FAULT TREES AND RELIABILITY BLOCK DIAGRAMS. Harry G. Kwatny. Department of Mechanical Engineering & Mechanics Drexel University

Transcription

1 SYSTEM FAULT AND Hrry G. Kwtny Deprtment of Mechnicl Engineering & Mechnics Drexel University

2 OUTLINE SYSTEM RBD Definition RBDs nd Fult Trees System Structure Structure Functions Pths nd Cutsets Reliility Redundncy

3 SYSTEM A reliility lock digrm (RBD) provides success oriented view of the system. RBD s provide frmework for understnding redundncy. RBDs fcilitte the computtion of system reliility from component reliilities. RBDs nd fult trees provide essentilly the sme informtion. Below is the RBD for the ckup power supply. Off-site power Trnsfer switch Dieselgenertor

4 DEFINITION SYSTEM A reliility lock digrm (RBD) is defined s follows: A reliility lock digrm is grph whose edges re the system components. There re pir of nodes clled terminl nodes () nd () in the ckup power supply digrm. If there is pth etween the terminl nodes which contins only edges with functionl components, the entire system is functionl. Otherwise it is not functionl.

5 SERIAL & PARALLEL CONFIGURATIONS A 1 A 2 Seril A 1 SYSTEM A 2 Prllel In the seril configurtion, filure of either component, A 1 or A 2 cuses system filure. In the prllel configurtion, oth components must fil in order for the system to fil redundncy

6 EXAMPLE: FIRE PUMP SYSTEM SYSTEM The reliility lock digrm for the fire pump system is shown elow. The redundncy of the pumps is clerly evident. Fire Pump 1 Engine Fire Pump2 Vlve

7 & FAULT 1 TOP TOP SYSTEM A 1 occurs A 2 occurs A 1 occurs A 2 occurs A 1 A 2 A 1 A 2 A 1 A 1 A 2 Seril A 2 Prllel Notice tht the fult tree tkes filure perspective, wheres the reliility lock digrm tkes success perspective.

8 & FAULT 2 TOP SYSTEM A 2 A 1 A 1 G1 A 3 A 2 A 3 A simple seril prllel composition.

9 SERIES SYSTEM 1 2 n A system composed of n susystems is clled series structure if the filure of ny one component cuses filure of the complete system.

10 PARALLEL 1 SYSTEM 2 n A system composed of n susystems is clled prllel structure if it opertes if ny one or more of its components opertes.

11 k-out-of-n S two-out-of-three structure 1 2 SYSTEM A system tht is functioning if nd only if t lest k of its n components is functioning is clled k-out-of-n structure. A 1-out-of-n structure is prllel structure.

12 DEFINITIONS - SYSTEM A system with n components is clled system of order n. x i denotes the stte of component or susystem i, x i = { 1 the component is functioning 0 the component is filed x denotes the stte of the entire system, { 1 the system is functioning x = 0 the system is filed, i = 1,..., n The structure function is function φ (x 1,..., x n ) ssocited with given system, such tht x = φ (x 1,..., x n )

13 seril structure SYSTEM prllel structure φ (x 1,..., x n ) = x 1 x 2 x n φ (x 1,..., x n ) = 1 (1 x 1 ) (1 x 2 ) (1 x n ) = 1 n (1 x i ) i=1 φ (x 1,..., x n ) = mx i {1,...,n} k-out-of-n structure x i { n 1 φ (x 1,..., x n ) = i=1 x i k n 0 i=1 x i < k

14 COHERENT S SYSTEM A component is irrelevnt if it hs no effect on the functioning of the system, i.e., the i th component is irrelevnt if φ (x 1,... x i 1, 0, x i+1,..., x n ) = φ (x 1,... x i 1, 1, x i+1,..., x n ) Otherwise it is clled relevnt. for ll x 1,... x i 1, x i+1,..., x n Assumption: the system will not run worse if filed component is replced y functionl component φ (x 1,..., x n ) is nondecresing function of ech of the vriles x 1,..., x n. A system is coherent if ll of its components re relevnt nd its structure function is nondecresing. By focusing on coherent structures we rule out certin pthologies.

15 SYSTEM The set of components of system of order n is C = {1, 2,..., n} A pth set, P, is suset of C which y functioning ensures tht the system is functioning. A pth set is miniml if it cnnot e reduced without losing its sttus s pth set. A cut set, K, is suset of C which y filing cuses the system to fil. A cut set is miniml if it cnnot e reduced without losing its sttus s cut set

16 DEFINITIONS - SYSTEM A system with n components is clled system of order n. A i denotes the event tht the component or susystem i, i = 1,..., n is functioning t time t. A denotes the event tht the entire system is functioning t time t. P (A i ) = R i (t) is the reliility of the i th susystem. P (A) = R (t) is the reliility of the entire system.

17 SERIES SYSTEM Note: A = A1 A 2 A n A c = A c 1 Ac 2 Ac n A Ai, i = 1,..., n R (t) R i (t) A seril system reliility is no greter thn the reliility of ny susystem! Suppose tht the filure events A i re mutully independent, then R (t) = P (A) = P ( n i=1 A i) = n i=1 P (A i) = n i=1 R i (t)

18 PARALLEL SYSTEM A = A 1 A 2 A n If the susystem filure events re independent: A c = A c 1 Ac 2 Ac n P (A c ) = P (A c 1 ) P (Ac 2 ) P (Ac n) P (A) = 1 P (A c ) = 1 n i=1 P (Ac i ) = 1 n i=1 [1 P (A i)] Consequently, for independent events: R (t) = 1 n i=1 (1 R i (t))

19 k-out-of-n S SYSTEM Consider identicl components, The proility filure of filure over specified time period for single component is p, so the component reliility is r = 1 p, Let M denote the numer of components tht fil in the specified time period, then P (M = m) = C n m }{{} # wys to get m/n p m }{{} m filures (1 p) n m = Cm c (1 r) m r n m } {{ } n m survivers The k/n system will survive if there re no more thn n k filures, so R = n k m=0 Cn m (1 r) m r n m

20 DEFINITIONS SYSTEM Redundncy is the dupliction of criticl components in system in order to improve reliility. Redundncy is normlly prllel connection of identicl components cn e ctive or stndy Active prllel Stndy prllel

21 OF ACTIVE AND STANDBY SYSTEMS SYSTEM Consider redundnt (prllel) comintion of two identicl components. Let T 1, T 2, T denote the filure times of component 1, component 2, nd the system, respectively. the units re identicl, so R1 (t) = R 2 (t) ssume the filures of the two units re independent events. The system reliility of the ctive prllel structure is R (t) = P (T 1 > t T 2 > t) = P (T 1 > t) + P (T 2 > t) P (T 1 > t) P (T 2 > t) R (t) = R 1 (t) + R 2 (t) R 1 (t) R 2 (t)

22 OF ACTIVE AND STANDBY SYSTEMS 2 SYSTEM The stndy system does not strt operting until the primry unit fils. The system cn survive until time t if the primry unit survives until time t or the primry unit fils efore time t, ut the second unit survives until time t: R s (t) = P (T 2 > t T 2 > T 1 ) = P (T 1 > t) + P (T 1 < t T 2 > t) The two possiilities cn e restted s T 1 > t or T 1 occurs t τ < t nd T 2 > t τ. Notice tht P (τ < T 1 < τ + dτ) = f 1 (τ) dτ P (τ < T 1 < τ + dτ T 2 > t) = R 2 (t τ) f 1 (τ) dτ P (T 1 < t T 2 > t) = t R2 (t τ) f1 (τ) dτ 0 t R s (t) = R 1 (t) + R 2 (t τ) f 1 (τ) dτ 0

23 EXAMPLE: CONSTANT FAILURE RATE SYSTEM Components with constnt filure rte λ hve reliility R (t) = e λt An ctive prllel structure hs reliility R (t) = 2e λt e 2λt A stndy prllel structure hs reliility R s (t) = (1 + λt) e λt RHtL Stndy prllel Active prllel 0.2 Single component λt

24 HIGH- AND LOW-LEVEL High (SYSTEM)-level Redundncy SYSTEM Low (COMPONENT)-level Redundncy

25 REVISITED SYSTEM Consider system with stte vector x = {x 1,..., x n} nd structure function φ 1 (x), nd second system with stte vector y = {y 1,..., y m} nd structure function φ 2 (y). if these systems re connected in series then the whole system is of order n + m, nd its structure function is φ (x, y) = φ 1 (x) φ 2 (y) if these systems re connected in prllel then the whole system is of order n + m, nd its structure function is φ (x, y) = mx {φ 1 (x), φ 2 (y)}

26 SYSTEM LEVEL VS COMPONENT LEVEL SYSTEM Suppose the two systems in the prllel structure re identicl, i.e., we hve system level redundnt configurtion. In this cse, φ S (x, y) = mx {φ 1 (x), φ 1 (y)} φ 1 (x 1y 1,..., x ny n) Consider component level redundnt configurtion of system one, the structure function is φ C (x, y) = φ 1 (mx {x 1, y 1},..., mx {x n, y n}) But, for coherent systems, φ 1 (mx {x 1, y 1},..., mx {x n, y n}) φ 1 (x 1y 1,..., x ny n) So, φ C (x, y) φ S (x, y) Thus, in generl, we get etter (more relile?) system through component redundncy rther thn system redundncy.

27 SYSTEM LEVEL VS. COMPONENT LEVEL 2 A 1 A 2 A K SYSTEM A B A B A B A B K A2 B1 A2 B2 A2 B3 A2 B K A3 B1 A3 B2 A3 B3 A3 B K AK B1 AK B2 AK B3 AK B K Consider system with K miniml cutsets, A 1, A 2,..., A K. For system level redundnt configurtion, plce n identicl system with cutsets leled B 1, B 2,..., B K in prllel with the originl. The redundnt system cutsets re ll comintions A i B j, i, j = 1,..., K. A component level redundnt system hs cut sets A i B i, i = 1,..., K. Consequently, R C R S.

28 EXAMPLE: SYSTEM VS COMPONENT 2 { 1},{ 2, 3} SYSTEM 3 { 1, 1},{ 1, 2, 3}, { 1, 2, 3},{ 2, 3, 2, 3} { 1, 1},{ 2, 3, 2, 3} 1 3 3

29 LIMITATIONS SYSTEM Common-mode filures re cused y dependencies tht cuse redundnt components to fil simultneously. common power supply shred environmentl stresses common mintennce issues Lod shring cn cuse reliility degrdtion in ctive prllel systems. filure of one unit cn cuse incresed stress on remining units, e.g., engines, pumps, etc., switching filures cn occur in stndy prllel systems