Introduction to Engineering Reliability

Transcription

1 3 Iroducio o Egieerig Reliabiliy 3. NEED FOR RELIABILITY The reliabiliy of egieerig sysems has become a impora issue durig heir desig because of he icreasig depedece of our daily lives ad schedules o he saisfacory fucioig of hese sysems. Some examples of hese sysems are aircraf, rais, compuers, auomobiles, space saellies, ad uclear power geeraig reacors. May of hese sysems have become highly complex ad sophisicaed. For example, oday a ypical Boeig 747 umbo airplae is made of approximaely 4.5 millio pars, icludig faseers []. Mos of hese pars mus fucio ormally for he aircraf o fly successfully. Normally, he required reliabiliy of egieerig sysems is specified i he desig specificaio, ad durig he desig phase every effor is made o fulfill his requireme effecively. Some of he facors ha play a ey role i icreasig he imporace of reliabiliy i desiged sysems are he icreasig umber of reliabiliy- ad qualiyrelaed lawsuis, compeiio, public pressures, high acquisiio cos, pas wellpublicized sysem failures, loss of presige, ad complex ad sophisicaed sysems. This chaper preses various iroducory aspecs of egieerig reliabiliy. 3.2 BATHTUB HAZARD RATE CONCEPT This is a well-ow cocep used o represe failure behavior of various egieerig iems because he failure rae of such iems is a fucio of ime (i.e., i chages wih ime). A bahub hazard rae curve is show i Figure 3.. I is divided io hree regios (i.e., Regio I, Regio II, ad Regio III). Regio I is ow as he bur-i regio, debuggig regio, ifa moraliy regio, or brea-i regio. Durig his period or regio he iem hazard rae (i.e., ime-depede failure rae) decreases because of failures occurrig for reasos such as lised i Table 3. [2]. Regio II is referred o as he useful life period, durig which he iem hazard rae remais cosa. Some of he reasos for he occurrece failure i his regio are preseed i Table 3.. Regio III is ow as he wear-ou period, durig which he hazard rae icreases because of failures occurrig for reasos such as preseed i Table 3.. Mahemaically, he bahub hazard rae curve show i Figure 3. ca be represeed by usig he followig fucio [3]: λ() θλβ β + ( θ) bb αe α b (3.) 23

2 24 Maiaiabiliy, Maieace, ad Reliabiliy for Egieers Hazard rae (ime depede failure rae) Regio I Regio II Regio III ime FIGURE 3. Bahub hazard rae curve. TABLE 3. Reasos for he Occurrece of Failures i he Three Regios of he Bahub Hazard Rae Curve Regio Reaso I: Bur-i period Poor maufacurig mehods Poor processes Poor qualiy corol Poor debuggig Huma error Subsadard maerials ad wormaship II: Useful life period Low safey facors Udeecable defecs Huma errors Abuse Higher radom sress ha expeced Naural failures III: Wear-ou period Wear caused by fricio Poor maieace Icorrec overhaul pracices Corrosio ad creep Shor desiged-i life of he iem Wear caused by agig

3 Iroducio o Egieerig Reliabiliy 25 for β, b, λ, ad α > ; θ ; β.5, b, ad ad where is ime, λ() is he hazard rae or ime depede failure rae, α ad λ are he scale parameers, ad β ad b are he shape parameers. 3.3 GENERAL RELIABILITY ANALYSIS FORMULAS A umber of formulas, based o he reliabiliy fucio, frequely are used o perform various ypes of reliabiliy aalysis. This secio preses four of hese formulas FAILURE DENSITY FUNCTION This is expressed by dr() f () d (3.2) where is ime, f() is he failure (or probabiliy) desiy fucio, ad R() is he iem reliabiliy a ime. Example 3. Assume ha he reliabiliy of a iem is defied by he followig fucio: R () e λ (3.3) where λ is he iem s cosa failure rae. Obai a expressio for he iem s failure desiy fucio. Subsiuig Equaio 3.3 io Equaio 3.2 yields de f () d λ e λ λ (3.4) HAZARD RATE FUNCTION This is defied by λ () f () R () (3.5) where λ () is he iem hazard rae or ime-depede failure rae. By iserig Equaio 3.2 io Equaio 3.5 we ge dr() λ () R () d (3.6)

4 26 Maiaiabiliy, Maieace, ad Reliabiliy for Egieers Example 3.2 Usig Equaio 3.3, obai a expressio for he iem s hazard rae ad comme o he resulig expressio. Subsiuig Equaio 3.3 io Equaio 3.6 yields λ () e λ λ de d λ (3.7) Thus, he iem s hazard rae is give by Equaio 3.7. As he righ side of Equaio 3.7 is o he fucio of ime, λ is ow as he cosa failure rae because i does o deped o ime GENERAL RELIABILITY FUNCTION This ca be obaied by usig Equaio 3.6. Thus, rearragig Equaio 3.6, we ge λ () d dr() R () (3.8) Iegraig boh sides of Equaio 3.8 over he ime ierval [, ], we ge R () λ () d dr() R () (3.9) because a, R(). Evaluaig he righ-had side of Equaio 3.9 yields l R () λ () d (3.) Thus, from Equaio 3., we ge he followig geeral expressio for reliabiliy fucio: R () e λ () d (3.) Equaio 3. ca be used o obai he reliabiliy of a iem whe is imes o failure follow ay ime-coiuous probabiliy disribuio.

5 Iroducio o Egieerig Reliabiliy 27 Example 3.3 Assume ha he ime o failures of a auomobile is expoeially disribued ad is failure rae is.3 failures per hour. Calculae he auomobile s reliabiliy for a -hour missio. Usig he daa values i Equaio 3. yields (. 3) d R( ) e e. 974 (. 3)( ) This meas here is a approximaely 97% chace ha he auomobile will o fail durig he -hour missio. More specifically, is reliabiliy will be MEAN TIME TO FAILURE This is a impora reliabiliy measure ad i ca be obaied by usig ay of he followig hree formulas [4,5]: MTTF R () d (3.2) or MTTF f () d (3.3) or MTTF s lim R () s (3.4) where s is he Laplace rasform variable, MTTF is he mea ime o failure, ad R(s) is he Laplace rasform of he reliabiliy fucio R(). Example 3.4 Prove by usig Equaio 3.3 ha Equaio 3.2 o Equaio 3.4 yield he same resul for MTTF. Thus, by iserig Equaio 3.3 io Equaio 3.2, we ge λ MTTF e d λ (3.5)

6 28 Maiaiabiliy, Maieace, ad Reliabiliy for Egieers Subsiuig Equaio 3.3 io Equaio 3.2 yields de f () d λ e λ λ (3.6) Thus, subsiuig Equaio 3.6 io Equaio 3.3 yields MTTF e λ λ d λ e e λ λ λ (3.7) Taig he Laplace rasform of Equaio 3.3, we ge Rs () e s e λ d s + λ (3.8) Subsiuig Equaio 3.8 io Equaio 3.4 yields MTTF s λ lim ( s + λ) (3.9) Equaio 3.5, Equaio 3.7, ad Equaio 3.9 are ideical, provig ha Equaio 3.2 o Equaio 3.4 give he same resul. 3.4 RELIABILITY NETWORKS A sysem ca form various cofiguraios i performig reliabiliy aalysis. This secio is cocered wih he reliabiliy evaluaio of such commoly occurrig cofiguraios or ewors.

7 Iroducio o Egieerig Reliabiliy FIGURE 3.2 A -ui series sysem SERIES NETWORK This is probably he mos commoly occurrig cofiguraio i egieerig sysems, ad is bloc diagram is show i Figure 3.2. The diagram represes a -ui sysem, ad each bloc i he diagram deoes a ui. All uis mus wor ormally for he successful operaio of he series sysem. The series sysem (show i Figure 3.2) reliabiliy is expressed by R P( E E E... E ) s 2 3 (3.2) where E deoes he successful operaio (i.e., success eve) of ui for, 2, 3,., ; R s is he series sysem reliabiliy; ad P( EE2 E3... E ) is he occurrece probabiliy of eves E, E 2, E 3,, ad E. For idepedely failig uis, Equaio 3.2 becomes R P( E ) P( E ) P( E )... P( E ) s 2 3 (3.2) where P(E ) is he probabiliy of occurrece of eve E for, 2, 3,,. If we le R P(E ) for, 2, 3,, i Equaio 3.2 becomes R R R 2 R 3... R s (3.22) where R is he ui reliabiliy for, 2, 3,,. For he cosa failure rae λ of ui from Equaio 3. (i.e., for λ () λ ), we ge R () e λ (3.23) where R () is he reliabiliy of ui a ime. Subsiuig Equaio 3.23 io Equaio 3.22 yields R () e s λ (3.24) where R s () is he series sysem reliabiliy a ime.

8 3 Maiaiabiliy, Maieace, ad Reliabiliy for Egieers Subsiuig Equaio 3.24 io Equaio 3.2 yields λ MTTF e d s λ (3.25) where MTTF s is he series sysem mea ime o failure. Example 3.5 Assume ha a sysem is composed of five idepede ad ideical subsysems i series. The cosa failure rae of each subsysem is.25 failures per hour. Calculae he reliabiliy of he sysem for a 5-hour missio ad he sysem mea ime o failure. By subsiuig he give daa io Equaio 3.24 we ge R ( s ) e (. 25)( 5) Usig he specified daa values i Equaio 3.25 yields MTTF s 5(. 25) 8 hours Thus, he sysem reliabiliy ad mea ime o failure are.5353 ad 8 hours, respecively PARALLEL NETWORK I his case, he sysem is composed of simulaeously operaig uis, ad a leas oe of hese uis mus operae ormally for sysem success. The bloc diagram of a -ui parallel sysem is show i Figure 3.3, ad each bloc i he diagram represes a ui. The parallel sysem (show i Figure 3.3) failure probabiliy is give by F P E E... ps 2 E ( ) (3.26) where F ps is he parallel sysem failure probabiliy, E deoes he failure (i.e., failure eve) of ui, for, 2,,, ad P( E E 2 E 3... E ) is he occurrece probabiliy of eves E, E,...,ad. 2 E

9 Iroducio o Egieerig Reliabiliy 3 2 FIGURE 3.3 Bloc diagram of a -ui parallel sysem. For idepedely failig parallel uis, Equaio 3.26 becomes F P( E ) P( E )... P( E ) ps 2 (3.27) where P( E ) is he probabiliy of occurrece of eve E for, 2,.,. If we le F P( E ) for, 2,.,, Equaio 3.27 becomes F FF 2... F ps (3.28) where F is he ui failure probabiliy for, 2,,. By subracig Equaio 3.28 from uiy we ge R ps F ps FF... 2 F (3.29) where R ps is he parallel sysem reliabiliy. For cosa failure rae λ of ui, subracig Equaio 3.23 from uiy ad he subsiuig i io Equaio 3.29 yields R ( ) e e 2... e ps ( )( ) ( ) λ λ λ where R ps () is he parallel sysem reliabiliy a ime. For ideical uis, subsiuig Equaio 3.3 io Equaio 3.2 yields (3.3) MTTFps e d λ λ ( ) (3.3)

10 32 Maiaiabiliy, Maieace, ad Reliabiliy for Egieers 2 m FIGURE 3.4 Bloc diagram of he m-ou-of- ui sysem. where MTTF ps is he parallel sysem mea ime o failure ad λ is he ui cosa failure rae. Example 3.6 A sysem is composed of hree idepede ad ideical subsysems. A leas oe of he subsysems mus operae ormally for he sysem o wor successfully. Calculae he sysem s reliabiliy if each subsysem s probabiliy of failure is.. By subsiuig he give daa io Equaio 3.29 we ge R ps (. )(. )(. ). 999 Thus, he sysem s reliabiliy is M-OUT-OF-N NETWORK I his case, he sysem is composed of a oal of acive uis, ad leas m uis mus operae ormally for sysem success. The bloc diagram of a m-ou-of- ui sysem is show i Figure 3.4, ad each bloc i he diagram deoes a ui. The series ad parallel ewors are special cases of he m-ou-of- ewors for m ad m, respecively. For idepede ad ideical uis, ad usig he biomial disribuio, we wrie dow he followig reliabiliy expressio for he Figure 3.4 diagram: R R ( ) R m m ( ) (3.32)

11 Iroducio o Egieerig Reliabiliy 33 where ( )!!! ( ) (3.33) where R is he ui reliabiliy ad R m/ is he m-ou-of- ewor reliabiliy. For cosa failure raes of he ideical uis, subsiuig Equaio 3.3 io Equaio 3.32 yields R () e ( ) e m λ ( λ ) m (3.34) where R m/ () is he m-ou-of- ewor reliabiliy a ime ad λ is he ui failure rae. Subsiuig Equaio 3.34 i Equaio 3.2 yields MTTFm e e ( ) m λ λ ( ) λ where MTTF m/ is he m-ou-of- ewor mea ime o failure. (3.35) Example 3.7 Assume ha a egieerig sysem is composed of four idepede ad ideical uis i parallel. A leas hree uis mus operae ormally for sysem success. Calculae he sysem mea ime o failure if he ui failure rae is.35 failures per hour. By subsiuig he specified daa values io Equaio 3.35 we ge MTTF Thus, he sysem mea ime o failure is hours. m m (. 35) ( 35) hours d STANDBY SYSTEM This is aoher impora reliabiliy cofiguraio i which oly oe ui operaes ad uis are ep i heir sadby mode. More specifically, he sysem coais a oal of + uis, ad as soo as he operaig ui fails, he swichig mechaisms

12 34 Maiaiabiliy, Maieace, ad Reliabiliy for Egieers 2 K FIGURE 3.5 Bloc diagram of a sadby sysem wih oe operaig ad sadby uis. or oher meas deec he failure ad he replace he failed ui wih oe of he sadby uis. Figure 3.5 shows he bloc diagram of a sadby sysem wih oe operaig ad sadby uis. Each bloc i he diagram deoes a ui. Usig he Figure 3.5 diagram for idepede ad ideical uis, perfec deecio, swichig mechaisms ad sadby uis, ad ime-depede ui failure rae, we wrie dow he followig expressio for sysem reliabiliy [6]: K Rsb () () d e λ λ () d /! (3.36) where R sb () is he sadby sysem reliabiliy a ime ad λ () is he ui imedepede failure rae. For cosa ui failure rae, (i.e., λ () λ), Equaio 3.36 becomes R () ( ) e /! sb K λ λ (3.37) Iserig Equaio 3.37 io Equaio 3.2 yields K MTTFsb e d ( λ ) λ /! K + λ (3.38) where MTTF sb is he sadby sysem mea ime o failure. Example 3.8 A sadby sysem is composed of wo idepede ad ideical uis: oe operaig ad he oher o sadby. The ui cosa failure rae is.45 failures per hour.

13 Iroducio o Egieerig Reliabiliy 35 Calculae he sysem reliabiliy for a -hour missio ad mea ime o failure if he sadby ui remais as good as ew i is sadby mode ad failure deecio ad ui replaceme mechaisms are % reliable. By subsiuig he give daa io Equaio 3.37 we ge R sb ( ). 45 ( )( ) K e ( )( ) Usig he specified daa values i Equaio 3.38 yields /! MTTF + sb. 45 ( ) hours Thus, he sadby sysem reliabiliy ad mea ime o failure are.9246 ad hours, respecively BRIDGE NETWORK Someimes uis of a egieerig sysem may form a bridge cofiguraio as show i Figure 3.6. The diagram is composed of five blocs, each of which deoes a ui. All blocs are labelled wih umerals. For idepedely failig uis, he Figure 3.6 diagram reliabiliy is expressed by R R R R R R + R R R + R R R + b R R + R R R2 R R R R R R R R R R R R R R R R R R R (3.39) where R i is he reliabiliy of ui i for i, 2, 3, 4, 5 ad R b is he bridge ewor reliabiliy FIGURE 3.6 A bridge ewor made up of five oideical uis.

14 36 Maiaiabiliy, Maieace, ad Reliabiliy for Egieers For ideical uis, Equaio 3.39 becomes R 2 R 5 b R + 2 R + 2 R (3.4) For cosa ui failure rae, subsiuig Equaio 3.3 io Equaio 3.4 yields R () 2e 5e + 2e + 2e b 5λ 4λ 3λ 2λ (3.4) where R b () is he bridge ewor reliabiliy a ime ad λ is he ui cosa failure rae. By subsiuig Equaio 3.4 io Equaio 3.2, we ge MTTF b 49 6 λ (3.42) where MTTF b is he bridge ewor mea ime o failure. Example 3.9 A sysem has five idepede ad ideical uis formig a bridge cofiguraio. The ui failure rae is.75 failures per hour. Calculae he ewor reliabiliy for a -hour missio ad mea ime o failure. Usig he give daa values i Equaio 3.4 yields R ( b ) 2 e 5 (. 75)( ) 5e 4 (. 75)( ) + 2e + 2e (. 75)( ) 2(. 75)( ) By subsiuig he specified daa value io Equaio 3.42, we ge MTTF 49 b 6(. 75) hours Thus, he bridge ewor s reliabiliy ad mea ime o failure are.4552 ad 8.89 hours, respecively. 3.5 RELIABILITY ALLOCATION This is he process of assigig reliabiliy requiremes o idividual compoes for achievig he specified sysem reliabiliy. Alhough here are may beefis of he reliabiliy allocaio, wo of he impora oes are as follows [,7]: I forces people ivolved i desig ad developme o udersad ad esablish he appropriae relaioships bewee reliabiliies of compoes ad pars, subsysems, ad sysems. I forces desig egieers o cosider reliabiliy equally wih oher desig parameers such as cos, performace, ad weigh.

15 Iroducio o Egieerig Reliabiliy 37 Over he year, may reliabiliy allocaio mehods have bee developed [8 2]. Oe of he commoly used mehods i he idusrial secor is described below [] HYBRID METHOD This mehod is he resul of combiig wo reliabiliy allocaio approaches ow as he similar familiar sysems mehod ad he facors of ifluece mehod. The resulig approach icorporaes beefis of hese wo mehods; hus, i is more aracive o use. The basis for he similar familiar sysems mehod is he desiger s familiariy wih similar sysems or subsysems. More specifically, durig he allocaio process he mehod uses he failure daa colleced o similar sysems, subsysems, ad iems from various sources. The mai disadvaage of his approach is o assume ha life cycle cos ad reliabiliy of pas similar desigs were saisfacory. The facors of ifluece mehod is based o he assumpio ha he facors show i Figure 3.7 effec he sysem reliabiliy. These are failure criicaliy, evirome, complexiy ad ime, ad he sae of he ar. The failure criicaliy facor cosiders he criicaliy of he failure of he iem i quesio o he sysem. For example, he failure of some auxiliary isrume i a aircraf may o be as criical as he egie failure. The evirome facor aes io accou he exposure or suscepibiliy of he iem or iems i quesio o eviromeal codiios such as vibraio, emperaure, ad humidiy. The complexiy ad ime facor relaes o he umber of subsysem pars ad he relaive operaig ime of he iem durig he fucioal period of he complee sysem. The sae-of-he-ar facor aes io accou he advaceme made i he saeof-he-ar for a cerai iem. I usig he above four facors, every iem uder cosideraio is raed wih respec o each of hese facors by beig assiged a umber from o. The Failure criicaliy Complexiy/ ime Facors Evirome Sae-of-hear FIGURE 3.7 Facors affecig sysem reliabiliy.

16 38 Maiaiabiliy, Maieace, ad Reliabiliy for Egieers assigme of meas he iem uder cosideraio is mos affeced by he ifluece facor i quesio, ad meas he iem is leas affeced by he same facor. Subsequely, he reliabiliy is allocaed o he basis of he weigh of hese assiged umbers for all ifluece facors cosidered. Fially, reliabiliy of a iem is allocaed by givig cerai weighs o boh similar familiar sysems ad facors of ifluece mehods. The hybrid mehod is more effecive ha boh hese mehods used aloe because i uses daa from boh of hem. 3.6 PROBLEMS. Wrie a essay o he eed for reliabiliy. 2. Describe he bahub hazard rae cocep. 3. Wrie dow a hazard rae fucio ha ca be used o represe a bahub hazard rae curve. 4. Defie hazard rae. 5. Wha is he differece bewee hazard rae ad cosa failure rae? 6. Defie failure desiy fucio. 7. Wha are he hree mahemaical approaches for obaiig a iem s mea ime o failure? 8. Discuss reliabiliy allocaio ad is beefis. 9. Usig Equaio 3.24, obai a expressio for hazard rae ad comme o he resulig expressio.. A sysem is composed of hree idepede ad ideical uis i parallel. A leas wo uis mus operae ormally for sysem success. Calculae he sysem mea ime o failure if he ui failure rae is.25 failures per hour. REFERENCES. Dhillo, B.S., Desig Reliabiliy: Fudameals ad Applicaios, CRC Press, Boca Rao, FL, Kapur, K.C., Reliabiliy ad maiaiabiliy, i Hadboo of Idusrial Egieerig, Salvedy, G., Ed., Joh Wiley & Sos, New Yor, 982, pp Dhillo, B.S., A hazard rae model, IEEE Trasacios o Reliabiliy, 28, 5, Shooma, M.L., Probabilisic Reliabiliy: A Egieerig Approach, McGraw-Hill, New Yor, Dhillo, B.S., Reliabiliy, Qualiy, ad Safey for Egieers, CRC Press, Boca Rao, FL, Sadler, G.H., Sysem Reliabiliy Egieerig, Preice Hall, Eglewood Cliffs, NJ, Gra Ireso, W., Coombs, C.F., ad Moss, R.Y., Eds., Hadboo of Reliabiliy Egieerig ad Maageme, McGraw-Hill, New Yor, Frederic, H.E., A reliabiliy predicio echique, Proceedigs of he Fourh Naioal Symposium o Reliabiliy ad Qualiy Corol, 34 37, 958.

17 Iroducio o Egieerig Reliabiliy Chipcha, J.S., A pracical mehod of maiaiabiliy allocaio, IEEE Trasacios o Aerospace ad Elecroic Sysems, 7, , 97.. Dhillo, B.S., Sysems Reliabiliy, Maiaiabiliy, ad Maageme, Perocelli Boos, New Yor, Balaba, H.S., Jeffers, H.R., ad Baechler, D.O., The Allocaio of Sysem Reliabiliy, Publicaio No , ARINC Research Corporaio, Chicago, Vo Alve, W.H., Ed., Reliabiliy Egieerig, Preice Hall, Eglewood Cliffs, NJ, 964.