DSpace at VNU: A new method for beam-damage-diagnosis using adaptive fuzzy neural structure and wavelet analysis

Mechanical Systems and Signal Processing 39 (2013) 181–194

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Mechanical Systems and Signal Processing
journal homepage: www.elsevier.com/locate/ymssp

A new method for beam-damage-diagnosis using adaptive
fuzzy neural structure and wavelet analysis
Sy Dzung Nguyen a,c, Kieu Nhi Ngo b, Quang Thinh Tran c, Seung-Bok Choi d,n
a

Inha University, Republic of Korea
Laboratory of Applied Mechanics (LAM) of Ho Chi Minh University of Technology, Vietnam
Ho Chi Minh University of Industry, HUI, Vietnam
d
Department of Mechanical Engineering, Smart Structures and Systems Laboratory, Inha University, Incheon 402-751, Republic of Korea
b
c

a r t i c l e i n f o

abstract

Article history:
Received 18 December 2012
Received in revised form
22 March 2013
Accepted 23 March 2013
Available online 16 April 2013

In this work, we present a new beam-damage-locating (BDL) method based on an algorithm
which is a combination of an adaptive fuzzy neural structure (AFNS) and an average
quantity solution to wavelet transform coefficient (AQWTC) of beam vibration signal. The
AFNS is used for remembering undamaged-beam dynamic properties, while the AQWTC is
used for signal analysis. Firstly, the beam is divided into elements and excited to be vibrated.
Vibrating signal at each element, which is displacement in this work, is measured, filtered
and transformed into wavelet signal with a used-scale-sheet to calculate the corresponding
difference of AQWTC between two cases: undamaged status and the status at the checked
time. Database about this difference is then used for finding out the elements having
strange features in wavelet quantitative analysis, which directly represents the beamdamage signs. The effectiveness of the proposed approach which combines fuzzy neural
structure and wavelet transform methods is demonstrated by experiment on measured data
sets in a vibrated beam-type steel frame structure.
& 2013 Elsevier Ltd. All rights reserved.

Keywords:
Fuzzy neural networks
Wavelet transform
Damage location
Damage diagnosis
Structure health monitor

1. Introduction
It is well known that the health monitoring of structures is one of the serious public issues which is concerned by many
researchers. The approach for the health monitoring of structures can be classified into two major groups: model-based and
data-driven methods [1]. In the first method group, the fault growth trend of structure is forecasted based on the
understanding of failure model progression. Since there are many different structures and many different types of failures,
it is difficult to develop accurate models in most practical instances. Moreover, in some cases the fault propagation of the
structures is quite complex and not fully understood. In the second method group, collected condition data is used for
building the fault propagation models. This group is based on mechanical properties in which some physical characteristics
of the structure such as structure stiffness affect to structure vibration characteristics. Damage appearing on structure

reduces its stiffness, or reduces horizontal section area, or both of these. Hence, the changes of the vibration characteristics
of the structure, such as the natural frequency, displacement or mode shape are signs to observe the damage in the
structure. These features are very important for structural health monitoring systems including structural damage detection,
location and quantification [2]. Many solutions to structure damage identification and prediction based on these signs have

n

Corresponding author. Tel.: +82 32 860 7319.
E-mail addresses: (S.D. Nguyen), (S.-B. Choi).

0888-3270/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
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S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

Nomenclature
the ith hyperplane of the NF
A(i)
a
scale parameter in wavelet transform
a–a
damage degree 3.56%
(.)
að:Þ
j ; j ¼ 0…n coefficient of A
a ¼ ½as1 as2 ⋯asm Š scale vector used for surveying
process
b

position index in wavelet transform
b–b
damage degree 6.9%
cdj
damage coefficient of the jth element
crj
relative damage index
c–c
damage degree 11.16%
d–d
damage degree 23.4%
Er ð:Þ
average-square error of the ANN
HAS
horizontal section area
M
number of fuzzy rules
N
number of neurons at the hidden layer of
the ANN
Ne
number of divided elements in the surveyed
beam
n
number of dimensions of the input data space
of the data set (or the factors relative to the
vibration-exciting statuses)
P
number of samples in the data set
Ptest

number of vibration exciting statuses used for
damage checking process

pHB
Rk
TΣ
T Σcheck
W
Wf ð:Þ
ðxi ; yi Þ
xi ¼ ½xi1
Yi
yðiÞ
j
zji
Γ ð:Þ
μikj ð:Þ
½ℵi ℘i Š
ψ
ψn

pure cluster typed a hyperbox
number of k-labeled fuzzy sets
set of training data sets corresponding to
undamaged time of beam
set of checking data sets
weight vector of the ANN
wavelet transform f ð:Þ
the ith input (xi )–output (yi ) data sample of
training data set

xi2 :::xin Š the ith input vector of training data
set (the ith vibration-exciting status)
displacement function at the ith position of
the surveyed beam
output value of the ith data sample corresponding to the jth hyperplane, A(j)
average quantity of wavelet transform
input-space-data group typed a hyperbox of
the NF system
membership value of the ith data sample in jth
k-labeled-fuzzy set
the ith input (ℵi )–output (℘i ) data sample of
the ANN
mother wavelet
complex conjugate of the mother wavelet
function ψ

been presented [3–8]. It is remarked that since the vibration signal of the real structures is the measured signal with noise
distortion, the filtering noise and the use of signal-analysis tools are to be carefully treated. One of effective approaches to
handle this problem is to use wavelet transform [9,10]. Since most of real physical systems are nonlinear, ill-defined and
uncertain, it is difficult to directly establish models by conventional mathematical means [11]. In addition, database for
structure-health-monitor system, which is based on data-driven methods, has to be correlatively established at two times:
structure-undamaged time and the checking time. This is really difficult if this process is based on traditional ways because
we are not able to exactly repeat an exciting status at two different times. Hence mathematical models such as artificial
neural networks technique (ANN) [6,12–14] or fuzzy logic (FL) [15], or combination of these models [1,11] are frequently
used to overcome the impending challenges with different degrees. In [12], ANN was used for predicting the onset of
corrosion in concrete bridge decks taking into account the parameter uncertainty. In [6], a solution to bridge-damaged
detection was proposed in which the location of the damage was obtained by using modal energy-based damage index
values.
The wavelet transform method analyzes the signal in two dimensions: time-frequency or space-frequency. In this use,
instead of using only a constant width window such as Fourier transform, wavelet transform uses a window-width-variable

parameter called scale a which can play a role similar to frequency. By this way, wavelet transform is able to locally analyze
signals to find out irregular events in dynamic response signal such as vibration signal of the damaged structure [16].
There have been many researches using wavelet transform for damage detection [3,9,10] in both types of damages: fatigue
or cracks. ANN technique and FL are types of artificial intelligence techniques. They have the potential to deliver effectively
solutions to problems which are difficult or impossible to be performed by conventional linear methods [11]. Since these
models can map any complicated functional relationship between independent and dependent variables, they can provide
better prediction and identification capabilities than traditional methods [17–19]. NF is a fuzzy system built based on ANN.
Since these mathematical models could combine to get advantages of FL and ANN, this has saliently strong points than ANN
or FL only. NF has usefully been used for identification and prediction [1,20,21].
To build NF, data space was classified to establish data clusters having common features [20,21]. This process creates the
data clusters having a role as a skeleton to build fuzzy sets of the fuzzy insurance system. On the other hand, this classifying
process could separate alien data samples from the data set to place them in distinct clusters. At these distinct clusters
membership values of other data samples in the data set are very small, or even are zero. As a result, this could reduce
influence of these alien data samples on calculated output of data samples to increase accuracy of NF system. Hence in this
side, the impact of the above data classifying process on system can be seen as a noise-filtering process for the data set.
This is one of featuring advantages of the NF built based on this way. However, the effectiveness of this issue depends on


S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

183

suitable degree of the NF-membership function for data set features. As usually, to adjust the role of each data cluster in
output-value-calculating process of the net, membership function is adjusted using an adjustment parameter, such as γ in
[20,21]. It is known that the choice of the adjustment parameter depends on features of each data space. Actually,
it significantly affects the prediction accuracy of the model. However it is very difficult to find out appropriate values of this
for each real application.
Consequently, the main contribution of this work is to propose a novel approach for beam-damage-diagnosis which is
easily applicable in practice. The proposed approach focuses on the combination of fuzzy neural structure and wavelet
transform methods. In order to achieve this goal, AFNS is used for building NF system with data clusters typed hyperboxes

which are created and joined as a part of input data space of the ANN. Subsequently, the role of each data cluster in outputvalue-calculating process of the net is adjusted by the training process of the ANN. By this way, the influence of
unsatisfactory-relationship-value quantity between fuzzy sets on calculating result of the fuzzy influence system is
compensated by the ANN when calculating the output value. This increases accuracy of the model. Furthermore, in order
to overcome the difficulty of signal analysis, a wavelet-quantitative analysis method is proposed. In this research, the
damage positions are investigated using two main steps. As a first step, database for the beam health monitor system, which
is based on the change of its dynamic property signals, is established at two times: the structure-undamaged time and the
checking time under same vibration exciting status (VES). Usually this is really difficult due to that we can not exactly repeat
VES at two different times. However, the use of the proposed AFNS could exactly interpolate data at any time when the
structure is not damaged. The second step is to analyze the created database using the wavelet-quantitative analysis based
on average wavelet transform of vibration signal with a used-scale-sheet to calculate the corresponding difference of
AQWTC between two cases: the undamaged status and the status at the checked time. Database about this difference is then
used for finding out the elements having strange features in wavelet quantitative analysis, which are beam-damage signs.
This could overcome the difficulty about finding out the optimal scale in wavelet transform for each application. The
proposed method is experimentally implemented on the beam-typed steel frame structure. Experiment results are analyzed
to verify the effectiveness of the proposed method as well as to evaluate the application possibility of the method for actual
structures.
2. Proposed fuzzy neural system
To well interact to a system its mathematical model needs to be built firstly. In structure damage location using datadriven methods as mentioned in the introduction, this model is usually established by identification based on measured
data sets. A given data set consisting of input–output samples is expressed by
ðxi ; yi Þ; xi ¼ ½xi1 xi2 :::xin Š;

i ¼ 1:::P:

ð1Þ

The above database is used for identifying the dynamic response of structure. In the above equation,xi is the input
vector,yi is corresponding output of the ith data sample in the set having P data samples. There are various ways to identify
system based on this database. In this paper, the proposed AFNS shown in Fig. 1 is used for this work. This AFNS contains
two substructures: an adaptive neuro-fuzzy system (NF) built by the neuro-fuzzy-building algorithm named HLM1of [20]
and cascade-forward neural networks (ANN). The ANN consists of one input layer, one hidden layer and one output layer.

Number of input signals depends on feature of training data set and structure of the NF; number of neurons at the hidden
layer of the net is adaptively established in training process.
The proposed structure of the AFNS in Fig. 1 is established with main steps as follows:
(1) Building data clusters of given data set
– Using the algorithm Hyperplane Clustering of [21] for building min–max hyperbox clusters Γ ðkÞ in input data
space and hyperplane classes AðkÞ ; k ¼ 1:::M; in output data space.
– Using the algorithm for cutting and separating the hyperbox having largest sample number named CSHL of [20]
to build pure data clusters, in which each pure cluster contains common-labeled samples.
Building NF system
(2) – Using the algorithm HLM1 for training process to build NF system.
Structure of the ANN
– Input vector of the ANN is M+n dimensions. The number of neuron N in the hidden layer is N ¼N0 at the beginning
time of the training process and adaptively adjusted in the AFNS training process. Where N0 is default number. In this
(3)
paper, we use N0 ¼ 100. The number of neuron in the output layer is 1. The ‘sum’ function is used for input of all of
neurons. The ‘purelin’ function, f ðsÞ ¼ s, is used for output of the neuron at output layer. Transfer function is used for
all of neurons at the hidden layer as follows:
f ðsÞ ¼

2
−1
1 þ exp ð−2sÞ

– The ith input–output sample of data set used for training the ANN is signed and established as follows:
½ℵi ℘i Š ¼ ½ðxi yðiÞ Þ; yi Š:

ð2Þ


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S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

Input layer

Hyper-planes of the NF
pHB11

The ANN layer

μi11

xi = [xi 1.... xin ]

y1( t )

(i = 1... P )
xi1

xi2

pHB1R1

μi1R1

pHBj1

μij1

Wi(in 2)

(i = 1... M )

b (2)

y ij

pHBjRj

Output

Wk(lr)

(k = 1... N )

μijR j

ˆyt

W j(in1 )

xin

( j = 1... M + n )

pHBM1

bk(1)

μiM 1

(t)
yM

pHBMRM

μi1MR M
Wi(in 2)
(i = 1... n)

Fig. 1. Structure of the AFNS based on an adaptive neuro-fuzzy system (NF) and cascade-forward neural networks (ANN).

This is combined by the ith input–output sample of the data set (1), ðxi ; yi Þ, and vector of values of corresponding
ðiÞ
ðiÞ
hyperplanesyðiÞ of the NF system trained by this data set (1), yðiÞ ¼ ½yðiÞ
1 y2 ::: yM Š. These values are calculated as
follows:
n

ðkÞ
¼ ∑ aðkÞ
yðiÞ
j xij þ a0 ;
k

k ¼ 1:::M; i ¼ 1:::P

j¼1

(k)

whereaðkÞ
created by training process of the NF.
p ; p ¼ 0…n, are coefficients of A
Training the ANN to adjust net parameters
Let W be the weight vector of the ANN, W ¼ ½w1 w2 :::wH ŠT . The error equation of the ANN can be written as follows:

(4)

Er ðWÞ ¼

1 P
^ i ðWÞÞ2
∑ ð℘ −℘
Pi¼1 i
P

¼ ∑ e2i ðWÞ
i¼1

¼ V T ðWÞVðWÞ
where
VðWÞ ¼ ½v1 ðWÞ v2 ðWÞ ::: vP ðWފT
¼ ½e1 ðWÞ e2 ðWÞ ::: eP ðWފT
Using the algorithm Levenberg–Marquardt, the weight vector W of the ANN at the (k+1)th loop, signed Wk+1,
is calculated as follows:
W kþ1 ¼ W k −½J T ðW k ÞJðW k Þ þ μIŠ−1 J T ðW k ÞVðW k Þ
where, I is unit-square matrix, size H; μ is an adaptive index; and J is the matrix Jacobian as follows:
2 ∂v
3
∂v1

∂v1
1
⋯ ∂w
∂w
∂w
H
6 ∂v 1 ∂v 2
7
∂v2 7
6 2
2
⋯ ∂w
6
H 7
JðW k Þ ¼ 6 ∂w1 ∂w2
7
6 ⋮
⋮ ⋱ ⋮ 7
4
5
∂vP
∂vP
∂vP
⋯ ∂wH
∂w1
∂w2
ðW k Þ


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185

3. Wavelet transform
3.1. Theory basis
Wavelet analysis provides a powerful tool to characterize local features of a signal. Unlike the Fourier transform, where
the function used as the basis of decomposition is always a sinusoidal wave, other basis functions can be selected for
wavelet shape according as the features of the signal. These basis functions named mother wavelet ψ. A mother wavelet is a
function of zero average:
Z þ∞
ψðtÞdt ¼ 0:
−∞

This can be dilated or compressed with a scale parameter a, and translated by a position parameter b as follows:


1
t−b
a 4 0; b∈R
ψ a;b ðtÞ ¼ pffiffiffi ψ
a
a
The wavelet transform of f at the scale a and position b is computed by correlating f(t) with a wavelet atom.
The continuous wavelet transform of f(t) is defined as follows [16]:


Z þ∞
1
t−b
Wf ða; bÞ ¼

dt
f ðtÞ pffiffiffi ψ n
a
a
−∞
where ψ n is complex conjugate of the mother wavelet functionψ.
When using the wavelet for signal analysis, if the scale parameter a is small, it results in very narrow windows and is
appropriate for high frequency components in the signal f(t). Oppositely if the scale parameter a is large, it results in wide
windows and is suitable for the low frequency components in the signal f(t). The choice of scale can be based on actual
analysis of the demand signal. The greater the scale is, the more details of the frequency division are. In fact, the advantages
of wavelet transform in signal analysis can be realized by selecting only appropriate wavelet function and wavelet scale.
3.2. Average quantity wavelet coefficient
3.2.1. Selection of mother wavelet function
In signal analysis using wavelet transform, selection of the most appropriate wavelet mother function is really essential.
In this work, the following considerations are used for this:
– The database used in this work is the displacement signal of the surveyed vibration beam which has a nearly symmetric
shape. In the database, it can be observed that signal value can usually reach zero at the initial and final points in a
period. Consequently, the wavelet mother function has to be compactly supported or nearly compactly supported with a
finite duration. In addition, both of orthogonal and biorthogonal properties are not required in this work since the
reconstruction of the original signal is not required.
– In signal decomposition and reconstruction, orthogonal wavelets such as Daubechies wavelets [22,23] are usually used
due to their efficiencies. Relation to the orthogonal wavelets, to satisfy the orthogonal property, integral of the wavelet
function and integral of the square of the wavelet function must respectively be equal to zero and one. However, the
above constraint makes the orthogonal wavelets non-differentiable [24]. In our research the wavelet NF model is
proposed, in which an optimization solution is utilized. By this approach, derivatives of the wavelet function are required
in order to minimize errors between the model outputs and the actual outputs. For this reason, in this use a nonorthogonal differentiable wavelet mother function is suitable.
Based on issues abovementioned, it can be seen that the Mexican hat function is an appropriate selection for this work.
In fact the Mexican hat wavelet function has several principal characteristics. Firstly, it is a rapidly vanishing function [25]
and is a computationally efficient function. Secondly, it can be analytically differentiated and can be used conveniently for
decomposing multidimensional time series. Equation of the selected wavelet is given as follows:

" 
"
2 #

2 #
2
t−b
1 t−b
ψ a;b ðtÞ ¼ pffiffiffi π −1=4 1−
exp −
ð3Þ
a
2 a
3

3.2.2. Sampling frequency
Actually, the appropriate sampling solution is usually selected according to the target of each use. For example, the
relation to digitally recorded noisy data decomposition and reconstruction, sampling at unequally spaced times is usually
used. In [26], to obtain information from noisy signals using wavelet methods, Hall and Penev suggested an adaptive
sampling rule. Differently, the relation to signal analysis to investigate crucial characteristics of the signal source without


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S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

reconstruction the initial signal, sampling at equally spaced times can be used. In [24], a time-frequency domain wavelet
analysis of acceleration signal of earthquake records was accomplished based on this solution.
In this study, the goal is to investigate signal characteristics to find out beam damage signs, without initial signal
reconstruction. Consequently, firstly the initial signal f ðtÞ is sampled at equally spaced times, corresponding to a constant

sampling frequency. A set of translation parameter b is then taken at the same points where the f ðtÞ is sampled. A scale
vector a ¼ ½as1 as2 ⋯asm Š, which is being depicted in detail in the next section, is then created to achieve an appropriate range
of frequency resolution. Subsequently, the created parameters a and b are used to dilate or compress the mother wavelet in
order to accomplish a family of wavelet ψ a;b ðtÞ. Next f ðtÞ is multiplied by the wavelets ψ a;b ðtÞ at different values of the scale a
and translation b. Finally, the continuous wavelet transform (CWT) coefficient Wf ða; bÞ is then obtained by summing the
created produces. Wf ða; bÞ indicates the correlation between the signal and the wavelet functions ψ a;b ðtÞ.
3.2.3. Building the scale vector
As abovementioned, the selection of the optimal scale for beam-damage diagnosis is very important to achieve best
effective result. However, the common way presented above is not enough to choose the best value of the scale. In addition,
due to the analyzed data the measured signals with noise distortion are occurred. Consequently, the result of using only one
value of the scale which is considered to be suitable for wavelet analysis to obtain information relative to the system is
sometimes unsatisfactory. Instead of getting the system information we may receive only features relative to noisy.
To overcome these, in this work the use of the scale vector a is proposed to build the AQWTC for finding out beam damage
signs. To create the scale vector a ¼ ½as1 as2 ⋯asm Š, the modified Gram–Schmidt algorithm of [27] is used, where m is number
of wavelets obtained. This work is performed as follows:
– Building element data sets: The surveyed beam is divided into Ne elements. By measuring vibration signal at each
element, which is displacement signal in this work, Ne element data sets called subsets are obtained.
– Building the element scale vector: Using the modified Gram–Schmidt algorithm for subsets, Ne element scale vectors,
signed K i ; i ¼ 1:::N e ; are correspondingly established. Relation to building K i ; the process can be summarized as follows.
Firstly, empty wavelets whose supports do not contain any data are deleted from the wavelet decomposition. Result of
this work is a nonzero wavelet coefficient frame to be created. Next, from this frame the wavelet which gives best
approximates to the measured data is selected. Subsequently, this wavelet is combined with the remainder of the
nonzero wavelet coefficient frame one at a time to determine the best combination. This procedure is repeated for all
nonzero wavelet coefficients and finally the element scale vector is created as a result of the accomplished process.
– Establishing the scale vector: The vector ais structured based on the following set:
K ¼ fK 1 ∪K 1 ∪:::∪K Ne g
where, ∪ denotes the union operator.

3.2.4. Average quantity wavelet coefficient
Based on Ne element data sets and the scale vector a created, AQWTC of the jth element at ith vibration status of the

beam, signed zji , is calculated as follows:
zji ¼

1 m
∑ Wf ji ðask ; bÞ; j ¼ 1…Ne
mk¼1

ð4Þ

where f ji is the function relating to jth element at ith vibration state of the beam. In this paper f is displacement function of
element.
4. Beam-damaged location algorithm (BDLA)
Based on the AFNS and using wavelet analysis as presented above, we propose an algorithm for beam-damaged location
named BDLA. Fig. 2 shows flow chart of the BDLA, and Fig. 3 presents relationship of parameters used in the algorithm.
At the beam-undamaged time:
Step 1. Building data sets
The beam is vibrated by different exciting statuses. Displacement is measured and filtered to build Ne data sets. The jth
data set, corresponding to the jth element, has P data samples ðxi ; zji Þ; in which xi ¼ ½xi1 :::; xin Š ; i ¼ 1:::P, expresses the ith
vibration-exciting status (VES); n depicts factors relative to the vibration-exciting statuses; zji is the AQWTC calculated based
on (4) as follows:
1
zji ¼ Bm
∑m
k ¼ 1 ∑ðbÞ Wf ji ðask ; bÞ, j¼1…Newhere B is number of sampled points of function f (it is also the size of vector b).
Step 2. Identifying elements at the beam-undamaged time
Ne the AFNS are trained to identify elements based on Ne corresponding data sets created in Step 1.
At the checking time:


S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

187

START
Building jthdata sets ( xi , z ji )
at undamaged time of the
beam, j=1...Ne ,i=1...P
AFNSj, j=1...Ne , trained

Step 1.
Undamaged
time
Step 2.

by ( xi z ji ) to identify the
beam at undamaged time

Set up Ptest ; k=0

k =: k +1

kP=

Step 3.

test ?

Y

N

Calculating Damage
Coefficients

- Making vibration of
the beam by xk
- Using the AFNSj for

Checking
time

cdj (6), crj(7)

calculating zˆ jk
Determine condition
of the beam based on

- Calculating z jk based
on (4), j = 1...Ne

Step 4.

c dj, c rj

Y

Continue ?

N

STOP

Fig. 2. Flow chart of the proposed algorithm BDLA.

The ith vibration
exciting status( xi , i = 1...Ptest )

xi1
AFNSj

xi2

zˆ ji

(j=1…Ne)

Calculate
crj , cdj

j = 1...Ne
zji

xin
- Measure vibration signal
- Calculate zji
Fig. 3. Principle for calculating damage indexes based on the AFNS.

Step 3. Calculating damage coefficients
– Perform Ptest vibration exciting statuses, which could be different from the vibration statuses used for beam-undamaged
identification as presented in Step 1. At each exciting status, vibration signal at elements is measured, filtered to calculate
AQWTC (zji ; ) based on (4) j ¼ 1:::Ne ; i ¼ 1:::P test . Based onzji ;calculate AQWTC at each element for all Ptest vibration
exciting statuses as follows:

zj ¼

P test m
1
∑ ∑ ∑ Wf ji ðask ; bÞ; j ¼ 1:::N e
mBP test i ¼ 1 k ¼ 1 ðbÞ

ð5Þ

where B is number of sampled points of function f (it is also the size of vector b).
– Based on Ptest vibration exciting statuses above, use Ne the AFNS built in Step 2 for establishing AQWTC corresponding to
each exciting status at each element, and then calculate AQWTC ðzðnotÞ
Þat each element for all Ptest vibration exciting
j


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S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

statuses as follows:
zðnotÞ
¼
j

1

P test

∑ z^ ; j ¼ 1:::Ne

P test i ¼ 1 ji

where ‘not’ expresses value of the index AQWTC corresponding to beam-undamaged time. Damage coefficient cdi of the
jth element is calculated as follows:










ðnotÞ

cdj ¼
zj −zj
; j ¼ 1…Ne
ð6Þ




– Relative change of AQWTC on each element at one time is expressed by a relative damage index, signed crj , calculated
based on (6) as follows:
crj ¼

cdj
;

max ½cdk Š

j ¼ 1:::N e

ð7Þ

k ¼ 1:::N e

Step 4. Locating damage of beam
Element having crj ¼ 1 (7) is the one most decreased in flexural rigidity, and degree of this damage is shown bycdj (6).
5. Experimental investigation
The proposed algorithm, BDLA, is used for data sets measured on a vibrated beam-typed steel frame to find out its
damaged locations.
5.1. Test rig and procedures
The experimental model is shown in Fig. 4. Motor D carrying a mass M is fixed on a beam-typed iron frame at many
different positions. The center deviation level of M, Md, is easily varied by varying distance d from position of fixed M to the
rotation axle center of D. The frequency converter V is used to change angular velocityωof D. Hence, the factors relative to
the vibration-exciting statuses n is 3. The frame, length L¼3 m, is divided into 12 equal parts by 13 nodes signed Y1,…,Y13.
Sensors (1), the vibration signal measuring system LAM_BRIDGE (2), and a computer (3) are used to measure vibration
signal at these nodes at different vibration-exciting-statuses (VES).
At the frame-undamaged time: By changing the position of D on the frame, changing the Md and angular velocity ω of
D, we created P ¼1500 VES to measure corresponding vibration signals of the structure at Y2,…,Y12. By this way, a data set
named T Σ consisting of Ne element subsets, each of these subsets having P¼ 1500 (n ¼3) input–output samples was
established. The T Σ was then used for training the AFNS to identify all elements of the frame at its undamaged time.
At the checking time: To make frame-damaged conditions, the frame was cut at one or two positions. In each case, the
reduction of horizontal section area (HSA) of the frame at the cut positions was performed by a group in 4 levels: a–a (3.56%
of HSA), b–b (6.9% of HSA), c–c (11.16% of HSA) and d–d (23.4% of HSA). Corresponding to each damage status above,
by exciting to produce vibration of the frame in order to measure vibration signal at its nodes we established a checking data
set, named T Σcheck used for checking process. Based on T Σcheck and the AFNSs trained by T Σ , the algorithm BDLA was then used
to find out beam-damaged locations corresponding to these conditions of the frame, upon this the effectiveness of the

proposed algorithm was estimated.
5.2. Results and discussion
5.2.1. Analysis of sensitivity
In order to estimate sensitivity of the signal used for the proposed algorithm, AQWTC zji (4), surveys are performed in
this section. In this survey, the beam was damaged at Y6 with 3 degrees, a–a, b–b, c–c. Analysis of vibration signal at this
position (Y6) was performed to calculate AQWTC zji (4) and damage index cd (6) in two cases: (1) using the ANN for
identifying undamaged status of the beam and a value a¼ 0.5 of the scale used for wavelet transform vibration signal, and
(2) using the proposed method with a used-scale vector a. In both of these cases, beam-vibration-exciting statuses were the
same. Results presented in Fig. 5a, b shows that although condition of the beam, beam-vibration-exciting statuses and
measuring position are the same, AQWTC zji and damage index cd in the first case is smaller than corresponding parameters
in the second case. Namely cd is 0.014, 0.02, and 0.028, in the first case; and is 0.0444, 0.0509, and 0.0542 in the second case,
corresponding to damage degrees a–a, b–b, and c–c, respectively. Besides, high increase of zji in Fig. 5b after each cycle more
clearly expresses damaged feature appearing in the beam than in Fig. 5a. These show that the proposed algorithm, BDLA,
in this work depicts better real condition of the beam at damaged location, even at small damaged degree 3.56%, a–a.


S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

Y13

Y12

Y11

Y10

Y9

Y8

Y7

Y6

Y5

Y4

Y3

Y2

189

Y1

L= 3m
Fig. 4. Experimental model: (a) photograph of the model; (b) the dividing nodes.

0.8

0.6

Average Wavelet Index, zji

Average Wavelet Index, zji

0.8

c-c, cd=0.028

b-b, cd=0.020
a-a, cd=0.014

0.4

0.2

c-c, cd=0.0542
b-b, cd=0.0509

0.6

a-a, cd=0.0444

0.4

0.2

0.0

0.0
0

200

400

600

800

Time (sampled points)

1000

0

200

400

600

800

1000

Time (sampled points)

Fig. 5. The beam was damaged at Y6 with 3 degrees: a–a, b–b, c–c. AQWTC zji (4) and damage index cd (6) were calculated based on analysis of vibration
signal at this position (Y6) of the beam in two cases: (1) using neural networks for identifying with a value of scale a ¼0.5 used for wavelet transform (a);
and (2) using the proposed method (b).

The above results can be analyzed based on the use of the scale parameter as follows. When using one value of the scale a
for wavelet transform (called the one-scale a, such as, a ¼ 0:5 as abovementioned), if the one-scale a does not correspond
with any frequency component of the signal, wavelet coefficient values are small for both damaged and undamaged statuses
of the structure. Hence no strange sign for damage diagnosis appears. However, it is different if the proposed scale vector ais
used. By this way, in the group of scales as1 ; as2 ; :::asm belong to the scale vector a; there is more probability of that at least
one ask ∈a corresponds with frequency component of the analyzed signal to create the marked sign for damage diagnosis.
In addition, it can be seen that when damage appears such as stiffness reduction of the beam, the natural frequency of the

vibration beam decreases. Consequently, natural frequencies of the beam at two statuses, damaged and undamaged
structure, are different. In this circumstance, if the scale parameter a is used for both, there may be two cases. The first is
that neither of these frequencies correspond with the scale a. Hence there is no clear sign for considering. In the second
case, the scale a corresponds with one of these frequencies or with both of them but different degrees. As a result, wavelet
coefficients corresponding to vibration signal of the beam at damaged and undamaged statuses are different. This different
feature is the positive sign for damage diagnosis. In this side, it can be clearly seen that if the scale vector a is used,
probability of appearing the second case is higher than using the one-scale a. Besides, if the scale vector a together with the
proposed average-quantity solution to wavelet transform coefficient (AQWTC) is used, the above different feature is
accumulated to create the more striking sign for damage diagnosis.


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S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

Based on the argument and verifying results as abovementioned, it can be observed that the proposed method has
advantage compared with the previous method as usually used. It is noted that the sensitivity of the signal used for the
proposed algorithm, AQWTC zji (4), is higher than the sensitivity of wavelet transform coefficient based on the one-scale a.
5.2.2. Analysis of vibration signal at elements
Here, the beam was damaged at 2 positions: Y4 and Y8 in degree 11.16%, c–c. Analysis of vibration signal is not only
performed at damaged positions but also at the other along the beam. Vibration signal at 6 positions: Y2, Y4, Y6, Y8, Y10 and
Y12, were all measured and filtered to calculate AQWTC zji (4) of each element. The results presented in Fig. 6 show that
when the beam is damaged, AQWTC at all elements in it are increased, even at undamaged elements Y2, Y6, Y10 and Y12.
However, at damaged positions the increase is faster. Namely at Y4 and Y8 AQWTC, zji, get the largest. In graphs ofzji , graphs
of z4i and z8i are highest. This feature is used for damaged location of the proposed algorithm, BDLA.
5.2.3. Single damage and the frame divided into 4 elements
In this experiment, the frame was divided into four elements and it was damaged at only one position. Four divided
elements were as follows: Y1–Y4, Y4–Y7, Y7–Y10, and Y10–Y13 having the equal lengths, L/4. We made single damage positions
on the frame, i.e. the frame reduced HSA at only one position, as follows:
– Or the frame was cut at Y4+ (the middle of Y4 and Y5) with 4 levels a–a, b–b, c–c, d–d;

– Or the frame was cut at Y6+ (the middle of Y6 and Y7) with 4 levels a–a, b–b, c–c, d–d.
Thus, in this test the different cut points all belong to the 2nd element. The algorithm BDLA was used in order to determine
these defected positions. The results are shown in Figs. 7 and 8, corresponding to the cut positions to be Y4+ and Y6+,
respectively. These figures show that the relative damage index gets the maximum value, cr2 ¼1, at the damaged element
(the 2nd element), and crð:Þ o 1 at the other elements. It means that in cases of the single damage as presented above, the
proposed algorithm BDLA rightly determines the positions in which HAS was reduced, even at low defected level (3.56%).
5.2.4. Case of double damage
In this case, the beam was divided into 6 elements and it was simultaneously damaged at two positions belonging to 2nd
and 4th elements with degree 6.9% (b–b). Damage coefficient cdi (6) of each element was calculated and depicted in Fig. 9.
Result shows that at two damaged elements, damage coefficient gets the largest values. This means that the proposed
method exactly locates the damaged positions appearing on the beam, even when damages simultaneously appeared and
damage degrees are quite small (6.9%).
5.2.5. Single/double damage and the frame divided into 3 elements
The frame was divided into 3 elements as following Y1–Y5, Y5–Y9, and Y9–Y13 having the equal lengths, L/3. BDLA and [6]
were used to determine the positions reduced HSA of the frame in two cases, single damage or double damage with three
levels b–b, c–c, d–d. In the first case, the frame was cut at Y6+. In the second case, the frame was simultaneously cut at two
positions Y6+ and Y10+ (the middle of Y10 and Y11). Figs. 10 and 11 show corresponding results. Fig. 10 presents the
experiment results in case single damage in the beam at Y6+ corresponding to three damage degrees, b–b (Fig. 10a), c–c
(Fig. 10b) and d–d (Fig. 10c). The diagrams show that the BDLA exactly determines the position reduced HSA at all these
levels. However, using [6], the damaged position is exactly determined only when damage level is higher than the level b–b
30

Average Wavelet Index, zji

z 2i
z 4i

25

z 6i

z 8i

20

z 10i
z 12i

15
10
5
0
0

100

200

300

400

500

600

Time (sampled points)
Fig. 6. The beam was damaged at Y4 and Y8, damage degree 11.16% (c–c). AQWTC zji (4) was calculated based on analysis of vibration signal at Y2, Y4, Y6, Y8,
Y10 and Y12.

1

1.0
0.8

Da. level 3.56% at
Y4+ (elem. 2)

0.670

0.6
0.330

0.4
0.190

0.2

Relative Damage Index, cr

Relative Damage Index, cr

S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

0.0

1

1.0
0.8

0.6
0.4

0.415

0.2

0.123

2

3

1

4

0.188

Da. level 11.16% at
Y4+ (elem. 2)

0.8
0.6
0.4
0.147

0.076

0.063

0.0
1

2

3

Relative Damage Index, cr

1

1.0

2

3

4

Element (1-4)

Element (1-4)

Relative Damage Index, cr

Da. level 6.9% at
Y4+ (elem. 2)

0.0

1

0.2

191

4

1

1.0

Da. level 23.4% at
Y4+ (elem. 2)

0.8
0.6
0.4
0.2

0.090

0.024

0.0
1

Element (1-4)

2

0.010

3

4

Element (1-4)

1

1.0
0.8

Da. level 3.56% at
Y6+ (elem. 2)
0.71

0.6
0.4
0.2

0.18
0.002

0.0
1

2

3

Relative Damage Index, cr

Relative Damage Index, cr

Fig. 7. Relative damage index cr (7) was calculated by BDLA in case the beam damaged at Y4+ (the middle of Y4 and Y5) belongs to the 2nd element with 4
different degrees a–a, b–b, c–c, d–d; results presented in a–d, respectively.

1

1.0
0.8

0.595

0.6
0.4
0.2

0.151
0.045

0.0
1

4

Da. level 11.16% at
Y6+ (elem. 2)

0.8
0.6
0.4
0.272
0.2

0.151

0.089

Relative Damage Index, cr

Relative Damage Index, cr

1

2

3

1

1.0

Da. level 23.4% at
Y6+ (elem. 2)

0.8
0.6

0.4
0.2

0.120

0.059

0.010

0.0

0.0
1

2

3

Element (1-4)

4

Element (1-4)

Element (1-4)

1.0

Da. level 6.9% at
Y6+ (elem. 2)

4

1

2

3

4

Element (1-4)

Fig. 8. Relative damage index cr (7) was calculated by BDLA in case the beam damaged at Y6+ (the middle of Y6 and Y7) belonging to the 2nd element with 4
different degrees a–a, b–b, c–c, d–d; results presented in a–d, respectively.

(6.9%). Fig. 10a shows that if it is based on [6], element damaged is the 3rd element due to its relative damage index gets the
largest value (cr3 ¼ 1). In fact, damage is at the 2nd element. Fig. 11 presents next experiment results in case double damage
at Y6+ and Y10+, corresponding to three damage degrees, b–b (Fig. 11a), c–c (Fig. 11b) and d–d (Fig. 11c). This figure shows that


192

S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

Damage Index, cd

10.0478

9.7681

10

Da. level 6.9%
at e.2 and e.4

8

6

5.2132
4.036

4

4.299

4.1022

5

6

2

0
1

2

3

4

Elements (1-6)

1

1.0

1

Relative damage index, crj (j=1...3)

Relative damage index, crj (j=1...3)

Fig. 9. Using the proposed algorithm, BDLA, in case the beam simultaneously damaged at two positions with degree b–b, belonging to 2nd and 4th
elements.

BDLA
[6]

0.8

0.6
0.38

0.4

0.28

0.2

0.17 0.14

1

1.0

1

BDLA
[6]

0.8

0.6
0.51
0.4

0.36

0.2

0.11 0.13

0.0

0.0
1

2

1

3

2

Relative damage index, crj, (j=1...3)

3

Elements (1-3)

Elements (1-3)

1

1.0

1

BDLA
[6]

0.8

0.6

0.4

0.29
0.21

0.2

0.14

0.10
0.0
1

2

3

Elements (1-3)
Fig. 10. Damage location results of the BDLA and [6] when the frame was divided into 3 elements, single damage position at Y6+ (belongs to the 2nd
element) with 3 different damage levels b–b (a), c–c (b), d–d (c).

in this case both of methods, BDLA and [6], could exactly determine the positions reduced HSA at all these levels since the
relative damage indexes of them get the largest at damaged elements 2nd and 3rd elements.
6. Conclusions
Reality shows that the effectiveness of the structure health monitors depends on many issues in which ability of
mathematical models and sensitivity of signal analysis tools are frequently treated. In order to improve accuracy degree of


S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

Relative damage index, crj (j=1...3)

1

0.8
0.66
0.6
0.50
0.4

0.2

1

1.0

BDLA
[6]

0.35

0.15

Relative damage index, crj (j=1...3)

1

1.0

193

1

BDLA
[6]

0.8

0.77

0.6
0.45
0.4
0.24
0.2

0.15

0.0

0.0
1

2

1

3

2

Elements (1-3)

1

1.0

Relative damage index, crj (j=1...3)

3

Elements (1-3)

BDLA
[6]

1

0.8

0.8
0.6

0.6
0.5
0.4

0.42

0.2

0.0

1

2

3

Elements (1-3)
Fig. 11. Damage location results of BDLA and [6] when the frame was divided into 3 elements, and the double damage was at Y6+, belonging to the 2th
element and at Y10+, belonging to the 3rd element with three damage degrees: b–b (a); c–c (b) and d–d (c).

structure health monitoring system, in this work we focus on two major issues: modeling method and signal analysis.
In order to achieve the research goal, two steps were performed based on the proposed beam-damage-locating algorithm
named BDLA which was developed based on a new AFNS, and a new solution using wavelet transform for signal analysis.
As a first step, the database for actual structure health monitor system, which is based on the change of its dynamic property
signals, has been established at two times: at structure-undamaged time and the checking time under same VES. In fact, this
is really difficult due to that we can not exactly repeat VES at two different times. In this work, the use of AFNS of algorithm
BDLA could exactly interpolate data at any time when the structure is not damaged, and hence it can resolve this difficulty.
The second step proposed in this work for signal analysis is average wavelet transform of vibration signal with a used-scalesheet to calculate the corresponding difference of AQWTC between two cases: undamaged status and the status at the
checked time. Database about this difference is then used for finding out the elements having strange features in wavelet
quantitative analysis, which are beam-damage signs. This could overcome the difficulty about finding out the optimal scale
in wavelet transform of vibration signal for each application.
Based on the analyzed arguments as well as verifying results via experimental investigation as abovementioned, it can be
observed that the proposed method has crucial advantages, such as the sensitivity and the ability to interpolate data.
The proposed algorithm, BDLA associated with AQWTC and the AFNS can predict or estimate the beam-damage location
much better than conventional methods. It is finally remarked that the proposed method can be extended to automaticstructure-health-monitoring systems.

Acknowledgment
This work was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea
government (MEST) (No. 2010-0015090).

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S.D. Nguyen et al. / Mechanical Systems and Signal Processing 39 (2013) 181–194

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