ANALYSIS OF A REDUNDANT SYSTEM WITH COMMON CAUSE FAILURES

Transcription

1 Dharmvir ingh Vahith et al. / International Journal of Engineering iene and Tehnology IJET ANALYI OF A REDUNDANT YTEM WITH OMMON AUE FAILURE Dharmvir ingh Vahith Department of Mathemati, R.N. Engg. ollege, Rohtak Haryana., India dharmvir.vahith@yahoo.in Rajeev Kumar Department of Mathemati t. N.R. Govt. ollege Rohtak Haryana India Rajivalgol@radiffmail.om Abtrat: The purpoe of the preent paper i to arry out the reliability of a redundant ytem with ommon aue failure. A tohati Model of two unit i developed in whih one unit i operative and other i kept a old tandby. The unit may fail either partially or ompletely. The failure time ditribution are exponential where a the repair ditribution i arbitrary. The expreion for reliability i derived uing upplementary variable tehnique. The numerial reult for a partiular ae have alo been made. Key word: tohati Model, Redundant ytem, upplementary variable tehnique.. Introdution All the reearher inluding Dhillon and Anude [], Goel, Gupta & Tyagi [] have onidered variou model of tranit ytem and tudied their reliability behaviour. However, the model onidered by them do not meet the hallenging requirement of modern omplexitie of variou equipment. Keeping thi in view, Goel, Kihore & ingh [] onidered a tranient ytem ompoed of idential unit in a tand by mode, whih an fail due to hardware and ommon aue failure. Initially, one unit i in operational mode and eond in tand by mode. The operator unit may fail partially or totally. In ae of total failure of the firt unit, tand by unit beome operational whih work with full effiieny. It i aumed that whenever there i a hardware failure, ytem goe to partial failure mode firt and then to total failure of both unit and ommon aue failure. The failure time ditribution are exponential while that of repair are arbitrary. Goel, ingh& Kihore [] tudied the analyi of two idential unit tand by redundant ytem with two type of failure namely hardware and ommon aue failure. upplementary variable tehnique ha been employed to obtain variou probability tate. The tand by, tate behaviour, peial aue and numerial example are deribed by them. Dhillon and Anude[] alo preented three Markov model and it wa hown that ytem may fail due to ommon aue failure only, where both the unit are ative. ome reearher have independently worked for two idential unit and the independent unit operating in hanged environment normal/tormy ondition. The repair faility i available when either one unit i in operation mode. If during the repair of the firt unit, the eond unit alo fail, then the ytem i aumed to go to down tate. Alo, it i aumed that the ytem may fail due to ommon aue failure from any of the operating tate, e.g., due to vibration, temperature, fire, flood, operator IN : Vol. No. Deember 87

2 Dharmvir ingh Vahith et al. / International Journal of Engineering iene and Tehnology IJET and maintenane error, deign effiieny, et. Alo, if ytem goe from normal loation to tormy loation and vie-vera by ontant tranition rate.. Notation a, : Normally-operating tate b, : Degraded tate, 5 : Failed tate d D / D / D / D / D / D d t x y z w: / / / / / dt t x y z w e / / / : ontant failure due to minor/major ommon aue. f λ : ontant failure due to ommon aue. g t : at time t the ytem in tate h x, t Δ : the ytem i in tate at time t due to minor failure and elaped repair time lie in the interval x, x+δ x i TO y, t Δ : the ytem i in tate at time t and elaped repair time lie in the interval y, y+δ y j TO, z t Δ : the ytem i in tate at time t and elaped repair time lie in the interval zz, z +Δ. K TT w, t Δ : the ytem i in tate at time t and elaped repair time lie in the interval ww, +Δ w. l x, t Δ : the ytem i in tate 5 at time t and elaped repair time lie in the interval X, X +Δ. x K xdx k i = K x e Where K = β, Ψ, φ, φ, w, i n =, =, otherwie tate. i= IN : Vol. No. Deember 88

3 Dharmvir ingh Vahith et al. / International Journal of Engineering iene and Tehnology IJET tate tranition of diagram w x O.. O t λ β x.. x, t φ z T. O. TO y, t T.. T τ, t T, T xt, 5 TT, wt, T Operable Degraded Failed. Bai Equation and Their olution Tehnique The et of differene differential equation aoiated with the mathematial model are a follow: d + + λ5 = β x xtdx, + w x xtdx, + Ψ y TO xtdx, dt d d + +, + β x x t = dx dt d d + +, +Ψ y TO y t = dy dt d d + +, + φ z T z t = dz dt d d + + φ, w TT w t = dw dt d d + + w x x, t = dx dt The boundary ondition are a follow:, t t φ z z, t dz = + = + φ T, t, x t dx w w, t dw TO TT IN : Vol. No. Deember 89

4 Dharmvir ingh Vahith et al. / International Journal of Engineering iene and Tehnology IJET T, t = T y, t dy TT, t = T z, t dz, t = λ t Initial ondition are We aume that ytem i initially in a normal tate i.e., =, otherwie zero.. olution of The Model Taking Laplae of the equation and and then applying B, we ee a = D B b = D d e + Ψ + A D β T =... + β + T TT = D φ = D f w = λ D The Laplae tranform of the probabilitie that at time t ytem i in up and down tate i given by up = + + T + T = β + + A + + D β Ψ.. + φ + w down = TT + = + λ D It i worth notiing that up + down = a Where, D = + + λ β + φ + β + A + β + Ψ. λ w β Ψ + φ Ψ IN : Vol. No. Deember 85

5 Dharmvir ingh Vahith et al. / International Journal of Engineering iene and Tehnology IJET β Ψ = φ A φ + + Ψ φ β + Ψ + β + B = + φ +. A φ + β + Ψ + =.. A Ergodi Behaviour of the ytem Uing Abel lemma in Laplae tranform equation and a the following term are obtained. up = + + T + T = + β + A β+ Ψ + D' 5 w down = TT + = μφ + λ M D ' Alo up + down = 6 d where, D' = D = d and k K M K = φ, w = 7 6. artiular ae A a partiular ae, let u aume that the repair follow exponential time ditribution θ Let θ = +θ Where θ i any one of thee Ψ, φ, β, φ, w a b O = E F E TO =.. G + + β + +Ψ H T = E H T = + φ E λ = + w E = 8 d e f Where a β φ β E. = + + λ6 + β G + + φ + + β + + β + +Ψ IN : Vol. No. Deember 85

6 Dharmvir ingh Vahith et al. / International Journal of Engineering iene and Tehnology IJET w F Ψ H φ Ψ λ + w + +Ψ + φ + φ +Ψ b F = + G + + φ + + β + +Ψ + + β φ φ G = + + φ + + β + +Ψ + φ + + φ + +Ψ d H = G + + φ + + β + +Ψ 7. Non-Repairable ytem The Laplae tranform of thi reliability, when all repair are zero of the ytem i given by A B D R = λ The reliability of the omplete ytem i R t Ae Be e De Where a + λ at at at = λ A = λ λ λ λ λ λ. And we have B =.. + λ =.. + λ D =.. + λ The mean failure time i given by Goel, ingh and Kihore []. MTTF = R tdt A B D = λ,,, andλ, et. We alulated value of A, B,, D, for partiular value of we find thee a follow: In all thee aue φ = φ =Ψ = β = If =., =., =., =., λ =. Then A =.55, B = 5, =., D = 7 And alo MTTF = 79. =., =., =., =., λ =.5 If 9 IN : Vol. No. Deember 85

7 Dharmvir ingh Vahith et al. / International Journal of Engineering iene and Tehnology IJET Then A = 8.99, B =, = 5., D =. And alo MTTF = 76.5 =., =., =., =., λ =.9 If Then A = 55, B =.5, = 7.7, D =. And alo MTTF = 6.5 From, and, we obtain Rt, repetively a follow: a b Rt = 5.5e + 5e +.e.7e.t.t.5t.t Rt 8.99e e 5.e.e.5t.t.t.t = + + Rt = 55.e +.5e + 7.7e.e.9t.t.t.9t From and at t =,,, 6, 8, we get a R =. b d R = 5.5e + 5e +.e.7e R 5.5e 5e.e.7e = R6 = 5.5e + 5e +.e.7e From and at t =,,, 6, 8 we obtain a R = 7.5 b d e R = 8.99e + e + 5.e.e R 8.99e e 5.e.e.6+ e.8..6 = R6 = 8.99e + e + 5.e.e R8 = 8.99e + e + 5.e.e..6.. On alulating thee value, we get from, 5 and 6 a follow: Table R R R R R onluion The above alulated value how that the reliability of the ytem dereae a the time progree. It i alo een that MTTF alo dereae due to variou failure rate. IN : Vol. No. Deember 85

8 Dharmvir ingh Vahith et al. / International Journal of Engineering iene and Tehnology IJET 9. Referene [] Dhillon B.. and Anude O.. 99: aue failure analye of a parallel ytem with warm tand by. Miroeletron and Reliability,, pp. 9-. [] Goel. K., ingh. B. and Jai Kihore : Reliability analye of two unit redundant ytem with three type of failure under waiting time to repair. Ata ienia India, XXIX, pp [] Goel L. R., Gupta R. and Tyagi. K. 995: Analye of two unit tand by ytem with preparation time and orrelated failure and repair. Miroeletron and Reliability, 5, pp [] Dharmvir ingh Vahith and A.. Gouia Banu: Reliability and ot-effetivene of a Redundant ytem with Three Type of Failure and Repair. International Journal of tatitika and Mathematika, Volume, Iue, pp 7-5 IN : Vol. No. Deember 85