reliability-based analytic models for fatigue lifetime...

Research ArticleReliability-Based Analytic Models for Fatigue LifetimeDistribution Estimation of Series MechanicalSystems under Random Load considering StrengthDegradation Path Dependence

Peng Gao1,2 and Liyang Xie3

1School of Mechanical Engineering, Liaoning Shihua University, Liaoning 113001, China2School of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China3School of Mechanical Engineering and Automation, Northeastern University, Liaoning 110819, China

Correspondence should be addressed to Peng Gao; [email protected]

Received 13 August 2017; Accepted 16 October 2017; Published 14 November 2017

Academic Editor: RomanWendner

Copyright © 2017 Peng Gao and Liyang Xie. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Time-dependent statistical characteristics of load process and strength degradation process are critical to lifetime distributionof series mechanical systems. Conventional rain-flow counting method for lifetime distribution estimation has limited practicalapplication due to its strict requirement for statistical properties of load process. Besides, dynamic interaction between loadprocess and strength degradation process results in strength degradation path dependence (SDPD). SDPD and failure dependenceof components jointly bring considerable difficulties in prediction of system lifetime distribution and system residual lifetimedistribution. To address these problems, reliability-based analytic models for estimation of whole lifetime distribution and residuallifetime distribution of series mechanical systems under random load are developed in this paper, which take the time-dependentstatistical parameters of load process and strength degradation process as the input of the models. Furthermore, SDPD and failuredependence of components are taken into account in the proposed models in an explicit mathematical expression. The resultsshow that SDPD, failure dependence of components, and initial strength dispersion have significant influences on system lifetimedistribution.

1. Introduction

With the function, structure, and working environment ofmechanical products becoming complicated, accurate assess-ment of fatigue lifetime distribution ofmechanical systems intheir full lifetime cycles is encountered with new challengesand has attracted extensive attention from researchers andengineers [1–3]. Various random factors, such as randomnessin material, geometry, and manufacturing of componentsand working load, make fatigue lifetime distribution analysismore complex and difficult. In the field of mechanicalengineering, analytic methods for quantitative estimation ofa systematic fatigue lifetime distribution are mainly basedon lifetime distributions of components in the system andthe systematic structure function. Lifetime distributions of

mechanical components are obtained from fatigue tests,which are usually performed under load with constant ampli-tude. To consider the randomness of working load, rain-flow counting (RFC) method is widely used in fatigue tests.Current fatigue test methods provide an important founda-tion for lifetime evaluation of mechanical products [4].

Some innovative research has been carried out for therandom lifetime distribution estimation with some specifieddistribution types. For instance, Elperin and Gertsbakhdeveloped a method for estimating the exponential meanlifetime in a random-censoringmodel with incomplete infor-mation. In the proposed method, the maximum-likelihoodmethod and the Monte Carlo simulations are combined toobtain the point and interval estimates of the mean valueof the lifetime [5]. Pan et al. suggested using a multivariate

HindawiMathematical Problems in EngineeringVolume 2017, Article ID 5291086, 15 pageshttps://doi.org/10.1155/2017/5291086

https://doi.org/10.1155/2017/5291086

2 Mathematical Problems in Engineering

•••

Stress process sample 1

Stre

ssSt

ress

PDF

of S

tress

Stresst

t

t0

IPDF of stress at t0

PPDF obtained from stress history sample nStress process sample n

Figure 1: Different expressions of load PDFs.

Birnbaum-Saunders distribution and its marginal distribu-tions for lifetime distribution analysis of systems that havemultiple dependent performance characteristics. In the pro-posedmodels, the system degradation was assumed to followthe gamma processes [6]. Asgharzadeh et al. presented amethod for two-parameter bathtub-shaped lifetime distribu-tion estimation based on upper record values by constructingexact confidence intervals and exact joint confidence regionsof parameters [7]. Furthermore, confidence intervals wereestimated via the asymptotic normality of the maximum-likelihood estimators. Kayid and Izadkhah analyzed therandom lifetime distribution by using the order statisticstheory and assuming that the residual lifetime followedthe exponential distribution [8]. Gong et al. proposed amethod for fatigue life estimation of the screw blade inthe screw sand washing machine. The random load wasassumed to follow the Gauss distribution [9]. Moreover, inthe statistics of random load, the Markov chain method,Monte Carlo simulations, and the finite elementmethodwerecombined for the purpose of reducing research cost.Then, therain-flow counting method, the Goodman stress correctionmethod, andMiner’s rules were jointly used to assess lifetimedistribution of the screw blade.

In some circumstances, difficulties could be encounteredin the process of systematic lifetime distribution modeling,which are listed as follows, and there needs an extension ofcurrent modeling methods to overcome these difficulties.(1) For mechanical components, fatigue tests under loadwith constant amplitude are aimed at obtaining lifetime dis-tributions resulting from the randomness of material param-eters. The RFC method is used to deal with the randomnessof load. Material parameters, such as strength, could exhibita dynamic stochastic degradation process which is governedby the time-dependent statistical properties of load [10, 11].According to the stochastic process theory, the randomnessof load can be expressed by two methods: (a) the probabilitydensity function (PDF) at each moment or at each loadapplication, which is always mathematically expressed viaa stochastic process and is referred to as instantaneous pdf(IPDF) in this paper; (b) the PDF obtained directly from aload history sample, which is referred to as the process PDF(PPDF) in this paper. The schematic illustration of IPDF andPPDFof load is shown in Figure 1. It can be learnt that the loadPDF obtained by using the RFCmethod is essentially a PPDF.

Unless the assumption that the load process is ergodic andstationary is strictly satisfied, IPDFs cannot be approximatedto PPDF. When the random load, characterized by PPDFfrom RFC method, is applied on the specimen in the fatiguetests, it infers that the IPDF is approximated by PPDF andthe information about the time dependence characteristics ofIPDFof the stochastic load processmight be lost, which couldresult in the error in the assessment of lifetime distributionand will be explained in detail in Section 2. Therefore, itis of considerable importance to establish analytical modelsfor lifetime distribution assessment of mechanical systemswith the load process and strength degradation process takenas the input of the models and considering the dynamicinteraction between the two processes.(2) As mentioned above, fatigue tests are mainly aimedat lifetime distributions of components. Systematic lifetimedistributions are calculated according to the system failurecriterion or structure function. However, lifetime distribu-tions of components in a mechanical system are alwaysmutually statistically correlative in the process of transferringload or motion. Therefore, it is difficult to provide anexplicit expression of the system structure function. Further-more, the implicit structure function is time-dependent dueto the time-dependent load process. The assumption thatcomponent lifetime distributions are mutually statisticallyindependent could lead to an error in the systematic lifetimedistribution estimation.(3) For mechanical systems, residual lifetime analysis isquite vital to their fault diagnosis and maintenance. Whenconsidering the statistical dependence of component lifetimedistributions and the complexity of the stochastic load pro-cess, it is necessary to improve the conventional methods forresidual lifetime analysis which are always performed basedon independent component lifetime distributions.

To address the aforementioned problems, analyticmodelsfor fatigue lifetime distribution estimation of mechanicalsystems under random load are developed in this paper,which can take the time-dependent statistical characteristicsof load and the failure dependence of components into con-sideration and provide an explicit expression of the PDF ofsystematic lifetime that characterized the systematic lifetimedistributions. Moreover, residual lifetime models of depen-dent mechanical systems are established. Besides, numericalexamples are given to validate the proposed models and

Mathematical Problems in Engineering 3

illustrate key factors that have great influences on systemlifetime distributions.

2. Randomness and Dependence of FatigueLifetime Distributions of Components

For electronic elements, lifetime always follows the exponen-tial distribution, which is represented by the parameter ofconstant failure rate [12]. The assumption of constant failurerate about the electronic elements facilitates the regularlifetime tests or the accelerated lifetime tests. However, formechanical components, time-dependent random vibrationload, random impact load, and randomness in the materialparameters caused by manufacturing are always encounteredin practice [13–15]. Moreover, the time-dependent stochas-tic load process and the corresponding time-dependentdegradation process of the material parameters are mutuallystatistical correlative.Meanwhile, the lifetime distributions ofmechanical components are quite sensitive to these randomfactors in the working load and the material parameters.Therefore, sometimes, it is impractical to analyze the impactsof these random factors on the lifetime distributions ofmechanical components and the systems only via fatiguetests. It is important to develop analytical models of lifetimedistributions of mechanical components and mechanicalsystems, which takes these random factors as their inputparameters and could be used to provider further informa-tion for improvement of design and manufacturing.

2.1. Influence Factors of Component Lifetime Distributions. Asa matter of fact, a PPDF is a comprehensive embodiment ofall the IPDFswithin the time periodwhen PPDF ismeasured.Only when the assumption that the load is ergodic andstationary is strictly satisfied, PPDF steadily approximate toIPDF. Otherwise, the PPDF mathematically have no directfunctional relationship with IPDFs and the IPDF cannot beapproximated by PPDF. RCF method is used to acquire aPPDF of load. When the load is not ergodic or stationary,fatigue tests based on a PPDF by using the RFCmethod couldlead to error in fatigue lifetime distributions. Furthermore,the statistical characteristics of strength degradation processare determined by the stochastic load process. The twocorrelative stochastic processes both contribute to the lifetimedistribution of components. The error in component lifetimedistribution mentioned above due to the incorrect usage ofRFC method could be magnified by the complex interactionbetween the two correlative stochastic processes.

For instance, suppose that the residual strength (MPa)of the components under load application for 𝑛 times isexpressed as follows:

𝑟 (𝑛) = 𝑟(1 − 𝑛∑𝑖=1

𝑥2𝑖109) , (1)

where 𝑥𝑖 is the magnitude of load at the 𝑖th load applicationand 𝑟 is the initial strength that is equal to 450MPa. In theload process, the magnitude of load at each load applicationfollows the normal distribution with the standard deviation

of 30MPa and the mean value of 𝜇𝑠(𝑛) (MPa) expressed asfollows:

𝜇𝑠 (𝑛) = 200 (1 + 0.001𝑛) . (2)

The actual working process is simulated by the Monte Carlosimulations (MCS) with the flow chart shown in Figure 2.In Figure 2, the load process is generated according to thegiven load IPDF and the PDF of the component lifetime isshown in Figure 3. Besides, the PPDF of load is obtained viathe statistics of the simulated IPDF of load in Figure 2. Then,with the PPDF taken as the approximate IPDF, the PDF of thecomponent lifetime according to the flow chart in Figure 2 isshown in Figure 3.

From Figure 3, it can be seen that it could result in largeerror in lifetime distribution estimation to approximate theIPDF by PPDF due to neglecting the information of time-dependent statistical characteristics of load. Thus, attentionshould be paid to the stochastic characteristics of loadprocess when using the RCF method for lifetime assessmentof mechanical components. Furthermore, it is necessary todevelop models for lifetime distribution estimation with thetime-dependent statistical characteristics of load considered,taken as the input of the models.

In addition, to consider the influences of the randomnessof strength degradation process on the component lifetimedistribution, suppose the initial strength 𝑟 follows the normaldistribution with the mean value of 450 (MPa). In the casewhere the standard deviation of 𝑟 is 10MPa, 20MPa, and30MPa, respectively, the PDF of the component lifetimebased on IPDF of load and that based on PPDF of load areshown in Figures 4 and 5, respectively.

From Figures 4 and 5, it can be learnt that, besidesthe statistical characteristics of load process, the statisti-cal characteristics of material parameters have significantinfluences on component lifetime distribution. Moreover,the dynamic interaction between the load process and thestochastic process of material parameters makes it moredifficult to acquire an accurate assessment of lifetime PDF.Therefore, it is necessary to develop a method for systemlifetime distribution evaluation, which takes the stochasticload process, the stochastic degradation process of materialparameters, and their interaction into consideration.

2.2. Dependence of Component Lifetime Distributions.Besides the factors that significantly influence componentlifetime distributions mentioned in Section 2.1, thedependence of component lifetime distributions has greatimpacts on system lifetime distribution. Mechanical systemsare designed for transferring load and motion. In thetransferring process of load and motion, components in thesystem always share the common load sources. The strengthdegradation processes of different components in thesystem are mutually statistically dependent because of theirinteraction with the common load sources. Thus, the systemlifetime distribution cannot be calculated based on thesystematic failure criterion and the lifetime distributions ofcomponents obtained from independent fatigue tests. Toillustrate the influences of the dependence of component


Start

Fit PDF according to

End

Yes

No

No

Yes

Set i = 1, j = 1, r1 = r

Generate load s

i = i + 1

Set j = 1, r1 = r

Calculate remaining strength rj

rj > s?

Set Ti = j

j = j + 1

i = N?

T1, T2, . . . , TN

Determine simulation times N and initial strength r

Figure 2: Flow chart for component PDF fitting.

lifetime distributions on system lifetime distribution, anumerical example is given as follows. The series mechanicalsystem is composed of two components with the statisticalproperties of the load and the material parameters identicalwith those listed in Section 2.1. The standard deviation of theinitial strength is 10MPa. The flow chart of the simulationfor lifetime PDF of an independent series system, in which

the two components are mutually independent, is shown inFigure 6. Besides, when the two components share the sameload, which constitute a dependent series system, the flowchart of simulation for lifetime PDF of the dependent systemis shown in Figure 7. The results from the two simulationsbased on IPDF of load and the simulation results based onPPDF of load are shown in Figures 8 and 9, respectively.


0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

IPDF-based life distributionPPDF-based life distribution

Com

pone

nt li

fetim

e PD

Ff(n

)

0 100 200 300 400 500 600 700 800 900Component life n

Figure 3: Lifetime PDF of components.

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Com

pone

nt li

fetim

e PD

Ff(n

)

500 550 600 650 700 750 800 850 900 950 1000Component life n

(r) of 5 MPa(r) of 10 MPa(r) of 15MPa

Figure 4: Lifetime PDF of components based on IPDF of load.

The failure dependence of components brings large dif-ficulty in system lifetime distribution estimation due to theimplicit system structure function. From Figures 8 and 9, itcan be seen that the time-dependent statistical characteristicsof load process have great influences on system lifetimedistribution. Conventional RCF method for component andsystem lifetime distribution analysis should be used withcaution. Compared with the component lifetime PDF, thesystem lifetime PDF shifts towards the left direction and thedispersion of the lifetime PDF decreases. Besides, the lifetimeof the dependent system is distributed in the interval oflonger lifetime comparedwith the lifetime of the independentsystem.

0.0000.0010.0020.0030.0040.0050.0060.0070.0080.0090.0100.0110.012

Com

pone

nt li

fetim

e PD

Ff(n

)

0 50 100 150 200 250 300 350 400 450 500Component life n

(r) of 5 MPa(r) of 10 MPa(r) of 15MPa

Figure 5: Lifetime PDF of components based on PPDF of load.

3. Reliability-Based Analytic Models forSystematic Lifetime Distribution

3.1. Whole Lifetime Distribution Models of Mechanical Sys-tems. To take into account the interaction between the loadprocess and strength degradation process and consider thedependence of component lifetime distribution, reliability-based analytic models for systematic lifetime distribution aredeveloped in this paper.The proposedmodels are establishedaccording to the relationship between dynamic reliability andthe failure probability. According to the reliability theory, thePDF of the system lifetime can be expressed as follows:

𝑓 (𝑡) = 𝜕𝜕𝑡 (1 − 𝑅 (𝑡)) = − 𝜕𝜕𝑡𝑅 (𝑡) , (3)

where 𝑅(𝑡) is the dynamic reliability of mechanical systems.Thus, the lifetime distribution of mechanical systems, char-acterized by the PDF, can be obtained from the dynamicreliability function. Theoretically, the system reliability 𝑅(𝑡)can be calculated based on the component reliability andthe system failure criterion. For instance, in a mechanicalsystem with 𝑘 components, the reliability of the componentsare denoted by 𝑅𝑖(𝑡) (𝑖 = 1, 2, 3, . . . , 𝑘). Then, the PDF of aseries mechanical system can be expressed as follows:

𝑓series (𝑡) = − 𝜕𝜕𝑡𝑅series (𝑡) = − 𝑘∑𝑖=1

𝜕𝜕𝑡𝑅𝑖 (𝑡)𝑘∏𝑗=1,𝑗 ̸=𝑖

𝑅𝑗 (𝑡) . (4)

The PDF of a parallel mechanical system can be given by

𝑓parallel (𝑡) = − 𝜕𝜕𝑡𝑅parallel (𝑡)= − 𝑘∑𝑖=1

𝜕𝜕𝑡𝑅𝑖 (𝑡)𝑘∏𝑗=1,𝑗 ̸=𝑖

[1 − 𝑅𝑗 (𝑡)] .(5)


Start


End

Yes

No

No

Yes

Yes

No

Set i = 1, j = 1

Set r11 = r1 and r21 = r2

i = i + 1

Set j = 1r1j > s1?

r2j > s2?

j = j + 1

Set Ti = j

i = N?

T1, T2, . . . , TN

Determine simulation times N

Generate initial strength r1 of component 1and initial strength r2 of component 1

Generate load s1 on component 1and load s2 on component 1

Calculate remaining strength r1j of component 1and r2j of component 2

Figure 6: Flow chart for PDF fitting of the independent system.


Start


End

Yes

No

No

Yes

Yes

No


Set i = 1, j = 1

Set r11 = r1 and r21 = r2

Generate common load s

i = i + 1

Set j = 1r1j > s?

r2j > s?

Calculate remaining strength r1j of component 1and r2j of component 2

j = j + 1

Set Ti = j

i = N?

T1, T2, . . . , TN

Generate initial strength r1 of component 1and initial strength r2 of component 1

Figure 7: Flow chart for PDF fitting of the dependent system.


0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

Dependent system Independent system

500 550 600 650 700 750 800 850 900 950 1000System life n

Syste

m li

fetim

e PD

Ff(n

)

Figure 8: Lifetime PDF of systems based on IPDF of load.

0.0000.0020.0040.0060.0080.0100.0120.0140.0160.0180.0200.022

Dependent systemIndependent system

0 50 100 150 200 250 300 350 400System life n

Syste

m li

fetim

e PD

Ff(n

)

Figure 9: Lifetime PDF of systems based on PPDF of load.

Equations (4) and (5) are derived from conventional reliabil-ity theory. However, there exist some problems in the calcula-tion of component reliability 𝑅𝑖(𝑡) and system reliability 𝑅(𝑡),which are listed as follows.(1) As described above, in the case of complex stochasticload process, it is difficult to propose an appropriate fatiguetest scheme to acquire the lifetime distribution of compo-nents. Consequently, the component reliability 𝑅𝑖(𝑡) cannotbe calculated directly via the component lifetime distribution.(2) Dynamic reliability models of mechanical compo-nents are mainly developed via extending the conventionalmoment-based reliability methods, such as the first-ordersecond-moments (FOSM) method [16–20] and the second-order second-moments (SOSM) method. In the conven-tional moment-based reliability models, reliability is calcu-lated according to the performance function, which can be

expressed as the difference between the strength 𝑟 and theload 𝑠 as follows:

𝑧 = 𝑟 − 𝑠. (6)

It should be noted that, in (6), 𝑠 represents generalized loadincluding force, stress, and temperature which is determinedby the failure criterion of the components. When the staticstrength and load are independent variables that follow thenormal distribution, reliability can be accurately computedvia the mean value and the standard deviation of 𝑧 as follows:

𝑅 = Φ(𝜇𝑧𝜎𝑧) , (7)

where Φ is standard normal distribution function. Providedthat the condition of independent normal variables is notsatisfied, a series of approximate reliability algorithm are pro-posed such as the aforementioned FOSMmethod and SOSMmethod. To calculate the dynamic reliability of components,static variables of 𝑟 and 𝑠 in (6) are replacedwith the stochasticstrength degradation process 𝑟(𝑡) and the stochastic loadprocess 𝑠(𝑡) and (6) can be rewritten as

𝑧 (𝑡) = 𝑟 (𝑡) − 𝑠 (𝑡) . (8)

Then dynamic reliability is further calculated accordingto 𝑧(𝑡). However, strength degradation path dependence(SDPD) is neglected when using this method, which couldcause large calculation error as is described in the authors’former research [21].

To illustrate the phenomenon of SDPD in reliabilitycalculation, two simulations are carried out. In the firstsimulation,which are independent of any analytical reliabilitymodels, practical operational processes of mechanical com-ponents are simulated with the flow chart shown in Figure 2.Besides, the distributions of residual strength at each loadapplication are obtained by recording the residual strengthin the first simulation, which is set as the input in the secondsimulation with the flow chart shown in Figure 10. Thus, thereliability obtained from the second simulation is based onthe residual strength distribution at each load application,which is essentially the method expressed by (8). In thesimulations, the deterministic strength degradation law isexpressed by (1) and the initial strength follows the normaldistributionwith themean value of 450MPa and the standarddeviation of 20MPa.The load at each load application followsthe normal distribution with the mean value of 350MPa andthe standard deviation of 20MPa. The component reliabilitywith the two simulations is shown in Figure 11.

It can be seen that it could result in computational errorin reliability when directly using the strength distribution ateach load application in (8) due to neglecting SDPD. Thedetailed explanation of the phenomenon of SDPD can bereferred to in [21].(3)Analogous to lifetime distribution ofmechanical com-ponents, the component reliability is mutually statisticallycorrelative. It is difficult to provide an explicit expressionof the time-dependent system structure function for systemreliability computation.


Start


End

Yes

No

No

Yes

according to the residual strength

distribution obtained in the �rst

simulation


Set i = 1, j = 1

Generate load s

Generate random residual strength r j = j + 1

i = i + 1

Set j = 1

r > s?

Set Ti = j

i = N?

T1, T2, . . . , TN

Figure 10: Flow chart for component reliability based on residual strength distributions.


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Reliability without considering SDPDReliability considering SDPD

Relia

bilit

yR(n

)

0 100 200 300 400 500 600 700 800 900 1000Application times of load n

Figure 11: Comparison between reliability considering SDPD andreliability based on residual strength distributions.

To address the problems above and take the load processand the strength degradation process as the input of proposedmodels, the lifetime distribution models of series mechanicalsystems are established based on the dynamic reliabilitymodels developed in the authors’ previous research [21]. Fora series mechanical system with 𝑘 components, the dynamicreliability can be calculated by

𝑅 (𝑛) = ∫∞0𝑘 [1 − ∫𝑟0

0𝑓𝑟0 (𝑟0) 𝑑𝑟0]

𝑘−1 𝑓𝑟0 (𝑟0)⋅ {𝑛−1∏𝑖=0

[∫𝑟0(1−𝑖 ∫∞

0𝑠𝑚𝑓𝑠(𝑠)𝑑𝑠/𝐶)

𝑎

0𝑓𝑠 (𝑠) 𝑑𝑠]}𝑑𝑟0,

(9)

where 𝑓𝑟0(𝑟0) is the PDF of initial strength 𝑟0, 𝑛 representsthe load application times, and 𝑓𝑠(𝑠) is load PDF. 𝐶 and 𝑎 arematerial parameters that describe the strength degradationprocess and can be referred to in [21].

In order to consider the relationship between the dynamicreliability and time, the number of load application 𝑛 in thetime duration 𝑡 is expressed as

𝑛 = 𝑤 (𝑡) 𝑡, (10)

where 𝑤(𝑡) is the frequency function of load. Then thedynamic reliability with respect to time can be further writtenas

𝑅1 (𝑡) = ∫∞0𝑘 [1 − ∫𝑟0

0𝑓𝑟0 (𝑟0) 𝑑𝑟0]

𝑘−1 𝑓𝑟0 (𝑟0)

⋅ {𝑤(𝑡)𝑡∏𝑖=1

[∫𝑟0(1−𝑖 ∫∞


𝑎

0𝑓𝑠 (𝑠) 𝑑𝑠]}𝑑𝑟0.

(11)

The PDF of the series system lifetime can be expressed asfollows:

𝑓1 (𝑡) = − 𝜕𝜕𝑡 ∫∞

0𝑘 [1 − ∫𝑟0

0𝑓𝑟0 (𝑟0) 𝑑𝑟0]

𝑘−1 𝑓𝑟0 (𝑟0)

⋅ {𝑤(𝑡)𝑡∏𝑖=1

[∫𝑟0(1−𝑖 ∫∞


𝑎

0𝑓𝑠 (𝑠) 𝑑𝑠]}𝑑𝑟0.

(12)

In addition, when the components are assumed to be mutu-ally independent, according to classic reliability theory, thedynamic reliability of independent series system is given byas follows:

𝑅2 (𝑡) = [∫∞0𝑓𝑟0 (𝑟0)

⋅ {𝑤(𝑡)𝑡∏𝑖=1

[∫𝑟0(1−𝑖 ∫∞


𝑎

0𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]}𝑑𝑟0]

𝑘

.(13)

The life PDF of the independent series system can becalculated by

𝑓2 (𝑡) = −𝑘[∫∞0𝑓𝑟0 (𝑟0)

⋅ {𝑤(𝑡)𝑡∏𝑖=1

[∫𝑟0(1−𝑖 ∫∞


𝑎

0𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]}𝑑𝑟0]

𝑘−1

× 𝜕𝜕𝑡 ∫∞

0𝑓𝑟0 (𝑟0)

⋅ {𝑤(𝑡)𝑡∏𝑖=1

[∫𝑟0(1−𝑖 ∫∞


𝑎

0𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]}𝑑𝑟0.

(14)

Provided that the distributions of residual strength at eachload application are adopted in the reliability calculation andSDPD is neglected, (12) can be rewritten as

𝑅3 (𝑡) = [𝑤(𝑡)𝑡∏𝑖=1

∫∞0𝑓𝑠𝑖 (𝑠𝑖) ∫

∞

𝑠𝑖𝑓𝑟𝑖 (𝑟𝑖) 𝑑𝑟𝑖𝑑𝑠𝑖]

𝑘

, (15)

where 𝑓𝑟𝑖(𝑟𝑖) is the PDF of residual strength at the 𝑖thload application. Correspondingly, the life PDF of the seriessystem can be calculated by

𝑓3 (𝑡) = −𝑘[𝑤(𝑡)𝑡∏𝑖=1

∫∞0𝑓𝑠𝑖 (𝑠𝑖)

⋅ ∫∞𝑠𝑖𝑓𝑟𝑖 (𝑟𝑖) 𝑑𝑟𝑖𝑑𝑠𝑖]

𝑘−1 𝜕𝜕𝑡𝑤(𝑡)𝑡∏𝑖=1

∫∞0𝑓𝑠𝑖 (𝑠𝑖) ∫

∞

𝑠𝑖𝑓𝑟𝑖 (𝑟𝑖) 𝑑𝑟𝑖𝑑𝑠𝑖.

(16)

3.2. Residual Lifetime Distribution Models of MechanicalSystems. Provided that a mechanical system still operatesnormally at the moment 𝑡, the survival operational timefrom 𝑡 is defined as its residual lifetime denoted by 𝑇. Whenthe randomness of load and material parameters is taken


into account, it is necessary to obtain the distribution of𝑇. The residual lifetime distribution prediction is of greatimportance to the fault diagnosis, preventive maintenance,and system health management of mechanical products. ThePDF of the residual lifetime is essentially the function of thePDF of the whole lifetime of mechanical systems. Thus, theproblems of SDPD and failure dependence of componentsin establishing analytical models for whole lifetime distri-bution of mechanical systems also bring large difficulties inmodeling PDF of the residual lifetime. In this section, thePDFs of residual lifetime of mechanical systems are derivedbased on the models provided in Section 3.1, which can take

the stochastic load process, SDPD, and failure dependence ofcomponents into consideration.

According to the conditional probability formula, thePDF of 𝑇 can be expressed by

𝑓𝑇 (𝑇) = 𝜕𝜕𝑇∫𝑡+𝑇𝑡 𝑓 (𝑡) 𝑑𝑡1 − ∫𝑡0 𝑓 (𝑡) 𝑑𝑡

, (17)

where 𝑓(𝑡) is the PDF of the system. Therefore, according to(12), the residual lifetime PDF of the dependent series systemcan be expressed as follows:

𝑓7 (𝑇) = 𝜕𝜕𝑇−∫𝑡+𝑇𝑡 (𝜕/𝜕𝑡) ∫∞0 𝑘 [1 − ∫𝑟00 𝑓𝑟0 (𝑟0) 𝑑𝑟0]𝑘−1 𝑓𝑟0 (𝑟0) {∏𝑤(𝑡)𝑡𝑖=1 [∫𝑟0{1−Φ(𝑖)}𝑎0 𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]} 𝑑𝑟0𝑑𝑡1 + ∫𝑡0 (𝜕/𝜕𝑡) ∫∞0 𝑘 [1 − ∫𝑟00 𝑓𝑟0 (𝑟0) 𝑑𝑟0]𝑘−1 𝑓𝑟0 (𝑟0) {∏𝑤(𝑡)𝑡𝑖=1 [∫𝑟0{1−Φ(𝑖)}𝑎0 𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]} 𝑑𝑟0𝑑𝑡

. (18)

From (14), the residual lifetime PDF of the independentseries system considering SDPD can be given by

𝑓8 (𝑇) = 𝜕𝜕𝑇{−∫𝑡+𝑇𝑡 𝑘 [∫∞0 𝑓𝑟0 (𝑟0) {∏𝑤(𝑡)𝑡𝑖=1 [∫𝑟0(1−𝑖 ∫∞0 𝑠𝑚𝑓𝑠(𝑠)𝑑𝑠/𝐶)𝑎0 𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]} 𝑑𝑟0]

𝑘−1 × (𝜕/𝜕𝑡) ∫∞0 𝑓𝑟0 (𝑟0) {∏𝑤(𝑡)𝑡𝑖=1 [∫𝑟0(1−𝑖 ∫∞0 𝑠𝑚𝑓𝑠(𝑠)𝑑𝑠/𝐶)𝑎0 𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]} 𝑑𝑟0𝑑𝑡}{1 + ∫𝑡0 𝑘 [∫∞0 𝑓𝑟0 (𝑟0) {∏𝑤(𝑡)𝑡𝑖=1 [∫𝑟0(1−𝑖 ∫∞0 𝑠𝑚𝑓𝑠(𝑠)𝑑𝑠/𝐶)𝑎0 𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]} 𝑑𝑟0]

𝑘−1 × (𝜕/𝜕𝑡) ∫∞0 𝑓𝑟0 (𝑟0) {∏𝑤(𝑡)𝑡𝑖=1 [∫𝑟0(1−𝑖 ∫∞0 𝑠𝑚𝑓𝑠(𝑠)𝑑𝑠/𝐶)𝑎0 𝑓𝑠𝑖 (𝑠𝑖) 𝑑𝑠𝑖]} 𝑑𝑟0𝑑𝑡}. (19)

The residual lifetime PDF of the independent series systemwithout SDPD taken into account can be written from (16)as follows:

𝑓9 (𝑇) = 𝜕𝜕𝑇−∫𝑡+𝑇𝑡 𝑘 [∏𝑤(𝑡)𝑡𝑖=1 ∫∞0 𝑓𝑠𝑖 (𝑠𝑖) ∫∞𝑠𝑖 𝑓𝑟𝑖 (𝑟𝑖) 𝑑𝑟𝑖𝑑𝑠𝑖]

𝑘−1 (𝜕/𝜕𝑡)∏𝑤(𝑡)𝑡𝑖=1 ∫∞0 𝑓𝑠𝑖 (𝑠𝑖) ∫∞𝑠𝑖 𝑓𝑟𝑖 (𝑟𝑖) 𝑑𝑟𝑖𝑑𝑠𝑖𝑑𝑡1 + ∫𝑡0 𝑘 [∏𝑤(𝑡)𝑡𝑖=1 ∫∞0 𝑓𝑠𝑖 (𝑠𝑖) ∫∞𝑠𝑖 𝑓𝑟𝑖 (𝑟𝑖) 𝑑𝑟𝑖𝑑𝑠𝑖]

𝑘−1 (𝜕/𝜕𝑡)∏𝑤(𝑡)𝑡𝑖=1 ∫∞0 𝑓𝑠𝑖 (𝑠𝑖) ∫∞𝑠𝑖 𝑓𝑟𝑖 (𝑟𝑖) 𝑑𝑟𝑖𝑑𝑠𝑖𝑑𝑡. (20)

4. Numerical Examples

In this section, numerical examples for series mechanicalsystems are provided to analyze the following problems:

(1) Validation of the proposedmodels for system lifetimedistribution estimation via Monte Carlo simulation(MCS).

(2) The influences of failure dependence of components,SDPD, and initial strength dispersion on whole life-time distribution of mechanical systems.

(3) The effects of failure dependence of components,SDPD, and initial strength dispersion on residuallifetime distribution of mechanical systems.

4.1. Monte Carlo Simulation Validation. Owing to the oper-ational mechanism of mechanical products, mechanical

components in mechanical systems are always logicallyconnected in series configuration. The practical examplesof series mechanical systems and parallel systems can bereferred to in [21]. In this section, for illustrative convenience,it is assumed that the components are identical in theseries system shown in Figure 12. The statistical parametersof load and material parameters in [21] are adopted forthe components in the two systems, which are listed inTable 1. To validate the lifetime distributionmodels,MCSs areperformedwith the flow chart shown in Figure 7. In theMCS,the deterministic residual strength under deterministic loadis calculated according to the strength degradation rule inphysical experiment with respect to component cumulativedamage𝐷(𝑛) as follows [21]:

𝑟 (𝑛) = 𝑟 [1 − 𝐷 (𝑛)]𝑎 . (21)


Component 1 Component 2

Component 1 Component 2

Common load Dependent system

Load 1 Load 2Independent system

Figure 12: Series mechanical systems.

Table 1: Stress parameters and material parameters of explosivebolts.

𝜇(𝑟0)[MPa]

𝜎(𝑟0)[MPa]

𝜇(𝑠)[MPa]

𝜎(𝑠)[MPa] 𝑚 𝑎 𝑤(𝑡)

[h−1]𝐶

[MPa2]400 30 300 20 2 1 0.5 108

Hence, the MCS, independent of any analytical model, isessentially identical with physical experiment. The resultsfrom the Monte Carlo simulation and the system lifetimePDF calculated by using the proposed models are shown inFigure 13.

From Figure 13, it can be seen that the lifetime PDFobtained from the proposed model shows good agreementwith the lifetime PDF from MCS. The model developed inthis paper provides a theory basis for analytical methodto quantitative lifetime distribution assessment. In addition,unlike the normal distribution curve, short system lifetimeaccounts for a certain proportion of the lifetime distribution,which is analogous to the situation in the infant mortalityperiod of the bathtub curve.

In addition, in order to illustrate the error in lifetimedistribution estimation by using conventional RCFmethod inthe case of nonergodic and nonstationary stress process, thetime-dependent mean value of the stress process is assumedto obey the following form:

𝜇𝑠 (𝑡) = 300 (1 + 0.0005𝑡) . (22)

Then, the comparison between the lifetime distribution of thedependent system by means of the proposed method and thelifetime distribution based on conventional RCFmethodwiththe PPDF of stress obtained via the statistics of the MCSs inthis section is shown in Figure 14.

From Figure 14, it can be seen that large calculationerror in system lifetime distribution estimation could becaused due to the usage of conventional RCF method whenthe stress is nonergodic or nonstationary. The incorrectusage of the conventional RCF method could result in theunderestimation of the system lifetime distribution in the

0

0.5

1

1.5

2

2.5

3

MCSProposed model

×10−3

Syste

m re

sidua

l life

time P

DFf(t)

(h−1)

0 100 200 300 400 500 600Time t (h)

Figure 13: LifetimePDF fromMCSand lifetimePDF fromproposedmodels.

0.0000

0.0025

0.0050

0.0075

0.0100

0.0125

0.0150

Distribution based on IPDF of stressDistribution based on PPDF of stress

Syste

m li

fetim

e PD

Ff(t)

(h−1)

0 50 100 150 200 250 300 350 400Time t (h)

Figure 14: System lifetime PDF based on different PDF of stress.

long lifespan regions. Furthermore, the increase in the meanvalue of the stress substantially reduces the system lifetime.

4.2. The Effects of SDPD and Failure Dependence of Compo-nents on System Whole Lifetime Distribution. For the seriessystem in Section 4.1, when the components are assumed tobe mutually independent, the system lifetime PDF is shownin Figure 15. Besides, provided that the strength distributionat each load application is used for lifetime distributionassessment without SDPD taken into account, the systemlifetime PDF is also shown in Figure 15. In addition, whenthe series system is composed of four components, the systemlifetime PDFs are shown in Figure 16. Moreover, in the casesof different initial strength dispersion, the lifetime PDFs


0.00000.00050.00100.00150.00200.00250.00300.00350.00400.00450.00500.00550.0060

Dependent system considering SDPDIndependent system considering SDPDIndependent system without considering SDPD

Syste

m li

fetim

e PD

Ff(t)

(h−1)

0 100 200 300 400 500 600 700 800Time t (h)

Figure 15: System lifetime PDF with two components.

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008


Syste

m li

fetim

e PD

Ff(t)

(h−1)

0 100 200 300 400 500 600Time t (h)

Figure 16: System lifetime PDF with four components.

of the dependent system considering SDPD are shown inFigure 17.

From Figures 15–17, it can be seen that the lifetime ofdependent series systems is distributed in the interval oflonger lifetime compared with the lifetime of independentsystems. The proposed system lifetime distribution modelsprovide the analytical methods for system lifetime distribu-tion assessment considering the failure dependence of com-ponents in an explicit form.The conventional system lifetimedistribution models, in which components are assumed to bemutually independent, could lead to error in system lifetimedistribution evaluation. Moreover, most mechanical systemsare in the series configuration, such as the drive systemsand the mechanisms for motion transmission. The series

0.00000.00050.00100.00150.00200.00250.00300.00350.00400.00450.00500.00550.00600.00650.0070

Syste

m li

fetim

e PD

Ff(t)

(h−1)

0 100 200 300 400 500 600 700 800Time t (h)

Standard deviation of 10 MPaStandard deviation of 20MPaStandard deviation of 30MPa

Figure 17: System lifetime in cases of different strength dispersion.

systemsmight be composed of a large amount of components.From Figures 15 and 17, it can be learnt that the increasein the number of components in the series systems resultsin the enhancement of the effects of failure dependence onsystem lifetime distribution. Thus, more attention should bepaid to system lifetime distribution in the case where seriessystems comprise a larger amount of components. Further-more, SDPD has much more significant influences on systemlifetime distribution than failure dependence of components.The mean value and dispersion of system lifetime couldbe underestimated due to neglecting the phenomenon ofSDPD. In addition, the dispersion of initial strength alsohas great impacts on system lifetime distribution. In general,the decrease in the dispersion of initial strength makes thesystem lifetime distribution more concentrated and causesthe increase in the mean value of system lifetime.

4.3. The Impacts of SDPD and Failure Dependence of Com-ponents on System Residual Lifetime Distribution. For thedependent series system in Section 4.1, the system residuallifetime PDFs from different time instant 𝑡 are shown inFigure 18. Besides, in the cases of different initial strengthdispersion, the residual lifetime PDFs of the dependentsystem from 𝑡 = 100 are shown in Figure 19. In addition, theresidual lifetime PDFs of independent system from 𝑡 = 100considering SDPDand that without SDPD taken into accountare shown in Figure 20.

It is difficult to quantitatively assess system residuallifetime distribution via an explicit mathematical expressionwith the failure dependence of components, SDPD, and thestatistical characteristics of both load process and materialparameters into consideration. To address these difficulties,analytical models for system residual lifetime distribution areproposed in this paper. From Figures 18–20, it can be learntthat the failure dependence, SDPD, and statistical charac-teristics of material parameters have considerable impacts


×10−3×10−3

Working time t (h)

8

7

6

5

4

3

2

10

7

6

5

4

3

2

1

0

300250

200150

10050

0 0 100 200 300400500600

700800

Residual lifetime T (h)

Syste

m re

sidua

l life

time P

DFf(T

)(h

−1)

Figure 18: System residual lifetime PDF at different moment.

0.00000.00050.00100.00150.00200.00250.00300.00350.00400.00450.00500.00550.00600.00650.0070

0 100 200 300 400 500 600Time t (h)

Standard deviation of 10 MPaStandard deviation of 20MPaStandard deviation of 30MPa

Syste

m re

sidua

l life

time P

DFf(t)

(h−1)

Figure 19: System residual lifetimePDF in cases of different strengthdispersion.

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010


0 100 200 300 400 500 600Time t (h)

Syste

m re

sidua

l life

time P

DFf(t)

(h−1)

Figure 20: System residual lifetime PDF.

on system residual lifetime distribution. The shape of theresidual lifetime PDF varies with the increase in the workingtime. Generally, the peak value of the residual lifetime PDFincreases and moves towards the left direction with theincrease in the working time. After working for a certainperiod of time, the local extremum of residual lifetime PDFin the interval of short residual lifetime disappears. Besides,compared with the failure dependence of components, SDPDhave more significant influences on residual lifetime distri-bution.The system lifetime could be underestimated withoutSDPD taken into consideration. In addition, the increase inthe initial strength dispersion results in larger dispersion andsmaller mean value of system residual lifetime.

5. Conclusions

For components in the seriesmechanical systems, the lifetimedistribution is determined by both the load process andstrength degradation process. When the load process is notergodic or stationary, conventional RCF method should beused with caution, which could lead to error in lifetime dis-tribution estimation of components and systems. Moreover,SDPD results from the dynamic interaction between the loadprocess and strength degradation process. It could cause largeerror in lifetime distribution assessment of components andsystems to neglect the phenomenon of SDPD. In addition,SDPD and failure dependence of components jointly bringconsiderable difficulties in prediction of system lifetimedistributions and system residual lifetime distributions.

To deal with these problems, reliability-based analyticmodels for estimation of whole lifetime distribution andresidual lifetime distribution of series mechanical systemsunder random load are developed in this paper, which takethe time-dependent statistical parameters of load process andstrength degradation process as the input of the models andconsider the dynamic interaction between load process andstrength degradation process. Moreover, SDPD and failuredependence of components are taken into account in theproposed models in an explicit mathematical expression.The results show that SDPD has significant influences onsystem lifetime distribution. The mean value and dispersionof system lifetime could be underestimated due to neglectingthe phenomenon of SDPD. Furthermore, increase in thenumber of components in the series systems results in theenhancement of the effects of failure dependence on systemlifetime distribution. Compared with the failure dependenceof components, SDPD have more significant influences onsystem lifetime distribution. Besides, decrease in the disper-sion of initial strength makes the system lifetime distributionmore concentrated and causes the increase in the mean valueof system lifetime. In addition, the shape of the residuallifetime PDF varies with the increase in the working time.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article.


Authors’ Contributions

All authors have read and approved this version of the article,and due care has been taken to ensure the integrity of thework.

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (Grant no. 51505207), Scientific ResearchFund of Liaoning Provincial Education Department ofChina (Grant no. L2015298), Liaoning Provincial NaturalScience Foundation of China (Grant no. 2015020152), andOpen Foundation of Zhejiang Provincial Top Key Aca-demic Discipline of Mechanical Engineering (Grant no.ZSTUME02A01).

References

[1] G. Hao and H. Li, “Conceptual designs of multi-degree of free-dom compliant parallel manipulators composed of wire-beambased compliant mechanisms,” Proceedings of the Institution ofMechanical Engineers, Part C: Journal ofMechanical EngineeringScience, vol. 229, no. 3, pp. 538–555, 2015.

[2] J. Wu and S. Yan, “An approach to system reliability predictionformechanical equipment using fuzzy reasoning Petri net,” Pro-ceedings of the Institution of Mechanical Engineers, Part O: Jour-nal of Risk and Reliability, vol. 228, no. 1, pp. 39–51, 2014.

[3] Y. Zhao, “A fatigue reliability analysis method including superlong life regime,” International Journal of Fatigue, vol. 35, no. 1,pp. 79–90, 2012.

[4] S. Suresh, Fatigue of Materials, Cambridge University Press,1998.

[5] T. I. Elperin and I. B. Gertsbakh, “Estimation in a randomcensoring model with incomplete information: exponentiallifetime distribution,” IEEE Transactions on Reliability, vol. 37,no. 2, pp. 223–229, 1988.

[6] Z. Pan, J. Feng, and Q. Sun, “Lifetime distribution and associ-ated inference of systems with multiple degradation measure-ments based on gamma processes,” Eksploatacja i Niezawodnosc- Maintenance and Reliability, vol. 18, no. 2, pp. 307–313, 2016.

[7] A. Asgharzadeh, M. Abdi, and S.-J. Wu, “Interval estimation forthe two-parameter bathtub-shaped lifetime distribution basedon records,”Hacettepe Journal ofMathematics and Statistics, vol.44, no. 2, pp. 399–416, 2015.

[8] M. Kayid and S. Izadkhah, “Characterizations of the exponen-tial distribution by the concept of residual life at random time,”Statistics & Probability Letters, vol. 107, pp. 164–169, 2015.

[9] J. Gong, Y.-F. Fu, W. Xia, J.-H. Li, and F. Zhan, “Fatigue lifeprediction of screw blade in screw sandwashingmachine underrandom load with gauss distribution,” American Journal ofEngineering and Applied Sciences, vol. 9, no. 4, pp. 1198–1212,2016.

[10] Y.Mori andB. R. Ellingwood, “Time-dependent system reliabil-ity analysis by adaptive importance sampling,” Structural Safety,vol. 12, no. 1, pp. 59–73, 1993.

[11] M. G. Stewart and D. V. Rosowsky, “Time-dependent reliabilityof deteriorating reinforced concrete bridge decks,” StructuralSafety, vol. 20, no. 1, pp. 91–109, 1998.

[12] R. E. Glaser, “Bathtub and related failure rate characterizations,”Journal of the American Statistical Association, vol. 75, no. 371,pp. 667–672, 1980.

[13] G. Stefanou and M. Papadrakakis, “Stochastic finite elementanalysis of shells with combined random material and geo-metric properties,” Computer Methods Applied Mechanics andEngineering, vol. 193, no. 1-2, pp. 139–160, 2004.

[14] M. Ostoja-Starzewski and X. Wang, “Stochastic finite elementsas a bridge between randommaterial microstructure and globalresponse,” Computer Methods Applied Mechanics and Engineer-ing, vol. 168, no. 1-4, pp. 35–49, 1999.

[15] B. N. Singh, D. Yadav, andN.G. R. Iyengar, “Natural frequenciesof composite plates with random material properties usinghigher-order shear deformation theory,” International Journalof Mechanical Sciences, vol. 43, no. 10, pp. 2193–2214, 2001.

[16] T.-H. Lee and K. M. Mosalam, “Seismic demand sensitivity ofreinforced concrete shear-wall building using FOSM method,”Earthquake Engineering & Structural Dynamics, vol. 34, no. 14,pp. 1719–1736, 2005.

[17] J. W. Baker and C. A. Cornell, Uncertainty Specification AndPropagation for Loss Estimation Using FOSM Method, PacificEarthquake Engineering Research Center, College of Engineer-ing, University of California, 2003.

[18] C. Guohua and D. Shuho, “Study on the reliability assessmentmethodology for pressure vessels containing defects,” Interna-tional Journal of Pressure Vessels and Piping, vol. 69, no. 3, pp.273–277, 1996.

[19] Y. G. Zhao, T. Ono, and M. Kato, “Second-order third-momentreliability method,” Journal of Structural Engineering, vol. 128,no. 8, pp. 1087–1090, 2002.

[20] Z. P. Qiu and J. Wang, “The interval estimation of reliability forprobabilistic and non-probabilistic hybrid structural system,”Engineering Failure Analysis, vol. 17, no. 5, pp. 1142–1154, 2010.

[21] P. Gao, L. Xie, and F. Liu, “Dynamic reliability analysis of seriesmechanical systems considering strength degradation pathdependence of components,” Journal of Advanced MechanicalDesign, Systems, and Manufacturing, vol. 9, no. 5, pp. 1–13, 2015.

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