ <sup>T</sup> <sup>k</sup>

The elements of the third group are subject to the exponential reliability law, in which the constant intensity of damage is assumed. The damage have a random nature and most often

The elements of the third group include bodies, glass housings and covers made of plastic,

If (T) time of correct operation to damage is recorded in a continuous manner and the intensity of damage λð Þt is constant and does not depend on time in the entire interval (0, t), that is,

The F(t) distribution function of this distribution of (T) time of the correct operation in the

and f(t) function of distribution density for t > 0 is calculated on the basis of the relationship:

<sup>λ</sup> and <sup>D</sup><sup>2</sup>

damage in the entire range of operation (0, t) is constant, and therefore, it does not increase or

ð Þ¼ <sup>T</sup> <sup>1</sup>

<sup>λ</sup> is true only for those elements of the device, for which the intensity of

dt <sup>¼</sup> <sup>λ</sup><sup>e</sup>

<sup>λ</sup><sup>2</sup> ; vTð Þ¼ <sup>D</sup><sup>2</sup>

as follows:

46 System of System Failures

come from:

where

The equality E Tð Þ¼ <sup>1</sup>

(T) variance and v(T) coefficient of variation for this distribution is

ð Þ T E Tð Þ <sup>¼</sup>

ffiffiffi 1 k r

λðÞ¼ t λ ¼ const (27)

�λ<sup>t</sup> <sup>¼</sup> <sup>1</sup> � exp ½ � �λ<sup>t</sup> (28)

�λ<sup>t</sup> <sup>¼</sup> <sup>λ</sup>exp ½ � �λ<sup>t</sup> (29)

<sup>λ</sup><sup>2</sup> (30)

(26)

p—means the probability of damage to the unit at K activation, it can be also calculated from the relationship providing the approximate values:

$$F(t) = P\{T \le K\} = 1 - e^{\frac{K}{E(T)}} = 1 - \exp\left[-\frac{K}{E(T)}\right] \tag{32}$$

where

E Tð Þ—expected value of time of proper operation to damage.

Therefore, for the purposes of operation, it is possible to use the following formula:

$$R(t) = \exp\left[-\lambda t\right] = \exp\left[-\frac{t}{E(T)}\right] \tag{33}$$

In relation to the fact that F(t) distribution function is the complement to the reliability function unity:

$$R(t) + F(t) = 1\tag{34}$$

Figure 4. Example courses of the exponential distribution density for different values of the parameter λ.

Figure 5. Reliability function courses; distribution: a—geometric, b—exponential.

then, R(t) for the geometric distribution is:

$$R(t) = 1 - \left[1 - \exp\left(-\frac{K}{E(T)}\right)\right] = \exp\left[-\frac{K}{E(T)}\right] \tag{35}$$

The following system of equations is right:

where

time interval of Δt;

following system of equations:

interval of Δt;

for n > 0

λ—proportionality factor that represents the damage risk;

P 0

P 0

P 0

For the system of equations (38), the initial conditions are as follows:

P0ð Þ¼ 0 1

By having the knowledge of P0ð Þt , then, P1ð Þt is determined, and so on.

P0ðÞ¼ t e

PnðÞ¼ t

�λ ðt

1 <sup>n</sup>! <sup>λ</sup> ðt

0

⋮

2 4

0

N tð Þdt

3 5

n

e �λ ðt

0

N Lð Þdt

The solution of the system of equations (39) takes the form of:

8

>>>>>>>>>>>>><

>>>>>>>>>>>>>:

P0ð Þ¼ t þ Δt P0ð Þt ð Þ 1 � λN tð ÞΔt

⋮

N tð Þ—number of operated aircraft, in which the considered damage may occur;

Δt—adopted time interval of aircraft operation (or the aircraft's flying time length).

<sup>0</sup> ¼ �λN tð ÞP0ð Þt

⋮

P1ð Þ¼ t þ Δt P1ð Þt ð1 � λN tð ÞΔtÞ þ P0ð Þt λN tð ÞΔt

Probabilistic Methods for Damage Assessment in Aviation Technology

http://dx.doi.org/10.5772/intechopen.72317

Pnð Þ¼ t þ Δt Pnð Þt ð1 � λN tð ÞΔtÞ þ Pn�<sup>1</sup>ð Þt λN tð ÞΔt

P0ð Þ t; t þ Δt —probability of non-occurrence of damage to basic and protection systems in the

Pið Þ t; t þ Δt , (i = 1,…n)—probability of the occurrence of 'i' number of damage in the time

By dividing Eq. (37) by Δt and going to the border with Δt ! 0, it is possible to obtain the

<sup>1</sup> ¼ �λN tð ÞP1ð Þþ t λN tð ÞP0ð Þt

<sup>n</sup> ¼ �λN tð ÞPnð Þþ t λN tð ÞPn�<sup>1</sup>ð Þt

Pnð Þ¼ 0 0 for n > 0

Equation (38) is a linear differential equation and it is solved recursively. First, P0ð Þt is found.

N tð Þdt

(

(37)

49

(38)

(39)

(40)

and for the exponential distribution, R(t) is:

$$R(t) = 1 - \left[1 - \exp\left(-\lambda t\right)\right] = \exp\left(-\lambda t\right) \tag{36}$$

In Figure 5, the reliability functions R tðÞ¼ P Tf g > t of the geometric distribution and the exponential distribution were presented. In the first case, the graph constitutes a step curve. However, the second graph constitutes a continuous curve. It was assumed that the parameters p and λ of both functions are the same, and their value is p = λ = 0.1.

Another way, which makes it possible to estimate the probability values of the occurrence of catastrophic damage in the aircraft devices, can include the use of models, including the limit state.
