8. Outline of the aircraft reliability assessment method taking into account signalled and catastrophic damage

#### 8.1. Description of operation conditions and adoption of assumptions

It is assumed that the aircraft operation is done in such a way that the following arrangements and assumptions are correct:

1. In order to assess the technical condition, n diagnostic parameters are used. So the technical condition vector adopts the following form [7]:

$$\mathbf{x} = (\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n) \tag{76}$$

2. Instead of the diagnostic parameter values in the reliability assessment, the following deviations are used:

$$z\_i = \mathbf{x}\_i - \mathbf{x}\_{i\,nm} \quad (i = 1, 2, \ldots, n) \tag{77}$$

where

(69)

(70)

(71)

(74)

qkð Þt —probability of the development process interruption (i.e., state changes). This probabil-

May Pkð Þt mean the probability that in the moment of t, the diagnostic parameter value

For these arrangements, it is possible to arrange the following system of the infinite number of

⋮ for k ¼ 1, 2, …

After division of both sides of k equation by Δt and transition to the border at Δt ! 0, the

⋮ for k ¼ 1, 2, …

�

The system of Eq. (70) is solved recursively. Having the results of the solved system of equations, it is possible to determine the probability (reliability) that in the time interval 0ð Þ ; t , the catastrophic damage will not occur. This relationship can be determined by adding up the obtained

R tðÞ¼ <sup>X</sup><sup>∞</sup>

k¼0

The probability of the fact that to the moment of t, the catastrophic damage will occur, can be

After the adding up operation performance, the following form of the solution is obtained [6]:

λ <sup>μ</sup>� <sup>λ</sup>þ<sup>μ</sup> ð Þ<sup>0</sup> <sup>t</sup>�<sup>λ</sup>

Hence, the time distribution density to the moment of the catastrophic damage occurrence.

�μ<sup>t</sup> � � � � e

Q tðÞ¼ 1 � e

f tðÞ¼ μ<sup>0</sup> þ λ 1 � e

Pkð Þt (72)

<sup>μ</sup> <sup>1</sup>�e�μ<sup>t</sup> ð Þ� <sup>λ</sup>þ<sup>μ</sup> ð Þ<sup>0</sup> <sup>t</sup> (75)

Q tðÞ¼ P Tf g¼ ≤ t 1 � R tð Þ (73)

μe�μ<sup>t</sup>

λ

1 dla i ¼ 0 0 dla i 6¼ 0

Pið Þ¼ 0

Pkð Þ¼ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> Pkð Þ<sup>t</sup> <sup>1</sup> � <sup>μ</sup><sup>0</sup> <sup>þ</sup> <sup>k</sup><sup>μ</sup> <sup>þ</sup> <sup>λ</sup> � �Δ<sup>t</sup> � � <sup>þ</sup> Pk�<sup>1</sup>ð Þ<sup>t</sup> <sup>λ</sup>Δ<sup>t</sup> <sup>þ</sup> <sup>0</sup>ð Þ <sup>Δ</sup><sup>t</sup>

h—average value of the diagnostic parameter increase in time Δt (one flight).

<sup>P</sup>0ð Þ¼ <sup>t</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> <sup>P</sup>0ð Þ<sup>t</sup> <sup>1</sup> � <sup>μ</sup><sup>0</sup> <sup>þ</sup> <sup>λ</sup> � �Δ<sup>t</sup> � � <sup>þ</sup> <sup>0</sup>ð Þ <sup>Δ</sup><sup>t</sup>

<sup>k</sup>ðÞ¼� <sup>t</sup> <sup>μ</sup><sup>0</sup> <sup>þ</sup> <sup>λ</sup> <sup>þ</sup> <sup>k</sup><sup>μ</sup> � �Pkð Þþ <sup>t</sup> <sup>λ</sup>Pkð Þ<sup>t</sup>

The initial condition for each of these equations is as follows:

ity depends on the state.

56 System of System Failures

equations:

achieved the state Ek (where k ¼ 1, 2, …).

following system of equations is obtained:

<sup>0</sup>ðÞ¼� <sup>t</sup> <sup>μ</sup><sup>0</sup> <sup>þ</sup> <sup>λ</sup> � �P0ð Þ<sup>t</sup>

P 0

P 0

relationships Pkð Þt . Hence

specified by the following relationship:

xi—i diagnostic parameter.

xi nom—nominal value of i parameter.


$$\frac{dz\_i}{dN} = g(z\_i, c\_i) \tag{78}$$

where

zi—diagnostic parameter deviation;

ci—indicators characterising the local operating conditions of elements, which the increase in the diagnostic parameter's deviation value depends on;

N—number of aircraft flights.

By using relationship (78), it is possible to determine the deviation value during one flight:

$$
\Delta z\_i = \mathbf{g}(z\_i, \ c\_i) \Delta \mathbf{N} \qquad \text{for } \Delta \mathbf{N} = 1 \tag{79}
$$

8. The intensity of the aircraft flights λ is determined by the following relationship:

$$
\lambda = \frac{P}{\Delta t} \tag{80}
$$

For the adopted arrangements, the dynamics of changes (increase) of i deviation can be

ð Þ 1 � λΔt —probability of the fact that in the time interval of Δt, the aircraft flight will not take

Eq. (83) expresses the following sense. The probability of the fact that in the moment of t þ Δt, the deviation value of i diagnostic parameter will be zi, if in t moment, it had this value and did not increase because of the lack of the aircraft flight or, in t moment, it had zi � Δzi value and in

Uzi,tþΔ<sup>t</sup> ¼ ð Þ 1 � λΔt Uzi,t þ λΔt Uzi�Δzi,t (83)

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59

u zi ð Þ¼ ; t þ Δt ð Þ 1 � λΔt u zi ð Þþ ; t λΔtu zi � Δzi ð Þ ; t (84)

characterised with the use of the following differential equation [3]:

λΔt—probability of the aircraft flight in the time interval of Δt.

u zi ð Þ¼ ; t þ Δt u zi ð Þþ ; t

u zi � Δzi ð Þ¼ ; t u zi ð Þ� ; t

By substituting Eqs. (85) to (84), it is possible to obtain:

Hence, after simplification, the following is obtained:

∂u zi ð Þ ; t

∂u zi ð Þ ; t ∂t

<sup>∂</sup><sup>z</sup> <sup>Δ</sup><sup>t</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>λ</sup>Δ<sup>t</sup> u zi ð Þþ ; <sup>t</sup> <sup>λ</sup>Δt uzð Þ� <sup>1</sup>; <sup>t</sup>

Δt ¼ �λΔziΔt

<sup>∂</sup><sup>t</sup> ¼ �bið Þ<sup>t</sup>

By dividing two sides of (87) equation by Δt, the following is obtained:

the time interval of Δt, Δzi increased, because the flight did not take place. Differential Eq. (83), in the function notation, adopts the following form:

u zi ð Þ ; t —deviation density function of i diagnostic parameter from the nominal value.

Differential Eq. (84) can be transformed to the partial differential equation, with the use of the

∂u zi ð Þ ; t ∂ zi

∂u zi ð Þ ; t ∂zi

∂u zi ð Þ ; t ∂zi

þ 1 2

þ 1 <sup>2</sup> aið Þ<sup>t</sup>

Δt

Δzi þ 1 2 ∂2

∂u zi ð Þ ; t ∂zi

λΔtð Þ Δzi

u zi ð Þ ; t ∂ z<sup>2</sup> i

> Δzi þ 1 2 ∂2 u zi ð Þ ; t ∂z<sup>2</sup> i

" #

<sup>2</sup> ∂<sup>2</sup>

∂2

u zi ð Þ ; t ∂z<sup>2</sup> i

u zi ð Þ ; t ∂z<sup>2</sup> i

ð Þ Δ zi 2

> ð Þ Δzi 2

(85)

(86)

(87)

(88)

∂u zi ð Þ ; t ∂ t

where

place;

where

u zi ð Þþ ; t

Hence 1ð Þþ � λΔt λΔt ¼ 1

following approximation:

∂u zi ð Þ ; t

where

Δt—is the time interval, in which the aircraft flight will take place with P probability. The time interval of Δt should be properly selected (depending on functioning of the aircraft operating system), for λΔt ≤ 1.

By using the intensity of flights λ, it is possible to determine the number of performed flights of the aircraft to the moment of t in accordance with the following relationship:

$$N = \lambda t \tag{81}$$


$$R(t) = R\_1(t)R\_2(t)\tag{82}$$

where

R1ð Þt —probability of the fact that to the moment t there will be no catastrophic (sudden) damage in the aircraft.

R2ð Þt —probability of the fact that to the moment t there will be no damage signalled in the aircraft.

Despite the attempts and great effort of technical services, currently, it is impossible to completely eliminate the risk of the catastrophic damage occurrence.

It is adopted that in case of a single flight of the aircraft, the probability determining the possibility of the catastrophic damage occurrence is Q. The progressive technical service of the aircraft is to prevent this probability increase together with an increase in operating time.

#### 8.2. Aircraft reliability determination including signalled and catastrophic damage

Under the adopted probabilistic assumptions, a description of the deviation increase of diagnostic parameters in the function of the aircraft operating time can be considered separately for each diagnostic parameter. In view of the above, it is assumed that the process of deviation changes of i diagnostic parameter is considered.

May Uzi,t mean the probability of the fact that in the moment t, the deviation of i parameter is zi.

For the adopted arrangements, the dynamics of changes (increase) of i deviation can be characterised with the use of the following differential equation [3]:

$$\mathcal{U}\_{z\_{\flat},t+\Lambda t} = (1 - \lambda \Delta t) \mathcal{U}\_{z\_{\flat},t} + \lambda \Delta t \; \mathcal{U}\_{z\_{\flat}-\Lambda z\_{\flat},t} \tag{83}$$

where

Δzi ¼ g zi ð Þ ; ci ΔN for ΔN ¼ 1 (79)

<sup>Δ</sup><sup>t</sup> (80)

N ¼ λt (81)

R tðÞ¼ R1ð Þt R2ð Þt (82)

8. The intensity of the aircraft flights λ is determined by the following relationship:

where

58 System of System Failures

operating system), for λΔt ≤ 1.

failures and crashes.

damage in the aircraft.

where

aircraft.

<sup>λ</sup> <sup>¼</sup> <sup>P</sup>

Δt—is the time interval, in which the aircraft flight will take place with P probability. The time interval of Δt should be properly selected (depending on functioning of the aircraft

By using the intensity of flights λ, it is possible to determine the number of performed flights of the aircraft to the moment of t in accordance with the following relationship:

9. It is assumed that the aircraft is operated. The task of the technical service is, among others, not to allow for the occurrence of signalled damage and to maximally limit the possibility of the occurrence of catastrophic damage, which is the cause of the aircraft

10. It is assumed that the sets of signalled and catastrophic damage to the aircraft are separate. Hence, the aircraft reliability in this case can be written in the following form:

R1ð Þt —probability of the fact that to the moment t there will be no catastrophic (sudden)

R2ð Þt —probability of the fact that to the moment t there will be no damage signalled in the

Despite the attempts and great effort of technical services, currently, it is impossible to

It is adopted that in case of a single flight of the aircraft, the probability determining the possibility of the catastrophic damage occurrence is Q. The progressive technical service of the aircraft is to prevent this probability increase together with an increase in operating time.

Under the adopted probabilistic assumptions, a description of the deviation increase of diagnostic parameters in the function of the aircraft operating time can be considered separately for each diagnostic parameter. In view of the above, it is assumed that the process of deviation

May Uzi,t mean the probability of the fact that in the moment t, the deviation of i parameter is zi.

8.2. Aircraft reliability determination including signalled and catastrophic damage

completely eliminate the risk of the catastrophic damage occurrence.

changes of i diagnostic parameter is considered.

ð Þ 1 � λΔt —probability of the fact that in the time interval of Δt, the aircraft flight will not take place;

λΔt—probability of the aircraft flight in the time interval of Δt.

Hence 1ð Þþ � λΔt λΔt ¼ 1

Eq. (83) expresses the following sense. The probability of the fact that in the moment of t þ Δt, the deviation value of i diagnostic parameter will be zi, if in t moment, it had this value and did not increase because of the lack of the aircraft flight or, in t moment, it had zi � Δzi value and in the time interval of Δt, Δzi increased, because the flight did not take place.

Differential Eq. (83), in the function notation, adopts the following form:

$$u(z\_i, \ t + \Delta t) = (1 - \lambda \Delta t) \quad u(z\_i, t) + \lambda \Delta t u(z\_i - \Delta z\_i, t) \tag{84}$$

where

u zi ð Þ ; t —deviation density function of i diagnostic parameter from the nominal value.

Differential Eq. (84) can be transformed to the partial differential equation, with the use of the following approximation:

$$\begin{aligned} u(z\_i, \ t + \Delta t) &= \quad u(z\_i, t) + \frac{\partial u(z\_i, t)}{\partial \ t} \ \Delta t\\ u(z\_i - \Delta z\_i, \ t) &= \quad u(z\_i, \ t) - \frac{\partial u(z\_i, t)}{\partial \ z\_i} \ \Delta z\_i + \frac{1}{2} \quad \frac{\partial^2 u(z\_i, \ t)}{\partial \ z\_i^2} \ (\Delta \ z\_i)^2 \end{aligned} \tag{85}$$

By substituting Eqs. (85) to (84), it is possible to obtain:

$$u(\mathbf{z}\_i, \ t) + \frac{\partial u(\mathbf{z}\_i, t)}{\partial \mathbf{z}} \ \Delta t = (1 - \lambda \Delta t) \ \ u(\mathbf{z}\_i, \ t) + \lambda \Delta t \left[ u(\mathbf{z}\_1, t) - \frac{\partial u(\mathbf{z}\_i, t)}{\partial \mathbf{z}\_i} \ \Delta \mathbf{z}\_i + \frac{1}{2} \frac{\partial^2 u(\mathbf{z}\_i, t)}{\partial \mathbf{z}\_i^2} (\Delta \mathbf{z}\_i)^2 \right] \tag{86}$$

Hence, after simplification, the following is obtained:

$$\frac{\partial u(\mathbf{z}\_i, t)}{\partial t} \Delta t = -\lambda \Delta \mathbf{z}\_i \Delta t \begin{array}{c} \partial u(\mathbf{z}\_i, t) \\ \partial \mathbf{z}\_i \end{array} + \frac{1}{2} \lambda \Delta t (\Delta \mathbf{z}\_i)^2 \quad \frac{\partial^2 u(\mathbf{z}\_i, t)}{\partial \mathbf{z}\_i^2} \tag{87}$$

By dividing two sides of (87) equation by Δt, the following is obtained:

$$\frac{\partial u(z\_i, t)}{\partial t}\_{} = -b\_i(t) \quad \frac{\partial u(z\_i, t)}{\partial z\_i} \quad + \frac{1}{2} \quad a\_i(t) \quad \frac{\partial^2 u(z\_i, t)}{\partial z\_i^2} \tag{88}$$

where

biðÞ¼ t λΔzi—means the average increase of i deviation of the diagnostic parameter from the normal value per time unit;

aiðÞ¼ t λð Þ Δzi 2 —means the average increase square of i deviation from the normal value per time unit;

Δzi—is determined by relationship (79) for ΔN = 1.

The solution of the specific Eq. (88), which meets the following conditions, is searched for:

when t ! 0, the equation is convergent to the Dirac function, that is, u zi ð Þ! ; t 0, dla z 6¼ 0 uð Þ! 0; t ∞ but in a way that the integral of u function is equal to the unity for t > 0.

For such an adopted condition, the equation solution (88) adopts the form:

$$\mu(z\_i, t) = \frac{1}{\sqrt{2\pi A\_i(t)}} \qquad e^{-\frac{\left(\frac{z\_i - B\_i(t)}{2A\_i(t)}\right)^2}{2A\_i(t)}} \tag{89}$$

where

$$B\_i(t) = \int\_0^t b\_i(t)dt\tag{90}$$

It results from the aircraft operation that a group of damage occurs as a result of sudden changes in measurable and non-measurable parameters due to the inability to observe the changes of their values. The exceeding of the applicable limits also affects an increase in the

The damage intensity plays a basic role in the probabilistic description of the occurrence of this

From relationship (94), after transformation, it is possible to obtain the following differential

� Ðt 0 χð Þt dt

χðÞ¼ t χ ¼ const, then:

In order to use relationship (95), it is important to estimate χ parameter. Based on the observations of operation of a specific type of aircraft, it is possible to obtain the times of the occur-

Time t<sup>k</sup> is time for the occurrence of the first of this type of damage in k aircraft calculated from

In order to estimate χ parameter, a method of moments will be used. The comparison of the expected value of operating time calculated from the theoretical relationship, with the average

The theoretical average time of given operation to the moment of the catastrophic damage

R1ð Þt dt ¼

ð ∞

0 e �χ<sup>t</sup> <sup>¼</sup> <sup>1</sup>

R1ðÞ¼ t e

P tð Þ < T < t þ ΔtjT > t

<sup>Δ</sup><sup>t</sup> (94)

(96)

61

<sup>1</sup>ð Þþ t χð Þt R2ðÞ¼ t 0 (95)

Probabilistic Methods for Damage Assessment in Aviation Technology

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<sup>R</sup>1ðÞ¼ <sup>t</sup> <sup>e</sup>�χ<sup>t</sup> (97)

<sup>χ</sup> (98)

risk of the aircraft catastrophic damage occurrence.

The intensity of damage is expressed by the following relationship:

χðÞ¼ t lim Δt!0<sup>þ</sup>

> R 0

Eq. (95), for the initial condition R1ð Þ¼ t ¼ 0 1, has the following solution:

rence of this type of damage tk, where k = 1, 2,…, ω.

value determined on the basis of the observation, will be made.

E T½ �¼ <sup>ð</sup> ∞

0

T—time random variable to the catastrophic damage occurrence;

type of damage.

t—aircraft operation time;

the beginning of operation.

occurrence is:

P(�)—contingent event probability.

where

equation:

If

$$A\_i(t) = \int\_0^t a\_i(t)dt\tag{91}$$

Relationship (90) determines the average value of the deviation, and relationship (91) determines the deviation variance.

By using relationship (89), the reliability in the aspect of the damage signalled for i diagnostic parameter can be written in the following form:

$$R\_i(t) \cong \int\_{-\nu}^{z\_i^d} \mu(z\_i, t) \, d \, \, z\_i \tag{92}$$

By taking into account all the diagnostic parameters and adopted assumptions, the reliability formula adopts the following form:

$$R\_2(t) = \prod\_{i=1}^n R\_i(t) \tag{93}$$

Now the relationship for the second component of R1(t) aircraft reliability is determined due to the catastrophic damage.

The catastrophic (sudden) damage is caused by incomplete control and knowledge of the aircraft technical condition.

It results from the aircraft operation that a group of damage occurs as a result of sudden changes in measurable and non-measurable parameters due to the inability to observe the changes of their values. The exceeding of the applicable limits also affects an increase in the risk of the aircraft catastrophic damage occurrence.

The damage intensity plays a basic role in the probabilistic description of the occurrence of this type of damage.

The intensity of damage is expressed by the following relationship:

$$\chi(t) = \lim\_{\Delta t \to 0^{+}} \frac{P(t < T < t + \Delta t | T > t)}{\Delta t} \tag{94}$$

where

where

aiðÞ¼ t λð Þ Δzi

60 System of System Failures

time unit;

where

normal value per time unit;

2

mines the deviation variance.

formula adopts the following form:

the catastrophic damage.

aircraft technical condition.

parameter can be written in the following form:

Δzi—is determined by relationship (79) for ΔN = 1.

biðÞ¼ t λΔzi—means the average increase of i deviation of the diagnostic parameter from the

The solution of the specific Eq. (88), which meets the following conditions, is searched for:

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>πAið Þ<sup>t</sup> <sup>p</sup> <sup>e</sup>

ðt

0

ðt

0

Relationship (90) determines the average value of the deviation, and relationship (91) deter-

By using relationship (89), the reliability in the aspect of the damage signalled for i diagnostic

ð zd i

�∞

<sup>R</sup>2ðÞ¼ <sup>t</sup> <sup>Y</sup><sup>n</sup>

By taking into account all the diagnostic parameters and adopted assumptions, the reliability

i¼1

Now the relationship for the second component of R1(t) aircraft reliability is determined due to

The catastrophic (sudden) damage is caused by incomplete control and knowledge of the

BiðÞ¼ t

AiðÞ¼ t

RiðÞffi t

uð Þ! 0; t ∞ but in a way that the integral of u function is equal to the unity for t > 0.

For such an adopted condition, the equation solution (88) adopts the form:

u zi ð Þ¼ ; t

when t ! 0, the equation is convergent to the Dirac function, that is, u zi ð Þ! ; t 0, dla z 6¼ 0

—means the average increase square of i deviation from the normal value per

� zi�Bi ð Þ ð Þ<sup>t</sup> <sup>2</sup>

<sup>2</sup>Aið Þ<sup>t</sup> (89)

bið Þt dt (90)

aið Þt dt (91)

u zi ð Þ ; t d zi (92)

Rið Þt (93)

T—time random variable to the catastrophic damage occurrence;

t—aircraft operation time;

P(�)—contingent event probability.

From relationship (94), after transformation, it is possible to obtain the following differential equation:

$$
\boldsymbol{R}\_1^\prime(t) + \chi(t)\boldsymbol{R}\_2(t) = \mathbf{0} \tag{95}
$$

Eq. (95), for the initial condition R1ð Þ¼ t ¼ 0 1, has the following solution:

$$R\_1(t) = e^{-\int\_0^t \chi(t)dt} \tag{96}$$

If

$$\begin{aligned} \chi(t) &= \chi = \text{const}, \text{ then:}\\ R\_1(t) &= e^{-\chi t} \end{aligned} \tag{97}$$

In order to use relationship (95), it is important to estimate χ parameter. Based on the observations of operation of a specific type of aircraft, it is possible to obtain the times of the occurrence of this type of damage tk, where k = 1, 2,…, ω.

Time t<sup>k</sup> is time for the occurrence of the first of this type of damage in k aircraft calculated from the beginning of operation.

In order to estimate χ parameter, a method of moments will be used. The comparison of the expected value of operating time calculated from the theoretical relationship, with the average value determined on the basis of the observation, will be made.

The theoretical average time of given operation to the moment of the catastrophic damage occurrence is:

$$E[T] = \bigcap\_{0}^{\infty} R\_1(t)dt = \bigcap\_{0}^{\infty} e^{-\chi t} = \frac{1}{\chi} \tag{98}$$

The average value of the aircraft operating time (from the moment of the catastrophic damage occurrence) calculated on the basis of the observation will be:

$$\stackrel{\text{e}}{E}[T] = \frac{\sum\_{k=1}^{\omega} t\_k}{\omega} \tag{99}$$

For such specified probabilities, the following equation is right:

regularities of its increase are the same as zi.

Eq. (109) in the function notation adopts the following form:

After completing these operations, the following is obtained:

Finally, the following partial differential equation is obtained:

þ λΔtð Þ 1 � Q u zð Þ� ; t Δz

∂u zð Þ ; t ∂t

∂u zð Þ ; t ∂t

But

Hence

deviation is z.

(85) and relationship (109).

u zð Þ� ; t u zð Þþ ; t

Hence

u zð Þþ ; t

ð Þ 1 � λΔt ð Þþ 1 � χΔt λΔtð Þþ 1 � χΔt ð Þ 1 � λΔt χΔt þ λΔtχΔt ¼ 1 (106)

ð Þ 1 � λΔt ð Þþ 1 � Q λΔtð Þþ 1 � Q ð Þ 1 � λΔt Q þ λΔt Q ¼ 1 (108)

ð Þ 1 � λΔt ð Þþ 1 � Q λΔtð Þþ 1 � Q Q ¼ 1 (109)

Uz,tþΔ<sup>t</sup> ¼ ð Þ 1 � λΔt ð Þ 1 � Q Uz,t þ λΔtð Þ 1 � Q Uz�Δz,t (110)

u zð Þ¼ ; t þ Δt ð Þ 1 � λΔt ð Þ 1 � Q u zð Þþ ; t λΔtð Þ 1 � Q u zð Þ � Δz; t (111)

Δt ¼ u zð Þ� ; t u zð Þþ ; t ð Þ 1 � λΔt ð Þ 1 � Q u zð Þþ ; t

þ 1 2 ð Þ Δz <sup>2</sup> ∂<sup>2</sup> u zð Þ ; t ∂z<sup>2</sup>

(112)

∂u zð Þ ; t ∂z

þ 1 2 ð Þ Δz <sup>2</sup> ∂<sup>2</sup> u zð Þ ; t ∂z<sup>2</sup>

(113)

∂u zð Þ ; t ∂z

Δt ¼ �ð Þ ð Þ 1 � λΔt ð Þþ 1 � Q λΔtð Þþ 1 � Q Q u zð Þþ ; t

<sup>Δ</sup><sup>t</sup> (107)

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Probabilistic Methods for Damage Assessment in Aviation Technology

<sup>χ</sup> <sup>¼</sup> <sup>Q</sup>

The description of deviation changes in the dominant diagnostic parameter currently marked with z will be started. The variable z has the same meaning as zi used in point '7.2' and the

May Uz,<sup>t</sup> mean the probability that in the moment t, the dominant diagnostic parameter

By using relationship (109) and assuming that z deviation increase is determined by first two components of this relationship, the differential Eq. (83) can be written in the following form:

Eq. (111) is transformed into the partial differential equation with the use of the approximation

For greater transparency, u(z,t) is added to and subtracted from the right side of Eq. (111).

þ λΔtð Þ 1 � Q u zð Þ� ; t Δz

Hence

$$\begin{aligned} \frac{1}{\chi^\*} &= \frac{\sum\_{k=1}^{\omega} t\_k}{\omega} \\ \chi^\* &= \frac{\omega}{\sum\_{k=1}^{\omega} t\_k} \end{aligned} \tag{100}$$

If the probability of the sudden damage occurrence during one flight is known, the intensity of this type of damage can be estimated by the following relationship:

$$
\chi^\* = \frac{\mathcal{Q}}{\Delta t} \tag{101}
$$

The relationship for estimation of R1(t) reliability will be:

$$R\_1(t) = e^{-\chi \ast t} \tag{102}$$

After taking into account relationships (93) and (102), the aircraft reliability formula will be:

$$R(t) = e^{-\chi t} \prod\_{i=1}^{n} R\_i(t) \tag{103}$$

#### 8.3. Modification of the applied method for the aircraft reliability determination including sudden and signalled damage

By starting the modification of the applied method in point '8.2', the following additional assumptions are adopted:


$$
\lambda \Delta t + (1 - \lambda \Delta t) = 1 \tag{104}
$$

$$
\chi \Delta t + (1 - \chi \Delta t) = 1 \tag{105}
$$

For such specified probabilities, the following equation is right:

$$(1 - \lambda \Delta t)(1 - \chi \Delta t) + \lambda \Delta t(1 - \chi \Delta t) + (1 - \lambda \Delta t)\chi \Delta t + \lambda \Delta t \chi \Delta t = 1\tag{106}$$

But

The average value of the aircraft operating time (from the moment of the catastrophic damage

Pω k¼1 tk

Xω k¼1 tk

ω

<sup>χ</sup><sup>∗</sup> <sup>¼</sup> <sup>ω</sup> Xω k¼1 tk

If the probability of the sudden damage occurrence during one flight is known, the intensity of

<sup>χ</sup><sup>∗</sup> <sup>¼</sup> <sup>Q</sup>

R1ðÞ¼ t e

After taking into account relationships (93) and (102), the aircraft reliability formula will be:

�χt Yn i¼1

By starting the modification of the applied method in point '8.2', the following additional

• It is assumed that there is one dominant parameter among diagnostic parameters. Its dynamics of changes is the greatest, and due to its causes, the signalled damage is formed

• The probabilities associated with the aircraft flight frequency and the possibility of its

withdrawal from operation constitutes separate, independent sets of events.

R tðÞ¼ e

8.3. Modification of the applied method for the aircraft reliability determination

• The aircraft catastrophic damage causes its withdrawal from operation;

<sup>ω</sup> (99)

<sup>Δ</sup><sup>t</sup> (101)

�χ∗<sup>t</sup> (102)

Rið Þt (103)

λΔt þ ð Þ¼ 1 � λΔt 1 (104)

χΔt þ ð Þ¼ 1 � χΔt 1 (105)

(100)

E \_ ½ �¼ T

> 1 χ<sup>∗</sup> ¼

occurrence) calculated on the basis of the observation will be:

this type of damage can be estimated by the following relationship:

The relationship for estimation of R1(t) reliability will be:

including sudden and signalled damage

assumptions are adopted:

in the quickest manner.

Hence

62 System of System Failures

$$
\chi = \frac{Q}{\Delta t} \tag{107}
$$

Hence

$$(1 - \lambda \Delta t)(1 - Q) + \lambda \Delta t (1 - Q) + (1 - \lambda \Delta t) \ Q + \lambda \Delta t \ Q = 1 \tag{108}$$

$$(1 - \lambda \Delta t)(1 - Q) + \lambda \Delta t (1 - Q) + \quad Q = 1\tag{109}$$

The description of deviation changes in the dominant diagnostic parameter currently marked with z will be started. The variable z has the same meaning as zi used in point '7.2' and the regularities of its increase are the same as zi.

May Uz,<sup>t</sup> mean the probability that in the moment t, the dominant diagnostic parameter deviation is z.

By using relationship (109) and assuming that z deviation increase is determined by first two components of this relationship, the differential Eq. (83) can be written in the following form:

$$dL\_{z,t+\Lambda t} = (1 - \lambda \Delta t)(1 - Q)L\_{z,t} + \lambda \Delta t (1 - Q)L\_{z-\Lambda z,t} \tag{110}$$

Eq. (109) in the function notation adopts the following form:

$$u(z, t + \Delta t) = (1 - \lambda \Delta t)(1 - Q)u(z, t) + \lambda \Delta t (1 - Q)u(z - \Delta z, t) \tag{111}$$

Eq. (111) is transformed into the partial differential equation with the use of the approximation (85) and relationship (109).

For greater transparency, u(z,t) is added to and subtracted from the right side of Eq. (111).

After completing these operations, the following is obtained:

$$\begin{split} u(z,t) + \frac{\partial u(z,t)}{\partial t} \Delta t &= u(z,t) - u(z,t) + (1 - \lambda \Delta t)(1 - Q)u(z,t) + \\ &+ \lambda \Delta t (1 - Q) \left( u(z,t) - \Delta z \frac{\partial u(z,t)}{\partial z} + \frac{1}{2} (\Delta z)^2 \frac{\partial^2 u(z,t)}{\partial z^2} \right) \end{split} \tag{112}$$

Hence

$$\begin{split} u(\boldsymbol{z},t) - u(\boldsymbol{z},t) + \frac{\partial u(\boldsymbol{z},t)}{\partial t} \Delta t &= -\left( (1 - \lambda \Delta t)(1 - \boldsymbol{Q}) + \lambda \Delta t (1 - \boldsymbol{Q}) + \boldsymbol{Q} \right) u(\boldsymbol{z},t) + \\ &+ \lambda \Delta t (1 - \boldsymbol{Q}) \left( u(\boldsymbol{z},t) - \lambda \boldsymbol{z} \frac{\partial u(\boldsymbol{z},t)}{\partial \boldsymbol{z}} + \frac{1}{2} (\Delta \boldsymbol{z})^2 \frac{\partial^2 u(\boldsymbol{z},t)}{\partial \boldsymbol{z}^2} \right) \end{split} \tag{113}$$

Finally, the following partial differential equation is obtained:

$$\frac{\partial u(z,t)}{\partial t} = -\chi u(z,t) - b(t) \frac{\partial u(z,t)}{\partial z} + \frac{1}{2} \left. a(t) \frac{\partial^2 u(z,t)}{\partial z^2} \right. \tag{114}$$

where

χ—intensity of the withdrawal of a specific type of aircraft due to the catastrophic damage occurrence:

$$\chi = \frac{Q}{\Delta t} \tag{115}$$

Function (114) has the density function characteristics, because:

ð ∞ ð ∞

u zð Þ ; t dz dt ¼

z ðd

2 4

�∞

ð ∞ 2 4 ð ∞

u zð Þ ; t dz

u zð Þ ; t dz

3

3

5dt ¼ 1 � e

Probabilistic Methods for Damage Assessment in Aviation Technology

Q tðÞ¼ Q1ð Þt R2ð Þþ t Q1ð Þt Q2ð Þt (122)

R tðÞ¼ R1ð Þ� t R2ð Þt (123)

R tð Þþ Q tðÞ¼ 1 (124)

<sup>2</sup>A tð Þ dz (125)

� ð Þ <sup>z</sup>�B tð Þ <sup>2</sup>

5dt ¼ 1 (120)

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65

�χ<sup>t</sup> (121)

�∞

ð ∞

zd

Q2ð Þt —unreliability caused by the deviation increase of the dominant parameter above the

0

u zð Þ ; t dz þ

0

By using relationship (118), the aircraft unreliability is determined

�∞

0 χe �χt

Q1ð Þt —unreliability caused by the aircraft catastrophic damage;

R1ð Þt —aircraft reliability referring to the catastrophic damage;

R2ð Þt —aircraft reliability referring to the dominant parameter;

R tðÞ¼ e

B(t) and A(t) are determined by relationships (90) and (91).

�χt z ðd

�∞

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>πA tð Þ <sup>p</sup> <sup>e</sup>

The above method applies to the cases, in which the effects of action of destructive processes cumulate, that are correlated with the aircraft operating time and this process is disrupted by the possibility of the occurrence of the sudden damage caused by, for example, overload

Q tðÞ¼ <sup>ð</sup><sup>t</sup>

Thus, it is possible to write that:

Thus, the aircraft reliability will be:

Thus, the aircraft reliability formula will be:

where

limit value.

Hence

where

pulses, hard landings, etc.

b(t)—average increase in the dominant parameter deviation per time unit:

$$b(t) = \lambda (1 - Q)\Delta z \tag{116}$$

a(t)—average increase square of the dominant parameter deviation per time unit:

$$a(t) = \lambda (1 - Q)\Delta z^2 \tag{117}$$

Δz—specified by relationship (79).

Eq. (114) is more general than the Fokker-Planck equation, written in the form of the relationship (88).

Eq. (114) has an additional element '-χu(z,t)'.

In order to present the equation solution (114), the equation solution (88) will be used. If the equation solution (88) constitutes the relationship (89), then, the equation solution (114) constitutes the following function:

$$
\overline{u}(z,t) = \chi e^{-\chi t} \ \overline{u}(z,t) \tag{118}
$$

where

u zð Þ ; t —is the equation solution (88) and is presented by the relationship (89). In this solution, in the integrals (90) and (91), it is important to use the relationships (116) and (117).

In order to justify that the function (118) is the equation solution (114), the following transformation is presented:

A derivative after the relationship time (118) is calculated.

$$\begin{split} \frac{\partial u(z,t)}{\partial t} &= \chi^2 e^{-\chi t} \overline{u}(z,t) + \chi e^{-\chi t} \frac{\partial \overline{u}(z,t)}{\partial t} = \\ &= \chi u(z,t) + \chi e^{-\chi t} \left( -b(t) \frac{\partial \overline{u}(z,t)}{\partial z} + \frac{1}{2} a(t) \frac{\partial^2 \overline{u}(z,t)}{\partial z^2} \right) = \\ &= \chi u(z,t) - b(t) \frac{\partial u(z,t)}{\partial z} + \frac{1}{2} a(t) \frac{\partial^2 u(z,t)}{\partial z^2} \end{split} \tag{119}$$

Hence, it can be observed that the function (118) is the equation solution (114).

Function (114) has the density function characteristics, because:

$$\int\_{-\infty}^{\infty} \int\_{0}^{\infty} \mu(z, t) dz \, dt = \int\_{0}^{\infty} \left[ \int\_{-\infty}^{\infty} \mu(z, t) dz \right] dt = 1 \tag{120}$$

By using relationship (118), the aircraft unreliability is determined

$$Q(t) = \int\_0^t \chi e^{-\chi t} \left[ \int\_{-\infty}^{z\_d} \overline{u}(z, t) dz + \int\_{zd}^{\infty} \overline{u}(z, t) dz \right] dt = 1 - e^{-\chi t} \tag{121}$$

Thus, it is possible to write that:

$$Q(t) = \overline{Q}\_1(t)\overline{R}\_2(t) + \overline{Q}\_1(t)\overline{Q}\_2(t) \tag{122}$$

where

∂u zð Þ ; t

where

occurrence:

64 System of System Failures

ship (88).

where

mation is presented:

Δz—specified by relationship (79).

stitutes the following function:

Eq. (114) has an additional element '-χu(z,t)'.

<sup>∂</sup><sup>t</sup> ¼ �χu zð Þ� ; <sup>t</sup> b tð Þ <sup>∂</sup>u zð Þ ; <sup>t</sup>

b(t)—average increase in the dominant parameter deviation per time unit:

a(t)—average increase square of the dominant parameter deviation per time unit:

∂z

χ—intensity of the withdrawal of a specific type of aircraft due to the catastrophic damage

<sup>χ</sup> <sup>¼</sup> <sup>Q</sup>

Eq. (114) is more general than the Fokker-Planck equation, written in the form of the relation-

In order to present the equation solution (114), the equation solution (88) will be used. If the equation solution (88) constitutes the relationship (89), then, the equation solution (114) con-

u zð Þ ; t —is the equation solution (88) and is presented by the relationship (89). In this solution,

In order to justify that the function (118) is the equation solution (114), the following transfor-

�χ<sup>t</sup> <sup>∂</sup>u zð Þ ; <sup>t</sup> ∂t ¼

> þ 1 2 a tð Þ <sup>∂</sup><sup>2</sup>

> > u zð Þ ; t ∂z<sup>2</sup>

u zð Þ ; t ∂z<sup>2</sup>

¼

(119)

∂z

þ 1 2 a tð Þ <sup>∂</sup><sup>2</sup>

u zð Þ¼ ; t χe

in the integrals (90) and (91), it is important to use the relationships (116) and (117).

u zð Þþ ; t χe

∂z

Hence, it can be observed that the function (118) is the equation solution (114).

A derivative after the relationship time (118) is calculated.

e �χt

<sup>¼</sup> <sup>χ</sup>u zð Þ� ; <sup>t</sup> b tð Þ <sup>∂</sup>u zð Þ ; <sup>t</sup>

<sup>¼</sup> <sup>χ</sup>u zð Þþ ; <sup>t</sup> <sup>χ</sup>e�χ<sup>t</sup> �b tð Þ <sup>∂</sup>u zð Þ ; <sup>t</sup>

∂u zð Þ ; t <sup>∂</sup><sup>t</sup> <sup>¼</sup> <sup>χ</sup><sup>2</sup> þ 1 <sup>2</sup> a tð Þ <sup>∂</sup><sup>2</sup>

u zð Þ ; t

<sup>Δ</sup><sup>t</sup> (115)

�χ<sup>t</sup> u zð Þ ; <sup>t</sup> (118)

b tðÞ¼ λð Þ 1 � Q Δz (116)

a tðÞ¼ <sup>λ</sup>ð Þ <sup>1</sup> � <sup>Q</sup> <sup>Δ</sup>z<sup>2</sup> (117)

<sup>∂</sup>z<sup>2</sup> (114)

Q1ð Þt —unreliability caused by the aircraft catastrophic damage;

Q2ð Þt —unreliability caused by the deviation increase of the dominant parameter above the limit value.

R1ð Þt —aircraft reliability referring to the catastrophic damage;

R2ð Þt —aircraft reliability referring to the dominant parameter;

Thus, the aircraft reliability will be:

$$R(t) = \overline{R}\_1(t) \cdot \overline{R}\_2(t) \tag{123}$$

Hence

$$R(t) + Q(t) = 1\tag{124}$$

Thus, the aircraft reliability formula will be:

$$R(t) = e^{-\chi t} \int\_{-\infty}^{z\_d} \frac{1}{\sqrt{2\pi A(t)}} \quad e^{-\frac{(z-\theta(t))^2}{2A(t)}} dz \tag{125}$$

where

B(t) and A(t) are determined by relationships (90) and (91).

The above method applies to the cases, in which the effects of action of destructive processes cumulate, that are correlated with the aircraft operating time and this process is disrupted by the possibility of the occurrence of the sudden damage caused by, for example, overload pulses, hard landings, etc.

This method may allow to estimate durability, due to individual diagnostic parameters. The data obtained in this way can be used in order to improve the technical service. The sequence of diagnostic controls adequately spread over operating time allows to prevent the signalled damage occurrence.

Therefore, it can be assumed that:

$$R\_2(t) = \prod\_{i=1}^n \int\_{-\infty}^{z\_d} \mu\_1(z\_i, t) dz\_i \approx 1 \tag{126}$$

The aircraft reliability, including the technical service, can be estimated by the following relationship:

$$R(t) = e^{-\chi^\* t} \tag{127}$$

Three factors determining identification errors can be mentioned:

interpreting the results

with q03.

These probabilities meet the condition:

The diagnosis process may include the following events:

statement is flawless. The probability of such an event is P11.

as well the percentage of features not subject to the inspection;

• monitoring susceptibility of the object: it shows the extent, to which the object is adapted to the inspection, and a way in which the inspection procedures identify the actual state,

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67

• technical equipment of the operator inspecting the state of the object and procedures of

• circumstances of the inspection, climatic conditions, time stress, information stress, etc.

As it results from the above considerations, the identification error is a parameter of the systemic nature. The object designer, the designer of diagnostic equipment, the operator equipped with diagnostic equipment of sufficient quality and the training of the operator conducting identification are responsible for the object state identification error. Despite the fact that the identification error depends on many factors, it is the person conducting the identification who is legally and morally responsible for the effects resulting from the identification error. The removal of responsibility from the operator follows the specified tests conducted by the specially appointed expert teams. These teams often include also experts from scientific and research institutions. These teams determine the causes of the erroneous qualification of the object condition. It results in a stressful situation for the operator, who does not always understand the essence of various sources of misidentification, blaming himself or herself for adverse events. The problem of the first and second type errors during the identifi-

The source of the error is sometimes unreliability of diagnosing units equipped with the necessary equipment and procedures of stating the usability state. With regard to the object, on which the condition is identified, it can be said that there are the following events on it:

A01—The event involving the fact that the object is in the state of usability and it will keep this

A02—The event involving the occurrence of damage detected in the identification process in

A03—The event involving the occurrence of damage not detected during the identification in the object until or during the identification process. The probability of such an event is marked

A11—The event involving the fact that the diagnosis is correctly carried out and the object state

P<sup>01</sup> þ q<sup>02</sup> þ q<sup>03</sup> ¼ 1 (128)

state during the identification. The probability of such an event is marked with P01.

the object until or during identification. The probability of such an event is q02.

• predispositions of the operator, his or her qualifications, personal characteristics;

cation of the usability state has a legal-moral, economic and technical aspect.

The presented methods for determining the relationships for the aircraft reliability are conditioned by the adopted assumptions. They can be modified in accordance with the assumptions. The more accurately the adopted assumptions will reflect the actual conditions, the more precisely the aircraft reliability will be estimated. The methods can be adapted to specific cases for determination of the catastrophic damage probability values, including the physics of occurring phenomena and operating conditions. The aircraft reliability forecasts can be used for consideration of specific problems with the reliability assessment and durability of elements, units and devices.

## 9. Reliability incorrect assessment results in air systems

#### 9.1. Analysis of errors of diagnosing and stating the usability state of technical systems

The reliability of diagnostic equipment and the ergonomics of technical systems affect the errors made by the operator. The chapter raised the problem of diagnosis errors and erroneous usability evaluation and describes the example of a real event of the aircraft landing without the released landing gear, as a consequence of erroneous diagnosing. The rescue process in a situation of an aviation accident hazard was briefly described in this chapter. A person equipped with diagnostic equipment can make two types of errors, whose measurements are the occurrence probabilities marked with symbols α and β.

α—means an error of the first type, which consists of qualifying the usable device as unfit;

β—means an error of the second type, which consists of qualifying the unfit device as usable.

Making the first type error in the identification process of the aircraft's usability may cause losses due to unplanned downtime and repeated inspection. In case of making the second type error, more dangerous consequences with the possibility of an aviation accident are often caused;

Three factors determining identification errors can be mentioned:

This method may allow to estimate durability, due to individual diagnostic parameters. The data obtained in this way can be used in order to improve the technical service. The sequence of diagnostic controls adequately spread over operating time allows to prevent the signalled

<sup>R</sup>2ðÞ¼ <sup>t</sup> <sup>Y</sup><sup>n</sup>

9. Reliability incorrect assessment results in air systems

the occurrence probabilities marked with symbols α and β.

i¼1

z ðd

�∞

The aircraft reliability, including the technical service, can be estimated by the following

R tðÞ¼ e

9.1. Analysis of errors of diagnosing and stating the usability state of technical systems

α—means an error of the first type, which consists of qualifying the usable device as unfit;

β—means an error of the second type, which consists of qualifying the unfit device as usable. Making the first type error in the identification process of the aircraft's usability may cause losses due to unplanned downtime and repeated inspection. In case of making the second type error, more dangerous consequences with the possibility of an aviation accident are often

The reliability of diagnostic equipment and the ergonomics of technical systems affect the errors made by the operator. The chapter raised the problem of diagnosis errors and erroneous usability evaluation and describes the example of a real event of the aircraft landing without the released landing gear, as a consequence of erroneous diagnosing. The rescue process in a situation of an aviation accident hazard was briefly described in this chapter. A person equipped with diagnostic equipment can make two types of errors, whose measurements are

The presented methods for determining the relationships for the aircraft reliability are conditioned by the adopted assumptions. They can be modified in accordance with the assumptions. The more accurately the adopted assumptions will reflect the actual conditions, the more precisely the aircraft reliability will be estimated. The methods can be adapted to specific cases for determination of the catastrophic damage probability values, including the physics of occurring phenomena and operating conditions. The aircraft reliability forecasts can be used for consideration of specific problems with the reliability assessment and durability of ele-

u<sup>1</sup> zi ð Þ ; t dzi ≈ 1 (126)

�χ∗<sup>t</sup> (127)

damage occurrence.

66 System of System Failures

relationship:

ments, units and devices.

caused;

Therefore, it can be assumed that:


As it results from the above considerations, the identification error is a parameter of the systemic nature. The object designer, the designer of diagnostic equipment, the operator equipped with diagnostic equipment of sufficient quality and the training of the operator conducting identification are responsible for the object state identification error. Despite the fact that the identification error depends on many factors, it is the person conducting the identification who is legally and morally responsible for the effects resulting from the identification error. The removal of responsibility from the operator follows the specified tests conducted by the specially appointed expert teams. These teams often include also experts from scientific and research institutions. These teams determine the causes of the erroneous qualification of the object condition. It results in a stressful situation for the operator, who does not always understand the essence of various sources of misidentification, blaming himself or herself for adverse events. The problem of the first and second type errors during the identification of the usability state has a legal-moral, economic and technical aspect.

The source of the error is sometimes unreliability of diagnosing units equipped with the necessary equipment and procedures of stating the usability state. With regard to the object, on which the condition is identified, it can be said that there are the following events on it:

A01—The event involving the fact that the object is in the state of usability and it will keep this state during the identification. The probability of such an event is marked with P01.

A02—The event involving the occurrence of damage detected in the identification process in the object until or during identification. The probability of such an event is q02.

A03—The event involving the occurrence of damage not detected during the identification in the object until or during the identification process. The probability of such an event is marked with q03.

These probabilities meet the condition:

$$P\_{01} + q\_{02} + q\_{03} = 1\tag{128}$$

The diagnosis process may include the following events:

A11—The event involving the fact that the diagnosis is correctly carried out and the object state statement is flawless. The probability of such an event is P11.

A12—The event involving the fact that the object was considered unfit regardless of its state. The probability of such an event is q12.

A13—The event involving the fact that the object was considered usable regardless of its state. The probability of such an event is q13.

A14—The event involving the fact that the object was considered unfit, whereas, in fact, it is usable, and the object was considered usable, whereas, in fact, it is unfit. The probability of such an event is marked with q14.

These probabilities meet the condition:

$$P\_{11} + q\_{12} + q\_{13} + q\_{14} = 1\tag{129}$$

The probability of an event involving the fact that the object considered unfit, in fact, is usable, that is, making the first type error is given by the following formula:

$$\alpha = 1 - \frac{P\_{01}}{1 - \frac{q\_{02} \left(P\_{11} - q\_{14}\right)}{P\_{11} + q\_{13}}} \tag{130}$$

9.3. Example result of erroneous diagnosing

Figure 7. The course of α<sup>m</sup> function for Cð Þ α various values for α ¼ 0, 1.

is a rare event in the aircraft operation.

9.4. Rescue process of the critical situation

landing at the Warsaw's Okęcie would not have happened.

The fact of a certain error in diagnosing can be stated on the example of the above-mentioned emergency landing of the PLL LOT plane, Boeing 767-300ER, on November 1, 2011 at Warsaw Chopin Airport. It should be reminded that the Boeing 767-300 plane of the Polish airlines LOT departed from the Newark airport (USA) after midnight on November 1, 2011. Thirty minutes after the takeoff from Newark, the crew of the Polish plane signalled a failure of the central hydraulic system. The machine had another system, the emergency and electrical one, which could retract the landing gear. After the departure, the plane was filled with fuel and despite the failure it would not be justified to fly around over the U.S. territory for many hours because only after fuel consumption, it would be possible to check the operation of the system extending the landing gear and to try to land. The captain decided to continue the flight, although he could not be sure as to the emergency system usability, and he intended to verify the operation in the territory of Poland. Over Warsaw, it occurred that the usability of the entire landing gear control system was evaluated erroneously because its extension failed, although the flaps had extended. Then, the decision on emergency landing was made. The result of the incorrect evaluation of the situation described above was the plane failure, which

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69

The members of the government committee investigating the circumstance of the emergency landing showed that the emergency system was efficient but the aircraft crew did not use it because one of the key fuses, which secured several aircraft's systems, including the emergency landing gear extension system, was disabled. If the fuse had been enabled, the dramatic

In the considered flight, an event involving consideration of the activating element as usable, regardless of its state, occurred, and it was not subject to diagnosing [2]. The unaware classification of the unfit device as usable without diagnosing is a systemic error of the second type.

The probability of an event involving the fact that the object will be considered usable, whereas it is, in fact, unfit, that is, making the second type error is given by the following formula:

$$\beta = 1 - \frac{1}{1 + \frac{P\_{01}}{1 - P\_{01} + q\_{02}\frac{P\_{11} - q\_{14}}{q\_{12} - q\_{14}}}} \tag{131}$$

The impact of possible events is the process of diagnosis on the values of the first and second type errors results from the provided formulas.

#### 9.2. Shaping of the first and second type errors by the operator teaching method

Figure 7 shows the course of function α<sup>m</sup> of reducing the error of the first type as a result of m repetition of action performed by the operator or diagnosing team for different values of the experimentally determined coefficient Cð Þ α .

These errors in the function of the number of m tests are given by the following formulas:

$$\alpha\_m = \alpha [1 - \alpha \text{ }\text{C}(\alpha)]^{m-1} \tag{132}$$

$$\beta\_m = \beta \left[1 - \beta \text{ C}(\beta)\right]^{m-1} \tag{133}$$

The intensity of learning has a significant impact on the reduction of the first and second type errors. As a result of the training, the operator learns using the controls, reading instrument indications and interpretation of symptoms of the object's usability and unfitness. For the purposes of teaching the operator, the specific states are modelled. As a result of conducted research and analyses, Cð Þ α , C β coefficients characterising the quantitative progress of the training and the intensity of reducing the first and second type errors are determined.

Figure 7. The course of α<sup>m</sup> function for Cð Þ α various values for α ¼ 0, 1.

#### 9.3. Example result of erroneous diagnosing

A12—The event involving the fact that the object was considered unfit regardless of its state.

A13—The event involving the fact that the object was considered usable regardless of its state.

A14—The event involving the fact that the object was considered unfit, whereas, in fact, it is usable, and the object was considered usable, whereas, in fact, it is unfit. The probability of

The probability of an event involving the fact that the object considered unfit, in fact, is usable,

The probability of an event involving the fact that the object will be considered usable, whereas it is, in fact, unfit, that is, making the second type error is given by the following formula:

> <sup>1</sup> <sup>þ</sup> <sup>P</sup><sup>01</sup> 1�P01þq<sup>02</sup>

The impact of possible events is the process of diagnosis on the values of the first and second

Figure 7 shows the course of function α<sup>m</sup> of reducing the error of the first type as a result of m repetition of action performed by the operator or diagnosing team for different values of the

The intensity of learning has a significant impact on the reduction of the first and second type errors. As a result of the training, the operator learns using the controls, reading instrument indications and interpretation of symptoms of the object's usability and unfitness. For the purposes of teaching the operator, the specific states are modelled. As a result of conducted

training and the intensity of reducing the first and second type errors are determined.

These errors in the function of the number of m tests are given by the following formulas:

<sup>P</sup>11�q<sup>14</sup> <sup>q</sup>12�q<sup>14</sup>

<sup>1</sup> � <sup>q</sup><sup>02</sup> <sup>P</sup>11�<sup>q</sup> ð Þ <sup>14</sup> P11þq<sup>13</sup>

<sup>α</sup> <sup>¼</sup> <sup>1</sup> � <sup>P</sup><sup>01</sup>

<sup>β</sup> <sup>¼</sup> <sup>1</sup> � <sup>1</sup>

9.2. Shaping of the first and second type errors by the operator teaching method

that is, making the first type error is given by the following formula:

P<sup>11</sup> þ q<sup>12</sup> þ q<sup>13</sup> þ q<sup>14</sup> ¼ 1 (129)

<sup>α</sup><sup>m</sup> <sup>¼</sup> <sup>α</sup>½ � <sup>1</sup> � <sup>α</sup> <sup>C</sup>ð Þ <sup>α</sup> <sup>m</sup>�<sup>1</sup> (132)

<sup>β</sup><sup>m</sup> <sup>¼</sup> <sup>β</sup> <sup>1</sup> � <sup>β</sup> <sup>C</sup> <sup>β</sup> <sup>m</sup>�<sup>1</sup> (133)

coefficients characterising the quantitative progress of the

(130)

(131)

The probability of such an event is q12.

68 System of System Failures

The probability of such an event is q13.

These probabilities meet the condition:

type errors results from the provided formulas.

experimentally determined coefficient Cð Þ α .

research and analyses, Cð Þ α , C β

such an event is marked with q14.

The fact of a certain error in diagnosing can be stated on the example of the above-mentioned emergency landing of the PLL LOT plane, Boeing 767-300ER, on November 1, 2011 at Warsaw Chopin Airport. It should be reminded that the Boeing 767-300 plane of the Polish airlines LOT departed from the Newark airport (USA) after midnight on November 1, 2011. Thirty minutes after the takeoff from Newark, the crew of the Polish plane signalled a failure of the central hydraulic system. The machine had another system, the emergency and electrical one, which could retract the landing gear. After the departure, the plane was filled with fuel and despite the failure it would not be justified to fly around over the U.S. territory for many hours because only after fuel consumption, it would be possible to check the operation of the system extending the landing gear and to try to land. The captain decided to continue the flight, although he could not be sure as to the emergency system usability, and he intended to verify the operation in the territory of Poland. Over Warsaw, it occurred that the usability of the entire landing gear control system was evaluated erroneously because its extension failed, although the flaps had extended. Then, the decision on emergency landing was made. The result of the incorrect evaluation of the situation described above was the plane failure, which is a rare event in the aircraft operation.

The members of the government committee investigating the circumstance of the emergency landing showed that the emergency system was efficient but the aircraft crew did not use it because one of the key fuses, which secured several aircraft's systems, including the emergency landing gear extension system, was disabled. If the fuse had been enabled, the dramatic landing at the Warsaw's Okęcie would not have happened.

#### 9.4. Rescue process of the critical situation

In the considered flight, an event involving consideration of the activating element as usable, regardless of its state, occurred, and it was not subject to diagnosing [2]. The unaware classification of the unfit device as usable without diagnosing is a systemic error of the second type.

In the model representing the situation of the emergency landing on November 1, 2011, it is possible to distinguish the following elements (Figure 8): protected object—landing gear extension system, protecting object—emergency landing gear extension system and the activating element.

After receiving the information about a defective protection system (electrical system) and inability to release the landing gear, there was the search for solutions, which had to take place at the available time—TD. After recognition of the erroneous evaluation of the emergency system, the only solution left was the use of a different emergency protection system in the form of the fuselage designed for this purpose. Owing to the pilot's wise action, great skills and precise operation, the implementation of the made decision on the emergency landing was successful. This type of situation can be described with the salvage equation, which designates the probability of the danger defuse at the available time through the convolution of distribution functions

of random variables of the available time and implementation time of the rescue task.

ð ∞

FDð Þ t þ τ dFOBð Þt (134)

Probabilistic Methods for Damage Assessment in Aviation Technology

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

71

0

The probability distribution of the available time depends on the type of event. For example, for a survivor at sea, it will be the time of survival dependent on circumstances (temperature of the water and his or her own equipment); for the aircraft, the remained flight persistence; for a parachutist, remaining height, etc. The probability distribution of the implementation time of an intervention task also depends on many factors—the type of the task, the degree of the

In the cited example, making the right decision and the precise landing proved to be the right action preventing from crash. The implementation of random variables in the considered event

The presented analysis of the diagnosis errors and the rescue process model were presented in

[1] Kaleta R. Consideration on estimation and research of operational/maintenance rates.

[2] Żurek J. Modelling of Keeping up Safety Systems. Radom: WNITE; 2010

[3] Wenzell AD. Lectures on the Theory of Stochastic Processes. Warsaw: PWN; 1980

P Tð Þ¼ OB < TD

rescue team or system's readiness, action efficiency.

\*Address all correspondence to: jozef.zurek@itwl.pl Air Force Institute of Technology, Warsaw, Poland

Author details

References

Józef Żurek\* and Ryszard Kaleta

ZEM Journal. 2004;4(140):95-115

in the relationship, (tOB < tD) fulfilled the salvage condition.

a shortened version due to the limited scope of the chapter.

In Figure 8, the probabilistic characteristics of the implementation time of the security task and available time were marked.

TOB—random variable of the security task implementation time,

FOBð Þt —distribution function of the random variable of the security task implementation time,

tOB, —random variable implementation: time of the flight over the airport and search for the solution,

TD—random variable of available time: time of the flight limited by fuel residues,

FDð Þt —distribution function of the random variable of available time,

tD—implementation of the random variable of available time: maximum time of the flight limited by fuel residues.

The available time determines the reasonable time necessary to prevent a dangerous situation. In general, this time may be determined by, for example, a fuel resource, a resource of an active substance or any other type of energy extending the system operation.

The analysis of the situation and taking activities at available time can be described as follows:


After receiving the information about a defective protection system (electrical system) and inability to release the landing gear, there was the search for solutions, which had to take place at the available time—TD. After recognition of the erroneous evaluation of the emergency system, the only solution left was the use of a different emergency protection system in the form of the fuselage designed for this purpose. Owing to the pilot's wise action, great skills and precise operation, the implementation of the made decision on the emergency landing was successful. This type of situation can be described with the salvage equation, which designates the probability of the danger defuse at the available time through the convolution of distribution functions of random variables of the available time and implementation time of the rescue task.

$$P(T\_{OB} < T\_D) = \bigcap\_{0}^{\approx} F\_D(t + \tau)dF\_{OB}(t) \tag{134}$$

The probability distribution of the available time depends on the type of event. For example, for a survivor at sea, it will be the time of survival dependent on circumstances (temperature of the water and his or her own equipment); for the aircraft, the remained flight persistence; for a parachutist, remaining height, etc. The probability distribution of the implementation time of an intervention task also depends on many factors—the type of the task, the degree of the rescue team or system's readiness, action efficiency.

In the cited example, making the right decision and the precise landing proved to be the right action preventing from crash. The implementation of random variables in the considered event in the relationship, (tOB < tD) fulfilled the salvage condition.

The presented analysis of the diagnosis errors and the rescue process model were presented in a shortened version due to the limited scope of the chapter.

## Author details

In the model representing the situation of the emergency landing on November 1, 2011, it is possible to distinguish the following elements (Figure 8): protected object—landing gear extension system, protecting object—emergency landing gear extension system and the activating element. In Figure 8, the probabilistic characteristics of the implementation time of the security task and

FOBð Þt —distribution function of the random variable of the security task implementation time, tOB, —random variable implementation: time of the flight over the airport and search for the

tD—implementation of the random variable of available time: maximum time of the flight

The available time determines the reasonable time necessary to prevent a dangerous situation. In general, this time may be determined by, for example, a fuel resource, a resource of an active

The analysis of the situation and taking activities at available time can be described as follows:

TD—random variable of available time: time of the flight limited by fuel residues,

TOB—random variable of the security task implementation time,

FDð Þt —distribution function of the random variable of available time,

substance or any other type of energy extending the system operation.

• receiving information about a faulty protection system (electrical system);

• making the decision about the emergency landing on the aircraft fabric covering;

• analysis of the obtained information and search for a solution;

• receiving information about the hydraulic system leak;

• making the decision to continue the flight;

• initiation of the landing procedure;

• implementation of the made decision;

Figure 8. Relief system model with the protection system.

• inspection of the made decision.

available time were marked.

70 System of System Failures

limited by fuel residues.

solution,

Józef Żurek\* and Ryszard Kaleta

\*Address all correspondence to: jozef.zurek@itwl.pl

Air Force Institute of Technology, Warsaw, Poland

### References


[4] Loroch L, Tomaszek H, Żurek J. Outline of the method of estimation reliability for aircraft's devices on conditions of small correlations of change of value diagnostic parameters in time of aircraft's flight. ZEM Journal. 2004;4(140):83-94

**Section 3**

**Exploring Purposes**


**Exploring Purposes**

[4] Loroch L, Tomaszek H, Żurek J. Outline of the method of estimation reliability for aircraft's devices on conditions of small correlations of change of value diagnostic parameters

[5] Tomaszek H, Żurek J, Jasztal M. Forecasting of Damage Being Hazardous for the Aircraft

[6] Gerebach JB, Kordoński CB. Models for Reliability of Technical Objects. Warsaw: WNT

[7] Żurek J. Modelling of the Protection Systems in Transport Devices. Warsaw: Publishing

House of Warsaw University of Technology, Scientific Papers, Transport; 1998

in time of aircraft's flight. ZEM Journal. 2004;4(140):83-94

Flight Safety. Warsaw: WNITE; 2008

72 System of System Failures

[Scientific and Technical Publishing]; 1968

**Chapter 5**

**Provisional chapter**

**Failures in a Critical Infrastructure System**

ture subsystems in circumstances involving emergencies.

**Failures in a Critical Infrastructure System**

DOI: 10.5772/intechopen.70446

The purpose of this chapter is to provide a comprehensive overview of a critical infrastructure system, of failures and impacts that occur within it and of the resilience, which effectively reduces the risk of these impacts spreading on to dependent subsystems. The chapter presents a basic description of a critical infrastructure system and of the hierarchic arrangement of its subsystems and linkages between them. Critical infrastructure system failures, including their causes and impacts on dependent subsystems and on society as a whole, are presented in the following section. Particular focus is given to the propagation of impacts in a critical infrastructure system and the current approaches to their modeling. The chapter concludes by expounding on the resilience of critical infrastructure subsystems and its impact on the minimization of failures in critical infrastruc-

**Keywords:** critical infrastructure, system, disruption, failure, impacts, resilience

Society has traditionally depended on a broad variety of services as much as on the infrastructures providing them. Over time, some of these infrastructures, or rather their elements considered to be of vital importance to society, began to be regarded as critical. At present, these infrastructures constitute the critical infrastructure system [1], which consists of individual subsystems, i.e., sectors, subsectors, and elements. There are dependencies between critical infrastructure subsystems which can, due to a disruption in the functionality of one subsystem, spread to dependent subsystems, and thereby escalate the impacts from emergen-

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

and reproduction in any medium, provided the original work is properly cited.

David Rehak and Martin Hromada

David Rehak and Martin Hromada

http://dx.doi.org/10.5772/intechopen.70446

**Abstract**

**1. Introduction**

cies on society.

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

**Provisional chapter**
