It. J. Mech. Eg. & Rob. Res. 2013 G Gurumahesh et al., 2013 Research Paper ISSN 2278 0149 www.ijmerr.com Vol. 2, No. 1, Jauary 2013 2013 IJMERR. All Rights Reserved RELIABILITY EVALUATION OF REPAIRABLE COMPLEX SYSTEMS AN ANALYZING FAILURE DATA P Vekataramaa 1 *, G Gurumahesh 1 ad V Ajay 1 *Correspodig Author: P Vekataramaa, veru8057@gmail.com Reliability is oe of the importat parameters, which cotributes to customer satisfactio. So every aspect of desig ad maufacturig, quality egieerig ad cotrol directly iflueces the reliability of a product. As the importace of reliability is growig i the market sceario, ad as the cost of maitaiig the equipmet is icreasig, it would ecessitate every orgaizatio to focus its iterest to icrease the reliability of their equipmet as well as their products. Before goig for the reliability improvig acts, it is ecessary ad good to measure the curret performace. As there are may more difficulties i aalyzig complex systems, which have repairable compoets, focus of the project work is cocetrated o it. I the preset work attempts is made to collectig data, idetifyig the data type ad evaluate the reliability of some systems. We have used for power-law for evaluate the reliability of some systems. The effort is put evaluate complex system reliability from the compoet reliabilities that have bee evaluated by aalyzig field failure data. The aalysis of reliability measuremet is maily focused o repairable uits, for which differet maiteace polices are available. By usig Bottom-Up approach it has bee tried to evaluate the system reliability of complex repairable systems with help of compoet reliabilities. For all these problems ad difficulties, a program i C-laguage is writte. Keywords: Reliability evaluato, Bottom-up approach, Failure data INTRODUCTION I today s techological world early everyoe depeds upo the cotiued fuctioig of a wide array of complex machiery ad equipmet for their everyday health, safety, mobility ad ecoomic welfare. We expect our cars, computers, electrical appliaces, lights, televisios, etc., to fuctio wheever we eed them-day after day, year after year. Whe they fail the results ca be catastrophic: ijury, loss 1 Madaapalle Istitute of Techology ad Sciece, Madaapalle, Chittoor (Dist.), Adhra Pradesh, Idia. 81
of life ad/or costly lawsuits ca occur. More ofte, repeated failure leads to aoyace, icoveiece ad a lastig customer dissatisfactio that ca play havoc with the resposible compay s marketplace positio. It takes a log time for a compay to build up a reputatio for reliability, ad oly a short time to be braded as ureliable after shippig a flawed product. Cotiual assessmet of ew product reliability ad ogoig cotrol of the reliability of everythig shipped are critical ecessities i today s competitive busiess area. BASIC TERMS AND MODELS USED FOR RELIABILITY EVALUATION Reliability theory developed apart from the maistream of probability ad statistics, ad was used primarily as a tool to help ieteeth cetury maritime ad life isurace compaies compute profitable rates to charge their customers. Eve today, the terms failure rate ad hazard rate are ofte used iterchageably. The followig sectios will defie some of the cocepts, terms, ad models we eed to describe, estimate ad predict reliability Reliability or Survival Fuctio The Reliability Fuctio R(t), also kow as the Survival Fuctio S(t), is defied by: R(t) = S(t) = The probability a uit survives beyod time t. Sice a uit either fails, or survives, ad oe of these two mutually exclusive alteratives must occur, we have R(t) = 1 F(t), F(t) = 1 R(t) Calculatios usig R(t) ofte occur whe buildig up from sigle compoets to subsystems with may compoets. For example, if oe microprocessor comes from a populatio with reliability fuctio R m (t) ad two of them are used for the CPU i a system, the the system CPU has a reliability fuctio give by R cpu t R 2 m t Survival is the Complemetary Evet to Failure A differet approach is used for modelig the rate of occurrece of failure icideces for a repairable system. I this chapter, these rates are called repair rates (ot to be cofused with the legth of time for a repair, which is ot discussed i this chapter). Time is measured by system power-o-hours from iitial tur-o at time zero, to the ed of system life. Failures occur at give system ages ad the system is repaired to a state that may be the same as ew, or better, or worse. The frequecy of repairs may be icreasig, decreasig, or stayig at a roughly costat rate. Let N(t) be a coutig fuctio that keeps track of the cumulative umber of failures a give system has had from time zero to time t. N(t) is a step fuctio that jumps up oe every time a failure occurs ad stays at the ew level util the ext failure. Every system will have its ow observed N(t) fuctio over time. If we observed the N(t) curves for a large umber of similar systems ad averaged these curves, we would have a estimate of M(t) = the expected umber (average umber) of cumulative failures by time t for these systems. 82
A o-repairable populatio is oe for which idividual items that fail are removed permaetly from the populatio. While the system may be repaired by replacig failed uits from either a similar or a differet populatio, the members of the origial populatio dwidle over time util all have evetually failed. We begi with models ad defiitios for o-repairable populatios. Repair rates for repairable populatios will be defied i a later sectio. The theoretical populatio models used to describe uit lifetimes are kow as Lifetime Distributio Models. The populatio is geerally cosidered to be all of the possible uit lifetimes for all of the uits that could be maufactured based o a particular desig ad choice of materials ad maufacturig process. A radom sample of size from this populatio is the collectio of failure times observed for a radomly selected group of uits. A lifetime distributio model ca be ay probability desity fuctio (or PDF) f(t) defied over the rage of time from t = 0 to t = ifiity. The correspodig cumulative Figure 1: The Relatio Betwee Fuctio (f(t)) ad Time distributio fuctio (or CDF) F(t) is a very useful fuctio, as it gives the probability that a radomly selected uit will fail by time t. The Figure 1 is shows the relatioship betwee f(t) ad F(t) ad gives three descriptios of F(t). 1. F(t) = The area uder the PDF f(t) to the left of t. 2. F(t) = The probability that a sigle radomly chose ew uit will fail by time t. 3. F(t) = The proportio of the etire populatio that fails by time t. The Figure 1 also shows a shaded area uder f(t) betwee the two times t 1 ad t 2. This area is [F(t 2 ) F(t 1 )] ad represets the proportio of the populatio that fails betwee times t 1 ad t 2 (or the probability that a brad ew radomly chose uit will survive to time t 1 but fail before time t 2 ). Note that the PDF f(t) has oly o-egative values ad evetually either becomes 0 as t icreases, or decreases towards 0. The CDF F(t) is mootoically icreasig ad goes from 0 to 1 as t approaches ifiity. I other words, the total area uder the curve is always 1. The 2-parameter Weibull distributio is a example of a popular F(t). It has the CDF ad PDF equatios give by: t t F 1 e, f t t e t t where is the shape parameter ad is a scale parameter called the characteristic life. Cesorig Whe ot all uits o test fail we have cesored data: 83
Cosider a situatio i which we are reliability testig (o repairable) uits take radomly from a populatio. W e are ivestigatig the populatio to determie if its failure rate is acceptable. I the typical test sceario, we have a fixed time T to ru the uits to see if they survive or fail. The data obtaied are called Cesored Type 1 data. Cesored Type 1 Data Durig the T hours of test we observe r failures (where r ca be ay umber from 0 to ). The (exact) failure times are t 1, t 2,..., t r ad there are ( r) uits that survived the etire T-hour test without failig. Note that T is fixed i advace ad r is radom, sice we do t kow how may failures will occur util the test is ru. Note also that we assume the exact times of failure are recorded whe there are failures. This type of cesorig is also called right cesored data sice the times of failure to the right (i.e., larger tha T) are missig. Aother (much less commo) way to test is to decide i advace that you wat to see exactly r failure times ad the test util they occur. For example, you might put 100 uits o test ad decide you wat to see at least half of them fail. The r = 50, but T is ukow util the 50 th fail occurs. This is called Cesored Type 2 data. Cesored Type 2 Data We observe t 1, t 2,..., t r, where r is specified i advace. The test eds at time T = t r, ad ( - r) uits have survived. Agai we assume it is possible to observe the exact time of failure for failed uits. Type 2 cesorig has the sigificat advatage that you kow i advace how may failure times your test will yield this helps eormously whe plaig adequate tests. However, a ope-eded radom test time is geerally impractical from a maagemet poit of view ad this type of testig is rarely see. Readout or Iterval Data Sometimes exact times of failure are ot kow; oly a iterval of time i which the failure occurred is recorded. This kid of data is called Readout or Iterval data ad the situatio was show i the Figure 2. Figure 2: Basics for Aalysig Failure Data PLOTTING RELIABILITY DATA Graphical plots of reliability data are quick, useful visual tests of whether a particular model is cosistet with the observed data. The basic idea behid virtually all graphical plottig techiques is the followig: Poits calculated from the data are placed o specially costructed graph paper ad, as log as they lie up approximately o a straight lie, the aalyst ca coclude that the data are cosistet with the particular model the paper is desiged to test. If the reliability data cosist of (possibly multicesored) failure data from a o repairable populatio (or a repairable populatio for which oly time to the first failure is cosidered) the the models are life 84
distributio models such as the expoetial, Weibull or logormal. If the data cosist of repair times for a repairable system, the the model might be the NHPP Power Law ad the plot would be a Duae Plot. The kids of plots we will cosider for failure data from orepairable populatios are: Probability (CDF) plots Hazard ad Cum Hazard plots Probability Plottig Probability plots are simple visual ways of summarizig reliability data by plottig CDF estimates vs. time o specially costructed probability paper. Commercial papers are available for all the typical life distributio models. Oe axis (some papers use the y-axis ad others the x-axis, so you have to check carefully) is labeled Time ad the other axis is labeled Cum Percet or Percetile. There are rules, idepedet of the model or type of paper, for calculatig plottig positios from the reliability data. These oly deped o the type of cesorig i the data ad whether exact times of failure are recorded or oly readout times. Whe the poits are plotted, the aalyst fits a straight lie through them (either by eye, or with the aid of a least squares fittig program). Every straight lie o, say, Weibull paper uiquely correspods to a particular Weibull life distributio model ad the same is true for logormal or expoetial paper. If the poits follow the lie reasoably well, the the model is cosistet with the data. If it was your previously chose model, there is o reaso to questio the choice. Depedig o the type of paper, there will be a simple way to fid the parameter estimates that correspod to the fitted straight lie. Plottig Positios: Cesored Data (Type 1 or Type 2). At the time t i of the i th failure, we eed a estimate of the CDF (or the Cum. Populatio Percet Failure). The simplest ad most obvious estimate is just 100 i/ (with a total of uits o test). This, however, is geerally a overestimate (i.e., biased). Various texts recommed correctios such as 100 (i 0.5)/ or 100 i/( + 1). Here, we recommed what are kow as (approximate) media rak estimates: Correspodig to the time t i of the i th failure, use a CDF or Percetile estimate of 100 (i 0.3)/( + 0.4). Plottig Positios: Readout DataLet the readout times be T 1, T 2,..., T k ad let the correspodig ew failures recorded at each readout be r 1, r 2,..., r k. Agai, there are uits o test. Correspodig to the readout time T j, use 100 r i i a CDF or Percetile estimate of 1. Plottig Positios: Multicesored Data Hazard ad Cum Hazard Plottig Just commercial probability paper is available for most life distributio models for probability plottig of reliability data, there are also special Cum Hazard Plottig papers available for may life distributio models. These papers plot estimates for the Cum Hazard H(t i ) vs. the time t i of the i th failure. As with probability plots, the plottig positios are calculated idepedetly of the model or paper used ad a reasoable straight-lie fit to the poits cofirms that the chose model ad the data are cosistet. j 85
Evaluatio of Reliability from the Bottom-Up (Compoet Failure Mode to System Failure Rate) This sectio deals with models ad methods that apply to o-repairable compoets ad systems. Models for failure rates (ad ot repair rates) are described. We use the Series Model to go from compoets to assemblies ad systems. These models assume idepedece ad first failure mode to reach failure causes both the compoet ad the system to fail. If some compoets are i parallel, so that the system ca survive oe (or possibly more) compoet failures, we have the parallel or redudat model. If a assembly has idetical compoets, at least r of which must be workig for the system to work, we have what is kow as the r out of model. The stadby model uses redudacy like the parallel model, except that the redudat uit is i a off-state (ot exercised) util called upo to replace a failed uit. Complex systems ca be evaluated usig the various models as buildig blocks. SERIES MODEL The series model is used to go from idividual compoets to the etire system, assumig the system fails whe the first compoet fails ad all compoets fail or survive idepedetly of oe aother The Series Model is used to build up from compoets to sub-assemblies ad systems. It oly applies to o-replaceable populatios (or first failures of populatios of systems). The assumptios ad formulas for the Series Model are idetical to those for the Competig Risk Model, with the k failure modes withi a compoet replaced by the compoets withi a system. The followig 3 assumptios are eeded: 1. Each compoet operates or fails idepedetly of every other oe, at least util the first compoet failure occurs. 2. The system fails whe the first compoet failure occurs. Each of the (possibly differet) compoets i the system has a kow life distributio model F i (t). Add failure rates ad multiply reliabilities i the Series Model. Whe the Series Model assumptios hold we have: with the subscript S referrig to the etire system ad the subscript i referrig to the i th compoet. Note that the above holds for ay arbitrary compoet life distributio models, as log as idepedece ad first compoet failure causes the system to fail both hold. The aalogy to a series circuit is useful. The etire system has compoets i series. The system fails whe curret o loger flows ad each compoet operates or fails idepedetly of all the others. The schematic below shows a system with 5 compoets i series replaced by a equivalet (as far as reliability is cocered) system with oly oe compoet). R F S S t t i 1 1 R i t 1 Fi t i 1 86
h S t i 1 h i t Figure 3: Series System Reduced to Equivalet Oe Compoet System The system operates as log as at least oe compoet is still operatig. System failure occurs at the time of the last compoet failure. The CDF for each compoet is kow Multiply compoet CDF s to get the system CDF for a parallel model For a parallel model, the CDF F s (t) for the system is just the product of the CDF s F i (t) for the compoets or F S t i 1 F i t Parallel or Redudat Model The parallel model assumes all compoets that make up a system operate idepedetly ad the system works as log as at least oe compoet still works. The opposite of a series model, for which the first compoet failure causes the system to fail, is a parallel model for which all the compoets have to fail before the system fails. If there are compoets, ay ( 1) of them may be cosidered redudat to the remaiig oe (eve if the compoets are all differet). Whe the system is tured o, all the compoets operate util they fail. The system reaches failure at the time of the last compoet failure. The assumptios for a parallel model are: All compoets operate idepedetly of oe aother, as far as reliability is cocered. R S (t) ad h S (t) ca be evaluated usig basic defiitios, oce we have F S (t). The schematic below represets a parallel system with 5 compoets ad the (reliability) equivalet 1 compoet system with a CDF F s equal to the product of the 5 compoet CDF s. Figure 4: Parallel System ad Equivalet Sigle Compoet STANDBY MODEL The Stadby Model evaluates improved reliability whe backup replacemets are switched o whe failures occur. 87
A Stadby Model refers to the case i which a key compoet (or assembly) has a idetical backup compoet i a off state util eeded. Whe the origial compoet fails, a switch turs o the stadby backup compoet ad the system cotiues to operate. I the simple case, assume the o-stadby part of the system has CDF F(t) ad there are ( 1) idetical backup uits that will operate i sequece util the last oe fails. At that poit, the system fially fails. The total system lifetime is the sum of idetically distributed radom lifetimes, each havig CDF F(t). Idetical backup Stadby model leads to covolutio formulas I other w ords,t = t 1 + t 2 +... + t, where each tihas CDF F(t) ad T has a CDF we deote by F (t). This ca be evaluated usig covolutio formulas: F F 2 t t F u f t u 0 t du t F 1 u f t u 0 where f(t) is the PDF F'(t) du I geeral, covolutios are solved umerically. However, for the special case whe F(t) is the expoetial model, the above itegratios ca be solved i closed form. Expoetial stadby systems lead to a gamm Special Case: The Expoetial (or Gamma) Stadby Model: If F(t) has the expoetial CDF (i.e., F(t) = 1 e lt ), the F f 2 t t 1 te t e t 2 2 te, ad f t 2 1 t e 1! t t ad the PDF f (t) is the well-kow gamma distributio. Stadby uits are a effective way of icreasig reliability ad reducig failure rates, especially durig the early stages of product life. Their improvemet effect is similar to, but greater tha, that of parallel redudacy. The drawback, from a practical stadpoit, is the expese of extra compoets that are ot eeded for fuctioality. Expoetial stadby systems lead to a gamma lifetime model. COMPLEX SYSTEMS Ofte the reliability of complex systems ca be evaluated by successive applicatios of Series ad/or Parallel model formulas. May complex systems ca be diagrammed as combiatios of Series compoets, parallel compoets, R out of N compoets ad Stadby compoets. By usig the formulas for these models, subsystems or sectios of the origial system ca be replaced by a equivalet sigle compoet with a kow CDF or Reliability fuctio. Proceedig like this, it may be possible to evetually reduce the etire system to oe compoet with a kow CDF. Below is a example of a complex system composed of both compoets i parallel ad compoets i series is reduced first to a series system ad fially to a oe-compoet system. 88
Figure 5: Complex System Reduced to Equivalet Oe compoet System CASE STUDY I our project work, the thermal power plat is cosidered as Repairable complex system. As the thermal power plat is electricity, which is a cotiuously produced product, it would be somewhat difficult to defie the failure. But, as the aim of ay thermalpower plat is to produce the cotiues flow of good quality of electricity, ay violatio to it ca be cosidered as the Failure for the system. But it is very difficult to cotiuously moitor the quality of the product ad also it is a cumbersome process, it ca t be defied as failure. The fact that fluctuatios i the quality of electricity is ievitable, will also add value to the above statemet. So the failure of the Table 1: Trippig Data of Each Uit System Sub System Compoet Sub Compoet Sub Compoet Reliability Values Compoet Sub System System Power Plat Uit-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 Furace 0.1 456 0.51044 Turbie Turbie 1 1.00 Ecoomiser 1 Geerator Potetial Trasformers 0.2 82233.35 0.762684 0.215 Bushes 1 Brushes 0.2 2199007 0.869 Rotor 1 Stator 1 Trasmissio Grid Failure 0.99 0.435 Rely 0.1 6 0.4795 Curret Trasformers 0.1 4835632480 0.9167 89
Table 1 (Cot.) System Sub System Compoet Sub Compoet Sub Compoet Reliability Values Compoet Sub System System Uit-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 Furace 0.1 46 0.4478 Turbie Turbie 1 1.00 Ecoomiser 0.1 6 1 Geerator Geerator Coditios 0.1 36 0.324 0.1712 Potetial Trasformers 0.762684 Bushes 0.3 16910.28 0.7972 Brushes 0.2 2199007 0.869 Rotor 1 Stator 1 Trasmissio Grid Failure 0.99 0.99 Rely 1 Curret Trasformers 1 system is cosidered as the obstacle i the productio of the electricity. The maiteace take by maiteace persoel may be obstacle to the productio of the electricity. But as it is ievitable to maitai loger life without this obstacle. It is ecessary to have measuremet techique of reliability so that effective aalysis ad evaluatio of the policies ca be made to obtai ecoomical solutio. W hile evaluatig the system reliability, the past field failure data ad the structure of various subsystems eed to be studied, so the failure data hare is the trippig data of each uit (Table 1). After idetificatio of basic compoets, subsystems ad systems, the ext system is to get the field data. The cotributio of each major ad critical sub compoets, which cause the failure of the system as a whole. CONCLUSION We ca achieved after calculatig the reliability of each compoet with respect to the system ad the usig the pareto aalysis, cause ad effect diagrams. This may result i quick improvemet i the reliability of the system, which may be best result obtaiable by ay orgaizatio. To obtai the effective reliability improvemet pla with less cost, by addig few lies to the program, the cost effective best 90
maiteace policy is achievable. We ca predict the reliability of system based o compoet reliability it ca be assessed effectively the compoet that causes the ext failure, so that the ivetory ca be maitaied less, which reduces the cost. The accurate results may be obtaied whe the field data is complete ad easily assessable. BIBLIOGRAPHY 1. Charles E Ebelig (2009), A Itroductio to Reliability ad Maiteace Egieerig, TMH. 2. http://www.itl.ist.gov 3. http://www.rac.org 4. http: //www reliasoft.com APPENDIX 1 91
APPENDIX 2 92