INSTITUTE OF PHYSICS PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 15 (2006) 1830–1836 doi:10.1088/0964-1726/15/6/037

Identifying damage using local flowvariation methodMing Liu and David Chelidze

Department of Mechanical Engineering and Applied Mechanics, University of Rhode Island,Kingston, RI, USA

E-mail: chelidze@egr.uri.edu

Received 24 March 2006, in final form 12 September 2006Published 2 November 2006Online at stacks.iop.org/SMS/15/1830

AbstractIn this paper, a damage identification method called local flow variation isintroduced. It is a practical implementation of a phase space warpingconcept. A hierarchical dynamical system is considered where a slow-timedamage process causes drifts in the parameters of a fast-time systemdescribing the measurable response of a structure. The method is based on ahypothesis that the probability distribution function of the fast-time trajectoryin its phase space is a function of a damage state. In this method, anensemble of estimated expectations of a trajectory in different locations ofthe reconstructed phase space is used as a damage feature vector. Using thesefeature vectors, damage identification is realized by a smooth orthogonaldecomposition. An experiment is conducted to validate the method. Atwo-dimensional slow-time damage process is identified from experimentalfast-time data. Although damage identification results from the local flowvariation are not as accurate as those from the direct application of phasespace warping tracking functions, the required computation time is about twoorders of magnitude shorter.

1. Introduction

In engineering fields, a gradual deterioration of componentsmay lead to the total failure of a system. The deteriorationthat adversely affects the system’s performance is defined asdamage. In today’s highly competitive market, engineersfind that conflict between skyrocketing costs of maintenanceand increasing demand for safety is extremely hard to handleusing traditional preventive maintenance approaches. One ofthe proposed solutions is condition-based maintenance, wheredamage is detected or predicted before it causes actual failures.To realize the condition-based maintenance, the ability toidentify damage and track its evolution in real-time is crucial.

Numerous solutions to damage identification and trackingproblems [1–3] are proposed in a large body of literature, butthis subject is far from mature. One of the main obstaclesis that the behaviour of a system may not be describedaccurately by available numerical models, especially whenthe system is nonlinear. In recent investigations [4–6],nonlinear characteristics show the capacity to provide valuableinformation about the system’s state without explicit use ofanalytical models. Researchers are attracted by this capacityand are applying related methods to damage identification.

Foong et al [7] studied the dynamical behaviour of a beamwith a propagating fatigue crack. Although the excitation isstationary and harmonic, the vibration of the beam is shownto change from periodic to chaotic with the propagation ofthe crack. Adams et al [8] propose a damage identificationapproach with the assumption that the undamaged system islinear, and nonlinearity is only introduced by damage. Forlinear systems, the response under some stationary chaoticexcitation is shown to be of benefit, and is used to detectdamage [9–11].

In previous work [12–14], a concept of phase spacewarping (PSW) is introduced. This concept describesdistortions in a system’s fast-time phase space caused by slow-time damage accumulation. A direct one-to-one relationshiphas been demonstrated between damage states and the PSWtracking function (PSWTF) based feature vectors. However,this procedure requires considerable computational time toestimate the short-time trajectory evolutions of a healthysystem from previously recorded data, since systems underconsideration are usually nonlinear.

To ease the computational complexity and to satisfy therequirements for on-line, real-time damage identification, alocal flow variation (LFV) method is developed as a practical

0964-1726/06/061830+07$30.00 © 2006 IOP Publishing Ltd Printed in the UK 1830

Local flow variation

implementation of the PSW concept. Here, expectation ofthe fast-time trajectories is estimated in a small volume of thephase space, and is employed as a damage tracking function.Thus, there is no need to predict the evolution of a fast-timetrajectory from a given point in the phase space, and the dataprocessing time is reduced considerably.

In the next section, the LFV method is introduced asa practical implementation of the PSW idea based on themain hypothesis which is illustrated using a simple simulation.Then an LFV-based damage identification method is presentedbased on LFV tracking functions and smooth orthogonaldecomposition (SOD). Following an experimental validationof the LFV-based method, the tracking results from differenttracking functions are compared. The difference between theLFV-based method and an earlier direct application of a PSWtracking function is also discussed, followed by a conclusion.

2. Description of the method: basic idea

In a dynamical systems approach, damage is consideredin a hierarchical dynamical system [15], where a fast-time subsystem describes the response of the system tosome excitation, and a slow-time subsystem characterizes thedamage evolution process. The damage evolution causesalternations in the phase space of the fast-time subsystemby causing drifts in the parameters of that system. Suchalternations are characterized as the PSW.

2.1. Local flow variation (LFV)

Two important phenomena are observed in both simulationsand experiments. Firstly, if the damage states are constant,in the absence of other external variabilities, the probabilitydistribution of trajectories in the fast-time phase space isstationary. Secondly, the change in damage states affectsthe distribution of trajectories. These phenomena have beenstudied by Hively [16] and Mcsharry [17] independently, andit is reasonable to advance a hypothesis:

A fast-time trajectory’s probability distribution in itsphase space is a function of slow-time damage states.

Local flow variation (LFV) is defined as the change in thedistribution of trajectories in the fast-time subsystem’s phasespace, which is caused by the evolution of damage.

2.2. LFV tracking function

For a small fixed volume B in the fast-time phase space,the distribution of trajectories can be described by a localprobability distribution function (LPDF), fB . Based on theabove hypothesis, the LPDF is a function of damage stateφ and coordinates of the phase space x with properties:∫B fB(x, φ) dx = 1 and fB(x, φ) = 0 if x /∈ B. Then the

first moment of the trajectory (EB[x]) in this volume can becalculated by

EB [x] =∫

Bx fB(x,φ) dx ≡ FB(φ) (1)

and is a function of the damage state φ. Here FB(φ) is usedto describe the first moment of the trajectories in volume B

with damage φ. Using (1), we define a LFV tracking function(LFVTF) lB(φi):

lB(φi) = FB(φi ) − FB(φ0), (2)

where φi and φ0 are current and reference (healthy) damagestates, respectively. Here the lB(φi ) describes the change in thefirst moment in the volume B caused by the change in damagestate from φ0 to φi .

If FB(φ) is continuous and differentiable for all possibledamage states, we can expect a linear observability ofdamage for small changes in the damage states (initial damageaccumulation):

lB(φi) = dFBdφ

∣∣∣∣φ=φ0

(φi − φ0) + O(|φi − φ0|2). (3)

In a practical context, the FLVTFs cannot be calculateddirectly. Firstly, continuous trajectories are not available inboth experiments and simulations. In numerical simulationsthe fast-time trajectories are discretely sampled at equal timeintervals ts. In experiments, only discrete samples of thereconstructed trajectories are available. Secondly, the LPDFsof trajectories can be very complicated when the response ofthe system is a non-periodic motion. However, if the samplingfrequency is kept constant and ergodicity is assumed, FB canbe estimated as

FB(φ) ∼= ‖B‖−1∑

x(n,φ)∈Bx(n,φ) (4)

where x(n,φ) are samples of the trajectory that are containedin this small volume B when the damage state equals φ,approximately; ‖B‖ can be approximated by the number ofsamples (x(n,φ)) in the volume B.

The LFVTF is not unique. If g(x) is a differentiablefunction for x ∈ B, the first moment of g(x) in the volumeB, EB[g(x)] is also a function of φ, since

EB [g(x)] = EB

[

g(x0) + dgdx

∣∣∣∣x=x0

(x − x0) + O(|�x|2)]

= g(x0) + dgdx

∣∣∣∣x=x0

(EB [x] − x0) + O(|�x|2)= AF(φ) + C, (5)

where x0 is the centre of the volume B, and A = dgdx

∣∣x=x0

and

C = g(x0) − dgdx

∣∣x=x0

x0 + O(|�x|2) are constants for B. So,any a function in the form

hB(φ i) = EB [g(x)]|φ=φi− EB [g(x)]|φ=φ0

(6)

is appropriate to be used as a tracking function. However,additional noise is introduced during the linearization of g(x)

in (5), so the estimated hB(φ) may not work as well as lB(φ)

in most situations.

2.3. An illustrative example

A simple two-well Duffing’s equation is used to illustrate themain hypothesis

x = kx − x3 − cx + F cos ωt (7)

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dx/d

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-1 0 1

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0.8

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1.2

1.4

1.6

Figure 1. Poincare section with c = 0.4 (a); Poincare section withc = 0.41 (b).

where k = 1, F = 1.1, ω = π/2, and c is regarded as adamage state. The phase spaces for c = 0.4 and c = 0.41 arecompared to show the effect of damage. Here, the system withc = 0.4 is regarded as a healthy system, and the correspondingphase space is used as the reference.

For the convenience of illustration, instead of phase spacevolumes, areas of Poincare sections at a fixed phase ωt =2mπ (m = 1, 2, 3, . . .) with c = 0.4 and c = 0.41are employed to show the change in the distribution of thetrajectories. Here, it is necessary to point out that the Poincaresections can be regarded as volumes which have an edge ofzero length. The Poincare section of the reference phase spaceis divided into 16 small areas using the equiprobable partitionmethod [18], and the grid is recorded for use on the otherPoincare section. The estimated first moments are emphasizedusing a dot for each small area (see figure 1). Each Poincaresection is generated using 9800 points. Therefore, there areabout 1666 points in each small area in the reference Poincaresection.

From a first look, the two Poincare sections are quitesimilar, and it is hard to describe the difference between them.However, if we take a closer look at one particular area labelled12 (see figure 2), the difference is clear and can be describedby the estimated first moments.

The simulations are repeated eight times using differentinitial conditions with each damage state, and the estimatedfirst moments of section 12 are plotted in figure 3. From thisplot, it can be confidently claimed that the observed differencebetween two Poincare sections is caused by the change in thedamage state instead of the change in the initial conditions. Sothe estimated first moments provide valuable information aboutthe damage states.

Although the number of points (1700) used to estimateeach first moment is larger than 302 (a ‘rule of thumb’ criterionfor a statistic estimation from chaotic trajectories in a phasespace with dimension d is 10d–30d data points [19]; in ourcase, d = 2), local fluctuations in the estimates are obvious.Thus, tests based on the LFVTF of a single volume may not berobust. Although more accurate LFVTF can be obtained when

x

dx/d

t

-0.8 -0.6 -0.4 -0.2 0-0.5

-0.45

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-0.25

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-0.05

x

dx/d

t

(a) (b)

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

-0.8 -0.6 -0.4 -0.2 0

Figure 2. Poincare section with c = 0.4 (a); Poincare section withc = 0.41 (b).

-0.58 -0.56 -0.54 -0.52 -0.5 -0.48

-0.26

-0.25

-0.24

-0.23

-0.22

-0.21

-0.19

-0.2

Figure 3. Estimated first moments with eight different initialconditions: • represents the estimated first moments with c = 0.4,and ∗ represents the estimated first moments with c = 0.41.

more data are used for estimation, it is not possible to collectsuch large amount of data in an experimental context.

2.4. Smooth orthogonal decomposition (SOD)

The observed local fluctuations may hinder the damageidentification using estimated first moments directly. However,this noisy information can still be handled using multivariatedata analysis. In previous studies, the SOD-basedanalysis [12, 20] has been proven to be a powerful tool foridentifying smooth trends in noisy multivariate data. Thus, theSOD is also employed in the LFV-based approach.

It is assumed that incipient damage evolves slowly. Thus,damage state φ is regarded as approximately constant for adata set collected over a short period of time. The LFVTFsare calculated for small phase space volumes of each data set,and are assembled together in a feature vector using the form[lB1; lB2; . . . ; lBNs

], where Ns is the number of volumes in thephase space. These feature vectors describe the evolution ofdamage state φ and are assembled together into a matrix Y in a

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Figure 4. The largest periodic span (about 10 min) present in theexperiment.

time sequence. This matrix Y records the influence of damageon the fast-time subsystem and is called a tracking matrix.

In the damage identification procedure, it is assumed thatthe damage state φ can be recovered by a linear projection ofY: ϕ = Yq. Assuming slow deterministic damage evolution,ϕ should vary smoothly with a maximum possible variance.The SOD is used to estimate the ϕ by solving a constrainedmaximization problem:

maxq

‖ϕ‖2 subject to ‖Dϕ‖2 = 1, (8)

where D is a differential operator.The (8) is equivalent to the following generalized

eigenvalue problem:

[YTY

]q = λ

[(DY)TDY

]q. (9)

The eigenvector or smooth orthogonal mode (SOM), q,corresponding to the largest eigenvalue or smooth orthogonalvalue (SOV), λ, of (9) yields the optimal projection of matrixY that maximizes the smoothness and the overall variation ofthe smooth orthogonal coordinate (SOC), ϕ. For practicalpurposes, the solution to (9) is obtained using generalizedsingular value decomposition (GSVD) of [Y, DY] matrices. Inparticular, these matrices are decomposed as:

Y = UCXT (10a)

DY = VSXT (10b)

CTC + STS = I (10c)

where matrices U and V are unitary, X is a square matrix, and Cand S are non-negative diagonal matrices. Then the SOMs arecolumn vectors of X−T, the SOVs are given by λi = C2

i i /S2i i ,

and the SOCs are given by UC.The SOD has many interesting properties that are

described in [21]. In particular, it is invariant with respect to thelinear transformation of data. In addition, the SOD is shownto be able to separate signals according to their frequencycontents.

5 10 15 20-0.2

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Times (Hours)

Y38

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Y38

1

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Y41

2

Figure 5. Several randomly chosen columns of the tracking matrix.

3. Experimental validation

To validate this new approach, a modified version of the well-known two-well magnetoelastic oscillator is used as a targetsystem, and two-dimensional slow-time damage is introduced.The details of this experimental system are described in [20],where data collected from this experimental system havebeen used to validate a short-time reference model predictionerror (STRMPE) based damage identification approach. Inthis experiment, a couple of electromagnets powered trougha computer-controlled power supply are used to cause aperturbation in the magnetic potential at the free end of avibrating cantilever beam. The maximum effect of theseperturbations manifests itself in approximately a 4% changein the natural frequencies of small oscillations in each energywell. In this experiment, the deflection of the beam is measuredby two laser vibrometers mounted near the clamped beamend. Vibration signals are low-pass filtered with 50 Hz cut-off frequency and data is collected at a 160 Hz samplingfrequency. The system is started in a nominally chaoticregime. However, observed response includes several windowsof periodic behaviour. The largest periodic span happens in the18th hour of the experiment and is shown in figure 4.

The voltages supplied to the electromagnets (v1(t) andv2(t)) are altered harmonically, in a way shown in figure 6(b).Because v1(t) (the supply voltage to the front electromagnet)and v2(t) (the supply voltage to the back electromagnet) areindependent of each other, a two-dimensional damage stateis present in the experiment. About 12 million data pointsare recorded in the experiment, which lasts about 20 h. Thesix-dimensional fast-time phase space is reconstructed usinga delay time of five time samples [22]. The first 215 pointsare employed as a reference phase space, and consecutivesets of 213 points are treated as data records correspondingto a particular damage state. The reference phase space ispartitioned into 81 small volumes (Ns = 81) using the firstfour dimensions [20], and lB(φ) is employed as the trackingfunction. Then a 486 × 1480 tracking matrix Y is assembledusing the feature vectors calculated for each data record.Several columns of the normalized tracking matrix are shownin figure 5. Although there are substantial local fluctuations in

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ized

Eig

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Times (Hours)S

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y V

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ge (

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tage

(V

)

(a) (b)

(c) (d)

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10

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Figure 6. Plot of the ten largest SOVs (a); actual damage states v1(t)(——) and v2(t) (– – –) versus time plot (b); plot of the first (——)and second (– – –) SOCs corresponding to the two largest SOVs (c);plots of v1(t) + v2(t) (——), v1(t) − v2(t) (– – –), and the scaledfirst two SOCs (·) (d).

these signals, some trends similar to the actual damage states(see figure 6(b)) are also observed.

Further damage identification is realized by applyingthe SOD to the obtained tracking matrix. The ten largestgeneralized eigenvalues or SOVs are plotted in figure 6(a). Thetwo largest eigenvalues are much larger than the rest, which isconsistent with the fact that there is a two-dimensional damageprocess present in the system. The third and forth largest onesare larger than expected, which can be explained by the noisein the experiment. The SOCs corresponding to the two largesteigenvalues are depicted in figure 6(c). The scaled first twoSOCs fit v1(t) + v2(t) and v1(t) − v2(t) well (see figure 6(d)).Thus, these SOCs can be regarded as linear combinations ofactual damage states.

4. Discussion

This section only focuses on two simple properties of theLFV-based approach: the influence of bifurcation noise andthe selection of LFVTFs. A comparison between the originalSTRMPE-based and the LFV-based tracking functions is alsoprovided.

4.1. Sensitivity to bifurcation noise

Our experimental system is structurally unstable for somevalues of damage states. Although chaotic motions dominatethe vibration most of the time during the experiment, aconsiderable amount of periodic motions is also observed.Since the amplitude and frequency of excitation are selectedcarefully, most of the periodic motions only last aboutseveral seconds, and do not cause problems to the damageidentification. However, large periodic bands still exist, as infigure 4, that need special discussion.

During bifurcations, the distribution of the trajectorieschanges dramatically. FB(φ) is no longer continuous or

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ized

Eig

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lues

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-1.5

-1

-0.5

0

0.5

1

1.5

2

Times(Hours)

SO

Cs

(a) (b)

2 4 6 8 10

Figure 7. Plot of the ten largest SOVs from the SOD analysisassociated with g(x(n)) = x(n + 1) (a); plot of the first (——) andsecond (– – –) SOCs corresponding to the two largest SOVs (b).

Table 1. SNR of the SOCs for different LFVTFs.

Tracking functions SNR (dB) Size of tracking matrix

g(x(n)) = STRMPE 19.54 486 × 1440g(x(n)) = x(n + 1) + x(n) 19.30 486 × 1480g(x(n)) = x(n) (eB(φ)) 19.16 486 × 1480g(x(n)) = x(n + 1) 19.05 486 × 1480g(x(n)) = x(n + 1) − x(n) 15.93 486 × 1480

differentiable, which is an important assumption in the LFV-based approach. Due to this, in figure 5, big jumps areobserved in several columns of the tracking matrix when theperiodic motions happen. These jumps are called bifurcationnoise. Because the periodic motions last less than 10 min,the bifurcation noise does not affect the SOCs to a significantextend (a small jump is still observed in the SOCs in figure 6(c)during the 18th hour of the experiment). If the lengths ofthe periodic bands are large, bifurcation noise may not benegligible.

4.2. Selection of LFVTFs

As mentioned before, LFVTFs are not unique and the selectionof the tracking function may affect the quality of the damagetracking results. For example, instead of lB(φ), other LFVTFs(hB(φ)) can be used to track damage.

Let us consider hB(φ) defined by

g(x(n)) = x(n + 1), (11)

where x(n) is a sample point of the reconstructed trajectoriesin the volume B, and x(n + 1) is the corresponding pointon the trajectory after one sampling period ts. Using thesame parameters and data processing procedures as before, atracking matrix is constructed and the SOD results are plottedin figure 7. When compared to the damage identificationresults for lB(φ), figure 7 looks similar. The two largest SOVsare much larger than the rest, and the SOCs can be regardedas linear combinations of damage states. However, since thelinearization of g(x) introduces additional noise, the signal-to-noise ratio (SNR) of the identified SOCs is not as high as thosewhen lB(φ) (see table 1).

As mentioned above, any differentiable functions can beused as g(x). The performances of several different LFVTFs

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v 2 (

V)

-2 0 2First SOC

Sec

ond

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C

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Sec

ond

SO

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-1

0

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First SOC

Sec

ond

SO

C

(a) (b)

(c) (d)

-2

-1

0

1

2

Figure 8. Actual damage phase portrait (a); the STRMPE-basedreconstruction of the phase portrait (b); the LFVTF-basedreconstruction of the phase portrait for lB(φ) (c); the LFVTF-basedreconstruction of the phase portrait for g(x(n)) = x(n + 1) (d).

are compared, and the corresponding SNR of identified SOCsare listed in table 1. Currently, the STRMPE-based [12]LFVTF provides the best identification outcome. However,there is no analytical evidence that it is the optimal LFVTF,and further research is necessary to find the optimal one.

4.3. STRMPE versus LFV

Generally, it is quite difficult to compare the performance ofthese two methods in an analytical way. Here several opinionsare given based on experiences and common sense.

4.3.1. Signal-to-noise ratio (SNR). If the STRMPE is usedfor g(x), it can be regarded as a special form of the LFV-based approach. In figure 8, the identified phase portraits fromthe STRMPE-based approach and those from the LFV-basedapproach are compared. Besides the geometric similarity ofthese phase portraits, figure 8 shows that the lowest level oflocal fluctuations happens when STRMPE is used. Generallyspeaking, SOCs from this approach always have a higher SNR.Although the difference in the SNR might be quite small (forexample, only 0.24 dB in the case of g(x(n)) = x(n+1)+x(n),table 1), we still cannot find an LFVTF that generates smootherSOCs than those from the STRMPE-based approach. The goodperformance of the STRMPE-based approach can be explainedby the fact that the STRMPEs are linear observers of damagestates themselves.

4.3.2. Robustness. Affects of environmental conditions areignored in our study. Therefore, the study of robustness focuseson the performances when bifurcation noise is prominent.Generally, the STRMPE-based approach is more robust withrespect to the bifurcation noise. This is explained by its focuson the short-time measures of dynamics, which are smoothfunctions of parameters. Thus, the STRMPE can still reflectthe change in damage states correctly during bifurcations.However, it does not mean that the STRMPE-based approach

is immune to bifurcation noise, since the population of pointsused in estimation changes drastically during bifurcations.

4.3.3. Data processing speed. The proposed LFV-basedapproach is two orders of magnitude faster than the STRMPE-based approach. For example, the LFV-based approach canprocess the data collected in our experiment within about4 min, whereas the STRMPE-based approach needs about12 h to process the same amount of data. On a workstationwith a 3.2 GHz Pentium® CPU and 1 GB of RAM, the LFV-based approach can process more than 60 000 data points persecond under a MATLAB® environment. If the algorithm isimplemented in hardware, a higher data processing speed canbe achieved.

As shown above, the LFV-based approach is not as robustas the STRMPE-based approach, and the corresponding SOCsare not as smooth. However, the damage identification resultsfrom the LFV-based approach still hold most of the informationabout damage states. Thus, the LFV-based approach showsgreat promise for on-line, real-time applications due to itsexcellent processing speed.

5. Conclusion

In this paper, the local flow variation (LFV)-based damageidentification approach is introduced as a practical implemen-tation of the phase space warping concept. This methodis based on a hypothesis that the probability distributionfunction of the fast-time trajectory is a function of thedamage state, and the estimated local expectation of thetrajectory in its phase space is used as a feature vector. Thendamage identification is realized by the smooth orthogonaldecomposition. To validate this method, an experiment isconducted and a two-dimensional damage process is identifiedfrom the collected data. Qualities of the identified damagecoordinates using different local flow variation functions arecompared. Although LFV-based damage tracking results arenot as good as those from the earlier short-time referencemodel prediction error based method, the LFV decreasesthe data processing time by about two orders of magnitudeand satisfies the requirement of real-time, on-line damageidentification.

Acknowledgment

This work was supported by the US National ScienceFoundation grant no. CMS-0237792.

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