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Int. J. Mech. Eng. & Rob. Res. 2013 G Gurumahesh et al., 2013

RELIABILITY EVALUATION OF REPAIRABLECOMPLEX SYSTEMS AN ANALYZING FAILURE

DATA

P Venkataramana1*, G Gurumahesh1 and V Ajay1

*Corresponding Author: P Venkataramana,[email protected]

Reliability is one of the important parameters, which contributes to customer satisfaction. Soevery aspect of design and manufacturing, quality engineering and control directly influencesthe reliability of a product. As the importance of reliability is growing in the market scenario, andas the cost of maintaining the equipment is increasing, it would necessitate every organizationto focus its interest to increase the reliability of their equipment as well as their products. Beforegoing for the reliability improving acts, it is necessary and good to measure the currentperformance. As there are many more difficulties in analyzing complex systems, which haverepairable components, focus of the project work is concentrated on it. In the present workattempts is made to collecting data, identifying the data type and evaluate the reliability of somesystems. We have used for power-law for evaluate the reliability of some systems. The effort isput evaluate complex system reliability from the component reliabilities that have been evaluatedby analyzing field failure data. The analysis of reliability measurement is mainly focused onrepairable units, for which different maintenance polices are available. By using ‘Bottom-Up’approach it has been tried to evaluate the system reliability of complex repairable systems withhelp of component reliabilities. For all these problems and difficulties, a program in C-languageis written.

Keywords: Reliability evaluaton, Bottom-up approach, Failure data

INTRODUCTIONIn today’s technological world nearly everyonedepends upon the continued functioning of awide array of complex machinery andequipment for their everyday health, safety,

ISSN 2278 – 0149 www.ijmerr.comVol. 2, No. 1, January 2013

© 2013 IJMERR. All Rights Reserved

Int. J. Mech. Eng. & Rob. Res. 2013

1 Madanapalle Institute of Technology and Science, Madanapalle, Chittoor (Dist.), Andhra Pradesh, India.

mobility and economic welfare. We expect ourcars, computers, electrical appliances, lights,televisions, etc., to function whenever we needthem-day after day, year after year. When theyfail the results can be catastrophic: injury, loss

Research Paper

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of life and/or costly lawsuits can occur. Moreoften, repeated failure leads to annoyance,inconvenience and a lasting customerdissatisfaction that can play havoc with theresponsible company’s marketplace position.

It takes a long time for a company to buildup a reputation for reliability, and only a shorttime to be branded as “unreliable” aftershipping a flawed product. Continualassessment of new product reliability andongoing control of the reliability of everythingshipped are critical necessities in today’scompetitive business arena.

BASIC TERMS AND MODELSUSED FOR RELIABILITYEVALUATIONReliability theory developed apart from themainstream of probability and statistics, andwas used primarily as a tool to help nineteenthcentury maritime and life insurance companiescompute profitable rates to charge theircustomers. Even today, the terms “failure rate”and “hazard rate” are often usedinterchangeably.

The following sections will define some ofthe concepts, terms, and models we need todescribe, estimate and predict reliability

Reliability or Survival Function

The Reliability Function R(t), also known asthe Survival Function S(t), is defined by:

R(t) = S(t) = The probability a unit survivesbeyond time t.

Since a unit either fails, or survives, and oneof these two mutually exclusive alternativesmust occur, we have

R(t) = 1 – F(t), F(t) = 1 – R(t)

Calculations using R(t) often occur whenbuilding up from single components tosubsystems with many components. Forexample, if one microprocessor comes froma population with reliability function R

m(t) and

two of them are used for the CPU in a system,then the system CPU has a reliability functiongiven by

tRtR mcpu2

Survival is the ComplementaryEvent to Failure

A different approach is used for modeling therate of occurrence of failure incidences for arepairable system. In this chapter, these ratesare called repair rates (not to be confused withthe length of time for a repair, which is notdiscussed in this chapter). Time is measuredby system power-on-hours from initial turn-onat time zero, to the end of system life. Failuresoccur at given system ages and the system isrepaired to a state that may be the same asnew, or better, or worse. The frequency ofrepairs may be increasing, decreasing, orstaying at a roughly constant rate.

Let N(t) be a counting function that keepstrack of the cumulative number of failures agiven system has had from time zero to time t.N(t) is a step function that jumps up one everytime a failure occurs and stays at the new leveluntil the next failure.

Every system will have its own observedN(t) function over time. If we observed the N(t)curves for a large number of similar systemsand “averaged” these curves, we would havean estimate of M(t) = the expected number(average number) of cumulative failures bytime t for these systems.

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A non-repairable population is one for whichindividual items that fail are removedpermanently from the population. While thesystem may be repaired by replacing failedunits from either a similar or a differentpopulation, the members of the originalpopulation dwindle over time until all haveeventually failed. We begin with models anddefinitions for non-repairable populations.Repair rates for repairable populations will bedefined in a later section.

The theoretical population models used todescribe unit lifetimes are known as LifetimeDistribution Models. The population isgenerally considered to be all of the possibleunit lifetimes for all of the units that could bemanufactured based on a particular designand choice of materials and manufacturingprocess. A random sample of size n from thispopulation is the collection of failure timesobserved for a randomly selected group of nunits.

A lifetime distribution model can be anyprobability density function (or PDF) f(t)defined over the range of time from t = 0 to t= infinity. The corresponding cumulative

distribution function (or CDF) F(t) is a veryuseful function, as it gives the probability thata randomly selected unit will fail by time t.The Figure 1 is shows the relationshipbetween f(t) and F(t) and gives threedescriptions of F(t).

1. F(t) = The area under the PDF f(t) to the leftof t.

2. F(t) = The probability that a single randomlychosen new unit will fail by time t.

3. F(t) = The proportion of the entire populationthat fails by time t.

The Figure 1 also shows a shaded areaunder f(t) between the two times t

1 and t

2. This

area is [F(t2) – F(t

1)] and represents the

proportion of the population that fails betweentimes t

1 and t

2 (or the probability that a brand

new randomly chosen unit will survive to timet1 but fail before time t

2).

Note that the PDF f(t) has only non-negativevalues and eventually either becomes 0 as tincreases, or decreases towards 0. The CDFF(t) is monotonically increasing and goes from0 to 1 as t approaches infinity. In other words,the total area under the curve is always 1.

The 2-parameter Weibull distribution is anexample of a popular F(t). It has the CDF andPDF equations given by:

tt

et

ttfetF ,1

where is the ‘shape’ parameter and is ascale parameter called the characteristic life.

Censoring

When not all units on test fail we have censoreddata:

Figure 1: The Relation Between Function(f(t)) and Time

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Consider a situation in which we arereliability testing n (non repairable) units takenrandomly from a population. We areinvestigating the population to determine if itsfailure rate is acceptable. In the typical testscenario, we have a fixed time T to run the unitsto see if they survive or fail. The data obtainedare called Censored Type 1 data.

Censored Type 1 Data

During the T hours of test we observe r failures(where r can be any number from 0 to n). The(exact) failure times are t

1, t

2, ..., t

r and there

are (n – r) units that survived the entire T-hourtest without failing. Note that T is fixed inadvance and r is random, since we don’t knowhow many failures will occur until the test is run.Note also that we assume the exact times offailure are recorded when there are failures.

This type of censoring is also called “rightcensored” data since the times of failure to theright (i.e., larger than T) are missing.

Another (much less common) way to test isto decide in advance that you want to seeexactly r failure times and then test until theyoccur. For example, you might put 100 unitson test and decide you want to see at leasthalf of them fail. Then r = 50, but T is unknownuntil the 50th fail occurs. This is called CensoredType 2 data.

Censored Type 2 Data

We observe t1, t

2, ..., t

r, where r is specified in

advance. The test ends at time T = tr, and

(n - r) units have survived. Again we assume itis possible to observe the exact time of failurefor failed units.

Type 2 censoring has the significantadvantage that you know in advance how many

failure times your test will yield—this helpsenormously when planning adequate tests.However, an open-ended random test time isgenerally impractical from a management pointof view and this type of testing is rarely seen.

Readout or Interval Data

Sometimes exact times of failure are notknown; only an interval of time in which thefailure occurred is recorded. This kind of datais called Readout or Interval data and thesituation was shown in the Figure 2.

Figure 2: Basics for Analysing Failure Data

PLOTTING RELIABILITYDATAGraphical plots of reliability data are quick,useful visual tests of whether a particular modelis consistent with the observed data. The basicidea behind virtually all graphical plottingtechniques is the following: Points calculatedfrom the data are placed on speciallyconstructed graph paper and, as long as theyline up approximately on a straight line, theanalyst can conclude that the data areconsistent with the particular model the paperis designed to test.

If the reliability data consist of (possiblymulticensored) failure data from a nonrepairable population (or a repairablepopulation for which only time to the first failureis considered) then the models are life

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distribution models such as the exponential,Weibull or lognormal. If the data consist ofrepair times for a repairable system, then themodel might be the NHPP Power Law and theplot would be a Duane Plot. The kinds of plotswe will consider for failure data from non-repairable populations are:

• Probability (CDF) plots

• Hazard and Cum Hazard plots

Probability Plotting

Probability plots are simple visual ways ofsummarizing reliability data by plotting CDFestimates vs. time on specially constructedprobability paper.

Commercial papers are available for allthe typical life distribution models. One axis(some papers use the y-axis and others thex-axis, so you have to check carefully) islabeled “Time” and the other axis is labeled“Cum Percent” or “Percentile”. There arerules, independent of the model or type ofpaper, for calculating plotting positions fromthe reliability data. These only depend on thetype of censoring in the data and whetherexact times of failure are recorded or onlyreadout times.

When the points are plotted, the analyst fitsa straight line through them (either by eye, orwith the aid of a least squares fitting program).Every straight line on, say, Weibull paperuniquely corresponds to a particular Weibulllife distribution model and the same is true forlognormal or exponential paper. If the pointsfollow the line reasonably well, then the modelis consistent with the data. If it was yourpreviously chosen model, there is no reasonto question the choice. Depending on the typeof paper, there will be a simple way to find the

parameter estimates that correspond to thefitted straight line.

Plotting Positions: Censored Data (Type 1or Type 2).

At the time ti of the ith failure, we need an

estimate of the CDF (or the Cum. PopulationPercent Failure). The simplest and mostobvious estimate is just 100 i/n (with a totalof n units on test). This, however, is generallyan overestimate (i.e., biased). Various textsrecommend corrections such as 100 (i –0.5)/n or 100 i/(n + 1). Here, we recommendwhat are known as (approximate) median rankestimates: Corresponding to the time t

i of the

ith failure, use a CDF or Percentile estimate of100 (i – 0.3)/(n + 0.4).

Plotting Positions: Readout DataLet thereadout times be T

1, T

2, ..., T

k and let the

corresponding new failures recorded at eachreadout be r

1, r

2, ..., r

k. Again, there are n units on

test. Corresponding to the readout time Tj, use

a CDF or Percentile estimate ofn

rj

i i

1100

.

Plotting Positions: Multicensored Data

Hazard and Cum Hazard Plotting

Just commercial probability paper is availablefor most life distribution models for probabilityplotting of reliability data, there are also specialCum Hazard Plotting papers available formany life distribution models. These papersplot estimates for the Cum Hazard H(t

i) vs. the

time ti of the ith failure. As with probability plots,

the plotting positions are calculatedindependently of the model or paper used anda reasonable straight-line fit to the pointsconfirms that the chosen model and the dataare consistent.

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Evaluation of Reliability from the“Bottom-Up” (Component FailureMode to System Failure Rate)

This section deals with models and methodsthat apply to non-repairable components andsystems. Models for failure rates (and notrepair rates) are described.

We use the Series Model to go fromcomponents to assemblies and systems.These models assume independence and“first failure mode to reach failure causes boththe component and the system to fail”.

If some components are “in parallel”, so thatthe system can survive one (or possibly more)component failures, we have the parallel orredundant model. If an assembly has nidentical components, at least r of which mustbe working for the system to work, we havewhat is known as the r out of n model.

The standby model uses redundancy likethe parallel model, except that the redundantunit is in an off-state (not exercised) until calledupon to replace a failed unit. Complex systemscan be evaluated using the various models asbuilding blocks.

SERIES MODELThe series model is used to go from individualcomponents to the entire system, assumingthe system fails when the first component failsand all components fail or surviveindependently of one another

The Series Model is used to build up fromcomponents to sub-assemblies and systems.It only applies to non-replaceable populations(or first failures of populations of systems). Theassumptions and formulas for the SeriesModel are identical to those for the Competing

Risk Model, with the k failure modes within acomponent replaced by the n componentswithin a system.

The following 3 assumptions are needed:

1. Each component operates or failsindependently of every other one, at leastuntil the first component failure occurs.

2. The system fails when the first componentfailure occurs.

Each of the n (possibly different)components in the system has a known lifedistribution model F

i(t).

Add failure rates and multiply reliabilities inthe Series Model.

When the Series Model assumptions holdwe have: with the subscript S referring to theentire system and the subscript i referring tothe ith component.

Note that the above holds for any arbitrarycomponent life distribution models, as long as“independence” and “first component failurecauses the system to fail” both hold.

The analogy to a series circuit is useful. Theentire system has n components in series. Thesystem fails when current no longer flows andeach component operates or failsindependently of all the others. The schematicbelow shows a system with 5 components inseries “replaced” by an “equivalent” (as far asreliability is concerned) system with only onecomponent).

n

iiS tRtR

1

n

iiS tFtF

1

11

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n

iiS thth

1

• The system operates as long as at least onecomponent is still operating. System failureoccurs at the time of the last componentfailure.

The CDF for each component is known

Multiply component CDF’s to get the systemCDF for a parallel model

For a parallel model, the CDF Fs(t) for the

system is just the product of the CDF’s Fi(t) for

the components or

n

iiS tFtF

1

RS(t) and h

S(t) can be evaluated using basic

definitions, once we have FS(t).

The schematic below represents a parallelsystem with 5 components and the (reliability)equivalent 1 component system with a CDFF

s equal to the product of the 5 component

CDF’s.

Figure 3: Series System Reducedto Equivalent One Component System

Parallel or Redundant Model

The parallel model assumes all n componentsthat make up a system operate independentlyand the system works as long as at least onecomponent still works.

The opposite of a series model, for whichthe first component failure causes the systemto fail, is a parallel model for which all thecomponents have to fail before the system fails.If there are n components, any (n – 1) of themmay be considered redundant to the remainingone (even if the components are all different).When the system is turned on, all thecomponents operate until they fail. The systemreaches failure at the time of the lastcomponent failure.

The assumptions for a parallel model are:

• All components operate independently ofone another, as far as reliability isconcerned.

Figure 4: Parallel System and EquivalentSingle Component

STANDBY MODELThe Standby Model evaluates improvedreliability when backup replacements areswitched on when failures occur.

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A Standby Model refers to the case in whicha key component (or assembly) has anidentical backup component in an “off” stateuntil needed. When the original componentfails, a switch turns on the “standby” backupcomponent and the system continues tooperate.

In the simple case, assume the non-standbypart of the system has CDF F(t) and there are(n – 1) identical backup units that will operatein sequence until the last one fails. At that point,the system finally fails.

The total system lifetime is the sum of nidentically distributed random lifetimes, eachhaving CDF F(t).

Identical backup Standby model leads toconvolution formulas

In other w ords, Tn = t

1 + t

2 + ... + t

n, where

each tihas CDF F(t) and Tn has a CDF wedenote by F

n(t). This can be evaluated using

convolution formulas:

t

duutfuFtF0

2

t

nn duutfuFtF0

1

where f(t) is the PDF F'(t)

In general, convolutions are solved numerically.However, for the special case when F(t) is theexponential model, the above integrations canbe solved in closed form.

Exponential standby systems lead to a gammSpecial Case: The Exponential (or Gamma)Standby Model:

If F(t) has the exponential CDF (i.e., F(t) = 1– e–lt), then

tt etetF 12

ttetf 22

, and

!1

12

n

ettf

tn

n

and the PDF fn(t) is the well-known gamma

distribution.

Standby units are an effective way ofincreasing reliability and reducing failure rates,especially during the early stages of productlife. Their improvement effect is similar to, butgreater than, that of parallel redundancy. Thedrawback, from a practical standpoint, is theexpense of extra components that are notneeded for functionality. Exponential standbysystems lead to a gamma lifetime model.

COMPLEX SYSTEMSOften the reliability of complex systems canbe evaluated by successive applications ofSeries and/or Parallel model formulas.

Many complex systems can bediagrammed as combinations of Seriescomponents, parallel components, R out of Ncomponents and Standby components. Byusing the formulas for these models,subsystems or sections of the original systemcan be replaced by an “equivalent” singlecomponent with a known CDF or Reliabilityfunction. Proceeding like this, it may bepossible to eventually reduce the entire systemto one component with a known CDF.

Below is an example of a complex systemcomposed of both components in parallel andcomponents in series is reduced first to aseries system and finally to a one-componentsystem.

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CASE STUDYIn our project work, the thermal power plant isconsidered as “Repairable complex system”.As the thermal power plant is electricity, whichis a continuously produced product, it wouldbe somewhat difficult to define the failure. But,as the aim of any thermalpower plant is toproduce the continues flow of good quality ofelectricity, any violation to it can beconsidered as the “Failure” for the system.But it is very difficult to continuously monitorthe quality of the product and also it is acumbersome process, it can’t be defined asfailure. The fact that fluctuations in the qualityof electricity is inevitable, will also add valueto the above statement. So the failure of the

Figure 5: Complex System Reduced toEquivalent One component System

Table 1: Tripping Data of Each Unit

SystemSub

SystemComponent Sub

Component

SubComponent

ComponentSub

System System

Reliability Values

Power Plant Unit-1 Boiler Tube

Failures 0.1 166 0.49265 0.09133 0.00854 0.03543

Boiler

Leakages 0.1 11 0.3632

Ash Hopper 1

Furnace 0.1 456 0.51044

Turbine Turbine 1 1.00

Economiser 1

Generator Potential

Transformers 0.2 82233.35 0.762684 0.215

Bushes 1

Brushes 0.2 2199007 0.869

Rotor 1

Stator 1

Transmission Grid Failure 0.99 0.435

Rely 0.1 6 0.4795

Current

Transformers 0.1 4835632480 0.9167

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system is considered as the obstacle in theproduction of the electricity. The maintenancetaken by maintenance personnel may beobstacle to the production of the electricity.But as it is inevitable to maintain longer lifewithout this obstacle. It is necessary to havemeasurement technique of reliability so thateffective analysis and evaluation of thepolicies can be made to obtain economicalsolution. While evaluating the systemreliability, the past field failure data and thestructure of various subsystems need to bestudied, so the failure data hare is the trippingdata of each unit (Table 1). After identificationof basic components, subsystems and

systems, the next system is to get the fielddata. The contribution of each major andcritical sub components, which cause thefailure of the system as a whole.

CONCLUSIONWe can achieved after calculating the reliabilityof each component with respect to the systemand then using the pareto analysis, cause andeffect diagrams. This may result in quickimprovement in the reliability of the system,which may be best result obtainable by anyorganization. To obtain the effective reliabilityimprovement plan with less cost, by addingfew lines to the program, the cost effective best

Table 1 (Cont.)

SystemSub

SystemComponent Sub

Component

SubComponent

ComponentSub

System System

Reliability Values

Unit-2 Boiler Tube

Failures 0.1 271 0.5662 0.16 0.02712

Boiler

Leakages 1.0

Ash Hopper 0.1 257556.15 0.6285

Furnace 0.1 46 0.4478

Turbine Turbine 1 1.00

Economiser 0.1 6 1

Generator Generator

Conditions 0.1 36 0.324 0.1712

Potential

Transformers 0.762684

Bushes 0.3 16910.28 0.7972

Brushes 0.2 2199007 0.869

Rotor 1

Stator 1

Transmission Grid Failure 0.99 0.99

Rely 1

Current

Transformers 1

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maintenance policy is achievable. We canpredict the reliability of system based oncomponent reliability it can be assessedeffectively the component that causes the nextfailure, so that the inventory can be maintainedless, which reduces the cost. The accurateresults may be obtained when the field data iscomplete and easily assessable.

BIBLIOGRAPHY1. Charles E Ebeling (2009), An

Introduction to Reliability andMaintenance Engineering, TMH.

2. http://www.Itl.nist.gov

3. http://www.RAC.org

4. http: //www reliasoft.com

APPENDIX 1

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APPENDIX 2

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