Reliability Engineering 19 (1987) 29-39

Condition Parameter Based Approach to Calculation of Reliability Characteristics

J. Knezev i c

University of Exeter, Department of Engineering Science, Engineering Building, North Park Road, Exeter EX4 4QL, Great Britain

(Received: 14 January 1987)

ABSTRACT

The classical approach to the calculation of reliability characteristics is based on the probability distribution of time to failure. The system under consideration is accepted as a 'black box' which performs the required function until it fails. This paper presents an 'engineering' approach to the calculation of reliability characteristics which attempts to obtain the same results at the same time providing information about 'what is going on inside the box'. According to the new approach, reliability characteristics have been determined using the probability distribution of a relevant condition parameter which fully describes the condition of the system in every instant of operating time. It is applicable to those systems whose components fail gradually. This approach offers greater potential for practical application in maintenance theory. The proposed approach can also be used as a method for accelerated testing for reliability of engineering systems and their components.

'Reliability presents the ability of a system to perform required functions under stated conditions for a stated period of time. This may be expressed as a probability'. 1

1 I N T R O D U C T I O N

In reliability theory it is commonly accepted that the reliability of a system is quantitatively expressed through some characteristics such as reliability function, hazard function, mean time to failure, etc.; these are called reliability characteristics.

29 Reliability Engineering 0143-8174/87/$03,50 © Elsevier Applied Science Publishers Ltd, England, 1987. Printed in Great Britain

30 .l. K n e z e v i c

The most common approach in reliability theory to the calculation of reliability characteristics is based on the probability distribution of random variable, T, which represents the operating time of the system to failure. According to this approach, the reliability function, R(t), is defined by the following equation:

R(t) = P (T> t)= f ( t )d t (1) t

where f ( t ) is the probability density function of T. This approach considers only two possible states of the system: a state of

functioning and a state of failure. It leaves little room, if any, for the user to follow the changes in the system during operating life. The system is accepted as a 'black box' which performs the required function until it fails. Such an approach is fully satisfactory from the point of view of mathematical statistics, but from the point of view of engineering it is not, because engineers, especially maintenance engineers, would like to know 'what is going on inside the box'. It is well known that the change in the condition of the system is a real continuous process during operating life, and that a system is able to perform its required function in spite of the fact that its condition could vary within the tolerance range.

Neither the time-to-failure approach, which defines a reliability function by the above equation, nor the stress-strength approach, 2'3 which is based on the probability that stress will be less than strength, are capable of taking into account the distinction of the satisfactory condition of the system. Designers and users would like to have more of an 'engineering' approach to reliability which is based on the actual condition of the system and its change during operating time. Therefore, there is room for the following question: Is it possible to incorporate the condition of the system into a reliability calculation?

The purpose of this paper is to present recent research which gives a positive answer to the above question, i.e. which proposes a new approach to the calculation of reliability of the system which is based on the condition of its components.

2 RELATIONSHIP BETWEEN CONDITION OF THE SYSTEM AND RELIABILITY

In order to provide the answer to the above question, let us analyse the relation between all the parameters which have influence on the system's transition from a state of functioning to a state of failure and the parameters which describe its condition and performance. For this we will consider a

Calculation of reliability characteristics 31

hypothetical engineering system whose diagrammatical presentation is given in Fig. 1, where:

1(11,12 . . . . ,Ira) denote the input parameters of a system which characterise inherent differences in raw materials and manufacture, together with operating conditions such as climate, terrain, the quality of the roads, traffic intensity, the technical education of users, misusage, storage, and so on;

O(O1, O2 . . . . . O,) denote the output parameters encompassing the system's performance: speed, fuel consumption, power, voltage, current, capacity, etc., and reliability characteristics;

R(R1,R 2 . . . . . Rj) denote the input parameters of the numerous subsystems and components of which the system is composed and indicate their operating regime: load, number of cycles per unit, intensity of use, frequency, and so on;

C(C~, C2, • • •, Ck) denote the output parameters of these subsystems and components which are the indices of their condition: dimensions, weight, geometry, etc.

Note that all the parameters presented are variable during operating time, storage and transport.

It can be seen from the diagram that the desired performance of the system is conditioned by the impact of all parameters from groups 'I' and 'R'. We assume that some relationship exists between them, which can be shown by the following expression:

O =f(I , R) (2)

The symbol ~ should be understood more as an indicator which physically describes that very complex relationship between parameters than as a specific function.

It is clear that the input parameters of both the system I and the components R have an impact which brings about a change in the condition of the components as defined by the parameters 'C'. Some functional

I I

12 =

I . 1

Im l

Fig. 1.

R 1

R 2 ~ "

R. 3

--- C 1

= C 2

C i

=i[] ~ C k

Parameters which influence the state of the system.

01

0 2

= o i

o n

32 J. Knezevic

dependence exists between them, which can be described by the following expression:

C =.1(I, R) (3)

It can be concluded from this expression that the condition of the system's components is determined by both the operating condition and the operating regime of the system.

By combining eqns (2) and (3) the following equation can be obtained, which defines the functional dependence between the output parameters of the system and the output parameters of the components, thus

O = f (C) (4)

The last expression shows that there is some relationship between performance and reliability of the system and its condition which allows us to investigate the reliability of the system through the condition of its components on one hand and provides the positive answer to the above question on the other hand. It is necessary to point out again that the above parameters are functions of time; therefore, according to the above expression, the output parameters of the system in any instant of time are in some way related to its condition parameters at that instant, thus

o(0 =f(c, t) (5)

In order to express reliability of the system through the condition of its components it is necessary to determine a way of describing the condition at any instant of time and also find a way of describing the mechanism of change in condition during operating life. Solutions are suggested according to this approach in the next few sections.

3 DESCRIPTION OF THE CONDITION OF THE SYSTEM

In order to describe the condition of the engineering system or its components in any instant of operating time, let us introduce the concept of condition parameter. A condition parameter could be any characteristic which is directly or indirectly connected with the system and its per- formance, and describes the condition of the system during operating life. In every engineering system it is possible to detect several such characteristics, only some of which will satisfy the following requirements:

(a) full description of the condition of a system at every instant of operating time;

(b) continuous and monotonic change during operating time; (c) numerical definition of the condition of the system.

Calculation of reliability characteristics 33

The condition parameter which satisfies all of these requirements we will call relevant condition parameter, RCP, because its numerical value fully describes and quantifies the condition of the system at every instant of operating time.

For a system as a whole, and its components, to be capable of functioning, their relevant condition parameter must lie between certain limits defined by the initial value, RCPin, and limiting value, RCPli m. Particular numerical values for these intervals are set by design and manufacture. When this parameter goes beyond the prescribed limits the component or system begins to operate unsatisfactorily, and this qualifies as failure. In this case it is assumed that failure occurs as soon as the relevant condition parameter exceeds the limit set. Therefore, the operating life of a system is determined by the time when the relevant condition parameter reaches this limiting level.

Our next task is to determine a way of describing the mechanism of that change during operating life and incorporate it in the calculation of reliability.

4 DESCRIPTION OF MECHANISM OF C H A N G E IN CONDITION

In order to determine a way of describing the mechanism of change in condition, let us for the moment analyse the nature of these processes by observing changes of relevant condition parameters during operating time.

Studies of processes of change in condition described by the relevant condition parameter show that they are random processes because it is impossible to predetermine how they will develop. 4 A particular process may thus be expressed by a series of curves, as shown in Fig. 2, each having a given probability of occurrence, hence it can only be described by using probability theory.

Let RCP(t) denote the random function of time which describes the random process of change in condition. Changes in RCP(t) with the passage of time are conditioned both by external factors and by the course of physical processes that take place inside the system. For each individual system or component the change in RCPi(t ) has a completely random nature which can only be described in a probabilistic way. At a given instant of time, say tk, the random function RCP(t) can be described through the relevant condition parameter which can have any value between the initial value, RCPt,, and the maximum possible value, RCPmax, which can be presented thus

RCPtn < RCP(tk) < RCPma X (6)

34 .I. K n e z e v i c

R(t)

R(tk)

RCPlim

RCPin

0

Fig. 2.

r(t)=r(~

~ J

Relation between related distributions: (m) values of RCP at instant tk; (O) instances when RCP exceeds the limit value.

Therefore, in every instant of operating time the relevant condition parameter, RCE is a random variable which can only be expressed through probability function and its distribution. Generally speaking, this function can have any continuous probability distribution.

One of the possible ways of expressing the probability distribution of continuous random variables is the probability density function, f(.). The probability density function of relevant condition parameter at the instant of operating time t is denoted byf(RCP, t). There is an infinite number of such distributions corresponding to the infinite number of possible instants of operating time.

As a result of this, in practice accurate calculation of these distributions is

Calculation of reliability characteristics 35

connected with many insoluble mathematical problems. cations are therefore necessary.

4.1 Formal description of change in condition

Some simplifi-

In order to simplify the calculation of the process of change in condition during operating time we will assume that all probability density functions f(RCP, t) belong to one family of probability distribution. In this case the distributive laws of vertical intersections of random function RCP(t) during the operating time do not change. 5

Even with this assumption it is impossible in engineering practice to determine parameters which define each of these distributions in every instant of time, for obvious reasons, but it is possible to find numerical values for some of them. Knowing the distribution of RCP in several instances of time we still do not know the relation between them throughout operating time.

In order to determine that relation we will consider parameters which define these probability distributions, say Pl and P2, whose numerical values can be approximated by some time-dependent function, e.g. pl(t) = ~l(t) and p2(t) = ~bz(t). These will be called displacement functions.

As it has already been assumed the relevant condition parameter changes continuously and monotonically during operating time, it is quite sensible to assume that the random process RCP(t) and its displacement functions ~b(t) also change monotonically.

As in engineering practice it is impossible to determine p~ and P2 at every instant of time, the solution is to find as many points as possible which present the values of p~i and p2 i of f(RCP, ti) at different instances of operating time, i = 1, n. The displacement functions can be obtained quite simply by using a points diagram, on the basis of which a regression line can be drawn.

The accuracy of results obtained for a description of the mechanism of change will directly depend on how close the assumptions are to real processes of change in condition. Up to now we have proposed the following:

(1) The reliability of the system can be expressed by the condition of the system.

(2) The condition of the system at any instant of time can be described by the relevant condition parameter.

(3) The state of failure is defined by the instant of time when the relevant condition parameter exceeds its limiting level.

(4) The change in the condition of the system during operating time can be defined by random function RCP(t).

36 J. Kne_-evic

(5) The relevant condition parameter in every instant of time is a random variable which is fully defined by its probability distribution function, e.g. f ( R C E t).

(6) The relation between probability density functions during operating time is determined by displacement functions q/l(t) and ~2(t).

As a consequence of this the following statement could be made: In the case considered, the probability of the value of the relevant condition parameter being within the tolerance range at the instant of time tk is also the probability of the reliable operation of the whole system at that instant, 6 thus

P(RCP~, < RCV(tk) < R C P u m ) = R(tk) (7)

5 NEW APPROACH TO DETERMINATION OF RELIABILITY CHARACTERISTICS

In order to express the reliability function through a relevant condition parameter which represents the real condition of the system, let us consider the vertical intersection of the random process RCP(t), say at time t k (see Fig. 2). The probability that RCP(t), at instant t k, will have a value within tolerance range, i.e. not exceeding the limiting value, can be defined as

RCPIim~ P(RCPin < RCP(t k) < RCPlim) = f(RCP, tk) dRCP (8)

RCPin .J

The above equation describes the probability that the random function RCP(t) at that instant of time will have a value in the acceptable interval. However, in the case considered, the probability of the value of the relevant condition parameter being within the tolerance range at any instant of time t is also the probability of the reliable operation of the whole system at that instant (see eqn (7)), thus

RCPlirn R({) ~ gcPin J f ( R C P , t) dRCP (9)

The above equation shows that reliability function can bc obtained, taking into consideration the mechanism of change in the condition of the system.

The integral on the right side of the above equation represents the cumulative distribution function of the relevant condition parameter at instant t, F(RCP, 0, within given limits. As the numerical value of the lower limit is obviously equal to zero, eqn (9) could be rewritten thus:

{RCPlim R(t) = F(RCP, t) aCP,n = F(RCP"m' t) (10)

Calculation of reliability characteristics 37

Applying the proposed approach, other reliability characteristics could easily be determined, for example mean time to failure, MTTF, can be expressed by the following equation:

MTTF= ~f F(RCP.m,t)dt (11)

Relating this approach to the classical one, the reliability function defined by eqn (1) presents the intersection of random process RCP(t) and the limiting value of relevant condition parameter, thus

; f R(tk) = P ( T > tk) = f(t, RCP = RCP.m)dt = f(Odt (12) tk tk

where f(t, RCP = RCPIim) is the probability density function of the time when the relevant condition parameter goes beyond its limiting value, which was defined as failure (see Fig. 2).

Thus it is shown that it is possible to obtain the same numerical values for reliability characteristics using the new approach which is based on the condition of the system.

Therefore, reliability characteristics could be obtained using either the probability distribution of time to failure or probability distribution of the relevant condition parameter of the system. These two distributions are related because one determines the other. The following equation demonstrates this relationship:

d [-RCPIim /~ RCP dRCP] J J( ,') (13)

Making use ofeqns (9) and (10), the above expression can be transformed into

d f(t) = --~- [F(RCPI, m, t)] (14)

In some cases it will be easier to establish parameters px(t) and pz(t) of f(RCP, t) than parameters off(t), which means that the proposed approach can also be used as a method for accelerated testing for reliability of engineering systems and their components.

6 CONCLUSIONS

The proposed approach presents novelty in reliability theory because the numerical values of the reliability characteristics obtained are based on the actual condition of the system and its change during the operating life. Such

38 d. Knezevic

an approach provides a fuller 'picture' of the condition of the system and its components during the whole life time because it is based on continuous process of change rather than the time-to-failure approach, which is based only on the moments of the system's transition to a state of failure. This information about changes in the condition of the system is very valuable for engineers, particularly to the maintenance engineer, who can base maintenance policy and strategy on the knowledge obtained by the application of the approach here presented.

The proposed approach is, in general, applicable to all engineering systems, but it is most likely that most of them will have a mechanical basis because they are subjected to wear processes, i.e. processes with a gradual deterioration of material.

The main difficulties in practical application of this approach are the selection of relevant condition parameters and the determination of the parameters of probability distribution of RCP(t). It cannot be taken for granted that a relevant condition parameter exists in every engineering system or that it will be always possible to find the function of change in condition due to the limitations of available equipment which could limit the application of this approach.

A C K N O W L E D G E M E N T S

This study was financially supported by the Research Fund Committee of Exeter University, UK. The author would like to thank the Department of Engineering Science of the same university, and particularly Professor J. O. Flower for his personal support and Dr J. L. Henshall for his assistance, which made this paper possible.

REFERENCES

1. BS 3811, Glossary of Maintenance Terms in Terotechnology, British Standards Institution, 1974.

2. Carter, A. D. S. Mechanical Reliability, Macmillan, London, 1973. 3. Kapur, K. C. and Lamberson, L. R. Reliability in Engineering Design, John Wiley

& Sons, New York, 1977. 4. Pronikov, A. S. Dependability and Durability of Engineering Products,

Butterworths, London, 1973. 5. Smirnov, N. N. and Itskovich, A. A. Obsluzivanie i Remont Aviatsionnoi Tehniki

po Sostoianiiu, Transport, Moskva, 1980. 6. Knezevic, J. Investigation of a strategy for control of maintenance processes in

engineering systems providing required reliability, PhD (in Serbo-Croatian), University of Belgrade, 1985.

Calculation of reliability characteristics 39

APPENDIX

In order to illustrate the methodology presented for the description of random function RCP(t), let us assume that the relevant condition parameter, as random variable in every instant of operating time, obeys the three-parameter Weibull distribution, with scale parameter n, shape parameter B and location parameter RCPi,.

This distribution was chosen in preference to other theoretical probability distributions because its range fully satisfies the range of relevant condition parameter (see eqn (5)), whereas the range of others goes below RCPi., which may introduce some numerical inaccuracies.

The probability density function f(RCP, t) will then have the following form:

8(t) • (RCP -_ RCP,.~"'"- ' f(RCP, t) = [n(t) - RCP.n ] \ n(t) - RCPin )

• exp {-[RCP-RCP'nl"'"~ (A') L ~-R--e-~,.j j

Taking into account eqn (10), the reliability function will be

)" /RCP,,m - RCP, n~B't"~ R(t) = 1 - exp t - \ n-~ Z RC---P~. -) J (A2)

According to eqn (11), mean time to failure will have the following form:

-- (RCPIim - RCP|n~ ~ l M T T F = 1 - e x p \ n ( t ) - R C P , . ] j j d t (A3)