J Intell Manuf (2012) 23:221–226DOI 10.1007/s10845-009-0348-9

Stochastic modeling of damage physics for mechanical componentprognostics using condition indicators

David He · Ruoyu Li · Eric Bechhoefer

Received: 23 August 2009 / Accepted: 19 October 2009 / Published online: 12 November 2009© Springer Science+Business Media, LLC 2009

Abstract The health of a mechanical component deterio-rates over time and its service life is randomly distributedand can be modeled by a stochastic deterioration process.For most of the mechanical components, the deteriorationprocess follows a certain physical laws and their mean lifeto failure can be determined approximately by these laws.However, it is not easy to apply these laws for mechanicalcomponent prognostics in current health monitoring applica-tions. In this paper, a stochastic modeling methodology formechanical component prognostics with condition indicatorsused in current health monitoring applications is presented.The effectiveness of the methodology is demonstrated witha real shaft fatigue life prediction case study.

Keywords Mechanical component prognostics ·Stochasticmodeling · Damage physics · Condition indicators

Introduction

The health of a mechanical component deteriorates over time.The service life of the mechanical component is randomlydistributed and can be modeled by a stochastic deteriorationprocess. For most of the mechanical components, the deterio-ration process follows a certain physical laws and their meanlife to failure can be determined approximately by these laws.

D. He (B) · R. LiIntelligent Systems Modeling and Development Laboratory,Department of Mechanical and Industrial Engineering,The University of Illinois-Chicago, Chicago, IL 60607, USAe-mail: davidhe@uic.edu

E. BechhoeferGoodrich Sensor and Integrated Systems, Vergennes, VT 05491, USAe-mail: Eric.Bechhoefer@Goodrich.com

For example, in the case where the deterioration process isdominated by crack growth, it follows Paris’ law (Paris andErdogan 1963). There are several challenges for using Paris’law to predict the remaining useful life of a damaged com-ponent, for example, a cracked shaft. First of all, in order toapply the law, one needs to know the actual damage of thecomponent, i.e., the crack size. In current mechanical com-ponent health monitoring applications, the actual damage ofthe component cannot be directly measured in real time. Sec-ond, in addition to crack size, other parameters of the law area function of several factors such as loading, geometry, andmaterial type. These factors are often unknown or vary acrossdifferent applications. Even one could estimate and updatethese parameters using state space models (Bechhoefer et al.2008), uncertainty still remain as the factors change due todifferent type of materials, geometry, loading conditions, andso on.

These challenges become problems if one wants to applythese laws to mechanical component prognostics in a healthmonitoring application such as health and usage monitor-ing systems (HUMS). Military and civil service helicoptersare currently equipped with HUMS for health monitoring offlight critical components. Typically, vibration data recordedduring a flight is processed to generate condition indica-tors (CIs). CIs from healthy components are normally usedto set thresholds such that there is a small probability ofthe CIs of nominal components exceeding the thresholds.If a CI exceeds the threshold, the component is declaredbad. Test stand data from the Navy, NASA, Westland, etc.,have been used to demonstrate these principals and there isantidotal evidence that as a component wears, the CI val-ues change from their nominal cohorts. It will be benefi-cial that methods and tools can be developed for mechanicalcomponents currently monitored by commercial applicationsusing CIs.

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Markov process is often used as a deterioration model inmaintenance modeling (Welte et al. 2006; Van Winden andDekker 1998; Endrenyi et al. 1998; Theil 2006; Lam and Yeh1994a,b; Jirutitijaroen and Singh 2004). In such a deteriora-tion model, the health condition of the system is often dividedinto a number of discrete states. When the sojourn times inthese discrete states are modeled by an exponential distribu-tion, the deterioration model is a discrete Markov model. Inthe context of prognostics, the sum of sojourn times repre-sents time to failure. Discrete Markov deterioration modelassumes that the condition or the amount of degradation canbe classified into discrete states and these states can be mod-eled by an exponential distribution.

However, in real applications, it cannot be guaranteed thatthe sojourn times in the deterioration states follow exactlyan exponential distribution. It is more reasonable that thesojourn times are modeled by a general distribution. In sucha case, the deterioration model is a semi-Markov model. Ingeneral, it is more difficult to solve semi-Markov models thanMarkov models. Therefore, it is suggested to divide a dete-rioration state into a sequence of exponentially distributedartificial states (EDASs) so that the semi-Markov process canbe approximated by a Markov process. It will be valuable ifthe deterioration model could be divided into discrete statesin terms of CIs in health monitoring applications. Hence, theobjective of mechanical component prognostics using CIs ina health-monitoring environment could be made possible.

In this paper, a methodology for mechanical componentprognostics using stochastic modeling of damage physicswith condition indicators used in health monitoring appli-cations is presented. The effectiveness of the methodologyis demonstrated with a real shaft fatigue life prediction casestudy. The remainder of the paper is organized as follows.In “The methodology” section, the stochastic modeling app-roach based on laws of damage physics is presented. “A shaftfatigue life prediction case study” section presents a casestudy on predicting fatigue life of cracked shaft with real run-to-failure test data using the stochastic modeling approach.Finally, the paper is concluded in “Conclusions” section.

The methodology

The general framework of the stochastic modeling approachfor mechanical component prognostics using condition indi-cators is presented in Fig. 1.

The framework of the stochastic modeling prognostic app-roach shown in Fig. 1 starts with identifying and selecting CIsused in component health monitoring applications. These CIsusually are indicators of component fatigue damage states.For example, in case of bearing health monitoring, as reportedin (He and Bechhoefer 2008), a strong correlation couldbe established between the bearing spall length and a linear

δ

(a) Parallel misalignment (b) Angular misalignment

(d) Bent shaft

δ

θ

(c) Combined parallel-angular misalignment

Laws of Damage Physics

Map CIs to Physical States of Component Damage

Solve Discrete Semi-Markov Models

Prognosis

Select CIs in Component Health Monitoring

1, 1 1, 2 i, j J, I Failure

( )TE 1,11,1

1=λ

( )TE 2,12,1

1=λ

( )TE ijij

,,

1=λ

( )TE IJIJ

,,

1=λ

Time or Usage

CIs

Artificial States

Failure

T1,1

T1,2

T2,1 T

2,2

T2,3 T

2,4

j=1, i=1 j=1, i=2

j=2, i=1

j=2, i=2

j=2, i=3

j=2, i=4

Fig. 1 The framework of the methodology

combination of CIs. These CIs include: (1) bearing passingfrequencies at the base frequency; (2) RMS of the vibra-tion signal between 0 and 1000 Hz; (3) envelop analysis ofthe bearing passing frequencies at 13–18 KHz, 20–25 KHzand 25–30 KHz windows; (4) RMS of the envelope analy-sis; (5) cepstrum analysis of the bearing passing frequen-cies. For another example, in the case of drive shaft healthmonitoring, the first three shaft order values could be usedeffectively to indicate the crack growth of on the surfaceof the drive shaft under different operating conditions (Wuand He 2008). Once these condition indicators are selectedthey will be mapped to physical damage states of the com-ponent deterioration process. Here, the condition indictorsCIs are used as a surrogate measurement of a physical dam-age state. Therefore, the values of CI correspond to certainphysical states that represent the physical damage conditionsof a component. The time that takes the CI correspondingto the initial condition to reach the CI corresponding to thefailure condition represents the time to failure. By dividingthe interval between any two known states, for example, aninitial healthy condition and a final failure condition, into alarge enough number of intervals, one creates some artifi-cial states. The time to move from one artificial state to thenext one will follow approximately exponential distribution.

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J Intell Manuf (2012) 23:221–226 223

Summing all the time intervals between the artificial states,one could estimate the time to failure. This task can be car-ried out by applying the underlying physical laws that gov-ern the damage process. The next step in the framework isto solve the discrete semi-Makov models by inserting opti-mal number of artificial states between the defined physicalstates in terms of CIs. The final step of the stochastic mod-eling approach is to predict the remaining useful life of themechanical components using condition indicators.

A shaft fatigue life prediction case study

In this paper, the application of the stochastic modelingmethodology for mechanical component prognostics is dem-onstrated with a real case study on drive shaft fatigue lifeprediction.

Shaft fatigue failures are the predominant failure mode ofproperly installed and maintained drive shafts. Fatigue failurestarts with an incipient crack that will grow until the shaft rup-tures, often with catastrophic consequences. There are twomain types of shaft cracks: surface and subsurface cracks.Surface cracks are more common than the subsurface cracks,and are much easier to observe and detect. Surface cracks canbe further classified as transverse, longitudinal, and slant,based on their geometry. Transverse cracks are perpendicu-lar to the shaft axis and are the most common. In addition,transverse cracks weaken shafts by reducing the cross-sectionand greatly increasing the likelihood of a catastrophic fail-ure. A transverse surface crack involves three stages: crackinitiation, crack growth, and final fractured stage. When thedegradation mechanism of a component is crack growth, itcan be described by Paris’ law:

da

d N= C(�K )m (1)

Here, da/d N is the crack growth rate and C and m are mate-rial constants. The term �K represents stress intensity factorrange and can be computed as:

�K = F ·�S·√πa (2)

Here, F is a geometry factor, �S the stress range, and a thecrack size.Define:

a0 = initial crack sizea f = final crack sizeN f = number of cycles for a crack to propagate from

a0 to a f

From Eqs. 1 and 2, N f can be computed as:

N f = a1−m/2f − a1−m/2

0

C(F�S

√π

)m(1 − m/2)

(3)

Note that in real on-line machinery health monitoringapplications such as Goodrich’s Integrated Mechanical Diag-nostics-Health and Usage Monitoring System (IMD-HUMS),physical damage such as crack size cannot be measured dir-ectly to determine the condition of the monitored systems.Rather, vibration features such as shaft orders are extracted ascondition indicators (CIs) for heath monitoring purpose. Pre-vious research results (He and Bechhoefer 2008; Wu and He2008) have suggested that certain CIs have shown a positivecorrelation with the crack size in the case of shafts and bear-ings. Therefore, we can express N f using the correspondingCIs.

Let CI f and CI0 be the CI value corresponding to cracksize a f and a0, respectively. Based on Eq. 3, then number ofcycles for a crack to propagate from a0 to a f can be written as:

N f = CI1−m/2f − CI1−m/2

0

WC(F�S

√π

)m(1 − m/2)

(4)

Note from Eq. 4, by replacing crack size with correspondingCIs, a new factor W is introduced. The factor W is determinedby the geometry and material properties of the system, andthe operating conditions.

Given a constant frequency of the stress cycles over time,the time needed to propagate from initial crack size a0 tofinal crack size a f can be computed as:

t f = N f

f(5)

Note that in real applications, the crack growth rate varies dueto differences in materials and variability of loading condi-tions, etc. Therefore, t f in Eq. 5 is considered as the averagetime taking to propagate from a0 to a f . Let random variableT f be the time needed to propagate from a0 to a f and followsan exponential probability distribution, then the distributionparameter of T f can be computed as:

λ = 1

t f= f

N f= f WC

(F�S

√π

)m(1 − m/2)

CI1−m/2f − CI1−m/2

0

(6)

Let the interval between CI0 and CI f be divided into I dis-crete artificial states. Then an artificial state i is characterizedby CI value CIi , CI0 ≤ CIi ≤ CI f . The CI of an artificialstate can be defined as:

CIi = CI0 + i ·CI f − CI0

I(7)

and

CIi−1 = CI0 + (i − 1) ·CI f − CI0

I(8)

Let random variable Ti be the time duration and ti the averagetime duration for CI to move from CIi−1 to size CIi , respec-tively. If Ti is exponentially distributed, then the distributionparameter λi can be computed as:

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λi = 1

ti= f

Ni= f WC

(F�S

√π

)m(1 − m/2)

CI1−m/2i − CI1−m/2

i−1

(9)

Let G = f WC(F�S

√π

)m(1 − m/2) and substitute Eqs. 7

and 8 into 9, we obtain:

λi(G, I

∣∣CI0, CI f , m

)

= G(

CI0 + iCI0−CI f

I

)1−m/2 −[CI0 + (i − 1)

CI f −CI0I

]1−m/2 (10)

Note from Eq. 10, if a0, a f , and m are given, then λi canbe obtained once G and I are estimated. The challenge is toestimate Gand I . The estimation of G and I is described asfollows.

Let μ and σ 2 be the mean and variance of the time durationfor crack to propagate from a0 to a f , respectively. Therefore,the estimates of G and I should be obtained such that the fol-lowing two conditions should be satisfied:

μ = E

(I∑

i=1

Ti

)

(11)

and

σ 2 = Var

(I∑

i=1

Ti

)

(12)

For any given I number of artificial states, the parameter Gis computed according to its definition and Eq. 5 as:

G(μ

∣∣CI0, CI f , m) = CI1−m/2

f − CI1−m/20

μ(13)

The estimation of I can be performed by solving an optimi-zation problem formulated as following:

Minimize :∣∣∣∣∣σ 2 − Var

(I∑

i=1

Ti

)∣∣∣∣∣

(14)

Subject to: Eqs. 7, 8, 10, 13

In this paper, the data collected from real shaft run-to-failure tests on industrial test stand is used to validate thestochastic modeling approach. The shaft run-to-failure testswere conducted and data was collected by Impact Technol-ogy at Goodrich shaft testing facility in Rome, NY. The testsetup is shown in Fig. 2.

As shown in Fig. 2, five channels of data were collected:(1) tachometer (once per rev); (2) tail stock, radial, 12 o’clock,bolt mounted; (3) test bed rail, vertical stud mounted, radial,12 o’clock, bolt mounted; (4) tail stock, radial, 3 o’clock,glue mounted; (5) tail stock, axial, glue mounted. At eachdata collection point, vibration data was collected over 0.5 swith a sampling frequency of 204,800 Hz. When an obviouscrack was observed on the coupling of the shaft the test wasstopped. Figure 3 shows the plot of the shaft order 3 valuesof vibration data collected from the four channels during the

Fig. 2 The setup of the shaft run-to-failure test

Fig. 3 Shaft order 3 values of the vibration of the 400 s

last 400 s of the test. The dramatic increase in all shaft order3 values right before the stop of the test clearly indicates theoccurrence of the crack.

In validating the stochastic models with the shaft run-to-failure test data, different ways for generating the artificialstates were tested. In the first experiment, artificial stateswere generated between the two known conditions: initialcondition and final failure condition. As a result, an opti-mal number of 14242 artificial states were generated. Theestimated SO3 values versus real SO3 values are plotted inFig. 4.

From Fig. 4, we can see that the SO3 was largely overes-timated. This was clearly due to the fact that only two trueconditions were used. Figure 5 shows the results obtainedby using three true conditions to generated artificial states.When three true conditions were used, the entire interval ofdata was divided into two sections. In this case, an optimalnumber of 11413 artificial states for section 1 and an optimal

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Fig. 4 Estimated versus real with two physical states

Fig. 5 Estimated versus real with three physical states

Fig. 6 Estimated versus real with 4 physical states

number of 4 artificial states for section 2 were generated. Theestimation in Fig. 5 clearly appears better than that in Fig. 4.

Figure 6 shows the results obtained by using four trueconditions to generate artificial states. When four true con-ditions were used, the entire interval of data was divided intothree sections. In this case, an optimal number of 1711 artifi-cial states for section 1, an optimal number of 1765 artificialstates for section 2, and an optimal number of 4 artificialstates for section 3 were generated. The estimation in Fig. 6clearly appears better than those in Figs. 5 and 4.

Conclusions

The health of a mechanical component deteriorates over timeand its service life is randomly distributed and can be mod-eled by a stochastic deterioration process. For most of themechanical components, the deterioration process follows acertain physical laws and their mean life to failure can bedetermined approximately by these laws. However, it is noteasy to apply these laws for mechanical component prog-nostics in current health monitoring applications. In addi-tion, there are some limitations in predicting crack propaga-tion strictly on empirical relationships such as Paris’ law andassociated models: (1) the crack grow is deterministic ratherthan probabilistic; (2) studies have showed that crack growthchanges across a set of identical specimens even under well-controlled experimental conditions and that the propagationof a fatigue crack beyond its initial appearance is a highlyvariable process.

In this paper, a stochastic modeling methodology formechanical component prognostics with condition indicatorsused in current health monitoring applications is presented.In this methodology, the condition indictors CIs are used asa surrogate measurement of a physical damage state. There-fore, the values of CI correspond to certain physical statesthat represent the physical damage conditions of a compo-nent. The time that takes the CI corresponding to the initialcondition to reach the CI corresponding to the failure con-dition represents the time to failure. By dividing the intervalbetween any two known states, for example, an initial healthycondition and a final failure condition, into a large enoughnumber of intervals, one creates some artificial states. Thetime to move from one artificial state to the next one will fol-low approximately exponential distribution. Summing all thetime intervals between the artificial states, one could estimatethe time to failure.

The effectiveness of the methodology is demonstratedwith a real shaft fatigue life prediction case study. In this casestudy, data collected from real industrial drive shaft run-to-failure tests were used to establish the stochastic model ofthe shaft crack propagation process and the time to failurewas estimated by a semi-Markov model. The results showedthat the method provides an accurate estimation of the crackgrowth process.

The methodology presented in this paper is a generalizedone. It can be applied to approximate other physical lawsof damage mechanics such as Miner’s law in describing thecrack initiation process.

References

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