4. Estimation of the average number of the aircraft failure within a given period

A quantitative description and probability evaluation of damage to the basic and protection systems of the aircraft can be carried out in accordance with the postulates of the Poisson process [4].

Assuming that:


The following system of equations is right:

$$\begin{aligned} P\_0(t + \Delta t) &= P\_0(t) \ (1 - \lambda N(t)\Delta t) \\ P\_1(t + \Delta t) &= P\_1(t) \ (1 - \lambda N(t)\Delta t) + P\_0(t)\lambda N(t)\Delta t \\ &\vdots \\ P\_n(t + \Delta t) &= P\_n(t) \ (1 - \lambda N(t)\Delta t) + P\_{n-1}(t)\lambda N(t)\Delta t \\ \text{for } n &> 0 \end{aligned} \tag{37}$$

where

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

48 System of System Failures

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

and the number of operated aircraft;

period

process [4].

Assuming that:

R tðÞ¼ <sup>1</sup> � <sup>1</sup> � exp � <sup>K</sup>

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

ters p and λ of both functions are the same, and their value is p = λ = 0.1.

• proportionality factor identifying the risk of damage is constant;

E Tð Þ

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 parame-

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.

4. Estimation of the average number of the aircraft failure within a given

A quantitative description and probability evaluation of damage to the basic and protection systems of the aircraft can be carried out in accordance with the postulates of the Poisson

• probability of damage is directly proportional to the length of the concerned time period

<sup>¼</sup> exp � <sup>K</sup>

R tðÞ¼ <sup>1</sup> � <sup>1</sup> � exp ð Þ �λ<sup>t</sup> <sup>¼</sup> exp ð Þ �λ<sup>t</sup> (36)

E Tð Þ 

(35)

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

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

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

λ—proportionality factor that represents the damage risk;

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

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

$$\begin{aligned} P\_0 &= -\lambda \mathcal{N}(t) P\_0(t) \\ P\_1' &= -\lambda \mathcal{N}(t) P\_1(t) + \lambda \mathcal{N}(t) P\_0(t) \\ &\vdots \\ P\_n' &= -\lambda \mathcal{N}(t) P\_n(t) + \lambda \mathcal{N}(t) P\_{n-1}(t) \end{aligned} \tag{38}$$

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

$$\begin{cases} P\_0(0) = 1 \\ P\_n(0) = 0 \end{cases} \begin{cases} \\ \\ \qquad \text{for } n > 0 \end{cases} \tag{39}$$

Equation (38) is a linear differential equation and it is solved recursively. First, P0ð Þt is found. By having the knowledge of P0ð Þt , then, P1ð Þt is determined, and so on.

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

$$\begin{cases} \begin{array}{c} \begin{array}{c} \begin{array}{c} t \\ \end{array} N(t)dt \end{array} \\ P\_0(t) = e \end{array} \\ \begin{array}{c} \begin{array}{c} \vdots \\ \end{array} \\ P\_n(t) = \frac{1}{n!} \left[ \begin{array}{c} t \\ \end{array} N(t)dt \right] \end{array} \end{cases} \tag{40}$$

The probability that n damage requiring the launch of protection systems will occur in the time interval (0,t) is described with the Poisson distribution, whereas the role of the expression 'λt' is replaced with the following magnitude λ Ðt 0 N tð Þdt due to the low frequency of the occurrence of this type of damage in the process of the aircraft operation.

The integral <sup>Ð</sup><sup>t</sup> 0 N tð Þdt can be replaced with the following total:

$$\int\_{0}^{t} N(t)dt \leftrightarrow \sum\_{i=1}^{N} t\_{i} \tag{41}$$

E n½ �¼ <sup>X</sup><sup>∞</sup>

ti—flying time within a given period of i aircraft

the observation, the following was obtained:

• in the interval (0, t1), n<sup>1</sup> of damage occurred; • in the interval (t1, t2), n<sup>2</sup> of damage occurred;

• in the interval (ti�1, ti), n<sup>i</sup> of damage occurred;

<sup>L</sup> <sup>¼</sup> ð Þ <sup>λ</sup>T<sup>1</sup> <sup>n</sup><sup>1</sup>

<sup>¼</sup> <sup>λ</sup><sup>n</sup>1þn2þ…þni

ni! <sup>e</sup>

equation in this manner, it is possible to find the relationship for λ.

maximum likelihood method is determined.

N—number of operated aircraft.

⋮

where

where

Hence

n¼1

We are often interested not only in the probability that n damage will occur for given flying time, but in the magnitude of λ coefficient characterising the intensity (risk) of the damage occurrence. In order to determine the estimator of λ parameter, a maximum likelihood method will be used. It should be supposed that we observed and recorded the occurrence of damage in several separate time intervals, when the aircraft's flying time was: t1, t2,…, ti. As a result of

The probability of the occurrence of the said number of damage, that is, n1+ n<sup>2</sup> +…+ n<sup>i</sup> during

ð Þ <sup>λ</sup>T<sup>2</sup> <sup>n</sup><sup>2</sup> <sup>n</sup>2! <sup>e</sup>

The above recorded probability, considered as a function of λ variable, at defined n1, n2, …, ni, T1,T2, …Ti is called the likelihood. Currently, such a value of λ, for which L likelihood adopts the greatest value, is found. For this purpose, relationship (47) is subjected to logarithms and a derivative in relation to λ, which is equated to zero, is calculated. By solving the obtained

> <sup>λ</sup><sup>b</sup> <sup>¼</sup> <sup>n</sup><sup>1</sup> <sup>þ</sup> <sup>n</sup><sup>2</sup> <sup>þ</sup> … <sup>þ</sup> ni T<sup>1</sup> þ T<sup>2</sup> þ … þ Ti

With the help of the relationship (49), the estimator of λ coefficient with the use of the

<sup>2</sup> … <sup>T</sup>ni i

�λT2<sup>⋯</sup> ð Þ <sup>λ</sup>Ti ni

�λð Þ T1þT2þ…þTi

ni! <sup>e</sup>

Ti ¼ ti � ti�<sup>1</sup> (48)

�λT<sup>1</sup>

(47)

(49)

operation with the intensity of their occurrence of λ is expressed by the relationship:

<sup>n</sup>1!n2! …ni! <sup>e</sup>

�λT<sup>1</sup> �

T<sup>n</sup><sup>1</sup> <sup>1</sup> <sup>T</sup><sup>n</sup><sup>2</sup> nPnðÞ¼ <sup>t</sup> <sup>λ</sup><sup>b</sup> <sup>X</sup>

N

Probabilistic Methods for Damage Assessment in Aviation Technology

ti (46)

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

51

i¼1

where

N—number of aircraft operated within the considered time;

ti—flying time of the aircraft within the considered time.

For a single aircraft, the probability of the damage occurrence during the considered t flying time will be:

$$q\_1 = 1 - e^{-\lambda t} \tag{42}$$

where

q <sup>1</sup>—probability of damage in one aircraft;

t—aircraft's flying time.

Since λ risk of damage to both systems (basic and protection) that causes the failure is low, the expression e�λ<sup>t</sup> can be expanded into a power series.

Hence

$$e^{-\lambda t} \cong \ 1 - \lambda t \tag{43}$$

By substituting Eq. (43) to (42), the following is obtained:

$$\mathcal{V}\_2 \cong \mathcal{U} \tag{44}$$

With the relationship (44), it is possible to estimate the probability of the failure occurrence in a single aircraft.

The probability of correct aircraft functioning is expressed by the following relationship:

$$P\_1 = 1 - \lambda t\tag{45}$$

In order to estimate the average number of failures during a given period for the operated aircraft park, the following relationship can be used:

Probabilistic Methods for Damage Assessment in Aviation Technology http://dx.doi.org/10.5772/intechopen.72317 51

$$E[n] = \sum\_{n=1}^{\infty} nP\_n(t) = \hat{\lambda} \sum\_{i=1}^{N} t\_i \tag{46}$$

where

The probability that n damage requiring the launch of protection systems will occur in the time interval (0,t) is described with the Poisson distribution, whereas the role of the expression 'λt'

N tð Þdt \$ <sup>X</sup>

For a single aircraft, the probability of the damage occurrence during the considered t flying

q<sup>1</sup> ¼ 1 � e

Since λ risk of damage to both systems (basic and protection) that causes the failure is low, the

e

q

The probability of correct aircraft functioning is expressed by the following relationship:

With the relationship (44), it is possible to estimate the probability of the failure occurrence in a

In order to estimate the average number of failures during a given period for the operated

N

i¼1

N tð Þdt due to the low frequency of the occurrence

ti (41)

�λ<sup>t</sup> (42)

�λ<sup>t</sup> ffi <sup>1</sup> � <sup>λ</sup><sup>t</sup> (43)

<sup>2</sup> ffi lt (44)

P<sup>1</sup> ¼ 1 � λbt (45)

Ðt 0

is replaced with the following magnitude λ

The integral <sup>Ð</sup><sup>t</sup>

50 System of System Failures

where

time will be:

t—aircraft's flying time.

where q

Hence

single aircraft.

0

of this type of damage in the process of the aircraft operation.

N—number of aircraft operated within the considered time;

ti—flying time of the aircraft within the considered time.

<sup>1</sup>—probability of damage in one aircraft;

expression e�λ<sup>t</sup> can be expanded into a power series.

By substituting Eq. (43) to (42), the following is obtained:

aircraft park, the following relationship can be used:

N tð Þdt can be replaced with the following total:

ðt

0

ti—flying time within a given period of i aircraft

N—number of operated aircraft.

⋮

We are often interested not only in the probability that n damage will occur for given flying time, but in the magnitude of λ coefficient characterising the intensity (risk) of the damage occurrence. In order to determine the estimator of λ parameter, a maximum likelihood method will be used. It should be supposed that we observed and recorded the occurrence of damage in several separate time intervals, when the aircraft's flying time was: t1, t2,…, ti. As a result of the observation, the following was obtained:


The probability of the occurrence of the said number of damage, that is, n1+ n<sup>2</sup> +…+ n<sup>i</sup> during operation with the intensity of their occurrence of λ is expressed by the relationship:

$$\begin{array}{rcl} L & = & \frac{\left(\lambda T\_1\right)^{n\_1}}{n\_i!} \quad e^{-\lambda T\_1} \cdot \frac{\left(\lambda T\_2\right)^{n\_2}}{n\_2!} \quad e^{-\lambda T\_2} \cdots & \frac{\left(\lambda T\_i\right)^{n\_i}}{n\_i!} \quad e^{-\lambda T\_1} \\ & = & \frac{\lambda^{n\_1 + n\_2 + \ldots + n\_i} T\_1^{n\_1} T\_2^{n\_2} \cdots}{n\_1! n\_2! \ldots n\_i!} \quad e^{-\lambda \left(T\_1 + T\_2 + \ldots + T\_i\right)} \end{array} \tag{47}$$

where

$$T\_i = t\_i - t\_{i-1} \tag{48}$$

The above recorded probability, considered as a function of λ variable, at defined n1, n2, …, ni, T1,T2, …Ti is called the likelihood. Currently, such a value of λ, for which L likelihood adopts the greatest value, is found. For this purpose, relationship (47) is subjected to logarithms and a derivative in relation to λ, which is equated to zero, is calculated. By solving the obtained equation in this manner, it is possible to find the relationship for λ.

Hence

$$\widehat{\lambda} = \frac{n\_1 + n\_2 + \dots + n\_i}{T\_1 + T\_2 + \dots + T\_i} \tag{49}$$

With the help of the relationship (49), the estimator of λ coefficient with the use of the maximum likelihood method is determined.

Hence, relationship (47) takes the following form:

$$
\widehat{q} = \widehat{\lambda}t\tag{50}
$$

Eq. (51) in the function notation adopts the following form:

u zð Þ ; t —density function of the diagnostic parameter z at the time of t.

u zð Þ¼ ; t

solving this equation, the following density function is obtained:

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

zd—diagnostic parameter value specifying the limit state.

determined after transformation of relationship (54) as follows [5]:

be presented in the following way:

a—average increase square of the diagnostic parameter per time unit;

Q tð Þ¼ ; zd

After taking into account the physics of the diagnostic parameter increase and appropriate transformation, the Fokker-Planck differential equation is obtained from Eq. (52). As a result of

> 1 ffiffiffiffiffiffiffiffiffi <sup>2</sup>πat <sup>p</sup> <sup>e</sup>

The probability of the catastrophic damage occurrence with the use of the relationship (53) can

1 ffiffiffiffiffiffiffiffiffi <sup>2</sup>πat <sup>p</sup> <sup>e</sup>

ð ∞

zd

The risk level of the catastrophic damage occurrence in the operating time function can be

ðt

0 f tð Þzd

1 ffiffiffiffiffiffiffiffiffi <sup>2</sup>πat <sup>p</sup> <sup>e</sup>

� <sup>z</sup>ð Þ <sup>d</sup>�bt <sup>2</sup>

Q tð Þzd ¼

f tð Þzd <sup>¼</sup> zd <sup>þ</sup> bt 2t

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

�ð Þ <sup>z</sup>�bt <sup>2</sup>

�ð Þ <sup>z</sup>�bt <sup>2</sup>

<sup>2</sup>at (53)

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

53

<sup>2</sup>at dz (54)

dt (55)

<sup>2</sup>at (56)

where

where

where

where

u zð Þ¼ ; t þ Δt ð Þ 1 � λΔt u zð Þþ ; t λΔtu zð Þ � Δz; t (52)

Probabilistic Methods for Damage Assessment in Aviation Technology

where

t—aircraft's flying time within the year.

Relationship (50) makes it possible to estimate the probability of the damage occurrence in a single aircraft within a given time interval.
