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Title: Damage detection using Andrew plots
Authors:
Fahit Gharibnezhad ,
Luis E. Mujica ,
Jose Rodellar
PAPER DEADLINE: **May 15, 2011**
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Aeronautics and Astronautics Department
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ABSTRACT
In current work, Andrew plot is used as a new index to detect any probable damage in
the structure. At the first step, using piezoelectric actuators and sensors, appropriate
lamb wave is propagated and received through the structure. Then Principal Component
Analysis is applied to the recorded data and prepares necessary data for Andrew curves.
Andrew plots are depicted based on calculated principal components. It has been shown
that comparing Andrew curves from baseline, structure without damage, with current
statues of structure can identify any probable damages in the structure.
INTRODUCTION
In recent years, there has been an increasing awareness of the importance of damage
prognosis systems in aerospace, civil and mechanical structures because of huge
expenses that are spent every year maintaining and repairing buildings, bridges, aircraft,
railroads, and other infrastructure. One such technique that has gained much attention in
the research and industrial communities over the past two decades is structural health
monitoring [1].
Structural health monitoring (SHM) is a damage detection technique that involves
placing intelligent sensors on a structure, periodically recording data from the sensors,
and using statistical methods to analyze the data in order to assess the condition of the
structure. The field of SHM developed through the combination of nondestructive
evaluation (NDE) methods and novel sensing and actuation techniques to create
intelligent monitoring systems permanently installed on structures. To achieve this aim
there are several potentially useful techniques, and their applicability to a particular
situation depends on the size of critical damage admissible in the structure. The
development of practical monitoring systems that can be implemented on real world
structures faces several design challenges including the development of stand alone
vibration sensors/actuators, effective damage detection algorithms, and low power
wireless transmission systems. Some SHM systems currently being developed include
vibration actuation, data acquisition, data processing, and data transmission capabilities
all in stand alone wireless sensor units attached at various locations along a structure.
All of these techniques follow the same general procedure: the pristine structure is
excited using appropriate actuators and the dynamical response is sensed at different
locations throughout the structure. Any damage will change this vibrational response,
as well as the transient by a wave that is spreading through the structure. Several
methods have been used to obtain this vibrational response, for instance: using
fiber-optic or piezoelectric transducers. In the next step, necessary data is collected and
then, the state of the structure is diagnosed by means of the processing of these data.
Correlating the signals to detect, locate and quantify these changes is a very complex
problem, but very significant progresses have been recently reported in conferences
new scientific journals and books. Among these methods, developing techniques using
or based on Principal Component Analysis (PCA) for feature discrimination has been
considered recently [2]. PCA is the most useful tool in dimensional reduction. The
central idea of PCA is to reduce the dimensionality of a data set consisting of a large
number of interrelated variables, while retaining as much as possible of the variation
present in the data set. This is achieved by transforming to a new set of variables, the
principal components (PCs), which are uncorrelated, and which are ordered so that the
first few retain most of the variation present in all of the original variables. If the data
compression is sufficient, the large number of variables is substituted by a small
number of uncorrelated latent factors, called principal components (PCs), which can
explain sufficiently the data structure. Obtained PCs could be used as feed for other
methods in SHM field. For instance, Andrew plots, as a tool to graphically interpret
multivariate data and classify them, could be used as an index to detect probable
damages in the structure using PCA. Experimental results in current work, confirm this
claim.
This paper is structured in the following way: First Andrew plots is introduced in more
details. Its properties and abilities such as case classification is reviewed. Then the
strategy to use Andrew plots based on PCA is explained. In next section, the
experimental setup is presented. then the algorithm to detect damages using Andrew
plots and PCA is scrutinized and result are presented. Finally, the obtained results are
summarized in conclusion.
ANDREW PLOTS
Graphical representation of multivariate data has been an important issue in exploratory
data analysis. Most data that are collected are multivariate in nature, and much of them
can be regarded as continuous. In the initial stages of analysis, graphic displays can be
used to explore the data, but for multivariate data, traditional histograms or two or
three-dimensional scatter plots may miss complex relationships that exist in the data set.
A number of methods for graphically displaying multivariate data have been suggested.
One of the most appealing methods is that of Andrews Plots[?].Andrews Plots provide a
means for the simultaneous display of several continuous variables. For a multivariate
observation 𝑦 = (𝑦1 , 𝑦2 , ⋯ , 𝑦𝑝 )′ Andrew suggests a representation of this
multivariate point by a curve in argument 𝑡 (−𝜋 ≤ 𝑡 ≤ 𝜋)
𝑓𝑦 (𝑡) =
𝑦1
√2
+ 𝑦2 𝑠𝑖𝑛(𝑡) + 𝑦3 𝑐𝑜𝑠(𝑡) + 𝑦4 𝑠𝑖𝑛(2𝑡) + 𝑦5 𝑐𝑜𝑠(2𝑡) + ⋯
(1)
In a two-dimensional space. The coefficient of 𝑦1 , 𝑦2,… are nothing but the terms in the
Fourier series. By storing them in a column vector
𝑎(𝑡) = (
1
√2
, 𝑠𝑖𝑛(𝑡), 𝑐𝑜𝑠(𝑡), … )′
(2)
we can write 𝑓𝑦 (𝑡) = 𝑎(𝑡)𝑦 thereby identifying 𝑓𝑦 (𝑡) as a particular linear
combination 𝑦 for a fixed 𝑡. Thus for a given 𝑡0 , (𝑡0 , 𝑓𝑦 (𝑡0 )) represents a point on
the curve, and by varying 𝑡 between −𝜋 and 𝜋 an Andrews curve is obtained as a
collection of all such points. Corresponding to 𝑛 different multivariate observations,
there will be 𝑛 different Andrews curves. A plot consisting of such curves is called an
Andrews plot.
Classifying cases with Andrew Plots
Andrews Plots can be used as a technique for classifying observations into groups.
Where unknown grouping of cases exists within a data set, an Andrews Plot can be used
to discover this effect. Cases within one group will have similar patterns of values for
the variables used in the Andrews Plot, and similar curves will be produced for these
cases. Cases in a second group will have a different profile of values for the variables
from those in the first group, and thus the curves produced for this second group will
show a different pattern from those for the first group.
Andrews Plots can also be used to allocate a case to one of a number of groups that are
known to exist in the data set.Curves for cases whose grouping is known are created and
then the curve for the unknown case superimposed. The unknown case is allocated to
the group whose set of curves its own curve most closely matches [4].
Andrews plots and Principal Components
One of the major shortcoming of Andrews plots is that while they are able to preserve
the distance and the average, they are not order preserving. In other words, the
technique places greater weight on those variables listed first (e.g., 𝒚𝟏 , 𝒚𝟐 ) and less
weight on variables listed later (𝒚𝒏−𝟏 , 𝒚𝒏 ). It is thus recommended that principal
components of the original data be produced and used in the order reflecting their
importance, unless there is a natural ordering that can be used. Consequently, their
shapes, patterns, clustering, etc., may be affected by interchanging the coefficients of
the terms in the Fourier series. Mathematically, if 𝑦 ∗ is a permutation of 𝑦, then
𝑓𝑦∗ (𝑡) = 𝑎(𝑡)𝑦 ∗ and 𝑓𝑦 (𝑡) = 𝑎(𝑡)𝑦 will represent different curves and their shapes
and/or specific patterns may be drastically altered or get hidden. As noted by
Andrews(1972), since low frequencies (large periods) are more readily caught by
human eyes than the high frequencies, it is useful to associate the most important
variables with high periods and thus arrange the variables in a multivariate vector 𝑦 by
the decreasing order of their importance. Due to mentioned reason , Principal
components could be a good solution to arrange the vector 𝑦 by the decreasing order of
their importance.
𝑦 = (𝑦1 , 𝑦2 , ⋯ , 𝑦𝑝 )′ . Imagine the data in a matrix form of 𝑌𝑛×𝑚 containing
information from 𝑛 experimental trials and 𝑚 sensors. Since physical variables have
different magnitudes and scales, each data-point is scaled using the mean of all
measurements of the sensor at the same time and the standard deviation of all
measurements of the sensor. Once the variables are normalized the covariance matrix
𝐶𝑦 calculated as follows:
1
𝐶𝑦 = 𝑛−1 𝑌 𝑇 𝑌
(3)
𝐶𝑦 is a square symmetric matrix (𝑛 × 𝑛) that measures the degree of linear
relationship within the data set between all possible pairs of variables (sensors). The
subspaces in PCA are defined by the eigenvectors and eigenvalues of the covariance
matrix as follow:
𝐶𝑦 𝑃 = 𝑃Λ
(4)
Where the eigenvectors of 𝐶𝑦 are the columns of 𝑃 and the eigenvalues are the
diagonal terms of Λ (the off-diagonal terms are zero). Columns of matrix 𝑃 are sorted
according to the eigenvalues by descending order and they are called the Principal
Components (PCs) of the data set. The eigenvector with the highest eigenvalue
represents the most important pattern in the data with the largest quantity of
information. Choosing only a reduced number 𝑟 of principal components, those
corresponding to the first eigenvalues, the reduced transformation matrix could be
imagined as a model for the structure. Geometrically, the transformed data matrix 𝑇
(score matrix) is the projection of the original data over the direction of the principal
components 𝑃.
𝑌 ♦ = 𝑌𝑃
(5)
now Andrew plot can be created based on the ordered data as 𝑓𝑦 (𝑡) = 𝑎(𝑡)𝑦 ♦ .
Alternatives for Andrew plots
Several authors, including Andrews himself, have suggested variations to the vector of
functions used in Andrews plots. In this work three different alternative for Andrew
plots are used as below. Their similarities and differences in distinguishing damaged
patterns from non damaged would be discussed. mentioned alternatives are numerated
as below. Equation (6) is original Andrew plot [3] and (7),(8) and (9) are from [5],[6]
and [7] respectively.
(1)
𝑓𝑦 (𝑡) =
𝑡≤𝜋
𝑦1
√2
+ 𝑦2 𝑠𝑖𝑛(𝑡) + 𝑦3 𝑐𝑜𝑠(𝑡) + 𝑦4 𝑠𝑖𝑛(2𝑡) + 𝑦5 𝑐𝑜𝑠(2𝑡) + ⋯ − 𝜋 ≤
(6)
(2)
𝑓𝑦 (𝑡) = 𝑦1 𝑐𝑜𝑠(𝑡) + 𝑦2 𝑐𝑜𝑠(√2𝑡) + 𝑦3 𝑐𝑜𝑠(√3𝑡) + 𝑦4 𝑐𝑜𝑠(√4𝑡) + ⋯ −
𝜋≤𝑡≤𝜋
(7)
(3)
𝜋
𝑓𝑦 (𝑡) = 𝑦1 𝑠𝑖𝑛(𝑡) + 𝑦2 𝑐𝑜𝑠(𝑡) + 𝑦3 𝑠𝑖𝑛(2𝑡) + 𝑦4 𝑐𝑜𝑠(2𝑡) + ⋯ − 𝜋 ≤ 𝑡 ≤
(8)
(4)
𝑓𝑦 (𝑡) =
1
√2
{𝑦1 + 𝑦2 (𝑠𝑖𝑛(𝑡) + 𝑐𝑜𝑠(𝑡)) + 𝑦3 (𝑠𝑖𝑛(𝑡) − 𝑐𝑜𝑠(𝑡)) +
𝑦4 (𝑠𝑖𝑛(2𝑡) + 𝑐𝑜𝑠(2𝑡)) + ⋯ }
(9)
EXPERIMENTAL SETUP
This specimen is a turbine blade of a commercial aircraft.It could be determined that the
blade is manufactured by a homogenous material with a similar density like titanium
(3.57 g/ml). Figure 1 shows the dimension of blade.
Seven PZT sensors are distributed over the surface to detect time varying strain
response data.Three of the sensors are on one face and four on the other face as can be
seen in Figure 1.To alleviate the influence of environment conditions and isolate the
experiments from environmental conditions the blade is suspended by elastic ropes.
Figure 1: Experimental setup - PZT's location and simulated damage(s)
The actuators are excited by a burst signal of three peaks and 350 KHz of frequency
(see Figure 2-a). A measured signal in one of the PZTs is shown in Figure 2-b.
Figure 2: a) Signal excitation, and b) Dynamical r esponse
Recorded data are arranged in this way that for healthy structure and each damage 10
experiments has been done.Damages are simulated by adding proper mass on the
surface of blade.
DAMAGE DETECTION ALGORITHM
The ability of Andrew plots to classify observations into groups and detect observation
that does not belong to a specific structure could be used as a method to detect
observations that are not belong to the healthy structure. In other side, it has been
mentioned that because of Andrew properties it would be necessary to associate the
most important variables with high periods and thus arrange the variables in a
multivariate vector 𝑦 by the decreasing order of their importance. To this aim, obtained
data from PZTs sensors are reduced in dimension and ordered using PCA. Primary PCs
are used as coefficients of Andrew plots its formulation. Andrew curves are calculated
for each observation and finally all curves are depicted in a same plot to demonstrate the
result. Using less number of PCs generate smoother function and more proper figure to
distinguish the damages from non damaged structure. In this work, moreover, to have a
better view curves were evaluated for values of 𝑡 from −2𝜋 to 2𝜋 a total range of
4𝜋 instead of original period. Equation (6) is re written using PCs as below
(1)
𝑓𝑦 (𝑡) =
(10)
𝑝𝑐1
√2
+ 𝑝𝑐2 𝑠𝑖𝑛(𝑡) + 𝑝𝑐3 𝑐𝑜𝑠(𝑡) + 𝑝𝑐4 𝑠𝑖𝑛(2𝑡) + 𝑝𝑐5 𝑐𝑜𝑠(2𝑡) + ⋯
Figure 4 shows the result of Andrew plots for 2 different damage with non-damaged
data using first five Principal Components. As it has been shown, there are three
different patterns. the pattern depicted in green is data from non-damaged structure and
other two patterns, depicted in red and blue, are related to damage 1 and damage 2
respectively. According to the figure, different patterns are distinguished clearly from
each other. This property not only can be used to detect probable damages in a structure
but also could classify any type of damages. To do this, enough experiment should be
done to create a baseline from the healthy structure using Andrew plot and PCA. after
collecting enough data from the non-damaged structure, available pattern could be used
as baseline to show that whether the next observation is belong to the healthy structure
pattern or there is a probable damage in the structure.
Figure 4: Andrew plots, non -damage with 2 different damages
Figure 3???
It is obvious if instead of first five PCs , the first seven PCs would be chosen the figures
will be less smooth as the higher frequencies are included. Figure 5 confirm this idea.
Figure 5: Andrew plots with classical PCA, first 17 Principal Components
Other alternatives for original Andrew plots also are used to see if they are able to
distinguish the patterns of damaged structure from non-damaged. Figure 6 shows the
result of equation (7). According to the figure, in this plot all functions belong to a
certain group intersect each other in the certain point. In other words, using this property
any new data could be categorized as non-damaged, known damaged pattern or new
type of damage.Therefore, both damage detection and identification are satisfied in this
method.
Figure 6: Gnanadesikan (left) and Paranjape(right) plot
Next figure is belong to equation (8) where the same behavior has been seen when
functions belong to the equal pattern meet each other in the same point (Figure 6).
The last one (figure 8) is depicted using equation (9) the behavior of this function is
very close to the original Andrew plot.
Figure 8: Ravindra plot
Figure 9: Different alternatives for Andrew plots
Finally , figure 9 shows all alternatives for Andrew plots in the same position.
CONCLUSIONS
Andrew plot as graphical representation of multivariate data could be used as an index
to identify any probable damage in the structure.As the order of coefficients of terms are
very important, these coefficients should be rearranged depending on their
importance.PCA can be used in this manner for bot ordering and reducing the
dimensionality.Obtained curves show that each observation is belong to which
group.Therefore,using depicted figures any damage can be separated from healthy
observation in which not only detect the damage but also clarify the type of
damage.Results from the experiments confirm this idea where two different simulated
damages are detected and classified.
ACKNOWLEDGMENTS
This work has been supported by the "Ministerio de Ciencia e Innovaciуn'' in Spain
through the coordinated research project DPI2008-06564-C02-02, and the "Formacion
de Personal Investigador" FPI doctoral scholarship. The authors would like to thank the
support from "Universitat Politйcnica de Catalunya" and "Universidad Politйcnica de
Madrid". We are also grateful to Professor Alfredo Gьemes and Dr. Antonio Fernandez
for his valuable suggestions and ideas during the experimental phase.
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