Lecture 2 - What are component and system reliability and how it can be improved?
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1 Lecture 2 - What are compoet ad system relablty ad how t ca be mproved? Relablty s a measure of the qualty of the product over the log ru. The cocept of relablty s a exteded tme perod over whch the expected operato of the product s cosdered ad we expect the product wll fucto accordg to certa expectatos over a stpulated perod of tme. Wth the customer ad warraty costs md, we must kow the chaces of successful operato of the product for at least a certa stpulated perod of tme. Such formato helps the maufacturer to select the parameters of a warraty polcy. Techcally, relablty s the probablty of a product performg ts teded fucto for a stated perod of tme uder certa specfed codtos. Four aspects of relablty are apparet from ths defto frst, relablty s a probablty of success-related cocept; the umercal value of ths probablty s always betwee 0 ad 1. Secod, the fuctoal performace of the product had to be measured uder certa stpulated codtos. Product desg s expected to esure developmet of a product that meets or exceeds the specfed requremets uder specfed operatg codtos. For example, f the breakg stregth of a ylo cord s expected to be 1000 kg, the the predefed operatoal codtos, the cord must be able to bear weghts of 1000 kg or more. Thrd, relablty mples successful operato over a certa perod of tme (t). Although o product s expected to last forever, the tme dmeso esures satsfactory performace over at least a mmal stated perod (say, 100 hours). I the cotext of these three aspects, the relablty of the ylo cord mght be descrbed as havg a probablty of successful performace of 0.92 bearg loads of 1000 kg for 1 year uder dry codtos. It s observed that most maufacturg products go through three dstct phases (see Fgure 3-8) from product cepto to wear-out. The lfe-cycle curve of Fgure 3-8 shows the varato the falure rate as a fucto of tme dfferet phases. Covetoally the falure rate ( λ ) s plotted as a fucto of tme. Ths curve s ofte referred to as the bathtub curve; t cossts of the debuggg (fatmortalty) phase, the chace-falure phase (useful lfe phase), ad the wear-out phase. The debuggg phase, also kow as the fat-mortalty phase, exhbts a drop the falure rate as tal problems detfed durg prototype testg are removed. The chace-falure phase, betwee tmes t 1 ad t 2, s the ecoutered; falures occur
2 radomly ad depedetly. Ths phase, whch the falure rate s costat, typcally represets the useful lfe of the product delvered to ed customer. I the wear-out phase, a crease the falure rate s expected due to wear ad tear of the product. Here, after the ed of ther useful lfe, parts age ad wear out. Fgure 3-8 Bathtub Curve For the radom chace-falure phase, whch represets the useful lfe of the product or compoet, the falure rate s assumed to be costat. As a result, the expoetal dstrbuto s selected to descrbe the tme-to-falure of the product for ths phase. A expoetal dstrbuto as a memory less property ad ts probablty desty fucto s gve by λ λ t f t = e, t 0 The mea-tme-to-falure (MTTF) for the expoetal dstrbuto ca be expressed as MTTF=1/λ The relablty, at tme t, say R(t), s the probablty of the product lastg up to tme t. It ca be
3 expressed as, = 1 F ( t) R t =1- t e dt = e 0 Here, F(t) represets the cumulatve dstrbuto fucto at ay tme t. Relablty decreases expoetally wth tme (Fgure 3-9) ad the falure-rate fucto, say r(t), s gve by the rato of the tme-to-falure probablty desty fucto to the relablty fucto. We have r t f t = R t Fgure 3-9 Relablty v/s Tme Thus, assumg a expoetal dstrbuto, r (t) mplyg a costat falure rate, as show below. r t λe = = λ e Let us cosder a resster compoet, whch follows a expoetal tme-to-falure dstrbuto wth a falure rate of 8% per 1000 hr. We are terested to calculate the relablty of the resster at 5000 hr, ad also we ted to calculate the mea-tme-to-falure. Here the costat falure rate λ s obtaed as
4 λ = 008 /1000 hr = /hr Thus, the relablty for 5000 hr of survval s R t = e =e =e = Thus there s about 67% chace of survval (success) of the resster uder stpulated codtos ad stpulated tme (5000 hr). The mea (average) tme-to-falure (assumg t caot be repar) of the resster wll be MTTF=1/ λ = 1 / = 12,500 h System Relablty Let us cosder a system wth three compoets (say three resster) seres as show Fgure Fgure 3-10 Seres System Wthout loss of geeralty, f the system compoets ca be assumed to have a tme-to-falure dstrbuto as expoetal wth each compoet has a costat falure rate, we ca easly compute the relablty of -system seres. Suppose the system has compoets ad seres, each wth expoetally dstrbuted tme-to-falure wth falure rates λ 1, λ 2,, λ. The system relablty s calculated as the product of the compoet relabltes:
5 R () t = e e e S =exp - λ1t λ2t = 1 λ t Ths mples that the tme-to-falure of the system s expoetally dstrbuted wth a equvalet falure rate of = 1 gve by λ. The mea tme to falure for the system s MTTF = 1 = 1 λ Systems wth Compoets Parallel System relablty ca be mproved by placg redudat compoets parallel. The system operates as log as at least oe of the compoets operates. A three compoet parallel system s represeted as gve Fgure Fgure 3-11 A Parallel Compoet System If the tme-to-falure of each compoet follows expoetal dstrbutos, each wth a costat falure rate, λ, = 1,...,, the system relablty, assumg depedece of compoet operato (falure of oe does ot mpact falure of ay other compoet), s gve by
6 ( ) R () t = 1-1 R() t S =1 λ t ( e ) =1-1 =1 I a specal case, where all compoets have the same falure rate, λ, relablty for parallel compoet s gve by the system λ t R () t = 1 1 e S For such specfc stuato, the mea-tme-to-falure for the system wth detcal compoets parallel, ad also assumg that each faled compoet s mmedately replaced by a detcal compoet, ca be expressed as MTTF = 1... λ Systems wth Compoets Seres ad Parallel Real lfe systems ofte cosst of compoets that are mxed ad cosst of both seres ad parallel cofgurato. For such system, relablty calculato s prmarly based o the prevously dscussed cocepts, ad assumpto of compoets operatg depedetly. Parallel systems are frst collated to get a composte relablty, ad the the overall compoets are cosdered as seres to calculate system relablty. Systems ca also cosst of stadby compoet, whch operates as ad whe base compoet fals. Reader may refer to book by Amtava Mtra (2008) or Besterfeld et al (2004 ) for further detals o. K-out-of-N system (parallel system s 1 out of N system) s aother possble system cofgurato.
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