5. Catastrophic damage model of the device including the limit state

These models can be used for determination of the probability of the occurrence of various negative events in the devices for the following cases:


It is assumed that:


May Uz,t mean the probability that in the moment of t, the diagnostic parameter value will be equal to z. For example, it can be assumed that z may mean, for example, the crack length or the surface wear value.

In order to describe an increase in the parameter value in the random basis, the following differential equation was adopted:

$$
\mathcal{U}\_{z,t+\Delta t} = (1 - \lambda \Delta t)\mathcal{U}\_{z,t} + \lambda \Delta t \mathcal{U}\_{z-\Delta z,t} \tag{51}
$$

where

Δz—increase in the diagnostic parameter value during one flight of the aircraft;

λΔt—probability of the aircraft flight in the time interval of Δt, whereas λΔt ≤ 1;

λ—intensity of the aircraft flights.

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

$$u(z, t + \Delta t) = (1 - \lambda \Delta t)u(z, t) + \lambda \Delta t u(z - \Delta z, t) \tag{52}$$

where

Hence, relationship (47) takes the following form:

t—aircraft's flying time within the year.

single aircraft within a given time interval.

negative events in the devices for the following cases:

parameter, which evaluates its state;

current value is determined by z.

• The parameter z is non-decreasing.

differential equation was adopted:

λ—intensity of the aircraft flights.

variables. It is assumed that:

the surface wear value.

where

in the parameter, which evaluates its state;

where

52 System of System Failures

<sup>b</sup><sup>q</sup> <sup>¼</sup> <sup>λ</sup>b<sup>t</sup> (50)

Relationship (50) makes it possible to estimate the probability of the damage occurrence in a

These models can be used for determination of the probability of the occurrence of various

• when a chance of the catastrophic damage occurrence is constant along the increasing

• when a chance of the catastrophic damage occurrence increases together with an increase

• when the parameters determining a chance of the damage occurrence constitute random

• The device's technical condition is determined by one dominant diagnostic parameter. Its

May Uz,t mean the probability that in the moment of t, the diagnostic parameter value will be equal to z. For example, it can be assumed that z may mean, for example, the crack length or

In order to describe an increase in the parameter value in the random basis, the following

Uz,tþΔ<sup>t</sup> ¼ ð Þ 1 � λΔt Uz,t þ λΔtUz�Δz,t (51)

• A change in the diagnostic parameter value occurs only during the aircraft flight:

Δz—increase in the diagnostic parameter value during one flight of the aircraft; λΔt—probability of the aircraft flight in the time interval of Δt, whereas λΔt ≤ 1;

5. Catastrophic damage model of the device including the limit state

• when the parameter, specifying their state, will exceed the limit state;

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

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 solving this equation, the following density function is obtained:

$$
\mu(z,t) = \frac{1}{\sqrt{2\pi at}} e^{-\frac{(z-bt)^2}{2at}} \tag{53}
$$

where

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

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

The probability of the catastrophic damage occurrence with the use of the relationship (53) can be presented in the following way:

$$Q(t, z\_d) = \bigcap\_{z\_d}^{\infty} \frac{1}{\sqrt{2\pi at}} e^{-\frac{(z-bt)^2}{2at}} dz \tag{54}$$

where

zd—diagnostic parameter value specifying the limit state.

The risk level of the catastrophic damage occurrence in the operating time function can be determined after transformation of relationship (54) as follows [5]:

$$\left(Q(t)\_{z\_d} = \int\_0^t f(t)\_{z\_d} dt\right) \tag{55}$$

where

$$(f(t)\_{z\_d} = \frac{z\_d + bt}{2t} \frac{1}{\sqrt{2\pi at}} e^{-\frac{(z\_d - bt)^2}{2at}} \tag{56}$$
