A Multivariate Cumulative Sum Method for Continuous Damage Monitoring with Lamb-wave Sensors

A Multivariate Cumulative Sum Method for
Continuous Damage Monitoring with
Lamb-wave Sensors
Spandan Mishra
O. Arda Vanli∗
Chiwoo Park
Department of Industrial and Manufacturing Engineering,
Florida A&M University, Florida State University,
Tallahassee, FL 32310-6046, USA
December 28, 2015
Abstract
This paper proposes a new damage monitoring method based on a multivariate
cumulative sum test statistic applied to Lamb-wave sensing data for health monitoring
in composites. The CUSUM monitoring method applied to the features extracted with
Principal Components Analysis was studied to improve robustness of detection and
sensitivity to small damages. The method is illustrated with real sensor data from
fatigue loading and impact tests of carbon ﬁber materials and the performance of the
proposed CUSUM approach was compared with existing Mahalanobis distance based
monitoring techniques commonly applied in the health monitoring literature. It was
shown that the CUSUM approach can signiﬁcantly improve the misdetection rate for
monitoring gradually developing damages.
Keywords: Structural Health Monitoring, Principal Components Analysis, Multi-
variate Cumulative Sum, Hotellings T2.
∗
Corresponding Author. Email: oavanli@eng.fsu.edu. Phone: 850-410-6354.
1

1 Introduction
Guided-wave structural health monitoring (SHM) is becoming increasingly popular for mon-
itoring large structures with sparsely distributed sensors. Active sensor and actuator piezo-
electric patches permanently attached or embedded in the structure are used to actively inter-
rogate structural integrity by imparting elastic waves and measuring the resulting structural
response (Raghavan and Cesnik, 2007). In contrast to metallic materials, composite materi-
als present additional challenges in guided-wave health monitoring due to their anisotropic
properties and complex failure characteristics. Therefore accurate and timely damage detec-
tion methods are crucial for reliable health monitoring of composites.
Load-carrying composite structures operating under tensile, fatigue or impact loading or
corrosive environments develop damages during service, including matrix cracks, debonding
and delamination. These damages are usually invisible to surface inspection and they do
not immediately result in failure. Before these damages reach critical size the structure can
continue to safely operate. However, it is important to continuously monitor the integrity
of the structure in order to detect these damages early and prevent them from exceeding
critical size and resulting in catastrophic failure. Continuous monitoring is also improtant
for scheduling maintenance only when needed and result in significant cost reduction.
Majority of the existing SHM methods utilize either univariate outlier analysis or multi-
variate Mahalanobis distance based approaches (Worden et al., 2000) which are very good for
detecting relatively large damages but may be ineffective for continuous monitoring of slowly
developing damages. This paper proposes a new multivariate cumulative sum (CUSUM)
damage monitoring method with Lamb wave sensors for improved detection time and mis-
detection rates. A principal component analysis is applied to raw sensor signal to extract
important features which are then used in the monitoring scheme. The effectiveness of the
approach is illustrated on fatigue loading and impact damage cases with Lamb wave sens-
ing data. It is shown that the multivariate CUSUM can significantly improve monitoring
accuracy of Hotteling’s T2
when the structure being monitored is undergoing fatigue loading.
The rest of the paper is organized as follows. Section 2 reviews relevant literature in
structural health monitoring and damage monitoring and detection. Section 3 shows the
development of the proposed multivariate Cumulative Sum for damage monitoring. Section
4 presents application of the method on a real Lamb wave health monitoring data and
comparison to existing damage quantification approaches. Section 5 provides the concluding
remarks of the paper.
2

2 Review of Relevant Literature: guided-wave sensing
and statistical methods for damage monitoring
Lamb waves are elastic perturbations that propagate in a solid plate in two dimensions
(Viktorov, 1967). The most commonly used transducers to excite Lamb waves are embedded
or surface-bonded piezoelectric sensors (Giurgiutiu, 2005). When operating as transmitter
they transform electrical energy into mechanical energy. Surface strains are generated when
voltage is applied to the piezoelectric patch and it expands and contracts parallel to the
surface. Similarly, when operating as a receiver they transform mechanical energy into
electrical energy. Voltage is generated on the piezoelectric due to local stress and strain
(Diamanti et al., 2004). Lamb-wave sensors operate either in pitch-catch or pulse-echo
modes. In a pitch-catch configuration, the diagnostic signal emitted from the actuator travels
across damaged area while the sensor on other side of area receives the signal. On the other
hand, in a pulse-echo configuration the actuator and sensor are placed on the same side of
the inspection area and the sensor receives back the signal echoed back from the damage
(Kessler et al., 2002). A useful characteristic of Lamb wave is that whenever it reaches a
region of discontinuity a portion of the wave is reflected proportionally to the difference in
the stiffness and density of the material. Analysis of the incident wave can therefore reveal
useful information about the location and size of damage. In this paper we use a pitch-catch
based actuator-sensor configuration (Ihn and Chang, 2004).
Feature extraction is an important data analysis step in Lamb-wave based damage mon-
itoring. Features of the waveforms that are sensitive to the damages need to be identified
and estimated from the raw sensor data. Damage detection then consists of comparing the
features of waveforms from damaged structure to an undamaged structure. Feature extrac-
tion is classified into model based and signal based methods. Model based approach uses
certain preestablished models to extract features from the signal while signal based approach
extracts features from the signal without applying any sort of deterministic model to the
signal (Su et al., 2006). Feature extraction process of Lamb wave being proposed in this
paper fall into second category.
Pullin et al., (2008) have applied principal component analysis on acoustic emission
signal to differentiate fatigue crack propagation from background noise of a landing gear
component. The authors have used first and second principal components of the sensor signal
to separate fatigue source signal from the landing gear noise. Pavloupoulou et al., (2013) used
nonlinear principal component analysis and principal curves for damage prognosis. Cross et
al., (2012) proposed a method to filter out environmental variations in Lamb wave sensors
by projecting the data into its minor components so that the dimensions of the data that
carry any dependence on environment factors can be discarded. Kessler and Agrawal (2007)
3

applied principal components to Lamb wave data to detect presence, type and severity of
various types of damage. Principal components of the original sensor data vector that explain
about 70% of the variability are used in a K-th nearest neighboor algorithm to classify the
damage mode.
Statistical process control methods have been used extensively for variation reduction
in manufacturing. The Shewhart control charts are employed as the main tool to detect
shifts from an in-control statistical model and to make sure the process continues to operate
in a stable manner (Montgomery, 2013). When multiple correlated quality characteristics
are of interest then multivariate control charts should be used to simultaneously monitor
all characteristics and detect deviations. Hotelling’s T2
is the multivariate counterpart of
the Shewart chart for monitoring the mean vector of a process (MacGregor and Kourti,
1995). While the Shewhart charts can detect large process upsets reasonably well, in order
to better detect small shifts cumulative sum procedures, a set of sequential procedures based
on likelihood ratios, are recommended (Woodall and Adams, 1993). Cumulative sum utilizes
the entire history of the observed data, in contrast to the Hotelling’s T2
or Shewhart charts
which utilize only the current data point, and are therefore more sensitive to gradually
developing small shifts in the signal mean. Multivariate cumulative sum methods have
been proposed by Woodall and Ncube (1985), who used multiple univariate CUSUMs to
test shifts in the mean of a multivariate normal, and by Crosier (1988), who accumulated
deviations of the vectors from the baseline and produce a quadratic form to find a scalar
monitoring statistic. Pignatiello and Runger (1990) compared and outline the benefits of
various multivariate CUSUM approaches.
Statistical process control techniques have been utilized in structural health monitoring
by many authors. Sohn et al., (2000) used Shewart ¯X charts to monitor coefficients of an
auto-regressive time series model fitted to the measured vibration time histories from an
undamaged structures. Control limits of the control charts are used to detect deviations
of the coefficients from the initial structures for damage detection. For monitoring multiple
features extracted from sensor signals, Worden et al. (2000) and Sohn et al., (2001) extended
the univariate control charts to multivariat charts by using a Mahalanobis distance to quan-
tify the distance between potential outlier vector and the in control sample mean vector, a
measure similar to the Hotelling T2
statistic. Mujica et al., (2010) used PCA in conjunction
with a T2
statistic to extract features from multi-sensory arrangement on a turbine blade to
detect the variation due to damages in the subspace of the dominant principal components
that are greater than what can be explained by the common cause variations. Deraemaeker
et al., (2008) applied factor analysis to subdue the effects of environmental fluctuation on
data and used multivariate control charts for damage detection. Kulla (2003) used missing
data model to eliminate environmental and operational variances and Hotelling T2
chart to
4

monitor changes in modal parameters and to detect the possible damage in the structure.
3 Proposed Methodology-Multivariate cumulative Sum
monitoring with Principle Components
Lamb-wave sensor data is a high dimensional vector (on the order of thousands, depending
on the sampling frequency) and some form of dimension reduction is required to practically
monitor fewer variables and to achieve robust and repeateable detection performance. Prin-
ciple component Analysis (PCA) is a popular multivariate statistical analysis method for
dimension reduction in process monitoring and fault diagnosis applications (Jackson, 1991).
The method transforms a set of correlated variables to a smaller number of uncorrelated new
variables. The original vector of variables x = (x1, . . . , xp) is projected into a vector of new
variables z = (z1, . . . , zr) called the principal components. In the new coordinate system,
z1 is a linear combination of the original variables x1, . . . , xp and explains the maximum
possible variance, z2, another linear combination, is orthogonal to z1 and explains most of
the remaining variance, and so on. It can be seen that if the original set of p variables are
actually a linear combination of r new variables, then the first r principal components will
be sufficient to explain all the variance and remaining p − r principal components are very
small.
The monitored area of the structure is assumed to incur some damage when the sen-
sor data vector deviates significantly from a baseline (no damage) condition. The baseline
condition is represented by collecting a set of N observations with the sensor under the no
damage condition and the data is given in a p × N matrix X = [x1, x2, . . . , xN ] in which
each p × 1 column vector xj represent an observation j = 1, 2, ..., N. In practice the sensor
data must be scaled in some meaningful way to account for differences in the measurement
units of the variables. A typical approach is scaling so that all variables have zero mean
and unit variance as xij = (˜xij − µi)/σi,where (i = 1, 2, . . . , p, j = 1, 2, . . . , N) and ˜xij is
the original data, µi and σi are the sample mean and standard deviations along the i-th
dimension. The covariance matrix of the sensor data C = 1/(n − 1)XXT
is decomposed,
using singular value decomposition, as C = V DV T
into an orthogonal eigenvector matrix
V = [v1 v2 . . . vp] and a diagonal eigenvalue matrix D = diag(λ1, λ2, . . . , λp), both matrices
are of size p × p and the eigenvalues are in descending order λ1 > λ2 > · · · > λp. It can be
seen that the i-th principal component is the linear combination:
zi = vT
i x = v1ix1 + v2ix2 + . . . + vpixp (1)
in which vi is the i-th column of the V matrix (i = 1, . . . , r) also called the i-th the princi-
pal component loading vector or eigenvector. Principal component scores of an observation
5

vector are the inner products of the observation vector with principle component loading vec-
tors. For example for the j-th (j = 1, ..., N) sensor data xj the score for the i-th (i = 1, ..., r)
PC is zij = vixj. In practice the first r principal components will be sufficient to represent
variance of original data, thus the eigenvectors associated with the eigenvalues λr+1, . . . , λp
are discarded and a reduced eigenvector matrix V of size p×r is formed. The transformation
to the principal component scores is obtained through the matrix multiplication z = V T
x
in which x is the vector of raw sensor data of size p×1 and z be the r ×1 vector of principle
components in the reduced dimension.
To detect a damage with a given confidence level, the PCA method is usually followed
by a decision making procedure based on Hotelling’s T2
, a statistic for testing differences
between mean values of two data groups (MacGregor and Kourti, 1995). The sensor data
collected from the baseline (undamaged) structure is used to establish an upper control limit
(UCL) for the T2
statistic under the 100(1 − α)% confidence level, which enables one to
control the false alarm probability (the probability that an alarm is generated when in fact
there is no damage) to α. It is assumed that under the baseline (no damage) condition the
raw sensor data x follows a p-dimensional vector Normal distribution with mean vector µ0
and covariance matrix C. To determine when the sensor data indicates damage, deviations
from baseline is monitored by calculating the Hotelling T2
statistic, defined as:
T2
= (x − µ0)T
C−1
(x − µ0) =
r
i=1
(vT
i x)2
λi
. (2)
in which, the first term on the right side represents the test statistic in terms the original
sensor vector x and the second term is the representation based on the r principal components
(MacGregor and Kourti, 1995). Under the hypothesis of no damage, the T2
statistic follows
an F distribution and to control the false alarm rate at α the upper control limit of the
monitoring statistic is set at
UCL =
(N − 1)(N + 1)r
N(N − r)
Fα(r, M − r) (3)
where N is the sample size, Fα(r, N −r) is the upper 100α% point of the F distribution with
r and N − r degrees of freedom. As soon as the test statistic exceeds the control limit, i.e.,
T2
> UCL, an alarm is given indicating a damage has initiated. Otherwise, i.e., T2
≤ UCL,
it is assumed that the structure still operates in the baseline no damage condition.
Many authors who have studied Hotelling’s T2
control charts have concluded that the
chart is quite effective for detecting large and sustained shifts (three standard deviations or
larger from baseline) however, it may take a long time to signal an alarm for relatively small
or gradually developing shifts (on the order of one or two standard deviations, Lowry and
Montgomery, 1995). This presents a limitation for structural health monitoring applications,
6

where it is important to be able to detect cracks or delaminations early on, from the onset,
to be able to continuously monitor as they grow and react on time by scheduling repair or
replacement, if the growth becomes rapid.
In this paper we propose a new multivariate cumulative sum (CUSUM) approach for
detecting and monitoring small damages with Lamb wave sensors. In order to detect small
shifts a commonly used approach is to use a cumulative sum statistic which utilizes not
only the most recent sensor measurement but also the past observations as well in order to
more quickly expose slowly accumulating changes (Woodall and Adams, 1993). A principal
component analysis is conducted on the raw sensor data to find the feature vector on which
the CUSUM chart operates. We follow the formulation studied by Pignatiello and Runger
(1990) in which a cumulative multivariate difference vector between the observed PC score
vector z and the expected (under baseline) scores at time t is defined as:
st =
t
i=t−nt+1
(zi − µz0) (4)
where µz0 = 0 for baseline PC scores. The cumulative sum (CUSUM) statistic to be moni-
tored is
MCt = max{0, ||st|| − knt} (5)
in which the norm of st is found as ||st|| = sT
t D−1
st
1/2
with D the matrix of eigen values
and the summation is found using nt, the number of measurements since the last renewal
(zero value) of the CUSUM, defined as:
nt =
nt−1 + 1 for MCt−1 > 0
1 otherwise .
(6)
The CUSUM statistic accumulates differences that are larger than k, which is a chart param-
eter that needs to be specified by the user, usually taken as one half of the desired change in
the mean vector we want to make the chart sensitive for. If the CUSUM statistic exceeds the
upper threshold h of the chart, that is, if MCt−1 > h, then an out of control alarm is signaled.
There is no closed form expression to the reference distribution for the statistic, therefore
the threshold to achieve a desired false alarm rate under the hypothesis of no damage is
found by simulation (Pignatiello and Runger, 1990). In our study, we conduct Monte Carlo
simulation, to generate replicated realizations of the in control process (under the hypothesis
of no damage) with increasing threshold h values to find the value that gives an average run
length (the average time to signal an alarm) of 200 samples (Lowry and Montgomery, 1995).
This value, which corresponds to the false alarm probability of α = 0.005, is a commonly
used value to tune control charts.
7

4 Application of the proposed approach
In this section we illustrate the application of the proposed damage monitoring method
under two different damage conditions. Both cases contain Lamb-wave condition data. The
first data set, available from public domain, corresponds to fatigue loading and contains
sensor data under cycled loading until failure for multiple specimens. The second data set
corresponds to impact tests that we have conducted for one specimen.
4.1 Fatigue tests:
The fatigue data set provided by the Prognostic Center of Excellence of NASA Ames Re-
search Center (Saxena et al., 2015) is used in this study. The data corresponds Lamb wave
sensor measurements from carbon fiber composite coupons during tension-tension fatigue
tests. Two sets of six piezoelectric sensors were attached on both ends of a dogbone shaped
coupon of size 6 in by 10 in. A notch (of size 0.20 in by 0.75 in) is introduced to induce stress
concentration and accelerate delamination growth at this site. Different layup configurations
were presented for the laminations, we used the data for the first layup configurations. X-ray
images taken from the samples show that after about 10 to 100 cycles delamination damage
starts to develop in the specimens. Data from three such coupons (labelled as L1S11, L1S12
and L1S19 in the data set) were used to illustrate our monitoring methods. Two X-ray
images for the first coupon are shown in Figure 1.
Lamb wave data are collected both before the test has started and after fatigue loading
has applied. We considered the path between actuator 1 and sensor 7 and the interrogation
frequency of 200 kHz. Figure 2 shows the Lamb wave signals measured at baseline and fatigue
loading conditions from the first coupon. A sampling rate of 1.2 MHz is used to acquire sensor
signals for 1667 microseconds long time series. This resulted in 2000 data points for each
sensor measurement, thus, x is a p = 2000 dimensional vector. Baseline state corresponds to
of 0-5 cycles of loading, consisting of 14 measurement points and fatigue loading corresponds
to between 10 and 10 million cycles of loading, consisting of 12 measurement points. Figure
1 shows the X-ray image of the first coupon at baseline and after 100K cycles of loading.
We find the principle component loading vectors v1, ..., vp of the 14 × 2000 baseline data
matrix X. Figure 3a shows the proportion of variance explained by additional principal
components, indicating that r = 5 principle components is sufficient to explain about 97%
of the variation and is selected as the reduced dimension representation of the sensor data.
Figure 3b gives the histogram of the T2
values found from the scores of the 5 principle
components of the baseline data. The superimposed curve corresponds to the F distribution
that the test statistic is expected to follow, which shows a reasonable fit to the data. The
UCL for the T2
control chart is found as the 99.5% percent point (vertical red line) of the
8

distribution which results in 0.005 false alarm rate (the area under the curve to the right of
the UCL).
Figure 4 shows the principle component scores of the five principal components computed
both from the baseline and fatigue loading conditions for the first coupon. Figures 5, 6 and
7 show the T2
and the multivariate CUSUM statistics computed using the PC scores for all
three coupons. All charts are obtained by applying the principal component analysis (also
5 PC’s are used) on the Lamb wave data from the tests. The vertical line shows the actual
separation between baseline and fatigue loading conditions. The horizontal red line is the
upper control limit for both charts. The T2
statistic is computed using Equation 2. For the
CUSUM chart the difference vector st was found from Equation 4 and the cumulative sum
statistic MCt is found by applying Equation 5, in which the subscript t denotes the sample
or measurement number. The control limit of the T2
chart is found as UCL = 18.21 using
Equation 3 and the F distribution with N = 14 and r = 5. For the CUSUM chart we set
k = 0.5 to make the chart sensitive to one half of one standard deviation shifts. The upper
control limit is found as h = 6.64 by running 1000 Monte Carlo simulations.
It can be seen that for all three coupons, the charts do not signal any false alarms under
the no damage condition: none of the test statistics plotted in Figures 5, 6 and 7 to the left
of the vertical line crosses the horizontal red line. Under the fatigue condition, the T2
chart
Figure 5a, for the first coupon, correctly detects the damage at the sample 15, however,
misses 2 conditions later on for the samples 16 and 25. The CUSUM statistic plotted in
Figure 5b first signals the alarm correctly at sample 15, and from that point onwards there
is no misdetections. For the second coupon there are 13 baseline samples and 24 fatigue
loading samples (for cycles 10 to 10,000). As shown in Figure 6, both T2
and CUSUM
charts detect the shift on time at sample 14. However, T2
has 1 misdetection later on at
sample 30, while the CUSUM statistic consistently increases with no misdetections. The
data from the third coupon, which contained 10 baseline samples and 26 fatigue loading
samples, was more challenging for both methods. From Figure 7 we see that for the damage
condition that occurred at sample 11, the T2
sounds the first alarm at sample 31, therefore
it takes 20 samples to detect the damage. By contrast the CUSUM chart signals the first
alarm at sample 13, thus it takes only 3 samples to detect the damage condition.
Overall, for the three specimens, it can be seen that the CUSUM chart has a much
faster reaction time (which results in a lower misdetection rate) than the traditional control
chart to small fatigue damages. No false alarms have been observed with either chart. A
summary of misdetection rates are given in Table 1. We note that the coupons have identical
layups and are expected to have similar damage propagation behavior, however, there are
some differences in the detection performance, especially for the T2
chart. The multivariate
CUSUM chart consistently performed better than the T2
chart; the approach was successfully
9

applied on three different specimens.
Coupon CUSUM T2
1 1 out of 24 (4.2%) 2 out of 24 (8.3%)
2 0 out of 24 (0.0%) 1 out of 24 (4.2%)
3 3 out of 26 (7.7 %) 20 out of 26 (76.9%)
Table 1: Misdetection rates for the CUSUM and T2
charts from fatigue loading tests
4.2 Impact tests:
In this section we present an experimental study to monitor impact damage with the proposed
method. A three-ply carbon fiber polymer composite panel of size 27 in × 10 in × 0.035 in
is used for the experiments. Two Lamb wave sensors are surface mounted on the panel 26
in apart in the length direction of the panel and mid-way from the width direction. A 2 in2
teflon sheet is inserted between first and second layer of carbon-fibre sheet halfway between
the two sensors at the time of curing. The composite was cured with teflon sheet in-between
the fibers so that adhesion of layers is prevented and to concentrate impact stress. The
teflon sheet helps to control the location and size of damage. Impact loading is generated
by dropping a 6.6 lb weight on the same laminate from increasing heights. Every impact of
the weight exerts higher force on the composite panel. The heights of the weight are 3.3 ft,
4.9 ft, 5.6 ft, 6.2 ft, and 6.6 ft. In order to verify the delamination damage, we inspected
the panel before and after the impact tests using ultrasound C-scan. Figure ?? shows the
C-scan image of the panel after the last tests and the Lamb wave measurements from the
tests (baseline and impacts).
We collected N = 10 measurements from the pristine panel before generating any impacts
to represent the baseline (no damage) condition. The excitation frequency of 400 KHz was
determined from a preliminary experiment (in which the specimen is scanned by frequencies
ranging from 50 KHz to 500 KHz) to minimize the amount of dispersion in the actuator
signal group velocities. The sampling rate of the 12 MHz is used to acquire the signals
for 500 microseconds long time series, which resulted in 6000 data points for each sensor
measurement, that is, x is a 6000 × 1 vector.
From the 10×6000 baseline data matrix X we found that 6 principal components explain
about 86% of the variability and used for the reduced dimensional representation of the
original data. The loading vectors for the baseline data before impacts and for the sensor
data for the 5 impacts are shown in Figure 8. The magnitudes of all principal components
are smaller under the baseline condition than those after impacts, illustrating the deviation
10

that occurs from the baseline in multiple dimensions under the impact damage. Figure
9 shows the control charts found from the principle component scores. The UCL for the
T2
control chart was found for the significance level of 5%, which was found to be 29.665.
For the multivariate cusum chart we used k = 0.5 and h was found to be 9.08. It can be
seen in Figure 9 that both T2
and MCUSUM have zero false alarms and zero misdetections.
However, it can be seen that the CUSUM statistic is monotonic which more precisely reflects
the irreversible deteriorating health of the structure.
5 Conclusion
Guided-wave sensing based health monitoring has received increased interest in recent years
due to the low cost implementation of these sensing systems and the ability of guided waves
to monitor large structures. However, challenges remain in processing high dimensional
sensor data and need for accurate and reliable damage detection and monitoring methods
especially for composite materials with unisotropic properties and multiple failure modes. In
this paper we studied a new multivariate damage monitoring method for Lamb-wave sensing
data. A multivariate cumulative sum test statistic was applied to the features extracted with
principal components analysis in order to improve the robustness of detection and sensitivity
to small damages.
Two case studies were presented using real sensor data from fatigue loading and im-
pact tests of carbon fiber materials. The monitoring performance of the proposed CUSUM
approach was compared with existing Mahalanobis distance based monitoring techniques
applied in the health monitoring literature. The results followed our expectation that the
existing monitoring methods work reasonably well for relatively large changes in the struc-
tural condition, however, supplementing them with a statistic that accumulates information
over time can make the monitoring much more sensitive to gradually developing damages.
This can enhance the ability to continuously monitor growing damages and react to them
by scheduling repair or replacement actions before they reach critical size and result in fail-
ure. Fatigue loading data from multiple specimens showed that CUSUM approach can have
significantly lower misdetection rates. Possible extensions of this research would include, ac-
commodating for environmental fluctuations in the monitoring method (Cross et al., 2012)
and modeling the degradation path as a stochastic process (Lu and Meeker, 1993) in order
to make inferences about remaining useful life.
11

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(a) (b)
Figure 1: (a) X-ray image of the ﬁrst coupon (L1S11) under baseline condition. (b) X-ray
image of the ﬁrst coupon (L1S11)after 100K cycles of fatigue loading, delamination can be
seen as light gray colored region centered around the notch (Saxena et al., 2015).
14

Figure 2: Lamb wave sensor data for fatigue tests (a) Data before loading has been applied.
14 sensor trajectories are collected. (b) Data after fatigue loading has been applied. 12
sensor trajectories were collected.
15

Figure 3: (a) Cumulative variance explained by additional principal components. (b) His-
togram of the T2
values found from the PC scores of baseline data and the expected F
distribution superimposed.
Figure 4: Principal component scores of the raw signal. 5 PC’s are used. (a) Baseline (b)
Fatigue loading applied.
16

Figure 5: Control charts plotted from the principal component scores (coupon 1). (a)
Hotelling T2
chart (b) Multivariate CUSUM.
Hotelling T2
17

Hotelling T2
(a) (b)
Figure 8: Principle component scores for baseline and impacts
18

Figure 9: Control charts plotted from the principal component scores for the impact test.
(a) Hotelling T2
19

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