Handbook of Reliability, Availability, Maintainability and Safety in Engineering Design - Part 38 ppsx

The measures used in maintainability analysis, besides the widely used mean time to repair MTTR, include concepts related mainly to maintenance, such as the expected mean preventive main

The following assumptions are associated with this multi-state model:

• System failures are statistically independent.

• A partially, or fully failed system is restored to a ‘good as new’ state.

• System failure rates are constant.

• System component failure times are random.

• The partially failed system repair rate is constant.

• Failed system repair times are arbitrarily distributed.

As with the two-state Markov model, the mathematical expressions for the multi-state Markov model, including supplementary variables indicating partial operation

or a reduced efficiency of the system, are given in the following Markov multi-state model equations, according to Fig 4.8:

P0(t +Δt ) = P0(t)(1 −λ1Δt )(1 −λ2Δt ) + P1(t)μpΔt (4.70)

+

⎡

⎣

∞

 0

P2(x,t)μf(x)dx

⎤

⎦Δt

P1(t +Δt ) = P1(t)(1 −λ3Δt )(1 −μpΔt ) + P0(t)λ1Δt (4.71)

P2(x +Δt;t+Δt ) = P2(x,t)[1 −μf(x)Δt] (4.72)

λj is the jth constant failure rate of the system with j= 1 (normal–partial

transition), j = 2 (normal to failed), j = 3 (partial to failed),

μp is the system constant repair rate from the partial operating state 1 to

the normal operating state 0,

μf(x) is the repair rate when the system is in the failed state and has the

elapsed repair time of x,

P0(t +Δt) is the probability that the system is in an operating state 0 at time t+Δt,

P1(t +Δt) is the probability that the system is in a partially failed state 1 at time

t+Δt,

P2(x +Δt;t+Δt ) is the probability that at time t, the system is in a failed state 2

and the elapsed repair time lies in the interval(x,x +Δx),

P0(t) is the probability that the system is in an operating state 0 at time t,

P1(t) is the probability that the system is in a partially failed state 1 at time t,

P2(x,t) is the probability that the system is in a failed state 2 after an elapsed

repair time of x,

(1 −λiΔ t) is the probability of no failure in time intervalΔt when the system is

in state i,

(1 −μpΔt) is the probability of no repair in time intervalΔt when the system is in

state 1,

(1 −μfΔt) is the probability of no repair in time intervalΔt when the system is in

state 2

The respective boundary and initial conditions are:

P (0,t) =λ P (t) +λ P (t)

and at t= 0

P0(0) = 1

P2(0) = 0

P2(x,0) = 0

The differential-difference equations with variable coefficients are

dP0(t)

dt + (λ1+λ2)P0(t) − P1(t)μp=

∞

 0

P2(x,t)μf(x)dx (4.73)

dP1(t)

dt + (λ3+μp)P1(t) − P0(t)λ1= 0 (4.74)

∂P2(x,t)

∂x +∂P2(x,t)

∂t +μf(x)P2(x,t) = 0 (4.75)

So far, the supplementary variable technique has been used to obtain the model’s partial differential-difference equations, or state equations, which describe the be-haviour of the system With the help of Laplace transforms, both transient and steady-state solutions for these state equations may now be found The Laplace transform of a function is given by the expression

E (t) =

∞

 0

Using Laplace transforms, and initial condition P0(0) = 1, the differential Eqs (4.73)

to (4.75) are transformed into steady-state solutions for these state equations, with the boundary condition of:

P2(0,s) =λ2P0(s) +λ3P1(s)

Then

sP0(s) − 1 + (λ1+λ2)P0(s) − P1(s)μp=

∞

 0

P2(x,s)μf(x)dx (4.77)

and

sP1(s) + (λ3+μp)P1(s) − P0(s)λ1= 0 (4.78) and

∂P2(x,s)

∂x + [s +μf(x)]P2(x,s) = 0 (4.79)

The steady-state values for P0(s), P1(s) and P2(s) can now be found through

inte-grating The steady-state solutions are independent of the type of waiting time and

repair time distributions, and only the expected values of these distributions become

apparent Furthermore, steady state is achieved under general conditions, and the solutions for steady state can be found without any exact knowledge about the dis-tributions of the system (Virtanen 1975)

4.2.2.2 Achieved Availability Modelling Subject to Maintenance

Achieved availability (A´s) is frequently used during development testing and initial production testing when a system or its equipment is not operating in its intended support environment Excluded are operator before-and-after maintenance checks and standby periods Achieved availability is much more of a system

hardware-oriented measure than is operational availability, which considers operating

envi-ronment factors

It is, however, dependent on a preventive maintenance policy, which can be

greatly influenced by non-hardware considerations The mathematical model for

achieved availability, according to the USA Department of Defence, is given by the

following expression (Eq 4.80), (Conlon et al 1982):

where:

OT = operating time

TCM= total corrective maintenance

TPM= total preventive maintenance

An alternative approach to modelling achieved availability is to consider the prob-ability that a system or its equipment, when used under designed conditions in an ideal support environment, will perform according to the specifications formulated

during the preliminary design phase The most significant characteristic of achieved

availability for both alternatives is that it includes maintenance time (corrective and preventive), and excludes logistic delay times The mathematical model for achieved availability in this context is given as (Dhillon 1999b):

where:

MTBM is the mean time between maintenance

This differs from inherent availability, A i, only in its inclusion of the

considera-tion for total preventive maintenance The measurement base for MTBM must be consistent when calculating achieved availability A MTBM is represented by the

following expression

MTBM=

 1

MTBPM

(4.82)

where:

MTBF is the mean time between failures

MTBM-LD is the mean time between maintenance less logistic delays

MTBPM is the mean time between preventive maintenance

The measurement base for MTBF, MTBM–LD and MTBPM must be consistent when calculating the MTBM parameter Consider further the values TCM and TPM

where:

MDT= mean active maintenance downtime

TCM= total corrective maintenance

TPM = total preventive maintenance

and

MDT=∑m i=1CM i CF i

∑m

i=1CF i +∑

n

j=1PM j PF j

∑n

j=1PF j

(4.84)

where:

n = total corrective tasks performed

m = total preventive tasks performed

CM i = elapsed time for corrective task i

PM j = elapsed time for preventive task j

CF i = estimated frequency for task i

PF j = estimated frequency for task j.

4.2.2.3 Maintainability Assessment with Maintenance Modelling

Maintainability and maintenance are closely interrelated, yet they are not the same Maintainability refers to the measures taken during the design, development and installation of a system or its equipment that will reduce the required maintenance effort, logistics and costs and, thus, also the operational downtime Maintenance refers to the measures taken to restore and keep the system or its equipment in an operable condition Maintenance is, in effect, the care of the physical and opera-tional condition of the system or its equipment Many mathematical models have been developed for both maintainability and for maintenance

However, maintenance models have mainly been developed to better define and predict certain aspects of maintenance, such as scheduled downtime, scheduled replacement, and optimal warranty periods, for installed systems and equipment

These models are usually based on certain probability distributions, predominantly the exponential distribution for representing corrective maintenance times, and the lognormal distribution for representing minimum operating times

a) Impact of Maintenance Assessment on Systems Design

A widely used probability distribution in predicting the impact of designing for maintainability on systems design, based upon defining constraints on the minimum operating time below which no maintenance activity will result in downtime, is the lognormal distribution

The lognormal distribution probability density function is defined by the follow-ing relationship

fr(t) = 1 (t −θ)σ√2πe−{1/2[ln(t−θ )−β]} (4.85) where:

t = maintenance time

θ = minimum operating time

β = mean time for maintenance

σ = standard deviation of the maintenance times

An estimate of the mean time for maintenance,β, is based on an estimate of the number of shutdowns (i.e planned downtimes that have an impact on production) that are required over a specific period, such as one year This is best approached from a calculation of the average of the sum of the natural logarithms of the

indi-vidually estimated downtimes, where m is the number of shutdowns over a specific

period

The relationship for the mean time for maintenance,β, considering the estimated downtimes and the number of shutdowns, is defined as

β = (lnt1+ lnt2+ lnt3+ + lnt m )/m (4.86) The standard deviation,σ, of the estimated mean time for maintenance,β, is given by

σ=

m

∑

i=1(lnt i −β)2/(m − 1)

1/2

For the lognormal distribution, the equation for the maintainability function M (t) is

given as the following expression

M (t) =

∞

 0

M (t) = 1/σ√2π

∞



e−1/2(lnt−β) 2

dt

This maintainability function serves primarily as a design parameter in designing for maintainability, whereby it defines the expected downtime over a specified period.

The measures used in maintainability analysis, besides the widely used mean

time to repair (MTTR), include concepts related mainly to maintenance, such as the expected mean preventive maintenance downtime, the median corrective mainte-nance downtime, the expected maximum corrective maintemainte-nance downtime and the expected mean maintenance downtime.

b) Maintainability Measures and Maintenance Assessment

The expected mean preventive maintenance downtime, Tp m, is a useful parameter in design for maintainability, in that it gives an indication of the expected scheduled downtime of a system over its life cycle The objective of defining the expected

mean preventive maintenance downtime is to estimate the impact of a preventive maintenance program on the system, whereby the system and its equipment

(as-semblies and components) are to be kept at a specified design performance level Such a preventive maintenance program is to affect the point in time at which the equipment wears out or fails, resulting in system downtime

A carefully planned preventive maintenance program can help to reduce system

downtime and improve its performance On the other hand, a poorly established preventive maintenance program can have a negative impact on system operations

The expected mean preventive maintenance downtime, Tp m, is expressed by the mathematical model (Dhillon 1999b):

Tp m=∑k i=1(T pti )(F pti)

∑k

where:

T pti = the estimated lapse time for preventive maintenance task i for i = 1,2,3, ,k

F pti = the estimated frequency of preventive maintenance task i for i = 1,2,3, ,k

k = number of preventive maintenance tasks

The median corrective maintenance downtime, Tc m, is a measure of the time within which 50% of all corrective maintenance can be completed Calculation of the me-dian corrective maintenance downtime depends upon the distribution of the times for corrective maintenance

For a lognormal distribution of repair time, the median corrective maintenance downtime, Tc m, is expressed as

σ2= the variance around the mean value of the natural logarithm of repair times

For an exponential distribution of corrective maintenance repair times, the median corrective maintenance downtime, Tc m, is expressed as

μ= the repair rate, which is the reciprocal of MTTR

The expected maximum corrective maintenance downtime Tc mis a measure of the time required to complete corrective maintenance repairs at the 90th or 95th per-centiles This implies that, for example, in the case of the 95th percentile, the ex-pected maximum corrective maintenance downtime is the time within which 95% of all corrective maintenance can be completed It indicates an estimation level of sig-nificance where no more than 5% of the expected corrective maintenance will take longer than the expected maximum corrective maintenance downtime Calculation

of the expected maximum corrective maintenance downtime also depends upon the distribution of the times for corrective maintenance

The expected maximum corrective maintenance downtime with a lognormal dis-tribution of corrective maintenance times is expressed as

Tc m= antilog(tm+ kσ) (4.92) where:

tm= the mean of the logarithms of repair times

k = the value 1.28 or 1.65 for the 90th or 95th percentiles

σ = the standard deviation of the logarithms of repair times

The expected maximum corrective maintenance downtime with an exponential dis-tribution of corrective maintenance times is expressed as

where:

MTTR= the mean time to repair, given by the following formula

MTTR=∑m i=1λi T i

∑m

i=1λi

(4.94)

λi = the constant failure rate of item i = 1,2,3, ,m

T i= the corrective maintenance or repair time needed to restore item

i = 1,2,3, ,m.

The expected mean maintenance downtime, MDT, is the total time needed to restore

the system or its equipment to a specified level of performance, and to maintain it at that level of performance It includes preventive and corrective maintenance times

but not administrative and logistic delay times In this regard, it is synonymous with achieved availability that includes maintenance time (corrective and preventive) but excludes administrative and logistic delays After determining T and T , the

expected mean maintenance downtime, MDT, is given by the following relation-ship

Substituting Eqs (4.89) and (4.93) with (4.94) into Eq (4.95) gives

MDT=∑k i=1(T pti )(F pti)

∑k

i=1(F pti) +

3∑m

i=1λi T i

∑m

i=1λi

(4.96)

where:

T pti = the estimated lapse time for preventive maintenance task i for i = 1,2,3, ,k

F pti = the estimated frequency of preventive maintenance task i for i = 1,2,3, ,k.

To determine the expected mean total downtime, DT, estimates of delays

(adminis-tration and logistic) need to be added to MDT These delays are usually estimated

as fractions of MDT

4.2.2.4 Maintenance Strategies and Cost Optimisation Modelling

So far, the interrelationships of maintainability and maintenance have been consid-ered with respect to measures used in maintainability analysis that include

mainte-nance concepts, such as preventive maintemainte-nance, corrective maintemainte-nance and down-time In designing for maintainability, it is important to understand the concepts of maintenance strategies.

In designing for maintainability, the up-front establishment of cost-effective

maintenance strategies has a significant impact on the final outcome of the

engi-neering design, particularly in considering built-in-testing (BIT), online fault di-agnostics, and the application of condition monitoring A proper understanding of the basic principles of maintenance thus becomes extremely important (in fact, it becomes essential) in the engineering design process, and includes not only mainte-nance and production people but design engineers as well Once the basic principles

of maintenance are fully understood, then the more sophisticated and complex as-pects essential to cost-effective maintenance strategies can be considered These aspects include an understanding of condition monitoring, condition measurement, fault diagnostics and predictive maintenance, and how and when they should be car-ried out in order to effectively care for the physical and operational condition of the system or its equipment

Designing for maintainability is not only a consideration of the measures taken during the design, development and installation of a system that will reduce the required maintenance effort and, thus, also the operational downtime, as well as lo-gistics and costs, but it is also a provision of the required maintenance strategies that complement these measures in order to ensure the as-designed system perfor-mance and related warranty All these aspects thus need to be carefully considered and placed in their correct perspective for establishing cost-effective maintenance strategies in designing for maintainability

a) The Basic Principles of Maintenance

Maintenance can be defined as “the continuous action of caring for the condition

of equipment” By definition, the concept of condition has been brought into the understanding of maintenance Equipment condition is the operational and physical

state of equipment on which the functions of the equipment depends

In order to understand equipment condition, it thus becomes necessary to

under-stand the concept of equipment function The function of equipment is the work and properties that the equipment is designed to perform and to have There are two

basic types of equipment functions:

• Operational function

• Physical function.

The operational functions can be grouped into primary and secondary functions The primary operational function of equipment is described by defining what work

the equipment primarily does The secondary operational functions of equipment

are the other activities that the equipment also does As an example, the primary

op-erational function of a heat exchanger would be to transmit heat through conduction from a hot fluid to a cooler fluid, thereby decreasing the temperature of the hot fluid, and increasing the temperature of the cooler fluid A secondary function of a heat exchanger is to reduce the occurrence of flash vapour in the liquid line (sometimes called flash gas, arising from a sudden change of the fluid to a vapour)

The physical functions of equipment are described by defining the design con-figuration and physical properties of the equipment Referring to the previous

ex-ample, the most significant physical function of a heat exchanger is the ability to provide efficient heat transfer at high temperature through a heat transfer surface that is large enough to transfer the heat sufficiently, and that is also able to resist expansion stresses that may cause cracks and dangerous leakages

Thus, the condition of equipment as described in the definition of maintenance

can now be reviewed It can be seen that the condition of equipment is directly re-lated to the equipment’s functions There are two types of equipment conditions,

related to the functions of the equipment and called the functional states of condi-tion The two types of equipment conditions are:

• Operational condition

• Physical condition.

The operational condition of equipment relates to its operational functions, and the physical condition of equipment relates to its physical functions.

Maintenance can now be redefined as “the continuous action of caring for the operational and physical conditions of equipment”.

The next concept to consider in this definition of maintenance is the “continuous action” There are predominantly two actions in maintenance:

• Corrective action

• Preventive action.

Corrective action, by definition, is “that action necessary to rectify or set right de-fects according to a standard” Corrective action is thus that maintenance work that fixes or repairs equipment after it has failed Preventive action, by definition, is “that action serving to hinder or stop defects” Preventive action is thus that maintenance work that prevents or stops defects from occurring in equipment before it has failed.

By progressive definition, the concept of failure has been brought into the

under-standing of maintenance action Thus, in order to fully understand maintenance, it

is essential to understand the concept of failure Equipment failure has already been

defined as “the inability of the equipment to function within its specified limits of performance” There are thus two descriptions of failure:

• Functional failure

• Potential failure.

Functional failure in equipment is “the inability of the equipment to carry out the work that it was designed to perform within specified limits of performance” This

inability has qualitative gradation, depending upon the severity of functional failure There are two degrees of severity in functional failure:

• A complete or total loss of function, where the equipment cannot carry out any

work that it was designed to perform

• A partial loss of function, where the item is unable to function within specified

limits of performance

Potential failure in equipment is “the identifiable condition of the equipment, indi-cating that functional failure can be expected” Potential failure is a condition or

state of condition of the equipment Functional failure is an occurrence or incident

The definition of preventive action in maintenance can now be reviewed From

the point of view of the two descriptions of failure, preventive action in

mainte-nance is “that action serving to hinder or to stop functional or potential failures”.

Thus, preventive action in maintenance is that action serving to hinder or stop the occurrences of defects in the function of equipment through the detection of an iden-tifiable condition arising in the equipment, indicating that it is unable to carry out the work that it was designed to perform within specified limits of performance

Maintenance can thus be comprehensively defined as “the continuous correc-tive and prevencorrec-tive action of caring for the operational and physical conditions of equipment”.

The different types of maintenance In order to convert the definition of

mainte-nance into practice, it is necessary to define how corrective and preventive action in maintenance is implemented These actions in maintenance are practically

imple-mented through different types of maintenance.

There are three basic types of maintenance:

• Defect maintenance.

• Routine maintenance.

• Preventive maintenance.

In designing... design, particularly in considering built -in- testing (BIT), online fault di-agnostics, and the application of condition monitoring A proper understanding of the basic principles of maintenance thus... Modelling

So far, the interrelationships of maintainability and maintenance have been consid-ered with respect to measures used in maintainability analysis that include

mainte-nance