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Journal of Mechanical and Technology 25 (7) (2011) 1717~1725 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0415-y

Reliability prediction of systems with competing failure modes due to component degradation† Young Kap Son* Department of Mechanical & Automotive Engineering, Andong National University, 760-749, Korea

(Manuscript Received April 8, 2010; Revised January 20, 2011; Accepted March 11, 2011)

------Abstract

Reliability of an engineering system depends on two reliability metrics: the mechanical reliability, considering component failures, that a functional system topology is maintained and the performance reliability of adequate system performance in each functional configura- tion. Component degradation explains not only the component aging processes leading to failure in function, but also system perform- ance change over time. Multiple competing failure modes for systems with degrading components in terms of system functionality and system performance are considered in this paper with the assumption that system functionality is not independent of system performance. To reduce errors in system reliability prediction, this paper tries to extend system performance reliability prediction methods in open literature through combining system mechanical reliability from component reliabilities and system performance reliability. The ex- tended reliability prediction method provides a useful way to compare designs as well as to determine effective maintenance policy for efficient reliability growth. Application of the method to an electro-mechanical system, as an illustrative example, is explained in detail, and the prediction results are discussed. Both mechanical reliability and performance reliability are compared to total system reliability in terms of reliability prediction errors.

Keywords: Component reliability; Degradation; Mechanical reliability; Performance reliability; Sample-based method; Total system reliability ------

not include hard failure. Condra [5] stated “Reliability is qual- 1. Introduction ity over time.” The definition is a much simpler customer- In , there are two major failure cases oriented definition, and it is mainly based on quality over time, of practical importance. These are a) hard failures that imply not functionality. Yang et al. [2] decomposed system failure complete breakdown in functionality, and b) soft failures mode into degradation failure mode and catastrophic failure whereby the system is functional but the performance meas- mode, and they defined total system reliability function RT(t) ures are out of conformance due to mainly component degra- at a time t as dations [1]. For hard failures, most traditional reliability stud- T ies have focused on so-called mechanical reliability and use R ()tRtRt= DC () () (1) binary state reliability models based on parallel/series/ com- plex configurations [2]. For soft failures, performance reliabil- where RD(t) is related to system response degradation, and ity is defined as the that system performance RC(t) to catastrophic failure in component. In addition, com- measures are within specification limits for the lifetime [3]. ponent or device reliability has been evaluated through inte- There have been research activities to relate reliability con- grating catastrophic failures and degradation failures in the cepts to performance reliability. Styblinski [4] defined “drift aspect of multiple competing failure modes [6, 7]. reliability” as a probability that the system will perform satis- Savage and Carr revisited definition of quality and reliabil- factorily (will not fail) for a specific period of time under the ity and related them through a framework that emphasized stated environmental conditions, provided that the only cause conformance and functionality [8]. They proposed that a struc- of failure is drift (or degradation) of component values in time. ture of reliability consisted of functionality and quality, Drift reliability considers soft failure in the circuit and does wherein functionality is related to component reliability and quality to system responses. They defined total system reli- † This paper was recommended for publication in revised form by Associate Editor ability as a probability that system performance standard is Tae Hee Lee *Corresponding author. Tel.: +82 54 820 5907, Fax.: +82 54 820 5044 satisfied, conditional on the system being in a functional to- E-mail address: [email protected] pology, and it can be rewritten as © KSME & Springer 2011

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ft analyzed using degrading characteristics of components and T R ()tStTtTt=⋅∑{} Pr(()|ii ())Pr( ()) (2) analytical system responses (i.e., strength of a structure). Sys- i=1 tem reliability was predicted through applying traditional stress-strength interference (SSI) reliability models presented where S(t) is the event of successful system performance and in Ref. [11] assuming that Pr(S(t)) = 1 in Eq. (2). In the mod-

Ti(t) is the event that the system exists in functional topology els, one-dimensional probability distribution was used to de- (ft). Pr(S(t)|Ti(t)) is the probability of conformance for each scribe each stochastic process such as stochastic loading and functional topology, and Pr(Ti(t)) is the topological probability strength aging, and both processes were assumed to be statisti- for each functional topology Ti. Therefore, total system reli- cally independent. As well, Andrieu-Renaud et al. proposed a ability becomes system performance reliability when the term time-variant FORM (first order reliability method) to evaluate

Pr(Ti(t)) in Eq. (2) is assumed to be unity, and it becomes sys- an out-crossing rate [12]. They predicted reliability of a struc- tem mechanical reliability when the term Pr(S(t)|Ti(t)) is as- tural system with strength aging due to corrosion. The reliabil- sumed to be unity. It is necessary to consider the term ity function was built numerically at discrete time intervals.

Pr(S(t)|Ti(t)) for accurate reliability evaluation if mechanical For Case (d), there are two different approaches: tracking- reliability is not independent of performance reliability. based approach and non-tracking based approach, assuming Consideration of component degradation links to reliability that there is no failure in system functionality due to degrada- problems. Modeled degradation data have been used to infer: tions i.e. Pr(Ti(t)) = 1 in Eq. (2). The tracking-based approach - Case (a): component lifetime distribution or reliability that uses Monte-Carlo (MCS) takes samples of the function, component distributions at t = 0 and traces their paths using - Case (b): system mechanical reliability from inferred their degradation models to provide time-variant system re- component reliabilities, sponses [4, 13, 14]. Through tracking and comparing the time- - Case (c): system mechanical reliability from component variant system responses with response specifications, system degradation data using analytical system response models, performance reliability is predicted. For the non-tracking - Case (d): system performance reliability using both com- based approach, system performance reliability has been pre- ponent degradation data and mechanistic system models for dicted through relating time-variant system responses to their multi-response systems, and response specifications, by time-variant limit-state functions. - Case (e): system dependability through integrating system Then, based on a set-theoretic concept, the incremental failure mechanical reliability and system performance reliability. region, emerging from success region in the limit-state func- tions during a time interval, is approximated using only two For Case (a), using both predicted degradation models X(t) contiguous discrete times. Finally, system cumulative distribu- and the pre-defined particular critical levels of degradation, tion function at a time is evaluated through summing up prob- either life-time distribution or reliability is inferred. Degrada- abilities of the incremental failure regions over time. Either tion data over time are fitted to degradation path curves or non-sample based method using FORM [1, 12, 13] or sam- degradation distributions, and then the fitted degradation pling method based on MCS [16] has been used to evaluate models are used to infer failure times. the incremental failure probability. For Case (b), component reliability functions are inferred as For Case (e), the fundamental idea of evaluating system de- mentioned in Case (a). Then, system reliability is predicted pendability comes from Case (d); system mechanical reliabil- using series/parallel system reliability models assuming that ity is evaluated considering a system-level response such as a Pr(S(t)) = 1 in Eq. (2). In the prediction, all components are mechanic stress that component interactions provide, not assumed to be independently working and degrading over component reliabilities related to degradation [17]. Cases (c) time. For examples, Liao and Elsayed [9] predicted system and (d) are integrated in system level to evaluate system de- mechanical reliability of a brake system using component pendability that considers both mechanical and performance reliabilities and parallel/series system reliability models. Coit reliability simultaneously. In general, system maintenance et al. [10] inferred component reliabilities using a distribution- policy is based on component reliability . Thus, it based approach, and then predicted system mechanical reli- would be difficult to determine maintenance periods to either ability function through applying a series system reliability fix or replace critical components using the system mechani- model. For example, predicted reliability RM(t) of a system cal reliability based on a system-level response. It follows that comprising two independent components (monotonically de- system reliability evaluation based on Cases (b) and (d) would creasing X1(t) and X2(t) with each lower critical level ζ1 and be required for efficient maintenance policy in order to per- ζ2) has the form form effective reliability growth. Therefore, it is necessary to evaluate system reliability through integrating both system M Rt()=> Pr{( Xt112 ()ζζ) I ( Xt () > 2)} (3) mechanical reliability from component reliabilities and system performance reliability. =>Pr{}{}Xt11 ()ζζ Pr X 2 () t > 2 . This paper, related to Cases (b) and (d), considers multiple For Case (c), mechanical reliability of a system has been competing failure modes for systems with degrading compo-

Y. K. Son / Journal of Mechanical Science and Technology 25 (7) (2011) 1717~1725 1719 nents connected in series in terms of system functionality and carried out to model degradation data [18]. Degradation data system performance, assuming that system functionality is not X(t) with distribution parameters versus time (p′(t)) are ex- independent of system performance. System performance pressed as a function of initial distribution parameters p and reliability prediction methods in open literature are extended time t, and have the form p′(t) = f(p, t). For example, Yang et to predict total system reliability of systems. The extended al. [19] used the s-normal random process to model degrada- method combines system mechanical reliability from compo- tion data under the assumption that at an individual time ti, the nent reliabilities and system performance reliability, and it observed degradation data follows a s-normal distribution with would provide an efficient way to evaluate Eq. (2). mean μ′(ti) and standard deviation σ′(ti) at time ti. Total system reliability modeling using component degrada- For the two types of degradation modeling, Son et al. pro- tions is described in Sec. 2. Specifically, mechanical reliability posed a unified general model based on Rosenblatt transfor- modeling using component reliability and performance reli- mation denoted as Γ(u, v, p) = 0 with standard normal vari- ability modeling using system responses are shown in Sec. 2.1 ables u and distribution parameters p of v [20]. From the uni- and Sec. 2.2, respectively, and total system reliability model- fied degradation model, the degradation sample x(t) of X(t) in ing is given in Sec. 2.3. The total system reliability assessment u-space is expressed as based on a sampling approach is explained in Sec. 3. Applica- tion of total system reliability prediction method to an electro- x()tf= (pu , , t). (4) mechanical system is discussed with some error analyses in Sec. 4. From the point of view of hard failure related to degradation, component reliability has been evaluated using a pre-defined 2. Total system reliability modeling using component particular critical level of degradation that would lead to hard degradations failure. Component degradation model X(t) with a critical level ζ provides a time-variant limit-state function, and the 2.1 System mechanical reliability modeling functions for a upper critical level ζU and a lower critical level Most components exhibit unavoidable degradations over ζL have the forms, respectively, time. Degradation arises from environmental conditions or g(())Xt=−ζ Xt () stresses under which a component operates. There are two U (5) gXt(())=− Xt ()ζ . general types of degradation modeling being widely used [18]: L the degradation path curve approach, and the degradation dis- tribution-based approach known as a graphical approach. Thus, reliability of a component with a degradation model X(t) The degradation path curve approach starts with a determi- can be defined as nistic description of degradation path (i.e., a known physical model of degradation over time). In the approach, it is widely Rt()= Pr( g ( X ()) t > 0) . (6) assumed that each sample degrades in the same way under fixed environmental conditions, and each degradation path has Let us consider a system comprising m degrading compo- an identical functional form [18]. Random coefficients are nents (X1(t), L Xm(t)) connected in series, and the compo- introduced to describe variations due to manufacturing proc- nents fail independently. The system mechanical reliability esses in the path curve. Then, statistical distribution parame- function RM(t) can be rewritten in terms of m limit-state func- ters of random coefficients are numerically estimated using tions as, for ∀τ∈[0, t] observed degradation data [10]. A single degradation path M curve X(t) of V (= X(t =0)) under a deterministic environ- Rt()=> Pr(() gX11 ( (ττ )) 0)ILI ( gXmm ( ( )) > 0) (7) mental condition c is a function of both a vector θ and time:

X(t) = f(θ(V, c), t), where the vector θ is composed of constant where gi represents the limit-state function for Xi(t). For ex- and random coefficients. For example, a single linear degrada- ample, system mechanical reliability function expressed as Eq. tion path model with fixed coefficient θ0 and random coeffi- (3), can be rewritten as, for ∀τ ∈[0, t] cient θ1, θ = [θ0, θ1], has the form X(t) = θ0 + θ1t. M The degradation distribution-based approach is based on Rt()=> Pr(() gXt11 ( ()) 0)I ( gXt 2 ( 2 ()) > 0) (8) statistical models, i.e., distribution parameters. In the approach, degradation is characterized by change of distribution parame- where gi = Xi(t) – ζi for i = 1, 2. ters versus time. A probability distribution function is chosen Now, system mechanical reliability RM(t) in terms of the to adequately describe the degradation data at each observa- unified degradation model denoted as Eq. (4) can be expressed tion time. A two-step statistical analysis: (a) estimating the as distribution parameters at each time supposing that the degradation data at each time follows the chosen dis- M ⎛⎞m R (tg )=>∀∈ Pr⎜⎟ (iii ((pu ) ,( ) ,ττ ) 0), for [0, t ] (9) tribution with time-variant distribution parameters, and then i=1 ⎝⎠I (b) fitting time-dependent distribution parameter functions is

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TMP where Xi(t) is mapped to the standard normal vector (u)i using Rt()=⋅ R () tRt (). (15) the distribution parameter vector (p)i. The vector (u)i are ele- M ments of u, and (p)i are also elements of p. Hence, R (t) has a In general, component degradation explains not only compo- general form nent aging processes leading to failure in function, but system performance change over time. It follows that system me- M ⎛⎞m chanical reliability would not be independent of system per- R (tg )=>∀∈ Pr (i (pu , ,ττ ) 0),for [0, t] . (10) ⎜⎟i=1 ⎝⎠I formance reliability. For more accurate reliability estimation, a total system reliability prediction method to combine system mechanical reliability from component reliabilities and system 2.2 System performance reliability modeling performance reliability is required. A system model relating outputs to inputs could be formed Let us consider a system wherein system mechanical reli- by either a mechanistic approach using the interactions of ability is not independent of system performance reliability. components, or an empirical approach using response surface Combination of system mechanical reliability RM(t) in Eq. methodology. In both approaches, the q uncertain perform- (10) with system performance reliability RP(t) in Eq. (14) pro- ance measures such as responses, Z = [Z1, Z2,… Zq] can be vides total system reliability. For notation convenience, setting written as functions of the m system variables V in the explicit gi (the limit-state function related to system performance) as form, gi+m for i = 1, 2,… n, we have the total system reliability in terms of (m + n) limit-state functions as

Zi = zi(V) for i = 1, 2, L , q. (11) T ⎧ nm+ ⎫ R ()tg=>∀∈ Pr⎨ ()i (pu , ,ττ ) 0, for [0, t ]⎬ . (16) The system variables represent component characteristics such i=1 ⎩⎭I as (a) resistance for a resistor in electrical circuits, and (b) either torque constant or winding resistance for a servo motor The reliability model denoted as Eq. (16) provides an efficient in electro-mechanical system. System responses depend on way to evaluate total system reliability in Eq. (2) since reli- time when component degradations X(t) for V are considered ability of both system functionality and system performance th in Eq. (11). Now, the i time-variant system response zi(x(t)) can be simultaneously investigated in the same space, i.e., u- has a form in terms of degradation models following Eq. (4) space. as [20]: 3. Total system reliability assessment ztii(())xpu= z (, , t). (12) Total system reliability expressed as Eq. (16), can be more easily evaluated by considering non-conformance, i.e., g(x(t)) Relating the response to a specification limit by a limit-state < 0, and thus we have the cumulative distribution function, function leads to a time-variant limit-state function of the form complement of total system reliability, as

TT gtii(())xpu=±{ z (,, t) −ζ } (13) Ft()=− 1 Rt () ⎧ nm+ ⎫ (17) =≤∃∈Pr⎨ ()gi (pu , , ττ)0 , for [0, t].⎬ i=1 where zi is a response and ζ is either a lower or an upper limit- ⎩⎭U specification. System performance reliability has been interpreted using a Probability in Eq. (17) herein is numerically approximated series system concept [1, 3-4]. That is, performance reliability using discrete time events based on a finite time step [1]. Con- at time t is the probability that all time-variant responses sat- sider a selected time tL using a fixed time step h. For a time th isfy their critical limits before time t. Thus, system perform- index l = 0, 1,… L, tl (= l·h) denotes the time at the l step. ance reliability for q responses with n limit-state functions can Now Eq. (17) becomes be expressed as [1]

T ⎪⎧ Lnm⎛⎞+ ⎪⎫ ⎛⎞n Ft()Lil=≤ Pr⎨ ⎜⎟((,,)0 gpu t ⎬ . (18) P ⎪ li==01⎝⎠⎪ R (tg )=>∀∈ Pr⎜⎟ (i (pu , ,ττ ) 0),for [0, t] . (14) ⎩⎭UU ⎝⎠Ii=1

For an instantaneous failure region of the ith limit-state func- 2.3 Total system reliability modeling tion at a discrete time tl, Egtli, =∈{uU: i ( pu , , l) ≤ 0} , we If components comprising a system degrade and fail with- have a system instantaneous failure region El at time tl as out leading to system performance change, system mechanical reliability is independent of system performance reliability, nm+ Ell==EE,1UULU l ,2 E lnm ,+ E li , . (19) and thus total system reliability has the form Ui=1

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System incremental failure probability from time tl during the Table 1. Mechanistic models and performance specifications of a servo system. time step h is written from Refs. [1, 16] as Response Mechanistic model Specification Δ=Ft(pEEE ,llll ) Pr(+1 U ) − Pr( ) . (20) 4JRmm( R+ R4 ) Smaller-is-best Z (t ) tc = 1 c 2 USL = 0.034 [sec] A random sampling method such as Monte Carlo simulation is K ()2RRRm ++43 1 herein applied to evaluate in Eq. (20) even rR34( Rm + R ) ω = v Target-is-best though FORM could be applicable [1, 15]. Probabilities in Eq. ss KR2 R++ R R 1 243()m LSL = 540, 2 (20) may be evaluated directly by sampling the sign of the Z2 (ωss) 2 rRmm() R+ R4 T2 = 570, appropriate limit-state functions [16]. To obtain the signs, two − τ15 USL = 590 [rad/sec] KRRR2 ()2 ++ 2 test functions for ∃∈inm [1, + ] are defined as m 43 KR3 Larger-is-best Z3 (τo) τo = v1 rRm R2 LSL3 = 0.24 [N-m] ⎧1 if gtil (pu , , )≤ 0 φ (,u t )= ⎨ , (21) 1 l 0 otherwise ⎩ ⎧1 if gtgtilil (pu , , ) or ( pu , , +1 ) ≤ 0 φ (,u t )= ⎨ . (22) 2 l 0 otherwise ⎩

Based on a random sampling method with N samples for u, the number of systems Nf (tl) that do not conform to the speci- fications at time tl, and the number of systems Nf (tl, tl+1) that do not conform to the specifications at times tl and tl+1 are Fig. 1. Schematic of mechatronic servo system. evaluated respectively as τ15 models the load arising from some arbitrary connected N subsystem. The electro-mechanical characteristics of the sys- k N f ()ttll= ∑φ1 (u , ), (23) tem arise from three subsystems; the difference amplifier, the k =1 motor and tacho-generator (M7,9 and G8,10), and the gear train N shown as G12,13. The difference amplifier consists of the three N (,tt )= φ (uk , t). (24) f ll+12∑ l resistors R2, R3 and R4 along with the operational amplifier k =1 (O5,6) of a very large closed-loop gain. System variables of interest comprise the torque constants K, the rotational inertias

Now, we have the approximate incremental failure over time h, J, the winding resistances Rm, the resistance for R2, R3 and R4, ˆ ΔFt()l , as and the gear ratio r (= r12/r13) for the gear train. The three system performance measures related system per- N (,tt )− Nt () formance reliability in this work are 1) the time constant t Δ=Ftˆ T () f ll+1 fl. (25) c l N related to the time for the shaft speed at S to reach steady-state

speed, 2) the steady-state shaft speed at S denoted as ωss, and 3) Then, total system reliability is evaluated using both Eq. (23) the required initial, or starting torque τo to supply the load at with l = 0 and Eq. (25) as point S. The mechanistic models in terms of the electro- mechanical characteristics with the response specification are ˆˆTT Rt()=− 1 Ft () given in Table 1. The value of voltage v1 supplied from a known power supply is uncertain owing to manufacturing ⎡⎤Nt() L (26) f 0 ˆ T =−1{()}.⎢⎥ + ΔFtl variations. The value for load torque τ15 of a known range is ⎢⎥N ∑ ⎣⎦l=0 uncertain owing to the particular end-use. Thus, both v1 and τ15 are designated as random variables. Also, the resistances of

Eq. (26) is also used to evaluate both mechanical reliability resistors R2, R3, and R4, are uncertain due to variation in their including component reliability and performance reliability in manufacturing process, and thus they are considered as random this paper since Nf (tl) and Nf (tl, tl+1) are determined by the variables. Moreover, values for the torque constant K, the mo- corresponding limit-state functions. tor resistance Rm, and the gear ratio r, respectively, that are critical system variables, are considered as random variables.

However, the value for the rotor inertia J of both the motor and 4. Case studies tacho-generator is fixed at the nominal value 1/1000000 kg-m2. The servo system of interest is shown in Fig. 1 with compo- According to Bonnett and Soukup [21], electric motor prob- nents and interconnection models taken from [15]. A voltage lems occur for a variety of reasons, ranging from basic design supply v1 acts as the input, and at the output an applied torque faults and poor manufacturing quality to problems caused by

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Table 2. Component information for a servo system. Thus, the degradation models for the winding resistance R , m System and resistors R2, R3, and R4 versus usage time are written as Distribution variable Distribution parameters Degradation model, t [year] (compo- p xtkkkk( )= μ′ ( t )+=σ ′ ( tu ) for k 2, 6, 7 and 8. (28) nent) i

μ = 0.008534 V 1 1 Normal tol = 2% None For other system variables that have no degradation, we use (K) 1 (σ1 = tol1μ1/300) x(t) = v. For component failure, critical upper specification μ = 2.9 Ω V 2 μμ′ ()tt=× exp(910−3 ) limits for resistance degradations X6(t), X7(t), and X8(t) are 2 Normal tol = 2% 22 (R ) 2 σσ′ (tt )=× exp(1 10−3 ) considered as 11000, 44000, and 11000 [Ω], respectively, and m (σ = tol μ /300) 22 2 2 2 the limit for initial winding resistance X (t) is as 3.103 [Ω]. μ = 0.52575 2 V 3 3 Normal tol = 2% None Thus, each limit-state function has is expressed as (r) 3 (σ3 = tol3μ3/300) μ = 12 volts V 4 gXt12( ( ))=− 3.103(μσ 2′′ ( t ) + 2 ( tu ) 2) , 4 Normal tol = 1% None (v ) 4 1 (σ = tol μ /300) gXt26( ( ))=−+ 11000()μσ 6′′ ( t ) 6 ( tu ) 6 , 4 4 4 (29) V5 μ5 = 1/100 N-m Uniform None g37(Xt ( ))=−+ 44000()μσ 7′′ ( t ) 7 ( tu ) 7 , (τ15) tol5 = 2%

μ6 = 10,000 Ω gXt48( ( ))=−+ 11000()μσ 8′′ ( t ) 8 ( tu ) 8 . V μμ′ (tt )=× exp(1.2 10−2 ) 6 Normal tol = 2% 66 (R ) 6 σσ′ ()tt=× exp(4.510−3 ) 2 (σ = tol μ /300) 66 6 6 6 Thus, component reliability from Eq. (6) with each specifica- μ7 = 40,000 Ω −2 V7 μμ77′ (tt )=× exp(1.3 10 ) tion limit ζ is defined as Normal tol7 = 2% −3 i (R3) σσ77′ ()tt=× exp(5.010 ) (σ7 = tol7μ7/300) μ8 = 10,000 Ω ⎛⎞ζμ− ′()t V μμ′ (tt )=× exp(1.2 10−2 ) ii 8 Normal tol = 2% 88 Rti ()=Φ⎜⎟ for i = 2, 6, 7, 8. (30) (R ) 8 σσ′ ()tt=× exp(4.510−3 ) ⎜⎟σ ′()t 4 (σ = tol μ /300) 66 ⎝⎠i 8 8 8

The system mechanical reliability RM(t) from Eq. (10) is de- application and site conditions. Specifically, they are most fined as, for ∀τ∈[0, t] likely to arise from bearing failures - probably the most com- mon cause - with stator winding insulation degradation a close M ⎛⎞((gX12 ())0)(ττ>>II g 26 ( X ())0) Rt()= Pr⎜⎟. (31) second. The critical degradation limit for motor winding resis- ((gX ())0)((ττ>> gX ())0) ⎝⎠37I 48 tance Rm is related to the insulation system in the motor. The degradation of motor winding resistance above a certain criti- The four time-variant limit-state functions for three responses cal limit is supposed to provide hard failure of the insulation have the forms system in the motor [22]. This hard failure would cause the motor to fail, and thus the servo system fails in system func- g51(pu, , tzt) =−0.045( pu , , ), tionality. For resistors, failure causes of resistors may be deg- g62()pu,, tzt=−589 () pu ,, , radation in resistance as well as dielectric breakdown and (32) gtztpu,, =− pu,, 551, either short or open circuit. Abrupt increase in resistance of a 72()() resistor might cause an open circuit problem. Thus, resistance gtzt83()()pu, , =− pu, , 0.22. degradation for each resistor R2, R3, and R4 above each certain critical limit is supposed to provide hard failure of the resistor. The terms zi(p, u, t) for i = 1, 2, and 3 in Eq. (32) represent a The servo system is assumed to fail in system functionality if time-variant response affecting system performance reliability. the motor and resistors fail due to degradation in resistance. For example, we have the time-variant response for starting Distribution and degradation information for each variable torque τo in g8 as is given in Table 2. Four resistances Rm, R2, R3, and R4 are assumed to degrade. The assumed degradation characteristics zt3(,,pu ) in terms of both means and tolerances (i.e. p = [μ1, tol1, μ2, tol2, ()μσ1117++ut() μ′′() σ 77 () tu =+()μσ444u . L μ8, tol8] ) are given in Table 2. The eight u-v mappings for ()μσ3332++uttuttu() μ′′() σ 226 ()() μ ′′ () + σ 66 () the system variables are given explicitly as (33)

vuiiiii=+μσ for = 1, 2, 3, 4, 6, 7 and 8 (27) System performance reliability is expressed using the limit- vtoltolu5=−μμ 5()()() 55 +2 55 μ Φ 5 state functions in Eq. (32) as

P ⎛⎞8 where Φ represents a cumulative standard normal distribution R (tg )=>∀∈ Pr (i (pu , ,ττ ) 0),for [0, t] . (34) ⎜⎟i=5 function. ⎝⎠I

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Fig. 2. System mechanical reliability with each component’s reliabil- ity. Fig. 3. Total system reliability with mechanical reliability and per- formance reliability. Now, total system reliability for the servo system can be writ- ten as

T ⎧⎫8 R (tg )=>∀∈ Pr⎨⎬()i (pu , ,ττ ) 0 , for [0, t ] . (35) i=1 ⎩⎭I

We predict total system reliability for p = [0.008534, 2, 2.9, 2, 0.52575, 2, 12, 1, 1/100, 2, 10000, 2, 40000, 2, 10000, 2] using N = 10,000 samples up to the time t = 14 [year] with the time step h = 0.2 [year]. First, predicted component reliability values for the motor (Rm), and resistors (R2, R3, R4) using Eq. (30) are shown in Fig. 2. Reliability of R2 is similar to one for R4 within the maximum probability difference of 0.008. The system mechanical reliability is evaluated using Eq. (31), and it is added to Fig. 2. System mechanical reliability (hereinafter M referred to as R (t)) is mainly determined by both Rm and R3 Fig. 4. Error due to consideration of either RM(t) or RP(t) as RT(t). up to 7 years, but resistance degradation of R3 has a greater effect on RM(t) after 7 years than other components. Thus, it M can be stated that reliability of R3 would be critical to R (t). 0.2 in probability. The error increases before about 6 years and Prediction results of total system reliability (hereinafter re- then decreases after that time, and then becomes zero at 8 ferred to as RT(t)) using Eq. (35) and performance reliability years. The error denoted as [RP(t) – RT(t)] is within 0.6 in (hereinafter referred to as RP(t)) using Eq. (34) are shown in probability, and it increases before around 8 years and then Fig. 3, where RP(t) is critical to RT(t) up to 6 years, but RM(t) is decreases after that time. These error characteristics would be critical to RT(t) mainly after 6 years. Thus, soft failures related explained by the points that (a) RP(t) has more influence on to system performance, causing system failures, are expected RT(t) mainly before 6 years, and (b) RM(t) does on RT(t) after 6 to occur in most cases up to 6 years. But, system hard failures years. related to system functionality are predicted to occur fre- Prediction of RT(t) using a simple product of RM(t) and RP(t), quently after 6 years. In this work, repair or replacement of the i.e., RM(t)⋅RP(t) assuming that RM(t) is independent of RP(t), resistor R3 would provide more efficient reliability growth instead of Eq. (35) leads to a prediction error. than any other components in the aspect of system mainte- Fig. 5 represents the prediction error that has a maximum M nance, since reliability of R3 is critical to R (t). value of about 0.019 in probability. The error is negligible There would be error in evaluating RT(t) with some assump- when RM(t) ≈ 0 or RM(t) ≈ 1 where the intersection probability tions. Prediction of RT(t) using RM(t) with the assumption of of failure regions in the limit-state functions related to soft and Pr(S(t)) = 1 in Eq. (2) causes errors in system reliability esti- hard failures from Eq. (35) is very small. However, the error is mation. And prediction of RT(t) as RP(t) with the assumption not negligible in other cases. Thus, we may conclude that RT(t) M P of Pr(Ti(t)) = 1 in Eq. (2) also leads to erroneous reliability is not a simple product of R (t) and R (t), and failure for the estimation results. These errors are evaluated using difference servo system in functionality is not independent of failure in in probability, [RM(t) – RT(t)] and [RP(t) – RT(t)], and they are performance. shown in Fig. 4. The error denoted as [RM(t) – RT(t)] is within

1724 Y. K. Son / Journal of Mechanical Science and Technology 25 (7) (2011) 1717~1725

and System Safety, 72 (2001) 35-45. [4] M. A. Styblinski, Formulation of the drift reliability optimi- zation problem, Microelectronics Reliability, 31 (1) (1991) 159-171. [5] L. W. Condra, Reliability improvement with design of ex- periments, Marcel Dekker Publishing, New York (1993). [6] W. Huang and R. G. Askin, Reliability analysis of electronic devices with multiple competing failure modes involving performance aging degradation, Quality and Reliability En- gineering International, 19 (3) (2003) 241-254. [7] Z. Pan, J. Zhou and P. Zhao, Joint accelerated failure mode modeling of degradation and traumatic times, Proceedings of the 2010 World Congress on Engineering, London, U.K. [8] G. J. Savage and S. M. Carr, Interrelating quality and reli- ability in engineering systems, Quality Engineering, 14 (1)

Fig. 5. Error due to the assumption of independence in failure. (2001) 137- 152. [9] H. Liao and E. A. Elsayed, Optimization of system reliabil- ity robustness using accelerated degradation testing, Proc. 5. Conclusions Annual Reliability and Maintainability Symposium, Alexan- This paper considered competing failure modes for systems dria, Virginia, USA (2005) 48-54. with degraded components in terms of system functionality [10] K. C. Kapur and L. R. Lamberson, Reliability in engineer- and system performance. To reduce errors in reliability predic- ing design, John Wiley & Sons, New York, USA (1977). tion, system performance reliability prediction methods in the [11] D. W. Coit and L. E. John, T. V. Nathan and James R. open literature were extended to predict total system reliability. Thompson, A method for correlating field life degradation The extended total system reliability prediction method com- with reliability prediction for electronic modules, Quality bines system mechanical reliability from component reliabil- and Reliability engineering international, 21 (2005) 713-726. ities and system performance reliability in case system func- [12] C. Andrieu-Renaud, B. Sudret and M. Lemaire, The PHI2 tionality is not independent of system performance. Applica- method: A way to compute time-variant reliability, Reliabil- tion of the prediction method to the servo system showed the ity Engineering and System Safety, 84 (1) (2004) 77-82. importance of this work in reducing reliability prediction er- [13] J. A. van den Bogaard, J. Shreeram and A. C. Brombacher, rors. A method for reliability optimization through degradation The extended reliability prediction method could provide an analysis and robust design, Proc. Annual Reliability and efficient way to evaluate total system reliability using not Eq. Maintainability Symposium, Tampa, Florida, USA (2003) (2) but Eq. (26). The prediction method might be extended to 55-62. reliability prediction of general systems with components [14] J. A. van den Bogaard, D. Shangguan, J. S. R. Jayaram, G. connected in series and/or parallel, and provide a useful way Hulsken, A. C. Brombacher and R. A. Ion, Using dynamic to compare designs as well as to determine both maintenance reliability models to extend the economic life of strongly in- policy and warranty time for efficient reliability growth. Total novative products, Proceedings of the 2004 IEEE Interna- system reliability prediction approach for general engineering tional Symposium on Electronics and the Environment, systems with multi-competing failure modes involving non- Scottsdale, Arizona, USA (2004) 220-225. degrading, stress-related failure mechanisms as well as com- [15] Y. K. Son, S.-W. Chang and G. J. Savage, Economic-based ponent degradation is an on-going research topic. design of engineering systems with degrading components using probabilistic loss of quality, Journal of Mechanical

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[20] Y. K. Son, J.-J. Kim and S.-J. Shin, Non-sample based Young Kap Son received his PhD from parameters design for system performance reliability im- the Department of Systems Design En- provement, Journal of Mechanical Science and Technology, gineering in 2006 at the University of 23 (2009) 2658-2667. Waterloo in Canada. Currently, he is an [21] A. H. Bonnett and G. C. Soukup, Cause and analysis of Assistant Professor in the Dept. of stator and rotor failures in three-phase squirrel-cage induc- Automotive & Mechanical Engineering tion motors, IEEE Transactions on Industry Applications, 28 at Andong National University. His (4) (1992) 921-937. research interests include storage life- [22] S. B. Lee and T. G. Habetler, An online stator winding time estimation of one-shot systems, probabilistic design of resistance estimation technique for temperature monitoring dynamic systems, and of failure for general engineer- of line-connected induction machines, IEEE Transactions on ing systems. Industry Applications, 39 (3) (2003) 685-694.