Reliability Calculation for Dormant k-out-of-n Systems with Periodic Maintenance

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1 Vol. 1, o. 2, 68 76, 216 Reliability Calculatio for Dormat k-out-of- Systems with Periodic Maiteace James Li Cetre of Competece for Mass rasit AME Bombardier rasportatio, Kigsto, Otario, Caada (Received Jue 2, 216; Accepted Jue 12, 216 Abstract I this paper, a dormat k-out-of- systems redudacy calculatio will be itroduced. Dormat failure is a failure that caot be detected whe it occurs because of the ature of the failure characteristic. herefore, a dormat failure becomes the blid poit to the desig for reliability ad maitaiability because of its iability to be detected. he most popular approach i detectig a dormat failure is to carry out a scheduled periodic ispectio, test or maiteace activity. he scheduled periodic maiteace is applied to prevet ad reduce the uexpected dormat failures that could lead to safety cosequeces, or costly corrective maiteace. his paper will itroduce a methodology o how to calculate the reliability parameter such as Mea ime Betwee Failure (MBF for the dormat k-out-of- redudat systems. he mathematical relatioship betwee the effective MBF ad the scheduled periodic ispectio/maiteace iterval is also elaborated. Case studies are adopted to illustrate how to apply the developed reliability calculatio methodology i the mass trasit trai reliability ad safety desig. Keywords -k-out-of- redudat systems, dormat failures, periodic maiteace, MBF 1. Itroductio I a k-out-of- redudat system, the uit failure ca be classified ito two categories: detectable ad o-detectable. A detectable failure is a failure that is detected ad /or auciated whe it occurs. I mass trasit trai subsystems ad uits, electrical, hydraulic ad peumatic subsystems failures are mostly liked to detectable failure ad is usually moitored by the trai o board health moitorig system. Whe such a failure occurs, the trai o board health moitorig system ca detect the degradatio ad abormal coditio ad subsequetly auciate a alarm to the trai operatio cotrol ceter. he trai operatio cotrol ceter will take corrective actios to maage these occurred failures. A o-detectable failure is a failure which is ot detected ad /or auciated whe it occurs. A o-detectable failure is also called a passive failure or dormat failure i some stadards ad documetatios. he most effective approach to detect a passive or dormat failure is to carry out a scheduled periodic ispectio or test to idetify them. he implemetatio of Failure Mode, Effects ad Criticality Aalysis (FMECA or Failure Mode Effect Aalysis (FMEA ca be utilized to idetify the detectio method of the failure mode. If a failure mode is idetified as a dormat failure, the a scheduled periodic ispectio or test is required as a mitigatig actio to detect these types of failures. I this paper, a reliability calculatio method is itroduced to speculate the relatioship betwee the effective Mea ime Betwee Failure (MBF ad the scheduled periodic ispectio iterval for the k-out-of- redudat system. he study idicates that the shorter the scheduled periodic ispectio iterval, the greater the effective MBF, vice versa, the loger the scheduled periodic ispectio iterval, the smaller the effective MBF. his paper will start with the itroductio of the reliability calculatio methodology for the k-out-of- redudat system which is periodically maitaied. he the study proceeds to apply the developed calculatio methodology i the brake discout 68

2 Vol. 1, o. 2, 68 76, 216 calculatio for the safe stoppig distace aalysis i the Sao Paulo Moorail. he coclusio ad boudary are summarized at the ed of this paper. I referece Klio (1977, systems periodically maitaied was itroduced. I referece Military stadard (198, failure mode, effects ad criticality aalysis was itroduced. I referece IEEE stadard (1999, the safe trai separatio ad typical safe brakig model were itroduced. I referece Vitr et al. (23, prevetive maiteace optimizatio o the basis of operatig data aalysis was preseted. I referece Military Stadard (25, Biomial distributio was itroduced. I referece utt et al. (29 ad (212, risk-iformed prevetive maiteace optimizatio was preseted. I referece Babishi et al. (216, maiteace ispectio optimizatio of k-out-of- redudat system was preseted. I referece Guo et al. (216, optimizatio of prevetive maiteace iterval o the aircraft idicators was preseted. 2. Reliability Calculatio Methodology for the Redudat Systems Periodically Maitaied I this sectio, a redudat calculatio approach is itroduced to a systems which is periodically ispected ad maitaied. he mass trasit trais ecompass various electrical ad mechaical subsystems. Most of the failures that occur i the electrical subsystems are usually detectable ad auciate with a alarm to the trai o board health moitorig system. otwithstadig, some of the failures that occur i the mechaical subsystem are dormat ad caot be detected. For example, a brake caliper stuck i the release positio is cosidered as a dormat failure, the failure caot be detected util the ext scheduled ispectio. For the subsystems with a potetial dormat failure, a maiteace team will visit these subsystems at every predetermied iterval ad repair all occurred failures. If we defie as the predetermied maiteace iterval or uatteded period of operatio; ad defie f(t as the failure desity fuctio. he he probability that the system will be o at the ed of is R 1 f ( t. If the system is still operatig at, the the operatig time for the system is. If the system fails at t i (,, the the operatig time for the system is t. herefore, the average uiterrupted operatig time of a system i (,, AU is give by AU tf ( t t(1 t (1 t t (1 It is possible for a system to fail before the first cycle (, is completed or it is possible that the system will ot fail util the th cycle is completed. herefore, if we had a large umber of such systems (X i the field ad iteded to maitai these systems over a log time period: = proportio of system survivig the first cycle with o failure 2 = proportio of system survivig the secod cycle with o failure 3 = proportio of system survivig the third cycle with o failure ad, 69

3 Vol. 1, o. 2, 68 76, 216 proportio of system survivig the first cycles with o failure. herefore, of the origial X systems. Cycle o. 1 X systems would operate uiterruptedly for a average of AU hours each before failure 2 X systems would operate uiterruptedly for a additioal AU hours 3 2 X systems would operate uiterruptedly for a additioal AU hours -1 X systems would operate uiterruptedly for a additioal AU hours Ad the average uiterrupted operatig time to first failure per system is AU 1 i X X X X AU X i i is a ifiite geometric series. is betwee ad 1. So the sum of this ifiite geometric series is show as the followig equatio. 1 i 1 ( i 1 Sice: R (3 i (2 R ( 1 Hece:, as gets arbitrarily large, the above equatio will be equal to 1 1 he average uiterrupted operatig time to the first failure FF: FF AU 1 t 1 (4 Where represets the uatteded period of operatio or predetermied maiteace iterval (i.e., every hours a maiteace team visits the system ad repairs all uit failures. 7

4 Vol. 1, o. 2, 68 76, 216 Fig. 1. Relatioship betwee Reliability Deterioratio ad Prevetive Maiteace he Fig.1 shows that a system is restored to its origial coditio followig prevetive maiteace, i.e., as good as ew. 3. Reliability Calculatio Methodology for k-out-of- Systems Periodically Maitaied I the able : Redudacy Equatio for Calculatio Reliability, System Reliability oolkit published by Reliability Aalysis Ceter, the reliability fuctio t for k-out-of- system is show as the followig equatio: km t k t e 1 e k t (5 k! Substitute equatio (5 ito the equatio (4 to obtai: MBF 1 k m k m k! k! t k t ( e 1 e k t k t ( e 1 e k (6 I practice, after a eormous reliability calculatio with a log time period, we have observed that the umerator of equatio (6: t e km! k t k! (1 e k is approximately equal to. herefore the equatio (6 ca be expressed as a followig closed-form equatio (7: MBF 1 km k! t k t ( e 1 e k (7 he advatage to equatio (7 compared with equatio (6 is that we ca save a lot of time by skippig the massive itegral calculatio ad obtaiig a approximated result. 71

5 Vol. 1, o. 2, 68 76, 216 Equatio (7 ca also be expressed as: MBF Where, 1 R km k! k 1 ( k t ( e for the expoetial distributio. (8 3.1 Determiig Effective MBF for a Sigle Uit For a sigle uit, the reliability fuctio t is e -λt ad the followig MBF equatio ca be applied, 1 t 1 e e 1 MBF (9 1 e 1 e It is uderstood that MBF is the reciprocal of the failure rate. he above equatio also idicates that the ispectio iterval will ot chage the failure rate of a sigle uit cofiguratio. 3.2 Determiig Effective MBF for 1-out-of-2 Redudat Systems I the able : Redudacy Equatio for Calculatio Reliability, System Reliability oolkit published by Reliability Aalysis Ceter. For 1-out-of-2 redudat system, the reliability fuctio t is expressed as 2e -λt e -2λt, ad therefore the effective MBF ca be calculated as: 2e MBF 1 t e 2t e 2 1 e 2 (1 It should be oted that the shorter the maiteace iterval (, the greater the effective MBF, vice versa, the loger the maiteace iterval (, the smaller the effective MBF. We provide the followig example to demostrate this cocept. For a repairable two uit redudat system with a idetical costat failure rate λ=1-5 failure per hour. If the periodic maiteace iterval is oe moth, the =1 moth=24x3=72 hours. Substitutig =72 ad λ=1-5 ito equatio ( e MBF 1.1 t.1 72 e e 2.1 t e ,988,949 If the periodic maiteace iterval is exteded to three moths, the =72x3=216 hours, ad 72

6 Vol. 1, o. 2, 68 76, e MBF 1.1 t.1216 e e 2.1 t e ,729,81. he above calculatio idicates that if the maiteace iterval is stretched out from oe moth to three moths, the effective MBF will be roughly shorteed to oe third. 4. Case Study-Brake Discout Calculatio i Safe Brakig Model rai collisio is cosidered as oe of the major safety cocers i the mass trasit idustry. Automatic rai Protectio (AP is a dedicated system to prevet oe trai from collidig with the other trai o the same lie by meas of maitaiig a safe separatio betwee trais. he safe separatio (brakig distace aalysis shall be based o brakig capacity (depedet o weight, the gradiet at the locatio cocered, the maximum possible speed of the trais usig the sectio, the allowace for system reactio ad a credible margi. he AP profile shall be govered by a safe brakig model show i Fig. 2 ad shall esure that uder o circumstaces (icludig failures the movemet authority limit will be exceeded by a AP equipped trai. With respect to the safe brakig model, a reliability egieer is required to aalyze the brake failure case ad determie the discouted brakes quatity. SPEED Max Allowable Speed Brake Delay Brakig Distace Safety Distace Brake Delay Brakig Distace Safety Distace (Service Brake (Service Brake rai rai rai Miimum Separatio Miimum Separatio Fig. 2. ypical safe brakig model For the Bombardier developed platform moorail, a trai is composed of two cars. A trai cosists of four brake axles; each braked axle is equipped with a sigle passive caliper ad disc pair. Fig. 3 shows the brake axle cofiguratio. 73

7 Vol. 1, o. 2, 68 76, 216 Fig. 3. Brake axle cofiguratio Each brake axle has a failure rate λ: 4.86 x 1-6 failure per hour. he trai missio time is 13.5 hours per day. Brake supplier has recommeded a three moth prevetive maiteace for brake axles. 4.1 Case Study 1: 1-out-of-4 Brake Axles Fails, 3 of 4 Brake Axles are workig I this case study, we cosider oe of four brake axles fails, ad the remaiig three brake axles are still workig ormally. Substitute k=3 (umber of brake axles workig i a trai, =4 (otal umber of brake axles i a trai, λ=4.86e-6 (failure rate of brake axle, uit: failure per hour, =3x3x13.5=1215 (three moth maiteace iterval x 3 days x daily missio, uit: hours ito equatio (7: 1215 MBF k (4k 1 e 1 e k3 k! ,888, (.9941 ( ( We covert the above calculated MBF ito the failure rate, ad the reciprocal failure rate =1.6983E-7 fph. I the mass trasit reliability regime, the threshold for the improbable probability is 1E-9 fph, which meas if the failure rate is lower tha 1-9 fph, it ca be assumed that the occurrece of such a failure may ot be experieced i the thirty year life time. Because the failure rate for 1-out-of-4 brake axles is E-7 fph, greater tha 1E-9 fph. It ca be cocluded that oe brake axle could fail i the moorail s thirty-year life time. herefore, oe brake axle failure shall be cosidered i the safe brakig model failure case. 4.2 Case Study 2: 2-out-of-4 Brake Axles Fail, 2 of 4 Brake Axles are workig I this case study, we cosider the situatio i which 2-out-of-4 brake axles fail, ad the remaiig two brake axles are still workig. Substitute k=2 (umber of brake axles workig i a trai, =4 (total umber of brake axles i a trai, λ=4.86e-6 (failure rate of brake axle, uit: failure per hour, =3x3x13.5=1215 (three moth maiteace iterval x 3 days x daily missio, uit: hours ito equatio (7: 74

8 Vol. 1, o. 2, 68 76, 216 MBF 4 1 k2 k! 1 6(.9941 ( k e 1 e (.9941 ( (.9941 (4k ,495,14,49 We covert the calculated MBF ito the failure rate, ad the reciprocal failure rate =6.6889E-1 fph. he failure rate for 2-out-of-4 brake axles failig is lower tha 1E-9 fph, thus it ca be cocluded that two brake axles failig simultaeously i [, 1215 hours] could ot be experieced i the moorail s thirty-year life time. herefore, the situatio that two of the four brake axles failig at the same time will ot be cosidered i the safe brakig model failure case study. 5. Coclusio ad Boudary he purpose of this paper is to determie the mathematical relatioship betwee the reliability parameter: Mea ime Betwee Failure (MBF or failure rate, ad the maiteace iterval or uatteded period of the operatio for the k-out-of- redudat systems. he developed formula ad methodology i this paper ca be utilized i the MBF approximatio practice for the k-outof- redudat system which is periodically maitaied. he approach preseted i this paper ca also be applied i reliability calculatios of the systems with the potetial dormat failures. As described i the paper, the approach itroduced i this paper is limited withi the applicatio of the expoetial distributio. Refereces Babishi, V. & aghipour. S. (216. Joit maiteace ad ispectio optimizatio of k-out-of system. I Proceedig Aual Reliability ad Maitaiability Symposium (RAMS, 216, ucso, USA. Guo, J., Li, Z., ad Wolf, J. (216. Reliability cetered prevetive maiteace optimizatio for aircraft idicators. I Proceedig Aual Reliability ad Maitaiability Symposium (RAMS, 216, ucso, USA. IEEE Stadard (1999. IEEE Std IEEE stadard for commuicatios-based trai cotrol (CBC performace ad fuctioal requiremets (pp he Istitute of Electrical ad Electroics Egieers, Ic. Publishig. Klio, J. (1977. System periodically maitaied. I A Redudacy otebook (pp Rome Air Developmet CeterPublishig. Laza, G., iggeschi, S. & Wemer, P. (29. Behavior of dyamic prevetive maiteace optimizatio for machie tools. I Proceedig Aual Reliability ad Maitaiability Symposium (RAMS, 29, Fort Worth, USA. Military Stadard (198. MIL-SD-1629A, Procedures for performig a failure mode, effects ad criticality aalysis (pp aval Air Egieerig Ceter Systems Egieerig Stadardizatio Departmet Publishig. Military Stadard (25. MIL-HDBK-338B military hadbook electroic reliability hadbook, otice 2 (pp Air Force Research Laboratory Iformatio Publishig. Reliability Iformatio Aalysis Ceter (25. Hardware reliability modelig. I System Reliability oolkit (pp Reliability Iformatio Aalysis Ceter (RIAC ad Data Aalysis Ceter for Software (DACS Publishig. 75

9 Vol. 1, o. 2, 68 76, 216 utt,., Sigh, I., Popova, E., & Kee E. (29. Risk-iformed prevetive maiteace optimizatio. I Proceedig Aual Reliability ad Maitaiability Symposium (RAMS, 29, Reo, USA. utt,., Sigh, I., Popova, E., & Kee, E. (212. Outage performace improvemet by prevetive maiteace optimizatio. I Proceedig Aual Reliability ad Maitaiability Symposium (RAMS, 212, Reo, USA. Vitr, Z. & Holub, R. (23. Prevetive maiteace optimizatio o the basis of operatig data aalysis. I Proceedig Aual Reliability ad Maitaiability Symposium (RAMS, 23, ampa, USA. 76

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