6. Assessment of a chance of the catastrophic damage occurrence with the constant level along the increasing diagnostic parameter value

In point 5, a case of the device operation, when the catastrophic damage occurred only after exceeding the limit state by the diagnostic parameter value, was considered. Currently, the next case is considered, when the opportunity of additional one (the second type of catastrophic damage), possible to be implemented in every moment of the aircraft operation, is added to the previous one.

Additionally, the intensity of the occurrence of this type of additional damage will be introduced:

$$
\mu = \frac{P}{\Delta t} \Rightarrow P = \mu \Delta t \le 1\tag{57}
$$

A tðÞ¼ <sup>ð</sup>

�∞

�μ<sup>t</sup> � � <sup>þ</sup> <sup>e</sup>

Hence, it is possible to write the relationship for the total probability of the occurrence of both

2 4

occurrence within the range of 0ð Þ ; t .

Q1ðÞ¼ t

types of catastrophic damage in the time interval 0ð Þ ; t :

where B tð Þ and A tð Þ specific relationships (64) and (65).

vided in Figure 6.

where

Ek—diagnostic parameter value states.

Figure 6. Discretisation diagram of the diagnostic parameter.

ðt

0 μe �μt ð ∞

Q tðÞ¼ 1 � e

R tðÞ¼ e

�μt z ðd

�∞

7. Model outline of the catastrophic damage occurrence with the increasing chance of its occurrence together with the diagnostic parameter increase

In order to solve the problem, The Yule's process will be used by carrying out its modification. The method of this modification is provided in Ref. [6]. In this case, it is necessary to perform the diagnostic parameter value discretisation. The discretisation method is pro-

λΔt—probability of the aircraft flight, as result of which a change in the state may occur.

The formula for the aircraft reliability adopts the following form:

By using relationship (62), it is possible to determine the additional catastrophic damage

u zð Þ ; t dz

�μt ð ∞

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

zd

�ð Þ <sup>z</sup>�B tð Þ <sup>2</sup>

3

5dt ¼ 1 � e

a tð Þdt (65)

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55

Probabilistic Methods for Damage Assessment in Aviation Technology

�μ<sup>t</sup> (66)

u zð Þ ; t dz (67)

<sup>2</sup>A tð Þ (68)

where

P—probability of the occurrence of this type of damage in a single aircraft flight;

Δt—time interval, in which the flight is to take place;

μ—additional damage intensity.

Other necessary parameters and magnitudes in this point will be the same as in point 4. The differential equation, in order to describe an increase in the value of the diagnostic parameter changes, adopts the following form (in the function notation):

$$u(z, t + \Delta t) = (1 - \lambda \Delta t)(1 - P)u(z, t) + \lambda \Delta t (1 - P)u(z - \Delta z, t) \tag{58}$$

From Eq. (58) after transformation, the following partial differential equation is obtained:

$$\frac{\partial u(z,t)}{\partial t} = -\mu u(z,t) - b(t) \frac{\partial u(z,t)}{\partial z} + \frac{1}{2} a(t) \frac{\partial^2 u(z,t)}{\partial z^2} \tag{59}$$

where

b tð Þ—average increase in the diagnostic parameter per time unit;

$$b(t) = \lambda (1 - P)\Delta z \tag{60}$$

a tð Þ—average increase in the diagnostic parameter's current value per time unit;

$$a(t) = \lambda (1 - P) \Delta z^2 \tag{61}$$

Δz—increase in the diagnostic parameter value during one flight (determined with the use of accuracy of changes in this parameter).

In Ref. [2], it was shown that the equation solution (59) adopts the following form:

$$
\mu(z,t) = \mu e^{-\mu t} \overline{\mu}(z,t) \tag{62}
$$

where

$$\overline{u}(z,t) = \frac{1}{\sqrt{2\pi A(t)}} e^{-\frac{(z-R(t))^2}{2A(t)}} \tag{63}$$

$$B(t) = \int\_{0}^{t} b(t)dt\tag{64}$$

Probabilistic Methods for Damage Assessment in Aviation Technology http://dx.doi.org/10.5772/intechopen.72317 55

$$A(t) = \int a(t)dt\tag{65}$$

By using relationship (62), it is possible to determine the additional catastrophic damage occurrence within the range of 0ð Þ ; t .

$$Q\_1(t) = \int\_0^t \mu e^{-\mu t} \left[ \int\_{-\infty}^\infty \overline{u}(z, t) dz \right] dt = 1 - e^{-\mu t} \tag{66}$$

Hence, it is possible to write the relationship for the total probability of the occurrence of both types of catastrophic damage in the time interval 0ð Þ ; t :

$$Q(t) = \left(1 - e^{-\mu t}\right) + e^{-\mu t} \int\_{z\_d}^{\infty} \overline{u}(z, t) dz \tag{67}$$

The formula for the aircraft reliability adopts the following form:

$$R(t) = e^{-\mu t} \int\_{-\infty}^{z\_d} \frac{1}{\sqrt{2\pi A(t)}} e^{-\frac{(z - R(t))^2}{2A(t)}} \tag{68}$$

where B tð Þ and A tð Þ specific relationships (64) and (65).

## 7. Model outline of the catastrophic damage occurrence with the increasing chance of its occurrence together with the diagnostic parameter increase

In order to solve the problem, The Yule's process will be used by carrying out its modification. The method of this modification is provided in Ref. [6]. In this case, it is necessary to perform the diagnostic parameter value discretisation. The discretisation method is provided in Figure 6.

Figure 6. Discretisation diagram of the diagnostic parameter.

where

damage), possible to be implemented in every moment of the aircraft operation, is added to the

Additionally, the intensity of the occurrence of this type of additional damage will be intro-

Other necessary parameters and magnitudes in this point will be the same as in point 4. The differential equation, in order to describe an increase in the value of the diagnostic parameter

From Eq. (58) after transformation, the following partial differential equation is obtained:

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

a tð Þ—average increase in the diagnostic parameter's current value per time unit;

In Ref. [2], it was shown that the equation solution (59) adopts the following form:

u zð Þ¼ ; t

u zð Þ¼ ; t μe

B tðÞ¼

Δz—increase in the diagnostic parameter value during one flight (determined with the use of

�μt

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

ðt

0

�ð Þ <sup>z</sup>�B tð Þ <sup>2</sup>

u zð Þ¼ ; t þ Δt ð Þ 1 � λΔt ð Þ 1 � P u zð Þþ ; t λΔtð Þ 1 � P u zð Þ � Δz; t (58)

þ 1 2 a tð Þ <sup>∂</sup><sup>2</sup>

∂z

) P ¼ μΔt ≤ 1 (57)

u zð Þ ; t

b tðÞ¼ λð Þ 1 � P Δz (60)

a tðÞ¼ <sup>λ</sup>ð Þ <sup>1</sup> � <sup>P</sup> <sup>Δ</sup>z<sup>2</sup> (61)

u zð Þ ; t (62)

<sup>2</sup>A tð Þ (63)

b tð Þdt (64)

<sup>∂</sup>z<sup>2</sup> (59)

<sup>μ</sup> <sup>¼</sup> <sup>P</sup> Δt

P—probability of the occurrence of this type of damage in a single aircraft flight;

Δt—time interval, in which the flight is to take place;

∂u zð Þ ; t

accuracy of changes in this parameter).

changes, adopts the following form (in the function notation):

b tð Þ—average increase in the diagnostic parameter per time unit;

μ—additional damage intensity.

previous one.

54 System of System Failures

duced:

where

where

where

Ek—diagnostic parameter value states.

λΔt—probability of the aircraft flight, as result of which a change in the state may occur.

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

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

May Pkð Þt mean the probability that in the moment of t, the diagnostic parameter value achieved the state Ek (where k ¼ 1, 2, …).

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

$$\begin{aligned} P\_0(t + \Delta t) &= P\_0(t) \left[ 1 - \left( \mu\_0 + \lambda \right) \Delta t \right] + 0(\Delta t) \\ \vdots & \quad \text{for } k = 1, 2, \dots \\ P\_k(t + \Delta t) &= P\_k(t) \left[ 1 - \left( \mu\_0 + k\mu + \lambda \right) \Delta t \right] + P\_{k-1}(t) \lambda \Delta t + 0(\Delta t) \end{aligned} \tag{69}$$

After division of both sides of k equation by Δt and transition to the border at Δt ! 0, the following system of equations is obtained:

$$\begin{aligned} P\_0'(t) &= - (\mu\_0 + \lambda) P\_0(t) \\ \vdots & \quad \text{for } k = 1, 2, \dots \\ P\_k'(t) &= - (\mu\_0 + \lambda + k\mu) P\_k(t) + \lambda P\_k(t) \end{aligned} \tag{70}$$

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

$$P\_i(0) = \begin{cases} 1 & dla \text{ } i = 0\\ 0 & dla \text{ } i \neq 0 \end{cases} \tag{71}$$

8. Outline of the aircraft reliability assessment method taking into

It is assumed that the aircraft operation is done in such a way that the following arrangements

1. In order to assess the technical condition, n diagnostic parameters are used. So the techni-

2. Instead of the diagnostic parameter values in the reliability assessment, the following

<sup>i</sup> . If 0 ≤ zi < zd

When at least one deviation exceeds the limit value, the aircraft is considered to be unfit

5. It is assumed that zi ð Þ i ¼ 1; 2;…; n deviations are independent random variables, that is, a change in the value of one deviation does not change the value of other deviation.

6. The change in zi deviation values occurs as a result of the aircraft operation, which is

7. The speed of changes in the deviation values can be described with the use of the

ci—indicators characterising the local operating conditions of elements, which the increase

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

dzi

in the diagnostic parameter's deviation value depends on;

x ¼ ð Þ x1; x2; …; xn (76)

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57

<sup>i</sup> ð Þ i ¼ 1; 2; …; n , the aircraft is operational.

dN <sup>¼</sup> g zi ð Þ ; ci (78)

zi ¼ xi � xi nom ð Þ i ¼ 1; 2;…; n (77)

8.1. Description of operation conditions and adoption of assumptions

account signalled and catastrophic damage

cal condition vector adopts the following form [7]:

and assumptions are correct:

deviations are used:

xi—i diagnostic parameter.

4. The deviation limit values are zd

during the aircraft flight.

following relationship:

zi—diagnostic parameter deviation;

N—number of aircraft flights.

where

for operation.

xi nom—nominal value of i parameter.

3. Deviation values zi (i = 1, 2,…, n) are positive.

where

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 relationships Pkð Þt . Hence

$$R(t) = \sum\_{k=0}^{\infty} P\_k(t) \tag{72}$$

The probability of the fact that to the moment of t, the catastrophic damage will occur, can be specified by the following relationship:

$$Q(t) = P\{T \le t\} = 1 - R(t) \tag{73}$$

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

$$Q(t) = 1 - e^{\frac{\lambda}{\mu} - \left(\lambda + \mu\_0\right)t - \frac{\lambda}{\mu}e^{-\mu t}}\tag{74}$$

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

$$f(t) = \left[\mu\_0 + \lambda\left(1 - e^{-\mu t}\right)\right]e^{\frac{\lambda}{\mu}(1 - e^{-\mu t}) - \left(\lambda + \mu\_0\right)t} \tag{75}$$
