Of a methOd fOr estimating the durability Of cOmpOnents Or device assemblies while maintaining the required reliability level

(*) Tekst artykułu w polskiej wersji językowej dostępny w elektronicznym wydaniu kwartalnika na stronie www.ein.org.pl ZIEJA M, WAŻNY M, STĘPIEŃ S. Outline of a method for estimating the durability of components or device assemblies while maintaining the required reliability level. Eksploatacja i Niezawodnosc – Maintenance and Reliability 2018; 20 (2): 260–266, http://dx.doi. org/10.17531/ein.2018.2.11.


Introduction
In the available literature, it is possible to find a number of papers, which demonstrate the problem of the impact of the external environment, ageing and wear processes on the technical system functioning [4,9,13,16,17,21].Due to technical advancement and a high degree of integration of the devices used on the board of military aircraft, the development of optimal operation models is a complex task.The methods for evaluating the reliability and durability of aviation equipment based on a change in diagnostic parameters are extremely useful within this area [6,7,8,12,15,20].
This paper includes a probabilistic method for evaluating the durability of components and the assemblies of the device that operates under the impact of ageing processes (corrosive, wear and other) in the aircraft devices [15,18,19].The technical condition of some aviation equipment can be assessed with the use of diagnostic parameters.This assessment requires knowledge of limit (acceptable) values, for which it is considered that the device or assembly is in the state of usability.
In the offered durability assessment model, the following assumptions are adopted: the device's technical condition is defined by one di-agnostic parameter "z" in the form of the parameter deviation from the nominal value: where: X -current value of the diagnostic parameter, X norm -nominal value of the diagnostic parameter; sciENcE aNd tEchNology change in the deviation value of the diagnostic parameter takes place in the entire operation period (operation and standstill); " -z" parameter is non-decreasing, because it is determined by the absolute value of the difference of the present and nominal values; increase speed of the diagnostic parameter deviation in case of random changes can be described by the following relationship: where: c -random variable which characterises the component's susceptibility to ageing changes depending on its features and operating conditions, t -calendar time.

Determination of the deviation density function taking into account the relationship (1)
The dynamics of changes in "z" deviation value in random terms will be characterised by the following differential equation: , z t

U
-probability of the fact that in the moment of t, the diagnostic parameter value adopts z value; P -probability of the event that the random wear occurs and that in the time interval of ∆t, the deviation value will be increased by ∆z value; ∆z -deviation increase.
In case, when P=1 equation (3) in the function notation will adopt the following form: The equation ( 4) has the following form: probability of the fact that in the moment of t +∆t, the deviation value will be z is equal to the probability of the fact that in the t moment, the deviation value was equal to z-∆z.It means that along with the probability equal to unity, in the time interval of ∆t, the deviation will be increased by ∆z value.
The equation ( 4) is transformed into the partial differential equation.Therefore, the following approximations are adopted [1,2]: 2 .(6) By using ( 5) and ( 6), the equation (4) adopts the following form: , , , 1 2 where: b=E[c] -average increase in the diagnostic parameter's deviation value per time unit; a=E[c 2 ] -average increase square of the diagnostic parameter's deviation per time unit.
We are searching for the solution of a particular equation (7), the one, which at t→0 is coergent to the so-called Dirac function, i.e.

( )
, 0 u zt → for z≠0 and ( ) 0, u t → +∞ , but in a way that the integral of u function is equal to "1" for all t>0.
The equation solution (7) adopts the following form for the above specified condition [3,11,14]: where: ( ) ( ) The value of 0 in lower limits of the integrals means the adopted initial moment of time, according to which the dynamics of changes in the diagnostic parameter's value is considered -it can be, e.g. the moment of putting a given device into operation.
The density function ( 8) of the diagnostic parameter's deviation increase can be used for assessing the reliability of the considered device component.

Determination of reliability and durability of the component or device assembly
By having a specific density function, it is possible to record the relationship on reliability and durability due to the time of the parameter's deviation increase to the limit value.The formula adopts the following form: where: ( ) , uzt -density function specified by the relationship (8); z d -acceptable value of the diagnostic parameter's deviation due to safety; t -calendar time of the device operation.
Figure 1 presents a diagram of the density function course and a way of determining the reliability and durability.
The relationship (9) taking into account (8), adopts the following form: By assuming the minimum, required value of R* reliability, it is possible to determine t * time, after which the reliability will decrease below the required level.The time t * can be treated as the durability of a given component for the required, acceptable reliability value.
In this case, it is possible to obtain: sciENcE aNd tEchNology ( ) 3. Method for estimating the durability with the use of the density function of the time exceeding the acceptable (limit) state

Determination of the time distribution of exceeding the acceptable (limit) state
The probability of exceeding the acceptable (limit) value by the diagnostic parameter with the use of the density function of changes in the diagnostic parameter's deviation ( 8) can be presented in the following form: The density function of the time distribution of the first transition beyond the acceptable value z d adopts the following form: Thus, ( ) By assuming (8) definition, it is possible to obtain: Furthermore, a derivative after the function time (8), adopts the following form: The relationship ( 16) was substituted to ( 14): The primary function for the integrand in the relationship ( 17) is searched for.It is expected that the function in the form of: is a primary function for the integrand of the relationship (17), where θ(z,t) is a sought unknown function.
That is: After transformations, the following equation is obtained: Homogeneous equation: Solution of the homogeneous equation: Thus, by calculating the integral (17) in the specified limits, it is possible to obtain: The relationship (19) determines the density function of the time of the first transition of the acceptable (limit) state by the diagnostic parameter's deviation.It should be checked, whether the function ( 19) is a density function of time of reaching the acceptable (limit) state.The function has the following form: The function (20) should meet the condition: In order to demonstrate the validity (21), the following justification is presented: In order to calculate the integral that occurs in the formula (22), the following substitution is used: Transformation of the limits of integration: After substituting to the output integral, it is possible to obtain: The above integral is an integral of N(0,1) normal distribution in the limits from -∞ to +∞ and is equal to unity.On this basis, it can be concluded that:

Evaluation of the durability of selected components of the aircraft construction with the use of the time distribution of obtaining the acceptable state
The formula for the aircraft's structural component reliability adopts the following form: where the density function f(t) is determined by the following formula (19).
However, the unreliability of the aircraft's structural component can be determined on the basis of the following relationship: The integral occurring in the relationships ( 27) and ( 28) must be transformed to the more convenient form: After changing the limits of integration, it is possible to obtain: The reliability of a given component will adopt the following form: 1 2 The integral occurring in the formula (31) is a value of N(0,1) normal distribution function for the argument occurring in the upper limit of integration.Again, by assuming the required minimum value of R * reliability, it is possible to determine t * durability.The use of ( 11) or (32) formula in the calculation requires estimation of the values of a and b coefficients.This estimation is carried out on the basis of the data obtained from the aircraft operation process.

Numerical example
In order to determine the durability of the considered component, it is important to determine (estimate) the values of a and b constants.Therefore, it is assumed that the observation of the tested device in the operation process results in the provision of data on the increase of the diagnostic parameter's deviation value in the form of: , , , , , , , , The best method for determining "b" and "a" values for the held data is a method that uses a likelihood function.Its form in the general case can be presented as the relationship: where: j=1,2,…,m; m -number of parameters characterising the wear process of a given technical object.
In this case, b * and a * estimates of unknown b and a parameters with the use of the maximum likelihood method consists in solving the system of equations: By solving the system of equations ( 29), b * and a * are found.
The component, which was chosen for a numerical example is 12-SAM-28 aircraft battery.Figure 2 shows a change in the time of the averaged battery capacity for held data.
In accordance with the relationship (1), the absolute value of the capacity difference and its nominal value were adopted as "z" diagnostic parameter.The change in time of "z" parameter was presented in Figure 3.
Thus, holding the data describing the values of the diagnostic parameter in the form of ( ) ( ) ( ) ( ) , , , , , , , , The parameter z d was determined with the use of technical documentation used for the implementation of maintenance works, in which the information on the acceptable value of the capacity of batteries was provided.Therefore, by holding the values of parameters * b ε , * a ε , z d , they were substituted to (11) or (32) equations by determining the relationship of t * time on R * probability -Figure 4. In both cases (relationship (11) or (32)) the same course was obtained.
By assuming the minimum value of R*=0.99 reliability, the time, to which the diagnostic parameter deviation will not exceed the limit state, in accordance with the assumed probability, was determined: The obtained value (39) can be used in the technical maintenance depending on the adopted strategy of maintenance.On the basis of the above methodology, it is possible to determine further periods, in which the control of the device diagnostic parameter should be carried out [5,10].

Final remarks
In this paper, an overview of the method for estimating the durability of components or assemblies, when the increase speed of changes was of random nature, was presented.However, the method of this change was described by the following simple relationship: where c was a random variable determining the possibility of the parameter's deviation increase.
It is possible to generalise this method, when the speed of the deviation increase will be described by the following relationships: In the first case, the increase speed of changes will be of random nature similar to the exponential one.In the second case, the increase nature of changes will be similar to the intensity of damage in the Weibull distribution.
In summary, it can be concluded that the presented method seems to be correct and right, and allows to analyse the device technical condition due to the nature of changes in the values of diagnostic parameters.The presented calculation example allowed to carry out the verification of the developed model, and emphasised the developed method's application advantages.This method may be useful in further works on the improvement of both the operational process and the method of using the aircraft with the use of its on-board systems, allowing for determining the time of the device's staying in the state of usability.
Furthermore, the presented method, owing to its universal nature, can be successfully used in order to specify the residual life of any technical object, the technical condition of which is determined on the basis of the analysis of the diagnostic parameters' values.
In this paper, the presented method can be further improved and extended to other cases of increase in random changes of the exponential type.It seems that it can be used for assessing the reliability of mechanical components, in case of considering the propagation of fatigue cracks in the components subjected to the random load, and in case of using the Paris formula in order to specify the crack velocity.
that the equation(18) fulfils the above solution.The general solution of the homogeneous equation: is the sought primary function of the integral(17) has the following form:

Fig. 2 .
Fig. 2. The course of changes in the averaged capacity of 12-SAM-28 battery