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On the detection of a nonlinear damage in an uncertain nonlinear beam using stochastic Volterra series

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Original Article
Structural Health Monitoring
1–14
The Author(s) 2019Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/1475921719876086 journals.sagepub.com/home/shm
On the detection of a nonlinear damage in an uncertain nonlinear beam using stochastic Volterra series
Luis GG Villani
1
, Samuel da Silva
1
, Americo Cunha
2
and Michael D Todd
3
Abstract
In the present work, two issues that can complicate a damage detection process are considered: the
uncertainties
andthe
intrinsically nonlinear behavior
. To deal with these issues, a
stochastic version
of the Volterra series is proposed as a base-line model, and novelty detection is applied to distinguish the condition of the structure between a reference baselinestate (presumed ‘‘healthy’’) and damaged. The studied system exhibits nonlinear behavior even in the reference condi-tion, and it is exposed to a type of damage that causes the structure to display a nonlinear behavior with a differentnature than the initial one. In addition, the uncertainties associated with data variation are taken into account in theapplication of the methodology. The results confirm that the monitoring of nonlinear coefficients and nonlinear compo-nents of the system response enables the method to detect the presence of the damage earlier than the application of some linear-based metrics. Besides that, the stochastic treatment enables the specification of a probabilistic interval of confidence for the system response in an uncertain ambient, thus providing more robust and reliable forecasts.
Keywords
Uncertainties, damage detection, stochastic Volterra model, nonlinear behavior
Introduction
Structural health monitoring (SHM) techniques aim toreduce the maintenance cost and increase the reliabilityand security of aerospace, civil, or mechanical engineer-ing structures.
1
Moreover, within the hierarchy of com-plexity that SHM methodologies may achieve, damage
detection
is the first step, and its performance is funda-mental to the success of the subsequent application of higher forms of SHM in the hierarchy.
2
In this sense,many authors have studied and developed various dam-age detection techniques to be implemented for differ-ent structures and applications.
2–7
Of course, there isno general approach that can be used to detect damagein all real systems, mainly when the intrinsically non-linear behavior of many systems
8
and the data variationrelated to uncertainties
9,10
are counted in the analysis.Otherwise, many linear structures can exhibit non-linear phenomena induced by the presence of damage,and in this situation, any form of nonlinearity detectoris akin to detecting damage.
11
Such damage thatproduces nonlinear behavior, for example, delamina-tion,
12,13
rubbing and unbalance in rotor systems,
14–16
and opening cracks,
17–19
may be detected through theobservation of nonlinear behavior in the measuredresponses. However, as mentioned before, many struc-tures fundamentally present nonlinear behavior even inthe reference condition,
20
causing confusion in the non-linearity (as a proxy for damage) detection process.
8
Inaddition, systems’ measured output can exhibit datavariability from sources such as environmental or inputload variation, aleatoric noise, changes in boundaryconditions, variations in the fabrication processes (i.e.
1
Departamento de Engenharia Mecaˆnica, Faculdade de Engenharia de IlhaSolteira, Universidade Estadual Paulista (UNESP), Sa˜o Paulo, Brazil
2
Nucleus of Modeling and Experimentation with Computers(NUMERICO), Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, Brazil
3
Department of Structural Engineering, University of California SanDiego (UCSD), La Jolla, CA, USA
Corresponding author:
Luis GG Villani, Departamento de Engenharia Mecaˆnica, Faculdade deEngenharia de Ilha Solteira, Universidade Estadual Paulista (UNESP), Av.Brasil 56, Ilha Solteira, Sa˜o Paulo 15385-000, Brazil.Email: luis.villani@unesp.br
unit-to-unit variability), and others.
21–23
These varia-tions all confound the damage detection process, sug-gesting the use of probabilistic tools,
24–26
regressionmodels,
5,27
machine learning algorithms,
28
probabilisticmodel selection approaches,
29–32
outlier analysis,
33,34
and novelty detection methods.
5
Considering the datavariation problem is important in reducing the numberof false alarms,
10,35
although there are situations wherevariability can mask positive detections as well.Villani et al.
36
introduced an extension of the deter-ministic Volterra series approach used by Shiki et al.
37
to detect damage in intrinsically nonlinear systemsbased on input/output measurements, recognizing datavariation. The authors examined the identification of the nonlinear data-driven model several times, usingMonte Carlo simulations, to create a stochastic refer-ence model capable of predicting the nonlinear perfor-mance and the fluctuations in the response at the sametime. Two main strategies were suggested to detect thepresence of a crack in an initially nonlinear beam, con-sidering simulated data, based on the random kernel’scontributions and random kernel’s coefficients. Theresults presented an adequate performance in distin-guishing the intrinsically nonlinear behavior and thedata variation from the nonlinearity caused by the dam-age. Although the simulated results showed a promisingperformance of the method, experimental practice sug-gested the use of a stochastic Volterra expansion toexplore the model space, particularly in the more chal-lenging present problem of inherent nonlinearity, ratherthan induced nonlinearity. The authors also hypothe-sized that a formulation using the kernels’ coefficientsand contributions concomitantly in the same indexcould be attractive to enhance the robustness of themethod, leaving the method to detect damage withdiverse characteristics without loss of performance. Inaddition, in the simulations performed, only the varia-tion of linear parameters was considered in the datavariation scenario, a simplification that might notreflect the behavior of structures that operate in thenonlinear regime of motion under the presence of uncertainties. This simplification caused a low variationof the high-order kernel coefficients, suggesting that amore realistic application encompassing the variationof the nonlinear components could complicate theapplication of the approach.Then, in Villani et al.,
38
the authors showed theexperimental application of the approach based onVolterra kernels’ contributions to detect damage in anintrinsically nonlinear beam, considering data variationrelated to the reassembly of the experimental setup.However, the damage simulated was associated withthe loss of mass that reflected in the variation of thenatural frequencies of the equivalent linear system. Inthis situation, the damage did not have an influence inthe estimation of kernels’ coefficients. These resultspointed to the demand for developing a hybridapproach capable of making use of the kernels’ contri-butions and coefficients together in a robust index.Moreover, an experimental application consideringdamage with direct influence on the nonlinear compo-nents of the response could improve the performanceof the approach with regard to the differentiationbetween the intrinsically nonlinear behavior anddamage-induced nonlinear behavior.Hence, this article aims to cover issues that were notyet considered in both previous works: (a) an experi-mental application of the stochastic Volterra seriesmethodology with the presence of damage that pro-duces nonlinear behavior to the system—a breathingcrack emulation; (b) the observance of variation in theintrinsic nonlinearity of the structure including the datavariability, thus not only considering its realization inthe linear components; (c) the generalization of theapproaches studied before with the development of anew hybrid method that considers both the kernels’coefficients and contributions simultaneously in thedamage index; and, finally, (d) the construction of atheoretical distribution of the damage index calculatedin the reference condition to reduce the number of experimental realizations needed to estimate the thresh-old value based on the kernel density method usedbefore (a practical problem when we consider real-world experiments). To the best of the authors’ knowl-edge, this is the first article that assumes nonlinearchanges associated with an experimental mechanism of damage with the assumption of reference already non-linear, but unlike Bornn et al.,
8
this paper considers theinherent uncertainties in the experimental setup to per-form a rigorous stochastic SHM method.In this context, an intrinsically nonlinear beam isanalyzed, with natural data variation in the full experi-mental context, to investigate the performance of theproposed methodology. The variation simulated inthe data reflects merged changes both in linearand nonlinear components of the system response.Furthermore, the novelty detection is reshaped to holdthe random kernels’ contributions and coefficients inthe same damage index, using principal componentanalysis (PCA) and Mahalanobis distance metrics. Aformal hypothesis test is presented to create a morerobust damage detection methodology, based on a the-oretical distribution for the Mahalanobis distancedetermined in the reference condition of the structure.The results exposed in this work have demonstrated thebeneficial performance of a nonlinear metric to detectdamage in this situation and the capability of the sto-chastic Volterra series to predict the data variation in aprobabilistic framework, improving the statistic confi-dence of the method.
2
Structural Health Monitoring 00(0)
To be familiar with the Volterra series model appli-cation in damage detection problems, find the motiva-tion to the study of the nonlinear phenomena in theprocedure and have more details about the Volterraseries reformulation to predict the nonlinear responsesconsidering data variation, the interested reader isreferred to seeing Shiki et al.,
37
Villani et al.,
36
andVillani et al.
38
This article fundamentally differentiatesfrom previous works by considering an initially uncer-tain nonlinear system subject to damage that itself induces nonlinear behavior in the structural response,which hasn’t been considered before.The present paper is organized as follows. Section‘‘The damage detection methodology based on stochas-tic Volterra series’’ describes the stochastic mathemati-cal model used to describe the nonlinear systemsresponse and the approach proposed to detect the dam-age considering the data variation related to uncertain-ties. Section ‘‘Experimental setup’’ shows the nonlinearstructure considered in this work, the damage simu-lated, and the main characteristics of its behavior.Section ‘‘Application of the proposed methodology’’shows the application of the methodology proposedand the main results obtained. Finally, section ‘‘Finalremarks’’ presents the conclusions of the work.
The damage detection methodologybased on stochastic Volterra series
This section outlines the methodology proposed to bepracticed in the damage detection problem in initiallynonlinear systems, taking into account the data varia-tion related to uncertainties. The development of thestochastic model is briefly described, with more detailin Villani et al.
36
In addition, the reader can obtainmore information about the deterministic Volterraseries expanded using the Kautz functions in Shikiet al.
37
The stochastic version of the Volterra series
In order to take into account the uncertainties in themodel formulation, a parametric probabilistic approachis assumed. Thus, the model parameters are assumed asrandom parameters and the model response as a ran-dom process.
21–23
Therefore, a probability space
(
Y
,
Σ
,
P
)
is considered, where
Y
represents the samplespace,
Σ
is a
s
-algebra over
Y
, and
P
is a probabilitymeasure.
36
In the discrete-time domain, assuming the presenceof uncertainties, single system output can be interpretedas a random process realization that is a consequenceof a single deterministic input. The relationship betweenthe deterministic input and the random output can bedescribed, using the convolution notion,
39
through thestochastic version of the Volterra series
y
(
u
,
k
) =
X
‘
h
=
1
X
N
1
1
n
1
=
0
. . .. . .
X
N
h
1
n
h
=
0
H
h
(
u
,
n
1
,
. . .
,
n
h
)
Y
h
i
=
1
u
(
k
n
i
)
ð
1
Þ
where
(
u
,
k
)
2
Y
3
Z
+
7!
y
(
u
,
k
)
represents the single ran-dom output that is consequence of the single determi-nistic input
k
2
Z
+
7!
u
(
k
)
,
(
u
,
n
1
,
::
,
n
h
)
2
Y
3
Z
h
7!
H
h
(
u
,
n
1
,
. . .
,
n
h
)
represents the random version of the
h
-order Volterra kernel, and
Z
+
represents the set of integer positive numbers.The principal benefit of the Volterra series model isthe capability to reproduce the system output as a sumof linear and nonlinear contributions
y
(
u
,
k
) =
X
‘
h
=
1
y
h
(
u
,
k
) ==
y
1
(
u
,
k
)
|ﬄﬄﬄ{zﬄﬄﬄ}
linear
+
y
2
(
u
,
k
) +
y
3
(
u
,
k
) +
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
nonlinear
ð
2
Þ
where
(
u
,
k
)
2
Y
3
Z
+
7!
y
1
(
u
,
k
)
is the random outputobtained using the first random kernel,
(
u
,
k
)
2
Y
3
Z
+
7!
y
2
(
u
,
k
)
is the random output obtainedusing the second random kernel, and so on. In thiswork, the series will be truncated in the third-order ker-nel because of the cubic characteristic of the nonlinearsystem response investigated, and the capability to sep-arate linear and nonlinear contributions in the totalresponse will be used in the damage detection proce-dure as feature sensitive to the presence of damage.On the other hand, as broadly addressed in previousworks,
36,37,40
the central disadvantage of the approachis the challenge in achieving the convergence when ahigh number of terms are used. To solve this problem,the Volterra series can be extended utilizing the Kautzfunctions,
41,42
and the system random output can berepresented as
y
(
u
,
k
)
’
X
‘
h
=
1
X
J
1
i
1
=
1
. . .. . .
X
J
h
i
h
=
1
B
h
u
,
i
1
,
. . .
,
i
h
Y
h
j
=
1
l
h
,
i
j
(
u
,
k
)
ð
3
Þ
where
J
1
,
. . .
,
J
h
represents the number of Kautz func-tions used in the kernels projections, the
h
-order ran-dom Volterra kernel expanded in the orthonormalbasis is represented by the random process
(
u
,
i
1
,
. . .
,
i
h
)
2
Y
3
Z
h
+
7!
B
h
(
u
,
i
1
,
. . .
,
i
h
)
, and the ran-dom process
(
u
,
k
)
2
Y
3
Z
+
7!
l
i
j
(
u
,
k
)
is a filtering of
Villani et al.
3
the deterministic input signal by the random Kautzfunctions. The Kautz functions are supposed randombecause their definition depends on the dynamics of thesystem and, as the system response is considered as arandom process, it is presumed that the Kautz func-tions will also randomly change.Conclusively, the coefficients of the kernels can becalculated using the least-squares approximation in adeterministic way,
37
and then, adopting Monte Carlosimulations, the process is repeated until the stochasticmodel converges. The Monte Carlo method was chosenbecause it is easier to perform when the deterministicalgorithm is known.
23,43
More information about therandom Kautz functions and the process of the randomVolterra kernels estimation may be found in Villaniet al.
36,38
Damage detection based on novelty detection
In the previous work, two points were analyzed sepa-rately in the damage detection process: the kernel’s coef-ficients and the kernel’s contributions. Nevertheless, itis difficult for practical applications to decide which fea-ture is better. Thus, the damage detection index usedhere examines both features jointly. Regarding the sto-chastic model identified with training data in the refer-ence condition, the damage-sensitive index may bedetermined in the reference status. First of all, recogniz-ing that the Volterra series will be truncated in thethird-order kernel, the kernel’s coefficients can be allo-cated together to be used as damage-sensitive feature
λ
lin
(
u
,
i
1
) =
B
1
(
u
,
1
)
B
1
(
u
,
2
)
...
B
1
(
u
,
J
1
)
8>>>><>>>>:9>>>>=>>>>;
λ
qua
(
u
,
i
1
=
i
2
) =
B
2
(
u
,
1
,
1
)
B
2
(
u
,
2
,
2
)
...
B
2
(
u
,
J
2
,
J
2
)
8>>>><>>>>:9>>>>=>>>>;
λ
cub
(
u
,
i
1
=
i
2
=
i
3
) =
B
3
(
u
,
1
,
1
,
1
)
B
3
(
u
,
2
,
2
,
2
)
...
B
3
(
u
,
J
3
,
J
3
,
J
3
)
8>>>><>>>>:9>>>>=>>>>;
ð
4
Þ
where the random process
(
u
,
i
1
)
2
Y
3
Z
+
7!
λ
lin
(
u
,
i
1
)
represents the coefficients of the first kernel,
(
u
,
i
1
=
i
2
)
2
Y
3
Z
+
7!
λ
qua
(
u
,
i
1
=
i
2
)
represents the coeffi-cients of the diagonal of the second kernel, and
(
u
,
i
1
=
i
2
=
i
3
)
2
Y
3
Z
+
7!
λ
cub
(
u
,
i
1
=
i
2
=
i
3
)
representsthe coefficients of the main diagonal of the thirdkernel.In addition, the contribution of the kernels can becalculated as
y
lin
(
u
,
k
)
’
X
J
1
i
1
=
1
B
1
u
,
i
1
ð Þ
l
1
,
i
1
(
u
,
k
)
y
qua
(
u
,
k
)
’
X
J
2
i
1
=
1
X
J
2
i
2
=
1
B
2
u
,
i
1
,
i
2
ð Þ
l
2
,
i
1
(
u
,
k
)
l
2
,
i
2
(
u
,
k
)
y
cub
(
u
,
k
)
’
X
J
3
i
1
=
1
X
J
3
i
2
=
1
X
J
3
i
3
=
1
B
3
u
,
i
1
,
i
2
,
i
3
ð Þ
..
l
3
,
i
1
(
u
,
k
)
l
3
,
i
2
(
u
,
k
)
l
3
,
i
3
(
u
,
k
)
ð
5
Þ
where
(
u
,
k
)
2
Y
3
Z
+
7!
y
lin
(
u
,
k
)
is the linear contribu-tion,
(
u
,
k
)
2
Y
3
Z
+
7!
y
qua
(
u
,
k
)
is the quadratic contri-bution, and
(
u
,
k
)
2
Y
3
Z
+
7!
y
cub
(
u
,
k
)
is the cubiccontribution. Therefore, to reduce the order of the clas-sification problem, the PCA
26,44–46
can be applied tothe kernel contributions
y
lin
(
u
,
k
)
PCA
C
lin
(
u
,
1
)
,
. . .
,
C
lin
(
u
,
n
pca
)
y
qua
(
u
,
k
)
PCA
C
qua
(
u
,
1
)
,
. . .
,
C
qua
(
u
,
n
pca
)
y
cub
(
u
,
k
)
PCA
C
cub
(
u
,
1
)
,
. . .
,
C
cub
(
u
,
n
pca
)
ð
6
Þ
where
(
u
,
n
pca
)
2
Y
3
Z
+
7!
C
lin
(
u
,
n
pca
)
represents theprincipal components of the linear contribution,
(
u
,
n
pca
)
2
Y
3
Z
+
7!
C
qua
(
u
,
n
pca
)
represents the principalcomponents of the quadratic contribution,
(
u
,
n
pca
)
2
Y
3
Z
+
7!
C
cub
(
u
,
n
pca
)
represents the principalcomponents of the cubic contribution, and
n
pca
is thenumber of principal components considered. The num-ber of components was defined based on the contribu-tion of each component in the construction of thecovariance matrix.
44
After the calculation of the kernel’s coefficients andthe principal components of the kernel’s contributions,the damage index can be defined in the referencecondition
I
lin
=
½
λ
lin
(
u
,
i
1
)
C
lin
(
u
,
n
pca
)
(
N
s
3
(
J
1
+
n
pca
))
I
nlin
=
½
λ
qua
(
u
,
i
1
=
i
2
)
λ
cub
(
u
,
i
1
=
i
2
=
i
3
)
..
C
qua
(
u
,
n
pca
)
C
cub
(
u
,
n
pca
)
(
N
s
3
(
J
2
+
J
3
+
2
n
pca
))
ð
7
Þ
where
I
lin
and
I
nlin
are the linear and nonlinear indicesin the reference situation, respectively. The linear andnonlinear indices will be assessed with the aim of com-parison between the linear and nonlinear methodolo-gies performance. Therefore, in the reference condition,
I
lin
is a
N
s
3
(
J
1
+
n
pca
)
matrix and
I
nlin
is a
N
s
3
(
J
2
+
J
3
+
2
n
pca
)
matrix, being
N
s
the number of
4
Structural Health Monitoring 00(0)
observations used in the training phase of the referencestochastic model.Now, with the structure in an unknown (‘‘test’’) situ-ation, a new deterministic model can be identified, andthe indices may be estimated in the unknown status
I
lin
=
½
l
lin
(
i
1
)
C
lin
(
n
pca
)
(
1
3
(
J
1
+
n
pca
))
I
nlin
=
½
l
qua
(
i
1
=
i
2
)
l
cub
(
i
1
=
i
2
=
i
3
)
..
C
qua
(
n
pca
)
C
cub
(
n
pca
)
(
1
3
(
J
2
þ
J
3
þ
2
n
pca
))
ð
8
Þ
where
I
lin
and
I
nlin
are, respectively, the linear and non-linear indices in an unknown condition. As the indicescalculated in the reference condition are matrices com-posed by more than one single feature, the noveltydetection has to be performed considering multivariatedata. Therefore, the comparison between the indicescalculated in the reference and an unknown conditioncan be made considering
D
2
Mahalanobis distance
33,47
D
2
m
¼ ½I
m
m
I
m
T
S
1
I
m
½I
m
m
I
m
ð
9
Þ
where
m
=
lin
or
m
=
nlin
;
m
I
m
and
S
I
m
are, respectively,the mean vector and the covariance matrix of the indexcalculated in the reference condition. The simplemachine learning method based on Mahalanobis dis-tance is used here because the classification is donebetween two possible conditions (healthy and dam-aged) and the goal is to examine the performance of the Volterra kernel characteristics as damage-sensitivefeatures and not to investigate the differences betweenrefined classification methods. Other classificationmethods can be used in the future to improve the meth-odology depending on the real application confronted.With the squared Mahalanobis distance calculated, ahypothesis test may be proposed. In this work, this dis-tance calculated in the reference condition is modeledwith a chi-square distribution, which may be calculatedbased on the assumption of independence and normal-ity in the underlying multivariate features from whichthe squared Mahalanobis distance is calculated.
48,49
Ideally,
sampling
distribution of the Mahalanobis dis-tance is desirable, but no such analytical form is knownto exist, so the theoretical model is fit to the (limited)empirical data obtained. This approximation has satis-factory performance, as will be shown further along.Therefore, the hypothesis is proposed
H
0
:
D
2
m
;
X
2
H
1
:
D
2
m
¿
X
2
ð
10
Þ
where
X
2
is the chi-square distribution,
H
0
is the null-hypothesis (healthy condition) and
H
1
is the alternativehypothesis (damaged condition).Besides, the probability of a distance value calcu-lated to be included in the theoretical chi-squaredistribution can be computed integrating its probabilitydensity function (PDF)
50
p
m
=
F
(
D
2
m
j
n
) =
ð
D
2
m
0
t
(
n
2
)
=
2
e
t
=
2
2
n
=
2
G
(
n
=
2
)
dt
ð
11
Þ
where
p
m
is the probability of the value
D
2
m
belonging tothe chi-square distribution,
G
(
:
)
is the Gamma function,and
n
is the number of degrees of freedom. Finally, asensitivity value can be determined depending on theapplication and probability of false alarms tolerated,and the hypothesis test can be rewritten
H
0
:
p
m
øb
H
1
:
p
m
\
b
ð
12
Þ
where
b
represents the sensitivity chosen for thehypothesis test. The definition of this parameterdepends on the practical application and, as an experi-mental laboratory setup is utilized, several values willbe examined to study the performance of the method.Figure 1 shows a flowchart of the damage detectionapproach. On the left-hand side of Figure 1, the train-ing phase is observed, with the identification of the sto-chastic reference model and the estimation of thedamage indices in the reference condition. On the right-hand side of Figure 1, the identification of a new modelin an unknown status and the calculation of the newindices are represented. Then, the indices are correlatedusing the squared Mahalanobis distance, and finally,the hypothesis test is applied to classify the conditionof the structure between healthy and damaged.
Experimental setup
The experimental setup used is presented in Figure 2.The structure monitored is formed by a clamped-freebeam, that is constructed by gluing four thin beams of Lexan together,
2
:
4
3
24
3
240
(mm
3
) each one, with theintention of emulating a damage propagation that isdescribed further on. At the free boundary, twosteel masses are affixed and interact with a magnet,generating a nonlinear behavior in the system response,even in the reference condition due to addedmagnetic potential. Moreover, the setup includes thefollowing:
A National Instruments acquisition system:CompactDAQ Chassis (NI cDAQ-9178);A C Series Sound and Vibration Input Modules(NI-9234);A C Series Voltage Output Module (NI-9263).
Villani et al.
5

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