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Scientific Research and Essays Vol. 6(7), pp. 1537-1545, 4 April, 2011
Available online at http://www.academicjournals.org/SRE
ISSN 1992-2248 ©2011 Academic Journals
Full Length Research Paper
Damage detection in beam-like structure based on
wavelet packet
M. Koohdaragh1, M. A. Lotfollahi-Yaghin2*, M. M. Ettefagh3, A. Mojtehedi2
and B. Beyghbabaye1
1Faculty of Civil Engineering, Islamic Azad University, Malekan Branch, Iran.
2Faculty of Civil Engineering, University of Tabriz, Iran.
3Faculty of Mechanical Engineering, University of Tabriz, Iran.
Accepted 13 October, 2010
Modern and efficient methods of structural damage detection focus on signal processing and have
drawn researchers’ attention in recent years. These methods mainly are based on Wavelet Packet
Transforms (WPT). The WPT is an extension of the Wavelet Transform, which provides a complete
level-by-level decomposition of signal. The wavelet packets are alternative bases formed by the linear
combinations of the usual wavelet functions. Therefore, the WPT enables the extraction of features
from the signals that combine the stationary and non-stationary characteristics with an arbitrary time-
frequency resolution. In this study, a damage detection index called Wavelet Packet Energy Rate Index
(WPERI) is proposed for the diagnosing of damage in beam-like structures with two different damage
scenarios. For this purpose, the measured vibration signals of an experimental model of the beam were
decomposed into the wavelet packet components and the wavelet energy rate index which were
computed to indicate the structural damages. It will be shown that, the advantage of this method is
applying minimum number of sensors which therefore makes this method practical and economical in
comparison with other methods, for example Continues Wavelet Transform and Discrete Wavelet
Transform. Also, in spite of low signal to noise ratio, the damage location and intensity will be
identified correctly.
Key words: Wavelet packet transform, damage, signal, sensor.
INTRODUCTION
During the service life of all structures such as beams,
large-scale frames, long-span bridges and high-rise build-
ings, local damage of their key positions may have been
continually accumulated, and finally results in sudden
failure of structures. Therefore damage identification of
the structures is very vital to protect against disastrous
collapses. A classification system of damage
identification commonly defines four levels of damage
assessment: (1) the presence of damage; (2) the location
of the damage; (3) quantification of the severity of the
damage; and (4) prediction of the remaining serviceability
of the structure (Doebling and Farrar, 1998). The effect of
*Corresponding author. E-mail: lotfollahi@tabrizu.ac.ir.
damage on the structural behaviour is the variation of its
stiffness that has great influences on dynamic properties
of structure. This subject is very significant in variation of
natural frequencies and modal shapes and these
differences make it possible to explore the changes of
damages. There are various methods for examination
and distinction of these differences, but each of them has
advantages and disadvantages. One of the methods is
based on the Fourier analysis, that transforms the signal
from a time-based or space-based domain to a
frequency-based one. Unfortunately, the time or space
information may be lost during performing this
transformation and it is sometimes impossible to
determine when or where a particular event takes place
(Ren and Roeck, 2002).
Zhong and Oyadiji (2007) presented a new approach
1538 Sci. Res. Essays
for crack detection in the beam-like structures when crack
is relatively small. This approach is based on finding the
difference between two sets of detail coefficients
obtained by the use of the Stationary Wavelet Transform
(SWT) of two sets of mode shape data of the beam-like
structure. In spite of detection of crack, this method has a
disadvantage, because the method can use only
stationary signal. Yan et al. (2006) suggested intelligent
damage diagnosis and its application prospects in
structural damage detection based on wavelet. Also, the
development trends of structural damage detection are
also put forward in their method. Xiang et al. (2006)
studied the model-based forward and inverse problems in
the diagnosis of structural crack location and size by
using the finite element method of a B-Spline Wavelet on
the Interval (FEM BSWI). Zhu and Law (2005) presented
a new method for crack identification of bridge under a
moving load based on wavelet analysis. Maosen and
Pizhong (2008) proposed a new technique, integrated
wavelet transform with taking synergistic advantages of
the Stationary Wavelet Transform (SWT) and the
Continuous Wavelet Transform (CWT) that, this
technique improved the robustness of abnormality
analysis of mode shapes in damage detection. Rucka
and Wilde (2006) presented a method for estimating the
damage location in beam and plate structures. In
addition, Zheng et al. (2006) suggested a damage
detection method based on a continuous wavelet
transform and applied to analyze flexural wave in a
cracked beam.
Most of the crack identification methods, which were
mentioned earlier, apply continuous wavelet transforms
based on the modal analysis and a large number of
sensors, so in practice this analysis can be very hard and
in some cases impossible. The present study aims to
devise a method of identifying the crack in a more
applicable and practical way. Hence, Wavelet Packet
Energy Rate Index is suggested, and the advantages of
the suggested method are compared with other ones.
Also, it will be shown that this index is sensitive to
increased depth of crack and this method used a low
number of sensors for detection of characteristics of
crack.
The theory of the wavelet packet transform
The Wavelet Packet Transform (WPT) is an extension of
the WT, which provides a complete level-by-level
decomposition of signal. This transform is formed by the
linear combination of the wavelet. So, the wavelet packet
transform can indicate the permanent and temporary
features of a signal with the desired frequency-time
separation (Han et al., 2005). Wavelet packets consist of
a set of linearly combined usual wavelet functions. The
wavelet packets inherit the properties such as
orthonormality and time-frequency localization from their
corresponding wavelet functions. A wavelet packet
)(
,t
i
kj
ψ
is a function with three indices where integers i, j
and k are the modulation, scale and translation
parameters, respectively:
)2(.
2
/
2)(
,kt
j
j
j
t
i
kj −=
ψψ
(1)
The wavelet function is derived from the following
rebound relation:
),2()(2)(
2ktkht i
k
j−=
∞
−∞=
ψψ
(2)
)2()(2)(
12 ktkgt
k
ij −=
∞
−∞=
+
ψψ
(3)
The first wavelet function is known as the mother
function:
)()(
1tt
ψψ
= (4)
The discrete filters h(k) and g(k) are the quadrate mirror
filters associated with the scaling function and the mother
wavelet function. There are quite a few mother wavelets
reported in the literature. Most of these mother wavelets
are developed to satisfy some key properties such as the
invisibility and orthogonally. Choosing the best wavelet
function to find the location of the damage is very
important. There have been different wavelets such as
bior6.8, sym6, Gaus4, Gaus6, and db5.
In the current research, the mentioned wavelets are
discussed and as the db5 functions are better than the
other wavelets, it is proposed in this paper as the main
function (Han et al., 2005). The transform of packet
wavelet includes a complete decomposition of each
surface, so it analyzes in an area with a high frequency.
The rebound relation between the components of ith and
i+1th surfaces are as follows:
)()()( 2
1
12
1tftftf i
j
i
j
i
j+
−
++= (5)
)()(
12
1tHftf i
j
j
j=
−
+ (6)
)()(
2
1tGftf i
j
i
j=
+ (7)
The amount of H, G is gained as follows:
∞
−∞=
−=
k
tkhH )2({.} (8)
∞
−∞=
−=
k
tkgG )2({.} (9)
After decomposition in "j" surface, the function of f(t) is
indicated as follows:
)()(
2
1
tftf
j
i
i
j
=
=
(10)
The )(tf i
jfunctions of linear combination of wavelet
packet functions are gained as follows:
)().()( ,, ttctf i
kj
k
i
kj
i
j
ψ
∞
−∞=
=
(11)
where the wavelet packet coefficients )(
,tci
kj can be
obtained from:
dtttfc i
kj
i
kj ).()( ,,
+∞
∞−
=
ψ
(12)
Providing that the wavelet packet functions are
orthogonal:
0)().( ,, =tt n
kj
m
kj
ψψ
(13)
Each component in the WPT tree can be viewed as the
output of a filter tuned to a particular basis function; thus,
the whole tree can be regarded as a filter bank. At the top
of WPT tree (lower level), the WPT yields a good
resolution in the time domain but a poor resolution in the
frequency domain. At the bottom of the WPT tree (higher
level), the WPT results in a good resolution in the
frequency domain yet a poor resolution in the time
domain.
In this study, the wavelet packet energy index is
proposed to identify the locations and severity of
damage. To do that, the signal energy i
f
Eat j level is first
defined as:
dttftfdttfE n
j
m n
m
jf
j j
j).()()(
2
1
2
1
2
= =
∞
∞−
∞
∞−
==
(14)
Substituting Equation (14) into (17) and using the
orthogonal condition Equation (16) yields:
=
=
j
i
j
j
i
f
fEE
2
1 (15)
Koohdaragh et al. 1539
Finally:
dttfE i
jf j
2
)(
∞
∞−
=
(16)
Equation (15) shows that, the overall energy of the signal
can be decomposed to wavelet packet energy
components in different frequency bands. Finally, the
WPERI is proposed as follows, to identify the location
and intensity of the crack (Han et al., 2005)
=
−
=∆
j
i
j
i
j
i
j
j
ia
f
a
f
b
f
fE
EE
E
2
1)(
)()(
)(
(17)
Where, a
fi
j
E)(
is the component signal energy i
j
f
E at
j level without damage, b
fi
j
E)( is the component signal
energy i
j
f
Eat j level with some damage.
It is postulated that structural damage would affect the
wavelet packet component energies and subsequently
would alter this damage indicator. It is desirable to select
the WPERI, that is sensitive to the changes in the signal
characteristics (Zeng et al., 2005).
Damage identification procedures
A damage identification procedure based on the
proposed WPERI is described here. Two assumptions
were adopted in this study:
(1) The reliable undamaged and damaged structural
models are available;
(2) The structure is excited by the same impulse load and
acts at the same location.
Vibration signals measured from the structure by sensors
are first processed using the WPT. The level of wavelet
packet decomposition is determined through a trial and
error sensitivity analysis, using the undamaged and
damaged structural models. Then the wavelet packet
energy rates of signals are calculated (Benffey, 1993). If
n stands for the total number of all sensors distributing in
structure, a total of n WPERIs can be obtained, after
performing the wavelet packet decomposition. When the
mean values and the standard deviations of these
WPERIs are expressed as WPERI
µ
and WPERI
σ
, the
one-side
(
)
α
−1 upper confidence limit for the WPERI
1540 Sci. Res. Essays
Figure 1. Flowchart of damage identification processing.
1 2 3 4 5 6 7 8 9 1 0 1 1
L = 8 2 0 m m
h = 1 0 m m
b = 2 0 m m
1 2 3 4 5 6 7 8 9 1 0 1 1
L = 8 2 0 m m
1
2
3
4
5
6
7
8
9
1 0
1 1
L = 8 2 0 m m
Figure 2. The experimental beam with its cross-section and crack positions.
can be obtained from;
)(
n
ZUL WPERI
WPERIWPERI
σ
µ
α
α
+= (18)
Where
α
Z is the value of a standard normal distribution
with zero mean and unit variance, such that the
cumulative probability is )1(100
α
−. This limit can be
considered as a threshold value for alarming of possible
abnormality in the damage indicator WPERI. One special
advantage of this damage identification is that, the setting
of the threshold value is based on the statistical
properties of the damage indicator which was measured
with sensors. Any indicator that exceeds the threshold
would cause damage alarming. The location of sensors
whose WPERI values exceed the threshold will indicate
where possible damage occurs (Ang and Tang, 1975).
The flowchart in Figure 1 illustrates the considered
processes schematically.
THE EXPERIMENTAL SETUP
Three Box-section aluminium beams with span length of 820 mm,
as shown in Figure 2, are used to evaluate the proposed damage
assessment index. The properties of the beams are as follows:
Koohdaragh et al. 1541
Table 1. Characteristics of local and intensity damage in all of the undamaged and damaged beams.
Specimen Location of damage Number of elements Intensity of damage
(percent of cutting relative to the height of the beam)
D0 Safe - -
D1 1/2 L 6 20
D2 1/2 L 6 30
D3 1/4 L 3 30
D4 1/4 L 3 40
Figure 3. Beam dynamic measurements in the laboratory.
Figure 4. Acceleration–time history for two specimens.
mass density 3
/2700 mkg=
ρ
, elastic modulus 2
/70 mGNE =,
cross section area 2
200mmA =, and the inertia moment of cross
section 44 67.6666,67.1666 mmImmI yx == . Undamaged beam
and two damaged ones with different locations and intensity of
damage were considered. All beams were divided into 11 segments
as shown in Figure 2. Also, intensity of all damages was calculated
based on percent of element cutting (crack) relative to the height of
the beam as shown in Table 1. Dynamic tests and measurements
have been carried out on all the beams in the laboratory as shown
in Figure 3. The excitation is provided by an impact hammer applied
at node 11th. An accelerometer is used to measure the dynamic
responses. The sampling frequency for all signals is 3000 Hz. The
acceleration responses of beams at the same node (node 11th) are
shown in Figure 4. It is shown that damage cannot be seen in the
acceleration-time responses.
It should be mentioned that, applying one sensor for identification
1542 Sci. Res. Essays
0
5
10
15
20
25
30
1 2 3 4 5 6 7 8 9 10 11
location
(
D
1
)
WPERI
Location (D1
)
Figure 5. Histogram of wavelet packet energy rate index for specimen D1.
0
5
10
15
20
25
30
35
40
45
50
1 2 3 4 5 6 7 8 9 10 11
location(D2)
WPERI
L
ocation
(
D2)
Figure 6. Histogram of wavelet packet energy rate index for specimen D2.
0
10
20
30
40
50
60
70
1 2 3 4 5 6 7 8 9 10 11
location
(
D
3
)
WPERI
Location (D3)
Figure 7. Histogram of wavelet packet energy rate index for specimen D3.
of crack is one of the main advantage of this method. That is, the
beam exited in the 11th point and the response is extracted in 1st
point. Then, the beam was exited in the 11th point and the sensor
moved to the 2nd point and this processing continues. Therefore,
the location of load is fixed but the position of sensor (one sensor)
changes in the length of beam.
RESULTS AND DISCUSSION
For the tested beams, the decomposition level is chosen
to be 5 where a total of 32 component energies are
generated because when scale and level of
decomposition are less than 4, the identification of
damage will be impossible. After decomposing the
signals, the WPERIs
(
)
j
f
E∆ of every node are calculated
using Equation (4). The histograms of results are shown
in Figures 5 to 8. As it can be seen in the stated figures,
the suggested index is in maximum amount in the
cracked position but in other positions, the index has low
Koohdaragh et al. 1543
0
20
40
60
80
100
120
140
1 2 3 4 5 6 7 8 9 10 11
location(D4)
WPERI
Location (D4)
Figure 8. Histogram of wavelet packet energy rate index for specimen D4.
0
2
4
6
8
10
12
1 2 3 4 5 6 7 8 9 10 11
location
(
D
1
)
Location (D1)
Figure 9. Histogram after considering crack threshold for specimen D1.
0
5
10
15
20
25
1 2 3 4 5 6 7 8 9 10 11
location (D2)
Location (D2)
Figure 10. Histogram after considering crack threshold specimen D2.
amounts. So, it can be used to identify the position of the
cracks. Furthermore, the present method not only
identifies the crack position but also is sensitive to the
crack depth.
With assuming 02.0
=
α
, the one-sided 98%
confidence upper limit
α
WPERI
UL for the WPERIs can be
calculated from Equation (5). For every damaged beam,
the histogram can be drawn when the
α
WPERI
UL value is
subtracted from the WPERI values. The histograms of
result are shown in Figures 9 to 12. As it can be seen in
the figures, the crack threshold is clearly visible and in
crack-free positions, this index is zero. For instance in
Figure 12, wavelet packet energy rate index in 6st nod
shows the exact position of the crack.
1544 Sci. Res. Essays
0
5
10
15
20
25
30
35
40
1 2 3 4 5 6 7 8 9 10 11
location (D3)
Location (D3)
Figure 11. Histogram after considering crack threshold specimen D3.
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
location
(
D
4
)
L
ocation
(
D4)
Figure 12. Histogram after considering crack threshold specimen D4.
CONCLUSION
Based on the analysis results of the simulated beam, it is
demonstrated that the proposed WPT-based energy rate
index is a good candidate index, that is sensitive to local
structural damage. These calculations are rather
straightforward and not time-consuming; hence, on-line
implementation is possible if the reference information is
available. The selection of scale and the level of suitable
decomposition are very important in wavelet packet
analysis.
As a result, if the scale and level of decomposition is
less than 4, the identification of damage will be
impossible. Also, increase in the depth of the crack raises
the suggested index based on the wavelet packet, but
does not have a linear relationship with high damage in
the structure. In addition, the last and main point which
can be concluded from this research is a possibility of
using minimum sensor for crack detection in comparing
with other methods. Therefore, this method is practicable
and applicable.
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