Research Article Damage Detection on Sudden Stiffness...
17
Research Article Damage Detection on Sudden Stiffness Reduction Based on Discrete Wavelet Transform Bo Chen, 1 Zhi-wei Chen, 1,2 Gan-jun Wang, 3 and Wei-ping Xie 1 1 Key Laboratory of Roadway Bridge and Structural Engineering, Wuhan University of Technology, Wuhan 430070, China 2 School of Architecture and Civil Engineering, Xiamen University, Xiamen 361005, China 3 Zhongshan Power Supply Bureau, Guangdong 528400, China Correspondence should be addressed to Bo Chen; [email protected] Received 26 December 2013; Accepted 5 May 2014; Published 1 June 2014 Academic Editor: Xiao-Wei Ye Copyright © 2014 Bo Chen et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e sudden stiffness reduction in a structure may cause the signal discontinuity in the acceleration responses close to the damage location at the damage time instant. To this end, the damage detection on sudden stiffness reduction of building structures has been actively investigated in this study. e signal discontinuity of the structural acceleration responses of an example building is extracted based on the discrete wavelet transform. It is proved that the variation of the first level detail coefficients of the wavelet transform at damage instant is linearly proportional to the magnitude of the stiffness reduction. A new damage index is proposed and implemented to detect the damage time instant, location, and severity of a structure due to a sudden change of structural stiffness. Numerical simulation using a five-story shear building under different types of excitation is carried out to assess the effectiveness and reliability of the proposed damage index for the building at different damage levels. e sensitivity of the damage index to the intensity and frequency range of measurement noise is also investigated. e made observations demonstrate that the proposed damage index can accurately identify the sudden damage events if the noise intensity is limited. 1. Introduction e widely used vibration-based damage assessment meth- ods require modal properties that are obtained from signals via the traditional Fourier transform (FT) [1, 2]. ere are a few inherent characteristics of the FT that might affect the accuracy of damage identification. e FT is not able to present the time dependency of signals and it cannot capture the evolutionary characteristics that are commonly observed in the signals measured from naturally excited structures [3, 4]. is factor adds difficulties to the implementation aspect of the FT-based damage detection techniques. Wavelet transform (WT) can be viewed as an extension of the traditional FT with the adjustable window location and size which has recently emerged as a promising tool for structural health monitoring (SHM) and damage detection due to its inherent properties [5–7]. e earliest work on applying wavelet analysis in SHM dated back to the work of Yamamato and his group in 1995. e cumulative damage of a building with bilinear restoring force subjected to a real earthquake ground motion was estimated in terms of the accumulated ductility ratio, which is related to the number of spikes in the wavelet results [8, 9]. e wavelet approach for online detection of a sudden stiffness loss was studied and the results were compared with other approaches such as a neural network based online approximation technique and the empirical mode decomposition (EMD) method. Hou et al. [10] proposed a wavelet-based approach to identify the damage time instant and damage location of a simple structural model with breakage springs. By decomposing a vibration signal in the time domain using wavelet analysis, the discontinuity in the signal will form a signal feature, termed damage spike, in the wavelet details. Sohn et al. [11] incorporated wavelet transforms with the Holder exponent to capture the time varying nature of discontinuities. Vincent et al. [12] and Yang et al. [13, 14] used empirical mode decomposition, developed by Huang et al. [15, 16], to decompose the vibration signal to capture the signal discontinuity. Xu and Chen [17] carried out experimental studies on the applicability of EMD for Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 807620, 16 pages http://dx.doi.org/10.1155/2014/807620
Transcript of Research Article Damage Detection on Sudden Stiffness...
Research ArticleDamage Detection on Sudden Stiffness ReductionBased on Discrete Wavelet Transform
Bo Chen1 Zhi-wei Chen12 Gan-jun Wang3 and Wei-ping Xie1
1 Key Laboratory of Roadway Bridge and Structural Engineering Wuhan University of Technology Wuhan 430070 China2 School of Architecture and Civil Engineering Xiamen University Xiamen 361005 China3 Zhongshan Power Supply Bureau Guangdong 528400 China
Correspondence should be addressed to Bo Chen cbsteven163com
Received 26 December 2013 Accepted 5 May 2014 Published 1 June 2014
Academic Editor Xiao-Wei Ye
Copyright copy 2014 Bo Chen et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The sudden stiffness reduction in a structure may cause the signal discontinuity in the acceleration responses close to the damagelocation at the damage time instant To this end the damage detection on sudden stiffness reduction of building structures hasbeen actively investigated in this study The signal discontinuity of the structural acceleration responses of an example building isextracted based on the discrete wavelet transform It is proved that the variation of the first level detail coefficients of the wavelettransform at damage instant is linearly proportional to the magnitude of the stiffness reduction A new damage index is proposedand implemented to detect the damage time instant location and severity of a structure due to a sudden change of structuralstiffness Numerical simulation using a five-story shear building under different types of excitation is carried out to assess theeffectiveness and reliability of the proposed damage index for the building at different damage levels The sensitivity of the damageindex to the intensity and frequency range of measurement noise is also investigated The made observations demonstrate that theproposed damage index can accurately identify the sudden damage events if the noise intensity is limited
1 Introduction
The widely used vibration-based damage assessment meth-ods require modal properties that are obtained from signalsvia the traditional Fourier transform (FT) [1 2] There area few inherent characteristics of the FT that might affectthe accuracy of damage identification The FT is not able topresent the time dependency of signals and it cannot capturethe evolutionary characteristics that are commonly observedin the signals measured from naturally excited structures[3 4] This factor adds difficulties to the implementationaspect of the FT-based damage detection techniquesWavelettransform (WT) can be viewed as an extension of thetraditional FT with the adjustable window location and sizewhich has recently emerged as a promising tool for structuralhealth monitoring (SHM) and damage detection due to itsinherent properties [5ndash7]
The earliest work on applying wavelet analysis in SHMdated back to the work of Yamamato and his group in 1995The cumulative damage of a building with bilinear restoring
force subjected to a real earthquake ground motion wasestimated in terms of the accumulated ductility ratio whichis related to the number of spikes in the wavelet results[8 9] The wavelet approach for online detection of a suddenstiffness loss was studied and the results were comparedwith other approaches such as a neural network basedonline approximation technique and the empirical modedecomposition (EMD) method Hou et al [10] proposed awavelet-based approach to identify the damage time instantand damage location of a simple structural model withbreakage springs By decomposing a vibration signal in thetime domain using wavelet analysis the discontinuity inthe signal will form a signal feature termed damage spikein the wavelet details Sohn et al [11] incorporated wavelettransforms with the Holder exponent to capture the timevarying nature of discontinuities Vincent et al [12] and Yanget al [13 14] used empirical mode decomposition developedby Huang et al [15 16] to decompose the vibration signalto capture the signal discontinuity Xu and Chen [17] carriedout experimental studies on the applicability of EMD for
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 807620 16 pageshttpdxdoiorg1011552014807620
2 The Scientific World Journal
detecting structural damage caused by a sudden change ofstructural stiffness Chen and Xu [18] proposed two onlinedetection approaches to the sudden damage detection
The sudden stiffness reduction in a structure may causethe signal discontinuity in the acceleration responses closeto the damage location at the damage time instant Inreality the signal discontinuity around damage instant dueto sudden stiffness loss can be taken as a kind of signalsingularity and can be detected by the WT However theseverity of damage events cannot be depicted by the currentdeveloped WT based detection approaches To this end thedamage detection on sudden stiffness reduction of buildingstructures has been actively investigated in this study Thesignal discontinuity of the structural acceleration responsesof an example building is extracted based on the discretewavelet transform (DWT) It is proved that the variationof the first level detail coefficients of the WT at damageinstant is linearly proportional to the magnitude of thestiffness reduction A new damage index is developed andimplemented in this paper to detect the damage time instantlocation and severity of a structure due to a sudden change ofstructural stiffness Numerical simulation using a five-storyshear building under different types of excitation is carriedout to assess the effectiveness and reliability of the proposeddamage index for the building at different damage levels Thesensitivity of the damage index to the intensity and frequencyrange of measurement noise is also investigated The madeobservations demonstrate that the proposed damage indexcan accurately identify the damage time instant and locationin the building due to a sudden loss of stiffness The relationbetween the damage severity and the proposed damage indexis linearThe proposed damage index can identify the damageevents from the contaminated acceleration responses if thenoise intensity is limited
2 Wavelet Transform
Morlet and Grossmann initially proposed wavelet theory andMeyer developed the mathematical foundations of waveletsThe two America-based researchers Daubechies [19 20] andMallat [21] changed this by defining the connection betweenwavelets and digital signal processing Wavelets have beenapplied to a number of areas including data compressionimage processing and time-frequency spectral estimation Amother wavelet 120595(119905) is a waveform that has limited durationand an average value of zero and the wavelet kernel can beexpressed by
120595119886119887(119905) =
1
radic119886120595(
119905 minus 119887
119886) (1)
where 119886 and 119887 are dilation and translation parametersrespectively Both are real numbers and 119886 must be positiveSimilar to the short time Fourier transform one can ana-lyze square-integrable function 119891(119905) with wavelet transform
which decomposes a signal in the time domain into a two-dimensional function in the time-scale plane (119886 119887) as follows
119862 (119886 119887) = int
+infin
minusinfin
119891 (119905) 120595119886119887(119905) 119889119905=
1
radic119886int
+infin
minusinfin
119891 (119905) 120595(119905 minus 119887
119886)119889119905
(2)
The term frequency instead of scale has been used in orderto aid in understanding since a wavelet with large-scaleparameter is related to low-frequency content componentand vice versa The mother wavelet 120595(119905) should satisfy thefollowing admissibility condition to ensure existence of theinverse wavelet transform such as
119862120595= int
+infin
minusinfin
1003816100381610038161003816 (120596)10038161003816100381610038162
|120596|119889120596 lt +infin (3)
where (120596) is the Fourier transform of 120595(119905) The existence ofthe integral in (3) requires that
(0) = 0 ie int+infin
minusinfin
120595 (119909) 119889119909 = 0 (4)
The signal 119891(119905) can be reconstructed by an inverse wavelettransform of 119862(119886 119887) as defined by
119891 (119905) =1
119862120595
int
+infin
119886=minusinfin
int
+infin
119887=minusinfin
119862 (119886 119887) 120595119886119887(119905 minus 119887
119886)1
1198862119889119886 119889119887 (5)
The calculatingwavelet coefficients at every possible scale willgenerate a lot of redundant data A discrete version of thewavelet is often utilized by discretizing the dilation parameter119886 and the translation parameter 119887 in real signal processingThe procedure becomes much more efficient if dyadic valuesof 119886 and 119887 are used That is
119886 = 2119895
119887 = 2119895
119896 119895 119896 isin 119885 (6)
where 119885 is a set of integers This sampling of the coordinates(119886 119887) is referred to as dyadic sampling because consecutivevalues of the discrete scales differ by a factor of 2 Using thediscrete scales of WT one can define the discrete wavelettransform (DWT) [21] as follows
119862119895119896= int
+infin
minusinfin
119891 (119905) 120595119895119896(119905) 119889119905 = 2
minus1198952
int
+infin
minusinfin
119891 (119905) 120595 (2minus119895
119905 minus 119896) 119889119905
(7)
The signal resolution is defined as the inverse of the scale1119886 = 2
minus119895 and the integer 119895 is referred to as the level Thesignal can be reconstructed from the wavelet coefficients 119862
119895119896
and the reconstruction algorithm is called the inverse discretewavelet transform as follows
119891 (119905) =
+infin
sum
119895=minusinfin
+infin
sum
119896=minusinfin
1198621198951198962minus1198952
120595 (2minus119895
119905 minus 119896) (8)
Another function 120601(119905) referred to as the scaling functionis important for the numerical implementation of the fastwavelet transform [19] Suppose now that the dyadic scale is
The Scientific World Journal 3
used for 119886 and 119887 and consider a reference level 119869 Applying(7) for this case one obtains a set of coefficients as follows
119888119863119869(119896) = int
+infin
minusinfin
119891 (119905) 120595119869119896(119905) 119889119905 (9)
The coefficient 119888119863119895(119896) is known as the level-119869 detail coef-
ficients Using the dyadic scale level 119869 yields the level-119869approximation coefficients as follows
119888119860119869(119896) = int
+infin
minusinfin
119891 (119905) 120601119869119896(119905) 119889119905 (10)
In theDWT a signal can be represented by its approximationsand details The detail at level 119895 is defined as
119863119895(119905) =
+infin
sum
119896=minusinfin
119888119863119895(119896) 120595119895119896(119905) (11)
and the approximation at level 119895 is defined as
119860119869(119905) =
+infin
sum
119896=minusinfin
119888119860119869(119896) 120601119895119896(119905) (12)
It becomes obvious that119860119869minus1
= 119860119869+ 119863119869
119891 (119905) = 119860119869(119905) + sum
119895le119869
119863119895(119905)
(13)
3 Signal Feature due to Sudden Damage
The dynamic responses of a five-story shear building sub-jected to a sudden stiffness reduction at its first story underthree different external excitations are computed The massand horizontal stiffness of the undamaged building areuniform for all stories as shown in Figure 1 The mass andhorizontal stiffness of the each floor are 119898 = 13 times 10
6 kgand 119896 = 40 times 10
9Nm respectively The Rayleigh dampingassumption is adopted to construct the structural dampingmatrix and the damping ratios in the first two modes ofvibration of the building are set as 005The original buildingis supposed to suffer a sudden 20 stiffness reduction inthe first story with the horizontal stiffness reducing from40times10
9Nm to 32times109Nm while the horizontal stiffnessin other stories remains unchangedThe frequency reductiondue to 20 stiffness reduction in the first story is small witha maximum reduction of nomore than 5 in the first naturalfrequency
The sinusoidal excitation seismic excitation and impulseexcitation are respectively utilized to calculate the accelera-tion responses of the example building to examine the signalfeatures due to sudden stiffness reductionThe seismic excita-tion used is the first 10-second portion of the El-Centro 1940earthquake ground acceleration (S-N component) with apeak amplitude of 10ms2 A sinusoidal excitation expressedby the following equationwith 10-second duration is assumedto act on each floor of the building
119891 (119905) = 1300 sdot sin (4120587119905) (0 le 119905 le 10 s) (kN) (14)
k1
k2
k3
k4
k5
m1
m2
m3
m4
m5
Figure 1 Elevation of a five-story building model
An impulse excitation represented by 01ms initial veloc-ity is supposed to occur at the first floor of the building Thedamage time instant of the building is set as 60 s for seismicexcitation and sinusoidal excitation and as 02 s for impulseexcitation The equation of motion of the example buildingwith a 20 sudden stiffness reduction at its first story atthe given time instant is established The dynamic responsesunder each type of external excitation are computed by usingthe Newmark-120573 method with a time interval of 0002 s Thetwo factors in theNewmark-120573method are selected as 120572 = 12and 120573 = 14 [22 23]
The computed acceleration time histories of the first floorunder seismic excitation are displayed in Figure 2 It is diffi-cult to find the signal feature due to sudden damage by directvisual inspection of the original acceleration responses The02-second portion of the acceleration responses is expandedto permit a close look at the signal feature due to suddendamage event It is seen that there exists a sudden jump inthe original signal at the damage time instant The structuralacceleration time histories of the first floor under sinusoidaland impulse excitation are also displayed in Figures 3 and 4respectively Similar to the observations made from seismicexcitation the direct inspection on original signals cannotdirectly find the signal feature due to sudden damage eventA detained investigation on small time portions indicates thesudden jump of original acceleration responses at damagetime instants as shown in the figures The sudden reductionof horizontal stiffness of the first floor causes a clear signaldiscontinuity in the acceleration response time history at thedamage time instant
Figure 5 displays the power spectrum of accelerationresponses with and without sudden damage events It isclear that the change in the spectrum amplitude induced bythe sudden damage is very small which cannot provide theenough information to capture the damage event Moreoverthe exact damage instant still cannot be determined inthe frequency domain based on the fast Fourier transformFurther inspection of the spectrum curves indicates that
4 The Scientific World Journal
0 2 4 6 8 10minus15
00
15 Acceleration response
Time (s)
(ms2)
(a)
Time (s)590 595 600 605 610
Acceleration response
minus15
00
15
(ms2)
(b)
Figure 2 Signal discontinuity due to sudden damage (seismic excitation)
0 2 4 6 8 10minus15
00
15 Acceleration response
Time (s)
(ms2)
(a)
Time (s)590 595 600 605 610
Acceleration response
minus15
00
15
(ms2)
(b)
Figure 3 Signal discontinuity due to sudden damage (sinusoidal excitation)
00 05 10 15 20minus10
0
10 Acceleration response
Time (s)
(ms2)
(a)
012 014 016 018 020 022 024minus2
0
2 Acceleration response
Time (s)
(ms2)
(b)
Figure 4 Signal discontinuity due to sudden damage (impulse excitation)
the structural acceleration responses present quite differentspectrum properties under different external excitations Ifthe building is subjected to El-Centro earthquake the powerspectrum has a relatively wide frequency range and the firsttwo natural frequencies can be effectively identified Theimpulse excitation signal however holds very short timeinterval and a very wide frequency range The accelerationresponses of the impulse excited building present very abun-dant frequency components and four natural frequenciescan be identified from the power spectrum To compare thespectrum components of sinusoidal seismic and impulseexcitation one can conclude that the structural responsessubjected to impulse excitation have the most abundant highfrequency components
Since the signal discontinuity is of very high frequencythe wavelet transform is applied to decompose the originalacceleration responses Figure 6 displays the first level detailcoefficients of wavelet transform for acceleration responsesunder seismic excitation It can be seen that the signaldiscontinuity is reserved in the first level detail coefficientonly instead of in the approximation components This isbecause the first level detail component often contains thehighest frequency component of the original signal To
extract inherent signal feature due to sudden damage fromthe signal discontinuity in the original acceleration responsetime history the acceleration responses of the building undereach type of excitation are computed for a sudden reductionof stiffness at the first story with different damage levels anddamage time instants Similar observations can bemade fromthe decomposed detail coefficients of the wavelet transformof the acceleration responses under sinusoidal and impulseexcitations
4 Damage Index
Let us consider a SDOF system subjected to a sudden stiffnessreduction under impulse excitationThemass of the system isdenoted as119898 the damping ratio 120585 of the system is supposed toremain unchanged before and after sudden damage and thestiffness is denoted as 119896 which will have a sudden reductionat time instant 119905
119894as follows
119896 = 119896119906
(0 le 119905 le 119905119894)
119896119889
(119905119894lt 119905)
(15)
The Scientific World Journal 5
0 5 10 15 20Frequency (Hz)
0
100
200
300
400Po
wer
spec
trum
(m2s3)
(a) Sinusoidal excitation
0 5 10 15 20Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
um (m
2s3)
(b) Seismic excitation
0 5 10 15 20Frequency (Hz)
00
25
50
75
100
Original structure20 damage
Pow
er sp
ectr
um (m
2s3)
(c) Impulse excitation
Figure 5 Power spectrum of acceleration responses of the first floor before and after sudden damage event
in which 119896119906and 119896
119889are the stiffness of undamaged and
damaged system respectively The initial velocity and dis-placement due to the impulse excitation are assumed to be0 and V
0 respectively The circular frequency of the system
before and after sudden damage can be expressed as
120596119906= radic
119896119906
119898 120596
119889= radic
119896119889
119898 (16)
Define a frequency reduction coefficient 120572 that varies from 0to 1 as follows
120596119889= 120572 sdot 120596
119906(0 lt 120572 lt 1) (17)
The stiffness reduction can be expressed as
Δ119896 = 119896119889minus 119896119906= 119898(120596
2
119889minus 1205962
119906) = 119898120596
2
119906(1205722
minus 1) (18)
The equation of motion of the SDOF system before suddendamage is
119910 + 2120585120596119906119910 + 1205962
119906119910 = 0 (19)
The above equation can be solved in terms of the given initialconditions and the structural dynamic responses are
119910 (119905) = 119860119906(119905) V0119890minus120585120596119906119905
sdot1
120596119906radic1 minus 1205852
119910 (119905) = V0119890minus120585120596119906119905
(119861119906(119905) minus
119860119906(119905) 120585
radic1 minus 1205852)
119910 (119905) = minus V0120596119906119890minus120585120596119906119905
sdot[119860119906(119905) (1 minus 2120585
2
) + 2120585119861119906(119905) radic1 minus 1205852]
radic1 minus 1205852
(20)
6 The Scientific World Journal
in which
119860119906(119905) = sin(120596
119906119905radic1 minus 1205852)
119861119906(119905) = cos(120596
119906119905radic1 minus 1205852)
(21)
Let us take the time instant 119905119894as the starting point of the SDOF
system after sudden damage and use a new time axis 1199051=
119905minus119905119894Then the equation ofmotion of the system after damage
becomes
119910119889+ 2120585120596
119889119910119889+ 1205962
119889119910119889= 0 (119905 gt 119905
119894) (22)
The initial conditions for (10) can be expressed as
119910119889(0) = 119910 (119905
119894) = 119860
119906(119905119894) V0119890minus120585120596119906119905119894 sdot
1
120596119906radic1 minus 1205852
119910119889(0) = 119910 (119905
119894) = V0119890minus120585120596119906119905119894 (119861119906(119905119894) minus
119860119906(119905119894) 120585
radic1 minus 1205852)
(23)
The damping ratio of a civil engineering structure is oftenvery small that is radic1 minus 1205852 asymp 1 The acceleration responseat the time instant 119905
1is
119910119889(1199051)
=V0120596119889119890minus120585(1205961198891199051+120596119906119905119894)
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894) [119861119889(1199051)radic1 minus 1205852 minus 119860
119889(1199051) 120585]
+ 120596119906119860119889(1199051) (21205852
minus 1) (119860119906(119905119894) 120585minus119861
119906(119905119894)radic1 minus 1205852)
minus 2120596119906120585119861119889(1199051) [119860119906(119905119894) 120585radic1 minus 1205852+119861
119906(119905119894) (1205852
minus 1)]
(24)
Furthermore the time interval should be very small todescribe the sudden stiffness reduction properly thus
Δ119905 = 119905119894+1
minus 119905119894997888rarr 0
119860119889(Δ119905) = sin(120596
119889Δ119905radic1 minus 1205852) asymp 0
119861119889(Δ119905) = cos(120596
119889Δ119905radic1 minus 1205852) asymp 1
(25)
Therefore the acceleration response at the time instant 119905119894+1
is
119910 (119905119894+1) = 119910119889(Δ119905)
=V0120596119889119890minus120585120596119906119905119894
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894)radic1 minus 1205852 minus 2120585120596
119906
times [119860119906(119905119894) 120585radic1 minus 1205852 + 119861
119906(119905119894) (1205852
minus 1)]
(26)
The first level detail coefficients of wavelet transform of theacceleration responses before sudden damage 119888119863 119910(119905)
1can be
expressed as
119888119863119910(119905)
1(119896) = int
+infin
minusinfin
119910 (119905) 1205951119896(119905) 119889119905 (27)
The first level detail coefficients of wavelet transform of theacceleration responses after sudden damage 119888119863 119910119889(1199051)
1can be
expressed as
119888119863119910119889(1199051)
1(119896) = int
+infin
minusinfin
119910119889(1199051) 1205951119896(1199051) 1198891199051 (28)
The variation of first level detail coefficients of theWT beforeand after the sudden damage event can be given as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896) = int
+infin
minusinfin
( 119910 (119905119894+1) minus 119910 (119905
119894)) 1205951119896(119905) 119889119905
(29)
Considering that the damping ratio of a civil engineeringstructure is often very small the above expression can besimplified as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
= minusΔ119896V0
119898120596119906
int
+infin
minusinfin
119890minus120585120596119906119905119894 sdot sin(120596
119906119905119894radic1 minus 1205852)120595
1119896(119905) 119889119905
(30)
The above equation reveals that the variation of first leveldetail coefficients of the WT before and after a suddendamage event is approximately linear to the sudden stiffnessreduction for given initial velocity damage instant andstructural parameters before damage If the time interval Δ119905for sudden damage is further regarded as a fixed value (30)indicates that the acceleration response discontinuity due tosudden stiffness reduction can be reflected by the variationrate of first level detail coefficients of the wavelet transform atdamage instant A damage index DI
119894 is defined to reflect the
signal discontinuity due to sudden damage at the time instant119905119894as follows
DI119894=
100381610038161003816100381610038161003816100381610038161003816
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
Δ119905
100381610038161003816100381610038161003816100381610038161003816
(119896 = 2 3 119899 minus 1) (31)
whereΔ119905 = 119905119896+1
minus119905119896and 119899 is the total number of time intervals
for the whole response time history This damage index iscomputed in the time domain and it is an instantaneous indexsuitable for online structural health monitoring applicationThe linear relationship between the proposed damage indexand the sudden stiffness reduction can be observed as follows
DI119894prop |Δ119896| (32)
5 Damage Detection
51 Selection of Mother Wavelet To examine the feasibil-ity of the proposed damage index and damage detection
The Scientific World Journal 7
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(a) db2
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(b) db4
Figure 6 Level-1 detail coefficients of wavelet transform for acceleration responses under seismic excitation
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
db1
db2
db3
db4
Figure 7 The Daubechies wavelets (db1ndashdb4)
approaches the first floor of the five-story building issupposed to suffer different levels of the sudden stiffnessreduction but the sudden damage occurs at the same timeSix damage scenarios are considered in the numerical inves-tigation Listed in Table 1 are the damage severities and thefive natural frequencies of the building before and after thesudden damage It is seen from Table 1 that the stiffnessreduction in the first story of the building affectsmainly lowernatural frequencies It is noted that if the stiffness reductionin the first floor is less than 10 the maximum frequencychange is no more than 2 In addition the variations of thehigher mode shapes are much smaller than those of the lowermode shapes
Wavelet transform can be utilized to detect the signalsingularity due to sudden stiffness change While the detec-tion efficiency depends on many factors such as waveletvanishing moments supporting length in the time domainfrequency components of original acceleration responsesand signal noise Thus three different Daubechies motherwavelets db1 db2 and db4 are utilized to examine the effectsof properties of mother wavelets on the detection on thestructural sudden damage The vanishing moments of thedb1 db2 and db4 wavelets are 1 2 and 4 respectivelyand they have the gradually increased supporting length asplotted in Figure 7 The basic principles of wavelet transformprove that the longer the wavelet supporting length is thefiner the distinguishing ability in the frequency domain isTherefore the mother wavelet with long supporting length ismore suitable for detecting the higher frequency componentsin the original signal
To examine the feasibility of the proposed damage indexand damage detection approaches for identifying damageevents the acceleration responses of the aforementioned five-story shear building to the seismic excitation sinusoidalexcitation and impulse excitation are computed respectivelyThe building is subject to a 20 sudden stiffness reductionat times 60 s 60 s and 02 s in the first story of thebuilding under seismic excitation sinusoidal excitation andimpulse excitation respectively The time step used in thecomputation is 0002 seconds
Shown in Figure 8 are the damage detection resultsusing db1 db2 and db4 for 20 sudden stiffness reductionrespectively It can be seen from Figure 8(a) that no matterwhich mother wavelet is used the damage index of the firstfloor is very large only at time 119905= 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 The damage indices of the first floor at allother time instants are very small so that the damage index attime 119905 = 60 seconds looks like a spikeTherefore the damagetime instant can be easily identified by the occurrence timeof the sharp damage index It is demonstrated that the DWTbased approach using all the three Daubechies wavelets canaccurately detect the damage time instant of the buildingsubjected to sinusoidal excitation For the building excitedby El-Centro ground motion DWT using db1 wavelet failto detect damage instant while the approach using db2 anddb4 wavelet successfully captures the damage events Forthe impulse excited case only the DWT using db4 waveletcan accurately detect the damage instant due to suddenstiffness change In reality the sudden stiffness loss will cause
8 The Scientific World Journal
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)200
100
0
Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
200
100
0
20 damage
20 damage
20 damage
(a) Sinusoidal excitation
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)
100
50
0
Dam
age i
ndex
(db2
)
100
50
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage
20 damage
20 damage
(b) Seismic excitation
Dam
age i
ndex
(db1
)
200
100
0 Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage 20 damage
20 damage
00 04 08 12 16 20
Time (s)00 04 08 12 16 20
Time (s)
00 04 08 12 16 20
Time (s)
(c) Impulse excitation
Figure 8 Damage detection using different db wavelets
Table 1 Natural frequency before and after sudden damage
Damage extent Frequencies (Hz)1198911
1198912
1198913
1198914
1198915
0 2513 7335 11563 14854 169411 2508 (minus018) 7324 (minus015) 1155 (minus011) 1485 (minus005) 1694 (minus001)2 2504 (minus036) 7313 (minus030) 1154 (minus021) 1484 (minus011) 1694 (minus003)5 2490 (minus094) 7278 (minus078) 1150 (minus053) 1481 (minus026) 1693 (minus007)10 2464 (minus197) 7218 (minus162) 1144 (minus106) 1478 (minus052) 1692 (minus014)20 2407 (minus441) 7088 (minus348) 1132 (minus218) 1471 (minus101) 1690 (minus026)40 2253 (minus116) 6781 (minus816) 1107 (minus450) 1457 (minus192) 1686 (minus046)Note values in brackets are the percentage of change in natural frequency
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 The Scientific World Journal
detecting structural damage caused by a sudden change ofstructural stiffness Chen and Xu [18] proposed two onlinedetection approaches to the sudden damage detection
The sudden stiffness reduction in a structure may causethe signal discontinuity in the acceleration responses closeto the damage location at the damage time instant Inreality the signal discontinuity around damage instant dueto sudden stiffness loss can be taken as a kind of signalsingularity and can be detected by the WT However theseverity of damage events cannot be depicted by the currentdeveloped WT based detection approaches To this end thedamage detection on sudden stiffness reduction of buildingstructures has been actively investigated in this study Thesignal discontinuity of the structural acceleration responsesof an example building is extracted based on the discretewavelet transform (DWT) It is proved that the variationof the first level detail coefficients of the WT at damageinstant is linearly proportional to the magnitude of thestiffness reduction A new damage index is developed andimplemented in this paper to detect the damage time instantlocation and severity of a structure due to a sudden change ofstructural stiffness Numerical simulation using a five-storyshear building under different types of excitation is carriedout to assess the effectiveness and reliability of the proposeddamage index for the building at different damage levels Thesensitivity of the damage index to the intensity and frequencyrange of measurement noise is also investigated The madeobservations demonstrate that the proposed damage indexcan accurately identify the damage time instant and locationin the building due to a sudden loss of stiffness The relationbetween the damage severity and the proposed damage indexis linearThe proposed damage index can identify the damageevents from the contaminated acceleration responses if thenoise intensity is limited
2 Wavelet Transform
Morlet and Grossmann initially proposed wavelet theory andMeyer developed the mathematical foundations of waveletsThe two America-based researchers Daubechies [19 20] andMallat [21] changed this by defining the connection betweenwavelets and digital signal processing Wavelets have beenapplied to a number of areas including data compressionimage processing and time-frequency spectral estimation Amother wavelet 120595(119905) is a waveform that has limited durationand an average value of zero and the wavelet kernel can beexpressed by
120595119886119887(119905) =
1
radic119886120595(
119905 minus 119887
119886) (1)
where 119886 and 119887 are dilation and translation parametersrespectively Both are real numbers and 119886 must be positiveSimilar to the short time Fourier transform one can ana-lyze square-integrable function 119891(119905) with wavelet transform
which decomposes a signal in the time domain into a two-dimensional function in the time-scale plane (119886 119887) as follows
119862 (119886 119887) = int
+infin
minusinfin
119891 (119905) 120595119886119887(119905) 119889119905=
1
radic119886int
+infin
minusinfin
119891 (119905) 120595(119905 minus 119887
119886)119889119905
(2)
The term frequency instead of scale has been used in orderto aid in understanding since a wavelet with large-scaleparameter is related to low-frequency content componentand vice versa The mother wavelet 120595(119905) should satisfy thefollowing admissibility condition to ensure existence of theinverse wavelet transform such as
119862120595= int
+infin
minusinfin
1003816100381610038161003816 (120596)10038161003816100381610038162
|120596|119889120596 lt +infin (3)
where (120596) is the Fourier transform of 120595(119905) The existence ofthe integral in (3) requires that
(0) = 0 ie int+infin
minusinfin
120595 (119909) 119889119909 = 0 (4)
The signal 119891(119905) can be reconstructed by an inverse wavelettransform of 119862(119886 119887) as defined by
119891 (119905) =1
119862120595
int
+infin
119886=minusinfin
int
+infin
119887=minusinfin
119862 (119886 119887) 120595119886119887(119905 minus 119887
119886)1
1198862119889119886 119889119887 (5)
The calculatingwavelet coefficients at every possible scale willgenerate a lot of redundant data A discrete version of thewavelet is often utilized by discretizing the dilation parameter119886 and the translation parameter 119887 in real signal processingThe procedure becomes much more efficient if dyadic valuesof 119886 and 119887 are used That is
119886 = 2119895
119887 = 2119895
119896 119895 119896 isin 119885 (6)
where 119885 is a set of integers This sampling of the coordinates(119886 119887) is referred to as dyadic sampling because consecutivevalues of the discrete scales differ by a factor of 2 Using thediscrete scales of WT one can define the discrete wavelettransform (DWT) [21] as follows
119862119895119896= int
+infin
minusinfin
119891 (119905) 120595119895119896(119905) 119889119905 = 2
minus1198952
int
+infin
minusinfin
119891 (119905) 120595 (2minus119895
119905 minus 119896) 119889119905
(7)
The signal resolution is defined as the inverse of the scale1119886 = 2
minus119895 and the integer 119895 is referred to as the level Thesignal can be reconstructed from the wavelet coefficients 119862
119895119896
and the reconstruction algorithm is called the inverse discretewavelet transform as follows
119891 (119905) =
+infin
sum
119895=minusinfin
+infin
sum
119896=minusinfin
1198621198951198962minus1198952
120595 (2minus119895
119905 minus 119896) (8)
Another function 120601(119905) referred to as the scaling functionis important for the numerical implementation of the fastwavelet transform [19] Suppose now that the dyadic scale is
The Scientific World Journal 3
used for 119886 and 119887 and consider a reference level 119869 Applying(7) for this case one obtains a set of coefficients as follows
119888119863119869(119896) = int
+infin
minusinfin
119891 (119905) 120595119869119896(119905) 119889119905 (9)
The coefficient 119888119863119895(119896) is known as the level-119869 detail coef-
ficients Using the dyadic scale level 119869 yields the level-119869approximation coefficients as follows
119888119860119869(119896) = int
+infin
minusinfin
119891 (119905) 120601119869119896(119905) 119889119905 (10)
In theDWT a signal can be represented by its approximationsand details The detail at level 119895 is defined as
119863119895(119905) =
+infin
sum
119896=minusinfin
119888119863119895(119896) 120595119895119896(119905) (11)
and the approximation at level 119895 is defined as
119860119869(119905) =
+infin
sum
119896=minusinfin
119888119860119869(119896) 120601119895119896(119905) (12)
It becomes obvious that119860119869minus1
= 119860119869+ 119863119869
119891 (119905) = 119860119869(119905) + sum
119895le119869
119863119895(119905)
(13)
3 Signal Feature due to Sudden Damage
The dynamic responses of a five-story shear building sub-jected to a sudden stiffness reduction at its first story underthree different external excitations are computed The massand horizontal stiffness of the undamaged building areuniform for all stories as shown in Figure 1 The mass andhorizontal stiffness of the each floor are 119898 = 13 times 10
6 kgand 119896 = 40 times 10
9Nm respectively The Rayleigh dampingassumption is adopted to construct the structural dampingmatrix and the damping ratios in the first two modes ofvibration of the building are set as 005The original buildingis supposed to suffer a sudden 20 stiffness reduction inthe first story with the horizontal stiffness reducing from40times10
9Nm to 32times109Nm while the horizontal stiffnessin other stories remains unchangedThe frequency reductiondue to 20 stiffness reduction in the first story is small witha maximum reduction of nomore than 5 in the first naturalfrequency
The sinusoidal excitation seismic excitation and impulseexcitation are respectively utilized to calculate the accelera-tion responses of the example building to examine the signalfeatures due to sudden stiffness reductionThe seismic excita-tion used is the first 10-second portion of the El-Centro 1940earthquake ground acceleration (S-N component) with apeak amplitude of 10ms2 A sinusoidal excitation expressedby the following equationwith 10-second duration is assumedto act on each floor of the building
119891 (119905) = 1300 sdot sin (4120587119905) (0 le 119905 le 10 s) (kN) (14)
k1
k2
k3
k4
k5
m1
m2
m3
m4
m5
Figure 1 Elevation of a five-story building model
An impulse excitation represented by 01ms initial veloc-ity is supposed to occur at the first floor of the building Thedamage time instant of the building is set as 60 s for seismicexcitation and sinusoidal excitation and as 02 s for impulseexcitation The equation of motion of the example buildingwith a 20 sudden stiffness reduction at its first story atthe given time instant is established The dynamic responsesunder each type of external excitation are computed by usingthe Newmark-120573 method with a time interval of 0002 s Thetwo factors in theNewmark-120573method are selected as 120572 = 12and 120573 = 14 [22 23]
The computed acceleration time histories of the first floorunder seismic excitation are displayed in Figure 2 It is diffi-cult to find the signal feature due to sudden damage by directvisual inspection of the original acceleration responses The02-second portion of the acceleration responses is expandedto permit a close look at the signal feature due to suddendamage event It is seen that there exists a sudden jump inthe original signal at the damage time instant The structuralacceleration time histories of the first floor under sinusoidaland impulse excitation are also displayed in Figures 3 and 4respectively Similar to the observations made from seismicexcitation the direct inspection on original signals cannotdirectly find the signal feature due to sudden damage eventA detained investigation on small time portions indicates thesudden jump of original acceleration responses at damagetime instants as shown in the figures The sudden reductionof horizontal stiffness of the first floor causes a clear signaldiscontinuity in the acceleration response time history at thedamage time instant
Figure 5 displays the power spectrum of accelerationresponses with and without sudden damage events It isclear that the change in the spectrum amplitude induced bythe sudden damage is very small which cannot provide theenough information to capture the damage event Moreoverthe exact damage instant still cannot be determined inthe frequency domain based on the fast Fourier transformFurther inspection of the spectrum curves indicates that
4 The Scientific World Journal
0 2 4 6 8 10minus15
00
15 Acceleration response
Time (s)
(ms2)
(a)
Time (s)590 595 600 605 610
Acceleration response
minus15
00
15
(ms2)
(b)
Figure 2 Signal discontinuity due to sudden damage (seismic excitation)
0 2 4 6 8 10minus15
00
15 Acceleration response
Time (s)
(ms2)
(a)
Time (s)590 595 600 605 610
Acceleration response
minus15
00
15
(ms2)
(b)
Figure 3 Signal discontinuity due to sudden damage (sinusoidal excitation)
00 05 10 15 20minus10
0
10 Acceleration response
Time (s)
(ms2)
(a)
012 014 016 018 020 022 024minus2
0
2 Acceleration response
Time (s)
(ms2)
(b)
Figure 4 Signal discontinuity due to sudden damage (impulse excitation)
the structural acceleration responses present quite differentspectrum properties under different external excitations Ifthe building is subjected to El-Centro earthquake the powerspectrum has a relatively wide frequency range and the firsttwo natural frequencies can be effectively identified Theimpulse excitation signal however holds very short timeinterval and a very wide frequency range The accelerationresponses of the impulse excited building present very abun-dant frequency components and four natural frequenciescan be identified from the power spectrum To compare thespectrum components of sinusoidal seismic and impulseexcitation one can conclude that the structural responsessubjected to impulse excitation have the most abundant highfrequency components
Since the signal discontinuity is of very high frequencythe wavelet transform is applied to decompose the originalacceleration responses Figure 6 displays the first level detailcoefficients of wavelet transform for acceleration responsesunder seismic excitation It can be seen that the signaldiscontinuity is reserved in the first level detail coefficientonly instead of in the approximation components This isbecause the first level detail component often contains thehighest frequency component of the original signal To
extract inherent signal feature due to sudden damage fromthe signal discontinuity in the original acceleration responsetime history the acceleration responses of the building undereach type of excitation are computed for a sudden reductionof stiffness at the first story with different damage levels anddamage time instants Similar observations can bemade fromthe decomposed detail coefficients of the wavelet transformof the acceleration responses under sinusoidal and impulseexcitations
4 Damage Index
Let us consider a SDOF system subjected to a sudden stiffnessreduction under impulse excitationThemass of the system isdenoted as119898 the damping ratio 120585 of the system is supposed toremain unchanged before and after sudden damage and thestiffness is denoted as 119896 which will have a sudden reductionat time instant 119905
119894as follows
119896 = 119896119906
(0 le 119905 le 119905119894)
119896119889
(119905119894lt 119905)
(15)
The Scientific World Journal 5
0 5 10 15 20Frequency (Hz)
0
100
200
300
400Po
wer
spec
trum
(m2s3)
(a) Sinusoidal excitation
0 5 10 15 20Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
um (m
2s3)
(b) Seismic excitation
0 5 10 15 20Frequency (Hz)
00
25
50
75
100
Original structure20 damage
Pow
er sp
ectr
um (m
2s3)
(c) Impulse excitation
Figure 5 Power spectrum of acceleration responses of the first floor before and after sudden damage event
in which 119896119906and 119896
119889are the stiffness of undamaged and
damaged system respectively The initial velocity and dis-placement due to the impulse excitation are assumed to be0 and V
0 respectively The circular frequency of the system
before and after sudden damage can be expressed as
120596119906= radic
119896119906
119898 120596
119889= radic
119896119889
119898 (16)
Define a frequency reduction coefficient 120572 that varies from 0to 1 as follows
120596119889= 120572 sdot 120596
119906(0 lt 120572 lt 1) (17)
The stiffness reduction can be expressed as
Δ119896 = 119896119889minus 119896119906= 119898(120596
2
119889minus 1205962
119906) = 119898120596
2
119906(1205722
minus 1) (18)
The equation of motion of the SDOF system before suddendamage is
119910 + 2120585120596119906119910 + 1205962
119906119910 = 0 (19)
The above equation can be solved in terms of the given initialconditions and the structural dynamic responses are
119910 (119905) = 119860119906(119905) V0119890minus120585120596119906119905
sdot1
120596119906radic1 minus 1205852
119910 (119905) = V0119890minus120585120596119906119905
(119861119906(119905) minus
119860119906(119905) 120585
radic1 minus 1205852)
119910 (119905) = minus V0120596119906119890minus120585120596119906119905
sdot[119860119906(119905) (1 minus 2120585
2
) + 2120585119861119906(119905) radic1 minus 1205852]
radic1 minus 1205852
(20)
6 The Scientific World Journal
in which
119860119906(119905) = sin(120596
119906119905radic1 minus 1205852)
119861119906(119905) = cos(120596
119906119905radic1 minus 1205852)
(21)
Let us take the time instant 119905119894as the starting point of the SDOF
system after sudden damage and use a new time axis 1199051=
119905minus119905119894Then the equation ofmotion of the system after damage
becomes
119910119889+ 2120585120596
119889119910119889+ 1205962
119889119910119889= 0 (119905 gt 119905
119894) (22)
The initial conditions for (10) can be expressed as
119910119889(0) = 119910 (119905
119894) = 119860
119906(119905119894) V0119890minus120585120596119906119905119894 sdot
1
120596119906radic1 minus 1205852
119910119889(0) = 119910 (119905
119894) = V0119890minus120585120596119906119905119894 (119861119906(119905119894) minus
119860119906(119905119894) 120585
radic1 minus 1205852)
(23)
The damping ratio of a civil engineering structure is oftenvery small that is radic1 minus 1205852 asymp 1 The acceleration responseat the time instant 119905
1is
119910119889(1199051)
=V0120596119889119890minus120585(1205961198891199051+120596119906119905119894)
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894) [119861119889(1199051)radic1 minus 1205852 minus 119860
119889(1199051) 120585]
+ 120596119906119860119889(1199051) (21205852
minus 1) (119860119906(119905119894) 120585minus119861
119906(119905119894)radic1 minus 1205852)
minus 2120596119906120585119861119889(1199051) [119860119906(119905119894) 120585radic1 minus 1205852+119861
119906(119905119894) (1205852
minus 1)]
(24)
Furthermore the time interval should be very small todescribe the sudden stiffness reduction properly thus
Δ119905 = 119905119894+1
minus 119905119894997888rarr 0
119860119889(Δ119905) = sin(120596
119889Δ119905radic1 minus 1205852) asymp 0
119861119889(Δ119905) = cos(120596
119889Δ119905radic1 minus 1205852) asymp 1
(25)
Therefore the acceleration response at the time instant 119905119894+1
is
119910 (119905119894+1) = 119910119889(Δ119905)
=V0120596119889119890minus120585120596119906119905119894
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894)radic1 minus 1205852 minus 2120585120596
119906
times [119860119906(119905119894) 120585radic1 minus 1205852 + 119861
119906(119905119894) (1205852
minus 1)]
(26)
The first level detail coefficients of wavelet transform of theacceleration responses before sudden damage 119888119863 119910(119905)
1can be
expressed as
119888119863119910(119905)
1(119896) = int
+infin
minusinfin
119910 (119905) 1205951119896(119905) 119889119905 (27)
The first level detail coefficients of wavelet transform of theacceleration responses after sudden damage 119888119863 119910119889(1199051)
1can be
expressed as
119888119863119910119889(1199051)
1(119896) = int
+infin
minusinfin
119910119889(1199051) 1205951119896(1199051) 1198891199051 (28)
The variation of first level detail coefficients of theWT beforeand after the sudden damage event can be given as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896) = int
+infin
minusinfin
( 119910 (119905119894+1) minus 119910 (119905
119894)) 1205951119896(119905) 119889119905
(29)
Considering that the damping ratio of a civil engineeringstructure is often very small the above expression can besimplified as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
= minusΔ119896V0
119898120596119906
int
+infin
minusinfin
119890minus120585120596119906119905119894 sdot sin(120596
119906119905119894radic1 minus 1205852)120595
1119896(119905) 119889119905
(30)
The above equation reveals that the variation of first leveldetail coefficients of the WT before and after a suddendamage event is approximately linear to the sudden stiffnessreduction for given initial velocity damage instant andstructural parameters before damage If the time interval Δ119905for sudden damage is further regarded as a fixed value (30)indicates that the acceleration response discontinuity due tosudden stiffness reduction can be reflected by the variationrate of first level detail coefficients of the wavelet transform atdamage instant A damage index DI
119894 is defined to reflect the
signal discontinuity due to sudden damage at the time instant119905119894as follows
DI119894=
100381610038161003816100381610038161003816100381610038161003816
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
Δ119905
100381610038161003816100381610038161003816100381610038161003816
(119896 = 2 3 119899 minus 1) (31)
whereΔ119905 = 119905119896+1
minus119905119896and 119899 is the total number of time intervals
for the whole response time history This damage index iscomputed in the time domain and it is an instantaneous indexsuitable for online structural health monitoring applicationThe linear relationship between the proposed damage indexand the sudden stiffness reduction can be observed as follows
DI119894prop |Δ119896| (32)
5 Damage Detection
51 Selection of Mother Wavelet To examine the feasibil-ity of the proposed damage index and damage detection
The Scientific World Journal 7
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(a) db2
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(b) db4
Figure 6 Level-1 detail coefficients of wavelet transform for acceleration responses under seismic excitation
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
db1
db2
db3
db4
Figure 7 The Daubechies wavelets (db1ndashdb4)
approaches the first floor of the five-story building issupposed to suffer different levels of the sudden stiffnessreduction but the sudden damage occurs at the same timeSix damage scenarios are considered in the numerical inves-tigation Listed in Table 1 are the damage severities and thefive natural frequencies of the building before and after thesudden damage It is seen from Table 1 that the stiffnessreduction in the first story of the building affectsmainly lowernatural frequencies It is noted that if the stiffness reductionin the first floor is less than 10 the maximum frequencychange is no more than 2 In addition the variations of thehigher mode shapes are much smaller than those of the lowermode shapes
Wavelet transform can be utilized to detect the signalsingularity due to sudden stiffness change While the detec-tion efficiency depends on many factors such as waveletvanishing moments supporting length in the time domainfrequency components of original acceleration responsesand signal noise Thus three different Daubechies motherwavelets db1 db2 and db4 are utilized to examine the effectsof properties of mother wavelets on the detection on thestructural sudden damage The vanishing moments of thedb1 db2 and db4 wavelets are 1 2 and 4 respectivelyand they have the gradually increased supporting length asplotted in Figure 7 The basic principles of wavelet transformprove that the longer the wavelet supporting length is thefiner the distinguishing ability in the frequency domain isTherefore the mother wavelet with long supporting length ismore suitable for detecting the higher frequency componentsin the original signal
To examine the feasibility of the proposed damage indexand damage detection approaches for identifying damageevents the acceleration responses of the aforementioned five-story shear building to the seismic excitation sinusoidalexcitation and impulse excitation are computed respectivelyThe building is subject to a 20 sudden stiffness reductionat times 60 s 60 s and 02 s in the first story of thebuilding under seismic excitation sinusoidal excitation andimpulse excitation respectively The time step used in thecomputation is 0002 seconds
Shown in Figure 8 are the damage detection resultsusing db1 db2 and db4 for 20 sudden stiffness reductionrespectively It can be seen from Figure 8(a) that no matterwhich mother wavelet is used the damage index of the firstfloor is very large only at time 119905= 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 The damage indices of the first floor at allother time instants are very small so that the damage index attime 119905 = 60 seconds looks like a spikeTherefore the damagetime instant can be easily identified by the occurrence timeof the sharp damage index It is demonstrated that the DWTbased approach using all the three Daubechies wavelets canaccurately detect the damage time instant of the buildingsubjected to sinusoidal excitation For the building excitedby El-Centro ground motion DWT using db1 wavelet failto detect damage instant while the approach using db2 anddb4 wavelet successfully captures the damage events Forthe impulse excited case only the DWT using db4 waveletcan accurately detect the damage instant due to suddenstiffness change In reality the sudden stiffness loss will cause
8 The Scientific World Journal
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)200
100
0
Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
200
100
0
20 damage
20 damage
20 damage
(a) Sinusoidal excitation
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)
100
50
0
Dam
age i
ndex
(db2
)
100
50
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage
20 damage
20 damage
(b) Seismic excitation
Dam
age i
ndex
(db1
)
200
100
0 Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage 20 damage
20 damage
00 04 08 12 16 20
Time (s)00 04 08 12 16 20
Time (s)
00 04 08 12 16 20
Time (s)
(c) Impulse excitation
Figure 8 Damage detection using different db wavelets
Table 1 Natural frequency before and after sudden damage
Damage extent Frequencies (Hz)1198911
1198912
1198913
1198914
1198915
0 2513 7335 11563 14854 169411 2508 (minus018) 7324 (minus015) 1155 (minus011) 1485 (minus005) 1694 (minus001)2 2504 (minus036) 7313 (minus030) 1154 (minus021) 1484 (minus011) 1694 (minus003)5 2490 (minus094) 7278 (minus078) 1150 (minus053) 1481 (minus026) 1693 (minus007)10 2464 (minus197) 7218 (minus162) 1144 (minus106) 1478 (minus052) 1692 (minus014)20 2407 (minus441) 7088 (minus348) 1132 (minus218) 1471 (minus101) 1690 (minus026)40 2253 (minus116) 6781 (minus816) 1107 (minus450) 1457 (minus192) 1686 (minus046)Note values in brackets are the percentage of change in natural frequency
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 3
used for 119886 and 119887 and consider a reference level 119869 Applying(7) for this case one obtains a set of coefficients as follows
119888119863119869(119896) = int
+infin
minusinfin
119891 (119905) 120595119869119896(119905) 119889119905 (9)
The coefficient 119888119863119895(119896) is known as the level-119869 detail coef-
ficients Using the dyadic scale level 119869 yields the level-119869approximation coefficients as follows
119888119860119869(119896) = int
+infin
minusinfin
119891 (119905) 120601119869119896(119905) 119889119905 (10)
In theDWT a signal can be represented by its approximationsand details The detail at level 119895 is defined as
119863119895(119905) =
+infin
sum
119896=minusinfin
119888119863119895(119896) 120595119895119896(119905) (11)
and the approximation at level 119895 is defined as
119860119869(119905) =
+infin
sum
119896=minusinfin
119888119860119869(119896) 120601119895119896(119905) (12)
It becomes obvious that119860119869minus1
= 119860119869+ 119863119869
119891 (119905) = 119860119869(119905) + sum
119895le119869
119863119895(119905)
(13)
3 Signal Feature due to Sudden Damage
The dynamic responses of a five-story shear building sub-jected to a sudden stiffness reduction at its first story underthree different external excitations are computed The massand horizontal stiffness of the undamaged building areuniform for all stories as shown in Figure 1 The mass andhorizontal stiffness of the each floor are 119898 = 13 times 10
6 kgand 119896 = 40 times 10
9Nm respectively The Rayleigh dampingassumption is adopted to construct the structural dampingmatrix and the damping ratios in the first two modes ofvibration of the building are set as 005The original buildingis supposed to suffer a sudden 20 stiffness reduction inthe first story with the horizontal stiffness reducing from40times10
9Nm to 32times109Nm while the horizontal stiffnessin other stories remains unchangedThe frequency reductiondue to 20 stiffness reduction in the first story is small witha maximum reduction of nomore than 5 in the first naturalfrequency
The sinusoidal excitation seismic excitation and impulseexcitation are respectively utilized to calculate the accelera-tion responses of the example building to examine the signalfeatures due to sudden stiffness reductionThe seismic excita-tion used is the first 10-second portion of the El-Centro 1940earthquake ground acceleration (S-N component) with apeak amplitude of 10ms2 A sinusoidal excitation expressedby the following equationwith 10-second duration is assumedto act on each floor of the building
119891 (119905) = 1300 sdot sin (4120587119905) (0 le 119905 le 10 s) (kN) (14)
k1
k2
k3
k4
k5
m1
m2
m3
m4
m5
Figure 1 Elevation of a five-story building model
An impulse excitation represented by 01ms initial veloc-ity is supposed to occur at the first floor of the building Thedamage time instant of the building is set as 60 s for seismicexcitation and sinusoidal excitation and as 02 s for impulseexcitation The equation of motion of the example buildingwith a 20 sudden stiffness reduction at its first story atthe given time instant is established The dynamic responsesunder each type of external excitation are computed by usingthe Newmark-120573 method with a time interval of 0002 s Thetwo factors in theNewmark-120573method are selected as 120572 = 12and 120573 = 14 [22 23]
The computed acceleration time histories of the first floorunder seismic excitation are displayed in Figure 2 It is diffi-cult to find the signal feature due to sudden damage by directvisual inspection of the original acceleration responses The02-second portion of the acceleration responses is expandedto permit a close look at the signal feature due to suddendamage event It is seen that there exists a sudden jump inthe original signal at the damage time instant The structuralacceleration time histories of the first floor under sinusoidaland impulse excitation are also displayed in Figures 3 and 4respectively Similar to the observations made from seismicexcitation the direct inspection on original signals cannotdirectly find the signal feature due to sudden damage eventA detained investigation on small time portions indicates thesudden jump of original acceleration responses at damagetime instants as shown in the figures The sudden reductionof horizontal stiffness of the first floor causes a clear signaldiscontinuity in the acceleration response time history at thedamage time instant
Figure 5 displays the power spectrum of accelerationresponses with and without sudden damage events It isclear that the change in the spectrum amplitude induced bythe sudden damage is very small which cannot provide theenough information to capture the damage event Moreoverthe exact damage instant still cannot be determined inthe frequency domain based on the fast Fourier transformFurther inspection of the spectrum curves indicates that
4 The Scientific World Journal
0 2 4 6 8 10minus15
00
15 Acceleration response
Time (s)
(ms2)
(a)
Time (s)590 595 600 605 610
Acceleration response
minus15
00
15
(ms2)
(b)
Figure 2 Signal discontinuity due to sudden damage (seismic excitation)
0 2 4 6 8 10minus15
00
15 Acceleration response
Time (s)
(ms2)
(a)
Time (s)590 595 600 605 610
Acceleration response
minus15
00
15
(ms2)
(b)
Figure 3 Signal discontinuity due to sudden damage (sinusoidal excitation)
00 05 10 15 20minus10
0
10 Acceleration response
Time (s)
(ms2)
(a)
012 014 016 018 020 022 024minus2
0
2 Acceleration response
Time (s)
(ms2)
(b)
Figure 4 Signal discontinuity due to sudden damage (impulse excitation)
the structural acceleration responses present quite differentspectrum properties under different external excitations Ifthe building is subjected to El-Centro earthquake the powerspectrum has a relatively wide frequency range and the firsttwo natural frequencies can be effectively identified Theimpulse excitation signal however holds very short timeinterval and a very wide frequency range The accelerationresponses of the impulse excited building present very abun-dant frequency components and four natural frequenciescan be identified from the power spectrum To compare thespectrum components of sinusoidal seismic and impulseexcitation one can conclude that the structural responsessubjected to impulse excitation have the most abundant highfrequency components
Since the signal discontinuity is of very high frequencythe wavelet transform is applied to decompose the originalacceleration responses Figure 6 displays the first level detailcoefficients of wavelet transform for acceleration responsesunder seismic excitation It can be seen that the signaldiscontinuity is reserved in the first level detail coefficientonly instead of in the approximation components This isbecause the first level detail component often contains thehighest frequency component of the original signal To
extract inherent signal feature due to sudden damage fromthe signal discontinuity in the original acceleration responsetime history the acceleration responses of the building undereach type of excitation are computed for a sudden reductionof stiffness at the first story with different damage levels anddamage time instants Similar observations can bemade fromthe decomposed detail coefficients of the wavelet transformof the acceleration responses under sinusoidal and impulseexcitations
4 Damage Index
Let us consider a SDOF system subjected to a sudden stiffnessreduction under impulse excitationThemass of the system isdenoted as119898 the damping ratio 120585 of the system is supposed toremain unchanged before and after sudden damage and thestiffness is denoted as 119896 which will have a sudden reductionat time instant 119905
119894as follows
119896 = 119896119906
(0 le 119905 le 119905119894)
119896119889
(119905119894lt 119905)
(15)
The Scientific World Journal 5
0 5 10 15 20Frequency (Hz)
0
100
200
300
400Po
wer
spec
trum
(m2s3)
(a) Sinusoidal excitation
0 5 10 15 20Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
um (m
2s3)
(b) Seismic excitation
0 5 10 15 20Frequency (Hz)
00
25
50
75
100
Original structure20 damage
Pow
er sp
ectr
um (m
2s3)
(c) Impulse excitation
Figure 5 Power spectrum of acceleration responses of the first floor before and after sudden damage event
in which 119896119906and 119896
119889are the stiffness of undamaged and
damaged system respectively The initial velocity and dis-placement due to the impulse excitation are assumed to be0 and V
0 respectively The circular frequency of the system
before and after sudden damage can be expressed as
120596119906= radic
119896119906
119898 120596
119889= radic
119896119889
119898 (16)
Define a frequency reduction coefficient 120572 that varies from 0to 1 as follows
120596119889= 120572 sdot 120596
119906(0 lt 120572 lt 1) (17)
The stiffness reduction can be expressed as
Δ119896 = 119896119889minus 119896119906= 119898(120596
2
119889minus 1205962
119906) = 119898120596
2
119906(1205722
minus 1) (18)
The equation of motion of the SDOF system before suddendamage is
119910 + 2120585120596119906119910 + 1205962
119906119910 = 0 (19)
The above equation can be solved in terms of the given initialconditions and the structural dynamic responses are
119910 (119905) = 119860119906(119905) V0119890minus120585120596119906119905
sdot1
120596119906radic1 minus 1205852
119910 (119905) = V0119890minus120585120596119906119905
(119861119906(119905) minus
119860119906(119905) 120585
radic1 minus 1205852)
119910 (119905) = minus V0120596119906119890minus120585120596119906119905
sdot[119860119906(119905) (1 minus 2120585
2
) + 2120585119861119906(119905) radic1 minus 1205852]
radic1 minus 1205852
(20)
6 The Scientific World Journal
in which
119860119906(119905) = sin(120596
119906119905radic1 minus 1205852)
119861119906(119905) = cos(120596
119906119905radic1 minus 1205852)
(21)
Let us take the time instant 119905119894as the starting point of the SDOF
system after sudden damage and use a new time axis 1199051=
119905minus119905119894Then the equation ofmotion of the system after damage
becomes
119910119889+ 2120585120596
119889119910119889+ 1205962
119889119910119889= 0 (119905 gt 119905
119894) (22)
The initial conditions for (10) can be expressed as
119910119889(0) = 119910 (119905
119894) = 119860
119906(119905119894) V0119890minus120585120596119906119905119894 sdot
1
120596119906radic1 minus 1205852
119910119889(0) = 119910 (119905
119894) = V0119890minus120585120596119906119905119894 (119861119906(119905119894) minus
119860119906(119905119894) 120585
radic1 minus 1205852)
(23)
The damping ratio of a civil engineering structure is oftenvery small that is radic1 minus 1205852 asymp 1 The acceleration responseat the time instant 119905
1is
119910119889(1199051)
=V0120596119889119890minus120585(1205961198891199051+120596119906119905119894)
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894) [119861119889(1199051)radic1 minus 1205852 minus 119860
119889(1199051) 120585]
+ 120596119906119860119889(1199051) (21205852
minus 1) (119860119906(119905119894) 120585minus119861
119906(119905119894)radic1 minus 1205852)
minus 2120596119906120585119861119889(1199051) [119860119906(119905119894) 120585radic1 minus 1205852+119861
119906(119905119894) (1205852
minus 1)]
(24)
Furthermore the time interval should be very small todescribe the sudden stiffness reduction properly thus
Δ119905 = 119905119894+1
minus 119905119894997888rarr 0
119860119889(Δ119905) = sin(120596
119889Δ119905radic1 minus 1205852) asymp 0
119861119889(Δ119905) = cos(120596
119889Δ119905radic1 minus 1205852) asymp 1
(25)
Therefore the acceleration response at the time instant 119905119894+1
is
119910 (119905119894+1) = 119910119889(Δ119905)
=V0120596119889119890minus120585120596119906119905119894
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894)radic1 minus 1205852 minus 2120585120596
119906
times [119860119906(119905119894) 120585radic1 minus 1205852 + 119861
119906(119905119894) (1205852
minus 1)]
(26)
The first level detail coefficients of wavelet transform of theacceleration responses before sudden damage 119888119863 119910(119905)
1can be
expressed as
119888119863119910(119905)
1(119896) = int
+infin
minusinfin
119910 (119905) 1205951119896(119905) 119889119905 (27)
The first level detail coefficients of wavelet transform of theacceleration responses after sudden damage 119888119863 119910119889(1199051)
1can be
expressed as
119888119863119910119889(1199051)
1(119896) = int
+infin
minusinfin
119910119889(1199051) 1205951119896(1199051) 1198891199051 (28)
The variation of first level detail coefficients of theWT beforeand after the sudden damage event can be given as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896) = int
+infin
minusinfin
( 119910 (119905119894+1) minus 119910 (119905
119894)) 1205951119896(119905) 119889119905
(29)
Considering that the damping ratio of a civil engineeringstructure is often very small the above expression can besimplified as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
= minusΔ119896V0
119898120596119906
int
+infin
minusinfin
119890minus120585120596119906119905119894 sdot sin(120596
119906119905119894radic1 minus 1205852)120595
1119896(119905) 119889119905
(30)
The above equation reveals that the variation of first leveldetail coefficients of the WT before and after a suddendamage event is approximately linear to the sudden stiffnessreduction for given initial velocity damage instant andstructural parameters before damage If the time interval Δ119905for sudden damage is further regarded as a fixed value (30)indicates that the acceleration response discontinuity due tosudden stiffness reduction can be reflected by the variationrate of first level detail coefficients of the wavelet transform atdamage instant A damage index DI
119894 is defined to reflect the
signal discontinuity due to sudden damage at the time instant119905119894as follows
DI119894=
100381610038161003816100381610038161003816100381610038161003816
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
Δ119905
100381610038161003816100381610038161003816100381610038161003816
(119896 = 2 3 119899 minus 1) (31)
whereΔ119905 = 119905119896+1
minus119905119896and 119899 is the total number of time intervals
for the whole response time history This damage index iscomputed in the time domain and it is an instantaneous indexsuitable for online structural health monitoring applicationThe linear relationship between the proposed damage indexand the sudden stiffness reduction can be observed as follows
DI119894prop |Δ119896| (32)
5 Damage Detection
51 Selection of Mother Wavelet To examine the feasibil-ity of the proposed damage index and damage detection
The Scientific World Journal 7
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(a) db2
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(b) db4
Figure 6 Level-1 detail coefficients of wavelet transform for acceleration responses under seismic excitation
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
db1
db2
db3
db4
Figure 7 The Daubechies wavelets (db1ndashdb4)
approaches the first floor of the five-story building issupposed to suffer different levels of the sudden stiffnessreduction but the sudden damage occurs at the same timeSix damage scenarios are considered in the numerical inves-tigation Listed in Table 1 are the damage severities and thefive natural frequencies of the building before and after thesudden damage It is seen from Table 1 that the stiffnessreduction in the first story of the building affectsmainly lowernatural frequencies It is noted that if the stiffness reductionin the first floor is less than 10 the maximum frequencychange is no more than 2 In addition the variations of thehigher mode shapes are much smaller than those of the lowermode shapes
Wavelet transform can be utilized to detect the signalsingularity due to sudden stiffness change While the detec-tion efficiency depends on many factors such as waveletvanishing moments supporting length in the time domainfrequency components of original acceleration responsesand signal noise Thus three different Daubechies motherwavelets db1 db2 and db4 are utilized to examine the effectsof properties of mother wavelets on the detection on thestructural sudden damage The vanishing moments of thedb1 db2 and db4 wavelets are 1 2 and 4 respectivelyand they have the gradually increased supporting length asplotted in Figure 7 The basic principles of wavelet transformprove that the longer the wavelet supporting length is thefiner the distinguishing ability in the frequency domain isTherefore the mother wavelet with long supporting length ismore suitable for detecting the higher frequency componentsin the original signal
To examine the feasibility of the proposed damage indexand damage detection approaches for identifying damageevents the acceleration responses of the aforementioned five-story shear building to the seismic excitation sinusoidalexcitation and impulse excitation are computed respectivelyThe building is subject to a 20 sudden stiffness reductionat times 60 s 60 s and 02 s in the first story of thebuilding under seismic excitation sinusoidal excitation andimpulse excitation respectively The time step used in thecomputation is 0002 seconds
Shown in Figure 8 are the damage detection resultsusing db1 db2 and db4 for 20 sudden stiffness reductionrespectively It can be seen from Figure 8(a) that no matterwhich mother wavelet is used the damage index of the firstfloor is very large only at time 119905= 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 The damage indices of the first floor at allother time instants are very small so that the damage index attime 119905 = 60 seconds looks like a spikeTherefore the damagetime instant can be easily identified by the occurrence timeof the sharp damage index It is demonstrated that the DWTbased approach using all the three Daubechies wavelets canaccurately detect the damage time instant of the buildingsubjected to sinusoidal excitation For the building excitedby El-Centro ground motion DWT using db1 wavelet failto detect damage instant while the approach using db2 anddb4 wavelet successfully captures the damage events Forthe impulse excited case only the DWT using db4 waveletcan accurately detect the damage instant due to suddenstiffness change In reality the sudden stiffness loss will cause
8 The Scientific World Journal
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)200
100
0
Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
200
100
0
20 damage
20 damage
20 damage
(a) Sinusoidal excitation
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)
100
50
0
Dam
age i
ndex
(db2
)
100
50
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage
20 damage
20 damage
(b) Seismic excitation
Dam
age i
ndex
(db1
)
200
100
0 Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage 20 damage
20 damage
00 04 08 12 16 20
Time (s)00 04 08 12 16 20
Time (s)
00 04 08 12 16 20
Time (s)
(c) Impulse excitation
Figure 8 Damage detection using different db wavelets
Table 1 Natural frequency before and after sudden damage
Damage extent Frequencies (Hz)1198911
1198912
1198913
1198914
1198915
0 2513 7335 11563 14854 169411 2508 (minus018) 7324 (minus015) 1155 (minus011) 1485 (minus005) 1694 (minus001)2 2504 (minus036) 7313 (minus030) 1154 (minus021) 1484 (minus011) 1694 (minus003)5 2490 (minus094) 7278 (minus078) 1150 (minus053) 1481 (minus026) 1693 (minus007)10 2464 (minus197) 7218 (minus162) 1144 (minus106) 1478 (minus052) 1692 (minus014)20 2407 (minus441) 7088 (minus348) 1132 (minus218) 1471 (minus101) 1690 (minus026)40 2253 (minus116) 6781 (minus816) 1107 (minus450) 1457 (minus192) 1686 (minus046)Note values in brackets are the percentage of change in natural frequency
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 The Scientific World Journal
0 2 4 6 8 10minus15
00
15 Acceleration response
Time (s)
(ms2)
(a)
Time (s)590 595 600 605 610
Acceleration response
minus15
00
15
(ms2)
(b)
Figure 2 Signal discontinuity due to sudden damage (seismic excitation)
0 2 4 6 8 10minus15
00
15 Acceleration response
Time (s)
(ms2)
(a)
Time (s)590 595 600 605 610
Acceleration response
minus15
00
15
(ms2)
(b)
Figure 3 Signal discontinuity due to sudden damage (sinusoidal excitation)
00 05 10 15 20minus10
0
10 Acceleration response
Time (s)
(ms2)
(a)
012 014 016 018 020 022 024minus2
0
2 Acceleration response
Time (s)
(ms2)
(b)
Figure 4 Signal discontinuity due to sudden damage (impulse excitation)
the structural acceleration responses present quite differentspectrum properties under different external excitations Ifthe building is subjected to El-Centro earthquake the powerspectrum has a relatively wide frequency range and the firsttwo natural frequencies can be effectively identified Theimpulse excitation signal however holds very short timeinterval and a very wide frequency range The accelerationresponses of the impulse excited building present very abun-dant frequency components and four natural frequenciescan be identified from the power spectrum To compare thespectrum components of sinusoidal seismic and impulseexcitation one can conclude that the structural responsessubjected to impulse excitation have the most abundant highfrequency components
Since the signal discontinuity is of very high frequencythe wavelet transform is applied to decompose the originalacceleration responses Figure 6 displays the first level detailcoefficients of wavelet transform for acceleration responsesunder seismic excitation It can be seen that the signaldiscontinuity is reserved in the first level detail coefficientonly instead of in the approximation components This isbecause the first level detail component often contains thehighest frequency component of the original signal To
extract inherent signal feature due to sudden damage fromthe signal discontinuity in the original acceleration responsetime history the acceleration responses of the building undereach type of excitation are computed for a sudden reductionof stiffness at the first story with different damage levels anddamage time instants Similar observations can bemade fromthe decomposed detail coefficients of the wavelet transformof the acceleration responses under sinusoidal and impulseexcitations
4 Damage Index
Let us consider a SDOF system subjected to a sudden stiffnessreduction under impulse excitationThemass of the system isdenoted as119898 the damping ratio 120585 of the system is supposed toremain unchanged before and after sudden damage and thestiffness is denoted as 119896 which will have a sudden reductionat time instant 119905
119894as follows
119896 = 119896119906
(0 le 119905 le 119905119894)
119896119889
(119905119894lt 119905)
(15)
The Scientific World Journal 5
0 5 10 15 20Frequency (Hz)
0
100
200
300
400Po
wer
spec
trum
(m2s3)
(a) Sinusoidal excitation
0 5 10 15 20Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
um (m
2s3)
(b) Seismic excitation
0 5 10 15 20Frequency (Hz)
00
25
50
75
100
Original structure20 damage
Pow
er sp
ectr
um (m
2s3)
(c) Impulse excitation
Figure 5 Power spectrum of acceleration responses of the first floor before and after sudden damage event
in which 119896119906and 119896
119889are the stiffness of undamaged and
damaged system respectively The initial velocity and dis-placement due to the impulse excitation are assumed to be0 and V
0 respectively The circular frequency of the system
before and after sudden damage can be expressed as
120596119906= radic
119896119906
119898 120596
119889= radic
119896119889
119898 (16)
Define a frequency reduction coefficient 120572 that varies from 0to 1 as follows
120596119889= 120572 sdot 120596
119906(0 lt 120572 lt 1) (17)
The stiffness reduction can be expressed as
Δ119896 = 119896119889minus 119896119906= 119898(120596
2
119889minus 1205962
119906) = 119898120596
2
119906(1205722
minus 1) (18)
The equation of motion of the SDOF system before suddendamage is
119910 + 2120585120596119906119910 + 1205962
119906119910 = 0 (19)
The above equation can be solved in terms of the given initialconditions and the structural dynamic responses are
119910 (119905) = 119860119906(119905) V0119890minus120585120596119906119905
sdot1
120596119906radic1 minus 1205852
119910 (119905) = V0119890minus120585120596119906119905
(119861119906(119905) minus
119860119906(119905) 120585
radic1 minus 1205852)
119910 (119905) = minus V0120596119906119890minus120585120596119906119905
sdot[119860119906(119905) (1 minus 2120585
2
) + 2120585119861119906(119905) radic1 minus 1205852]
radic1 minus 1205852
(20)
6 The Scientific World Journal
in which
119860119906(119905) = sin(120596
119906119905radic1 minus 1205852)
119861119906(119905) = cos(120596
119906119905radic1 minus 1205852)
(21)
Let us take the time instant 119905119894as the starting point of the SDOF
system after sudden damage and use a new time axis 1199051=
119905minus119905119894Then the equation ofmotion of the system after damage
becomes
119910119889+ 2120585120596
119889119910119889+ 1205962
119889119910119889= 0 (119905 gt 119905
119894) (22)
The initial conditions for (10) can be expressed as
119910119889(0) = 119910 (119905
119894) = 119860
119906(119905119894) V0119890minus120585120596119906119905119894 sdot
1
120596119906radic1 minus 1205852
119910119889(0) = 119910 (119905
119894) = V0119890minus120585120596119906119905119894 (119861119906(119905119894) minus
119860119906(119905119894) 120585
radic1 minus 1205852)
(23)
The damping ratio of a civil engineering structure is oftenvery small that is radic1 minus 1205852 asymp 1 The acceleration responseat the time instant 119905
1is
119910119889(1199051)
=V0120596119889119890minus120585(1205961198891199051+120596119906119905119894)
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894) [119861119889(1199051)radic1 minus 1205852 minus 119860
119889(1199051) 120585]
+ 120596119906119860119889(1199051) (21205852
minus 1) (119860119906(119905119894) 120585minus119861
119906(119905119894)radic1 minus 1205852)
minus 2120596119906120585119861119889(1199051) [119860119906(119905119894) 120585radic1 minus 1205852+119861
119906(119905119894) (1205852
minus 1)]
(24)
Furthermore the time interval should be very small todescribe the sudden stiffness reduction properly thus
Δ119905 = 119905119894+1
minus 119905119894997888rarr 0
119860119889(Δ119905) = sin(120596
119889Δ119905radic1 minus 1205852) asymp 0
119861119889(Δ119905) = cos(120596
119889Δ119905radic1 minus 1205852) asymp 1
(25)
Therefore the acceleration response at the time instant 119905119894+1
is
119910 (119905119894+1) = 119910119889(Δ119905)
=V0120596119889119890minus120585120596119906119905119894
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894)radic1 minus 1205852 minus 2120585120596
119906
times [119860119906(119905119894) 120585radic1 minus 1205852 + 119861
119906(119905119894) (1205852
minus 1)]
(26)
The first level detail coefficients of wavelet transform of theacceleration responses before sudden damage 119888119863 119910(119905)
1can be
expressed as
119888119863119910(119905)
1(119896) = int
+infin
minusinfin
119910 (119905) 1205951119896(119905) 119889119905 (27)
The first level detail coefficients of wavelet transform of theacceleration responses after sudden damage 119888119863 119910119889(1199051)
1can be
expressed as
119888119863119910119889(1199051)
1(119896) = int
+infin
minusinfin
119910119889(1199051) 1205951119896(1199051) 1198891199051 (28)
The variation of first level detail coefficients of theWT beforeand after the sudden damage event can be given as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896) = int
+infin
minusinfin
( 119910 (119905119894+1) minus 119910 (119905
119894)) 1205951119896(119905) 119889119905
(29)
Considering that the damping ratio of a civil engineeringstructure is often very small the above expression can besimplified as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
= minusΔ119896V0
119898120596119906
int
+infin
minusinfin
119890minus120585120596119906119905119894 sdot sin(120596
119906119905119894radic1 minus 1205852)120595
1119896(119905) 119889119905
(30)
The above equation reveals that the variation of first leveldetail coefficients of the WT before and after a suddendamage event is approximately linear to the sudden stiffnessreduction for given initial velocity damage instant andstructural parameters before damage If the time interval Δ119905for sudden damage is further regarded as a fixed value (30)indicates that the acceleration response discontinuity due tosudden stiffness reduction can be reflected by the variationrate of first level detail coefficients of the wavelet transform atdamage instant A damage index DI
119894 is defined to reflect the
signal discontinuity due to sudden damage at the time instant119905119894as follows
DI119894=
100381610038161003816100381610038161003816100381610038161003816
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
Δ119905
100381610038161003816100381610038161003816100381610038161003816
(119896 = 2 3 119899 minus 1) (31)
whereΔ119905 = 119905119896+1
minus119905119896and 119899 is the total number of time intervals
for the whole response time history This damage index iscomputed in the time domain and it is an instantaneous indexsuitable for online structural health monitoring applicationThe linear relationship between the proposed damage indexand the sudden stiffness reduction can be observed as follows
DI119894prop |Δ119896| (32)
5 Damage Detection
51 Selection of Mother Wavelet To examine the feasibil-ity of the proposed damage index and damage detection
The Scientific World Journal 7
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(a) db2
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(b) db4
Figure 6 Level-1 detail coefficients of wavelet transform for acceleration responses under seismic excitation
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
db1
db2
db3
db4
Figure 7 The Daubechies wavelets (db1ndashdb4)
approaches the first floor of the five-story building issupposed to suffer different levels of the sudden stiffnessreduction but the sudden damage occurs at the same timeSix damage scenarios are considered in the numerical inves-tigation Listed in Table 1 are the damage severities and thefive natural frequencies of the building before and after thesudden damage It is seen from Table 1 that the stiffnessreduction in the first story of the building affectsmainly lowernatural frequencies It is noted that if the stiffness reductionin the first floor is less than 10 the maximum frequencychange is no more than 2 In addition the variations of thehigher mode shapes are much smaller than those of the lowermode shapes
Wavelet transform can be utilized to detect the signalsingularity due to sudden stiffness change While the detec-tion efficiency depends on many factors such as waveletvanishing moments supporting length in the time domainfrequency components of original acceleration responsesand signal noise Thus three different Daubechies motherwavelets db1 db2 and db4 are utilized to examine the effectsof properties of mother wavelets on the detection on thestructural sudden damage The vanishing moments of thedb1 db2 and db4 wavelets are 1 2 and 4 respectivelyand they have the gradually increased supporting length asplotted in Figure 7 The basic principles of wavelet transformprove that the longer the wavelet supporting length is thefiner the distinguishing ability in the frequency domain isTherefore the mother wavelet with long supporting length ismore suitable for detecting the higher frequency componentsin the original signal
To examine the feasibility of the proposed damage indexand damage detection approaches for identifying damageevents the acceleration responses of the aforementioned five-story shear building to the seismic excitation sinusoidalexcitation and impulse excitation are computed respectivelyThe building is subject to a 20 sudden stiffness reductionat times 60 s 60 s and 02 s in the first story of thebuilding under seismic excitation sinusoidal excitation andimpulse excitation respectively The time step used in thecomputation is 0002 seconds
Shown in Figure 8 are the damage detection resultsusing db1 db2 and db4 for 20 sudden stiffness reductionrespectively It can be seen from Figure 8(a) that no matterwhich mother wavelet is used the damage index of the firstfloor is very large only at time 119905= 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 The damage indices of the first floor at allother time instants are very small so that the damage index attime 119905 = 60 seconds looks like a spikeTherefore the damagetime instant can be easily identified by the occurrence timeof the sharp damage index It is demonstrated that the DWTbased approach using all the three Daubechies wavelets canaccurately detect the damage time instant of the buildingsubjected to sinusoidal excitation For the building excitedby El-Centro ground motion DWT using db1 wavelet failto detect damage instant while the approach using db2 anddb4 wavelet successfully captures the damage events Forthe impulse excited case only the DWT using db4 waveletcan accurately detect the damage instant due to suddenstiffness change In reality the sudden stiffness loss will cause
8 The Scientific World Journal
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)200
100
0
Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
200
100
0
20 damage
20 damage
20 damage
(a) Sinusoidal excitation
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)
100
50
0
Dam
age i
ndex
(db2
)
100
50
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage
20 damage
20 damage
(b) Seismic excitation
Dam
age i
ndex
(db1
)
200
100
0 Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage 20 damage
20 damage
00 04 08 12 16 20
Time (s)00 04 08 12 16 20
Time (s)
00 04 08 12 16 20
Time (s)
(c) Impulse excitation
Figure 8 Damage detection using different db wavelets
Table 1 Natural frequency before and after sudden damage
Damage extent Frequencies (Hz)1198911
1198912
1198913
1198914
1198915
0 2513 7335 11563 14854 169411 2508 (minus018) 7324 (minus015) 1155 (minus011) 1485 (minus005) 1694 (minus001)2 2504 (minus036) 7313 (minus030) 1154 (minus021) 1484 (minus011) 1694 (minus003)5 2490 (minus094) 7278 (minus078) 1150 (minus053) 1481 (minus026) 1693 (minus007)10 2464 (minus197) 7218 (minus162) 1144 (minus106) 1478 (minus052) 1692 (minus014)20 2407 (minus441) 7088 (minus348) 1132 (minus218) 1471 (minus101) 1690 (minus026)40 2253 (minus116) 6781 (minus816) 1107 (minus450) 1457 (minus192) 1686 (minus046)Note values in brackets are the percentage of change in natural frequency
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 5
0 5 10 15 20Frequency (Hz)
0
100
200
300
400Po
wer
spec
trum
(m2s3)
(a) Sinusoidal excitation
0 5 10 15 20Frequency (Hz)
0
10
20
30
40
Pow
er sp
ectr
um (m
2s3)
(b) Seismic excitation
0 5 10 15 20Frequency (Hz)
00
25
50
75
100
Original structure20 damage
Pow
er sp
ectr
um (m
2s3)
(c) Impulse excitation
Figure 5 Power spectrum of acceleration responses of the first floor before and after sudden damage event
in which 119896119906and 119896
119889are the stiffness of undamaged and
damaged system respectively The initial velocity and dis-placement due to the impulse excitation are assumed to be0 and V
0 respectively The circular frequency of the system
before and after sudden damage can be expressed as
120596119906= radic
119896119906
119898 120596
119889= radic
119896119889
119898 (16)
Define a frequency reduction coefficient 120572 that varies from 0to 1 as follows
120596119889= 120572 sdot 120596
119906(0 lt 120572 lt 1) (17)
The stiffness reduction can be expressed as
Δ119896 = 119896119889minus 119896119906= 119898(120596
2
119889minus 1205962
119906) = 119898120596
2
119906(1205722
minus 1) (18)
The equation of motion of the SDOF system before suddendamage is
119910 + 2120585120596119906119910 + 1205962
119906119910 = 0 (19)
The above equation can be solved in terms of the given initialconditions and the structural dynamic responses are
119910 (119905) = 119860119906(119905) V0119890minus120585120596119906119905
sdot1
120596119906radic1 minus 1205852
119910 (119905) = V0119890minus120585120596119906119905
(119861119906(119905) minus
119860119906(119905) 120585
radic1 minus 1205852)
119910 (119905) = minus V0120596119906119890minus120585120596119906119905
sdot[119860119906(119905) (1 minus 2120585
2
) + 2120585119861119906(119905) radic1 minus 1205852]
radic1 minus 1205852
(20)
6 The Scientific World Journal
in which
119860119906(119905) = sin(120596
119906119905radic1 minus 1205852)
119861119906(119905) = cos(120596
119906119905radic1 minus 1205852)
(21)
Let us take the time instant 119905119894as the starting point of the SDOF
system after sudden damage and use a new time axis 1199051=
119905minus119905119894Then the equation ofmotion of the system after damage
becomes
119910119889+ 2120585120596
119889119910119889+ 1205962
119889119910119889= 0 (119905 gt 119905
119894) (22)
The initial conditions for (10) can be expressed as
119910119889(0) = 119910 (119905
119894) = 119860
119906(119905119894) V0119890minus120585120596119906119905119894 sdot
1
120596119906radic1 minus 1205852
119910119889(0) = 119910 (119905
119894) = V0119890minus120585120596119906119905119894 (119861119906(119905119894) minus
119860119906(119905119894) 120585
radic1 minus 1205852)
(23)
The damping ratio of a civil engineering structure is oftenvery small that is radic1 minus 1205852 asymp 1 The acceleration responseat the time instant 119905
1is
119910119889(1199051)
=V0120596119889119890minus120585(1205961198891199051+120596119906119905119894)
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894) [119861119889(1199051)radic1 minus 1205852 minus 119860
119889(1199051) 120585]
+ 120596119906119860119889(1199051) (21205852
minus 1) (119860119906(119905119894) 120585minus119861
119906(119905119894)radic1 minus 1205852)
minus 2120596119906120585119861119889(1199051) [119860119906(119905119894) 120585radic1 minus 1205852+119861
119906(119905119894) (1205852
minus 1)]
(24)
Furthermore the time interval should be very small todescribe the sudden stiffness reduction properly thus
Δ119905 = 119905119894+1
minus 119905119894997888rarr 0
119860119889(Δ119905) = sin(120596
119889Δ119905radic1 minus 1205852) asymp 0
119861119889(Δ119905) = cos(120596
119889Δ119905radic1 minus 1205852) asymp 1
(25)
Therefore the acceleration response at the time instant 119905119894+1
is
119910 (119905119894+1) = 119910119889(Δ119905)
=V0120596119889119890minus120585120596119906119905119894
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894)radic1 minus 1205852 minus 2120585120596
119906
times [119860119906(119905119894) 120585radic1 minus 1205852 + 119861
119906(119905119894) (1205852
minus 1)]
(26)
The first level detail coefficients of wavelet transform of theacceleration responses before sudden damage 119888119863 119910(119905)
1can be
expressed as
119888119863119910(119905)
1(119896) = int
+infin
minusinfin
119910 (119905) 1205951119896(119905) 119889119905 (27)
The first level detail coefficients of wavelet transform of theacceleration responses after sudden damage 119888119863 119910119889(1199051)
1can be
expressed as
119888119863119910119889(1199051)
1(119896) = int
+infin
minusinfin
119910119889(1199051) 1205951119896(1199051) 1198891199051 (28)
The variation of first level detail coefficients of theWT beforeand after the sudden damage event can be given as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896) = int
+infin
minusinfin
( 119910 (119905119894+1) minus 119910 (119905
119894)) 1205951119896(119905) 119889119905
(29)
Considering that the damping ratio of a civil engineeringstructure is often very small the above expression can besimplified as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
= minusΔ119896V0
119898120596119906
int
+infin
minusinfin
119890minus120585120596119906119905119894 sdot sin(120596
119906119905119894radic1 minus 1205852)120595
1119896(119905) 119889119905
(30)
The above equation reveals that the variation of first leveldetail coefficients of the WT before and after a suddendamage event is approximately linear to the sudden stiffnessreduction for given initial velocity damage instant andstructural parameters before damage If the time interval Δ119905for sudden damage is further regarded as a fixed value (30)indicates that the acceleration response discontinuity due tosudden stiffness reduction can be reflected by the variationrate of first level detail coefficients of the wavelet transform atdamage instant A damage index DI
119894 is defined to reflect the
signal discontinuity due to sudden damage at the time instant119905119894as follows
DI119894=
100381610038161003816100381610038161003816100381610038161003816
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
Δ119905
100381610038161003816100381610038161003816100381610038161003816
(119896 = 2 3 119899 minus 1) (31)
whereΔ119905 = 119905119896+1
minus119905119896and 119899 is the total number of time intervals
for the whole response time history This damage index iscomputed in the time domain and it is an instantaneous indexsuitable for online structural health monitoring applicationThe linear relationship between the proposed damage indexand the sudden stiffness reduction can be observed as follows
DI119894prop |Δ119896| (32)
5 Damage Detection
51 Selection of Mother Wavelet To examine the feasibil-ity of the proposed damage index and damage detection
The Scientific World Journal 7
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(a) db2
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(b) db4
Figure 6 Level-1 detail coefficients of wavelet transform for acceleration responses under seismic excitation
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
db1
db2
db3
db4
Figure 7 The Daubechies wavelets (db1ndashdb4)
approaches the first floor of the five-story building issupposed to suffer different levels of the sudden stiffnessreduction but the sudden damage occurs at the same timeSix damage scenarios are considered in the numerical inves-tigation Listed in Table 1 are the damage severities and thefive natural frequencies of the building before and after thesudden damage It is seen from Table 1 that the stiffnessreduction in the first story of the building affectsmainly lowernatural frequencies It is noted that if the stiffness reductionin the first floor is less than 10 the maximum frequencychange is no more than 2 In addition the variations of thehigher mode shapes are much smaller than those of the lowermode shapes
Wavelet transform can be utilized to detect the signalsingularity due to sudden stiffness change While the detec-tion efficiency depends on many factors such as waveletvanishing moments supporting length in the time domainfrequency components of original acceleration responsesand signal noise Thus three different Daubechies motherwavelets db1 db2 and db4 are utilized to examine the effectsof properties of mother wavelets on the detection on thestructural sudden damage The vanishing moments of thedb1 db2 and db4 wavelets are 1 2 and 4 respectivelyand they have the gradually increased supporting length asplotted in Figure 7 The basic principles of wavelet transformprove that the longer the wavelet supporting length is thefiner the distinguishing ability in the frequency domain isTherefore the mother wavelet with long supporting length ismore suitable for detecting the higher frequency componentsin the original signal
To examine the feasibility of the proposed damage indexand damage detection approaches for identifying damageevents the acceleration responses of the aforementioned five-story shear building to the seismic excitation sinusoidalexcitation and impulse excitation are computed respectivelyThe building is subject to a 20 sudden stiffness reductionat times 60 s 60 s and 02 s in the first story of thebuilding under seismic excitation sinusoidal excitation andimpulse excitation respectively The time step used in thecomputation is 0002 seconds
Shown in Figure 8 are the damage detection resultsusing db1 db2 and db4 for 20 sudden stiffness reductionrespectively It can be seen from Figure 8(a) that no matterwhich mother wavelet is used the damage index of the firstfloor is very large only at time 119905= 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 The damage indices of the first floor at allother time instants are very small so that the damage index attime 119905 = 60 seconds looks like a spikeTherefore the damagetime instant can be easily identified by the occurrence timeof the sharp damage index It is demonstrated that the DWTbased approach using all the three Daubechies wavelets canaccurately detect the damage time instant of the buildingsubjected to sinusoidal excitation For the building excitedby El-Centro ground motion DWT using db1 wavelet failto detect damage instant while the approach using db2 anddb4 wavelet successfully captures the damage events Forthe impulse excited case only the DWT using db4 waveletcan accurately detect the damage instant due to suddenstiffness change In reality the sudden stiffness loss will cause
8 The Scientific World Journal
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)200
100
0
Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
200
100
0
20 damage
20 damage
20 damage
(a) Sinusoidal excitation
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)
100
50
0
Dam
age i
ndex
(db2
)
100
50
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage
20 damage
20 damage
(b) Seismic excitation
Dam
age i
ndex
(db1
)
200
100
0 Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage 20 damage
20 damage
00 04 08 12 16 20
Time (s)00 04 08 12 16 20
Time (s)
00 04 08 12 16 20
Time (s)
(c) Impulse excitation
Figure 8 Damage detection using different db wavelets
Table 1 Natural frequency before and after sudden damage
Damage extent Frequencies (Hz)1198911
1198912
1198913
1198914
1198915
0 2513 7335 11563 14854 169411 2508 (minus018) 7324 (minus015) 1155 (minus011) 1485 (minus005) 1694 (minus001)2 2504 (minus036) 7313 (minus030) 1154 (minus021) 1484 (minus011) 1694 (minus003)5 2490 (minus094) 7278 (minus078) 1150 (minus053) 1481 (minus026) 1693 (minus007)10 2464 (minus197) 7218 (minus162) 1144 (minus106) 1478 (minus052) 1692 (minus014)20 2407 (minus441) 7088 (minus348) 1132 (minus218) 1471 (minus101) 1690 (minus026)40 2253 (minus116) 6781 (minus816) 1107 (minus450) 1457 (minus192) 1686 (minus046)Note values in brackets are the percentage of change in natural frequency
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 The Scientific World Journal
in which
119860119906(119905) = sin(120596
119906119905radic1 minus 1205852)
119861119906(119905) = cos(120596
119906119905radic1 minus 1205852)
(21)
Let us take the time instant 119905119894as the starting point of the SDOF
system after sudden damage and use a new time axis 1199051=
119905minus119905119894Then the equation ofmotion of the system after damage
becomes
119910119889+ 2120585120596
119889119910119889+ 1205962
119889119910119889= 0 (119905 gt 119905
119894) (22)
The initial conditions for (10) can be expressed as
119910119889(0) = 119910 (119905
119894) = 119860
119906(119905119894) V0119890minus120585120596119906119905119894 sdot
1
120596119906radic1 minus 1205852
119910119889(0) = 119910 (119905
119894) = V0119890minus120585120596119906119905119894 (119861119906(119905119894) minus
119860119906(119905119894) 120585
radic1 minus 1205852)
(23)
The damping ratio of a civil engineering structure is oftenvery small that is radic1 minus 1205852 asymp 1 The acceleration responseat the time instant 119905
1is
119910119889(1199051)
=V0120596119889119890minus120585(1205961198891199051+120596119906119905119894)
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894) [119861119889(1199051)radic1 minus 1205852 minus 119860
119889(1199051) 120585]
+ 120596119906119860119889(1199051) (21205852
minus 1) (119860119906(119905119894) 120585minus119861
119906(119905119894)radic1 minus 1205852)
minus 2120596119906120585119861119889(1199051) [119860119906(119905119894) 120585radic1 minus 1205852+119861
119906(119905119894) (1205852
minus 1)]
(24)
Furthermore the time interval should be very small todescribe the sudden stiffness reduction properly thus
Δ119905 = 119905119894+1
minus 119905119894997888rarr 0
119860119889(Δ119905) = sin(120596
119889Δ119905radic1 minus 1205852) asymp 0
119861119889(Δ119905) = cos(120596
119889Δ119905radic1 minus 1205852) asymp 1
(25)
Therefore the acceleration response at the time instant 119905119894+1
is
119910 (119905119894+1) = 119910119889(Δ119905)
=V0120596119889119890minus120585120596119906119905119894
(1205852 minus 1) 120596119906
times 120596119889119860119906(119905119894)radic1 minus 1205852 minus 2120585120596
119906
times [119860119906(119905119894) 120585radic1 minus 1205852 + 119861
119906(119905119894) (1205852
minus 1)]
(26)
The first level detail coefficients of wavelet transform of theacceleration responses before sudden damage 119888119863 119910(119905)
1can be
expressed as
119888119863119910(119905)
1(119896) = int
+infin
minusinfin
119910 (119905) 1205951119896(119905) 119889119905 (27)
The first level detail coefficients of wavelet transform of theacceleration responses after sudden damage 119888119863 119910119889(1199051)
1can be
expressed as
119888119863119910119889(1199051)
1(119896) = int
+infin
minusinfin
119910119889(1199051) 1205951119896(1199051) 1198891199051 (28)
The variation of first level detail coefficients of theWT beforeand after the sudden damage event can be given as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896) = int
+infin
minusinfin
( 119910 (119905119894+1) minus 119910 (119905
119894)) 1205951119896(119905) 119889119905
(29)
Considering that the damping ratio of a civil engineeringstructure is often very small the above expression can besimplified as
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
= minusΔ119896V0
119898120596119906
int
+infin
minusinfin
119890minus120585120596119906119905119894 sdot sin(120596
119906119905119894radic1 minus 1205852)120595
1119896(119905) 119889119905
(30)
The above equation reveals that the variation of first leveldetail coefficients of the WT before and after a suddendamage event is approximately linear to the sudden stiffnessreduction for given initial velocity damage instant andstructural parameters before damage If the time interval Δ119905for sudden damage is further regarded as a fixed value (30)indicates that the acceleration response discontinuity due tosudden stiffness reduction can be reflected by the variationrate of first level detail coefficients of the wavelet transform atdamage instant A damage index DI
119894 is defined to reflect the
signal discontinuity due to sudden damage at the time instant119905119894as follows
DI119894=
100381610038161003816100381610038161003816100381610038161003816
119888119863119894+1
1(119896) minus 119888119863
119894
1(119896)
Δ119905
100381610038161003816100381610038161003816100381610038161003816
(119896 = 2 3 119899 minus 1) (31)
whereΔ119905 = 119905119896+1
minus119905119896and 119899 is the total number of time intervals
for the whole response time history This damage index iscomputed in the time domain and it is an instantaneous indexsuitable for online structural health monitoring applicationThe linear relationship between the proposed damage indexand the sudden stiffness reduction can be observed as follows
DI119894prop |Δ119896| (32)
5 Damage Detection
51 Selection of Mother Wavelet To examine the feasibil-ity of the proposed damage index and damage detection
The Scientific World Journal 7
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(a) db2
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(b) db4
Figure 6 Level-1 detail coefficients of wavelet transform for acceleration responses under seismic excitation
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
db1
db2
db3
db4
Figure 7 The Daubechies wavelets (db1ndashdb4)
approaches the first floor of the five-story building issupposed to suffer different levels of the sudden stiffnessreduction but the sudden damage occurs at the same timeSix damage scenarios are considered in the numerical inves-tigation Listed in Table 1 are the damage severities and thefive natural frequencies of the building before and after thesudden damage It is seen from Table 1 that the stiffnessreduction in the first story of the building affectsmainly lowernatural frequencies It is noted that if the stiffness reductionin the first floor is less than 10 the maximum frequencychange is no more than 2 In addition the variations of thehigher mode shapes are much smaller than those of the lowermode shapes
Wavelet transform can be utilized to detect the signalsingularity due to sudden stiffness change While the detec-tion efficiency depends on many factors such as waveletvanishing moments supporting length in the time domainfrequency components of original acceleration responsesand signal noise Thus three different Daubechies motherwavelets db1 db2 and db4 are utilized to examine the effectsof properties of mother wavelets on the detection on thestructural sudden damage The vanishing moments of thedb1 db2 and db4 wavelets are 1 2 and 4 respectivelyand they have the gradually increased supporting length asplotted in Figure 7 The basic principles of wavelet transformprove that the longer the wavelet supporting length is thefiner the distinguishing ability in the frequency domain isTherefore the mother wavelet with long supporting length ismore suitable for detecting the higher frequency componentsin the original signal
To examine the feasibility of the proposed damage indexand damage detection approaches for identifying damageevents the acceleration responses of the aforementioned five-story shear building to the seismic excitation sinusoidalexcitation and impulse excitation are computed respectivelyThe building is subject to a 20 sudden stiffness reductionat times 60 s 60 s and 02 s in the first story of thebuilding under seismic excitation sinusoidal excitation andimpulse excitation respectively The time step used in thecomputation is 0002 seconds
Shown in Figure 8 are the damage detection resultsusing db1 db2 and db4 for 20 sudden stiffness reductionrespectively It can be seen from Figure 8(a) that no matterwhich mother wavelet is used the damage index of the firstfloor is very large only at time 119905= 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 The damage indices of the first floor at allother time instants are very small so that the damage index attime 119905 = 60 seconds looks like a spikeTherefore the damagetime instant can be easily identified by the occurrence timeof the sharp damage index It is demonstrated that the DWTbased approach using all the three Daubechies wavelets canaccurately detect the damage time instant of the buildingsubjected to sinusoidal excitation For the building excitedby El-Centro ground motion DWT using db1 wavelet failto detect damage instant while the approach using db2 anddb4 wavelet successfully captures the damage events Forthe impulse excited case only the DWT using db4 waveletcan accurately detect the damage instant due to suddenstiffness change In reality the sudden stiffness loss will cause
8 The Scientific World Journal
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)200
100
0
Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
200
100
0
20 damage
20 damage
20 damage
(a) Sinusoidal excitation
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)
100
50
0
Dam
age i
ndex
(db2
)
100
50
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage
20 damage
20 damage
(b) Seismic excitation
Dam
age i
ndex
(db1
)
200
100
0 Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage 20 damage
20 damage
00 04 08 12 16 20
Time (s)00 04 08 12 16 20
Time (s)
00 04 08 12 16 20
Time (s)
(c) Impulse excitation
Figure 8 Damage detection using different db wavelets
Table 1 Natural frequency before and after sudden damage
Damage extent Frequencies (Hz)1198911
1198912
1198913
1198914
1198915
0 2513 7335 11563 14854 169411 2508 (minus018) 7324 (minus015) 1155 (minus011) 1485 (minus005) 1694 (minus001)2 2504 (minus036) 7313 (minus030) 1154 (minus021) 1484 (minus011) 1694 (minus003)5 2490 (minus094) 7278 (minus078) 1150 (minus053) 1481 (minus026) 1693 (minus007)10 2464 (minus197) 7218 (minus162) 1144 (minus106) 1478 (minus052) 1692 (minus014)20 2407 (minus441) 7088 (minus348) 1132 (minus218) 1471 (minus101) 1690 (minus026)40 2253 (minus116) 6781 (minus816) 1107 (minus450) 1457 (minus192) 1686 (minus046)Note values in brackets are the percentage of change in natural frequency
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 7
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(a) db2
cD1
02
00
minus02
20 damage
0 2 4 6 8 10
Time (s)
(b) db4
Figure 6 Level-1 detail coefficients of wavelet transform for acceleration responses under seismic excitation
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
2
0
minus20 5 10 15
db1
db2
db3
db4
Figure 7 The Daubechies wavelets (db1ndashdb4)
approaches the first floor of the five-story building issupposed to suffer different levels of the sudden stiffnessreduction but the sudden damage occurs at the same timeSix damage scenarios are considered in the numerical inves-tigation Listed in Table 1 are the damage severities and thefive natural frequencies of the building before and after thesudden damage It is seen from Table 1 that the stiffnessreduction in the first story of the building affectsmainly lowernatural frequencies It is noted that if the stiffness reductionin the first floor is less than 10 the maximum frequencychange is no more than 2 In addition the variations of thehigher mode shapes are much smaller than those of the lowermode shapes
Wavelet transform can be utilized to detect the signalsingularity due to sudden stiffness change While the detec-tion efficiency depends on many factors such as waveletvanishing moments supporting length in the time domainfrequency components of original acceleration responsesand signal noise Thus three different Daubechies motherwavelets db1 db2 and db4 are utilized to examine the effectsof properties of mother wavelets on the detection on thestructural sudden damage The vanishing moments of thedb1 db2 and db4 wavelets are 1 2 and 4 respectivelyand they have the gradually increased supporting length asplotted in Figure 7 The basic principles of wavelet transformprove that the longer the wavelet supporting length is thefiner the distinguishing ability in the frequency domain isTherefore the mother wavelet with long supporting length ismore suitable for detecting the higher frequency componentsin the original signal
To examine the feasibility of the proposed damage indexand damage detection approaches for identifying damageevents the acceleration responses of the aforementioned five-story shear building to the seismic excitation sinusoidalexcitation and impulse excitation are computed respectivelyThe building is subject to a 20 sudden stiffness reductionat times 60 s 60 s and 02 s in the first story of thebuilding under seismic excitation sinusoidal excitation andimpulse excitation respectively The time step used in thecomputation is 0002 seconds
Shown in Figure 8 are the damage detection resultsusing db1 db2 and db4 for 20 sudden stiffness reductionrespectively It can be seen from Figure 8(a) that no matterwhich mother wavelet is used the damage index of the firstfloor is very large only at time 119905= 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 The damage indices of the first floor at allother time instants are very small so that the damage index attime 119905 = 60 seconds looks like a spikeTherefore the damagetime instant can be easily identified by the occurrence timeof the sharp damage index It is demonstrated that the DWTbased approach using all the three Daubechies wavelets canaccurately detect the damage time instant of the buildingsubjected to sinusoidal excitation For the building excitedby El-Centro ground motion DWT using db1 wavelet failto detect damage instant while the approach using db2 anddb4 wavelet successfully captures the damage events Forthe impulse excited case only the DWT using db4 waveletcan accurately detect the damage instant due to suddenstiffness change In reality the sudden stiffness loss will cause
8 The Scientific World Journal
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)200
100
0
Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
200
100
0
20 damage
20 damage
20 damage
(a) Sinusoidal excitation
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)
100
50
0
Dam
age i
ndex
(db2
)
100
50
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage
20 damage
20 damage
(b) Seismic excitation
Dam
age i
ndex
(db1
)
200
100
0 Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage 20 damage
20 damage
00 04 08 12 16 20
Time (s)00 04 08 12 16 20
Time (s)
00 04 08 12 16 20
Time (s)
(c) Impulse excitation
Figure 8 Damage detection using different db wavelets
Table 1 Natural frequency before and after sudden damage
Damage extent Frequencies (Hz)1198911
1198912
1198913
1198914
1198915
0 2513 7335 11563 14854 169411 2508 (minus018) 7324 (minus015) 1155 (minus011) 1485 (minus005) 1694 (minus001)2 2504 (minus036) 7313 (minus030) 1154 (minus021) 1484 (minus011) 1694 (minus003)5 2490 (minus094) 7278 (minus078) 1150 (minus053) 1481 (minus026) 1693 (minus007)10 2464 (minus197) 7218 (minus162) 1144 (minus106) 1478 (minus052) 1692 (minus014)20 2407 (minus441) 7088 (minus348) 1132 (minus218) 1471 (minus101) 1690 (minus026)40 2253 (minus116) 6781 (minus816) 1107 (minus450) 1457 (minus192) 1686 (minus046)Note values in brackets are the percentage of change in natural frequency
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 The Scientific World Journal
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)200
100
0
Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
200
100
0
20 damage
20 damage
20 damage
(a) Sinusoidal excitation
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
(db1
)
100
50
0
Dam
age i
ndex
(db2
)
100
50
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage
20 damage
20 damage
(b) Seismic excitation
Dam
age i
ndex
(db1
)
200
100
0 Dam
age i
ndex
(db2
)
200
100
0
Dam
age i
ndex
(db4
)
100
50
0
20 damage 20 damage
20 damage
00 04 08 12 16 20
Time (s)00 04 08 12 16 20
Time (s)
00 04 08 12 16 20
Time (s)
(c) Impulse excitation
Figure 8 Damage detection using different db wavelets
Table 1 Natural frequency before and after sudden damage
Damage extent Frequencies (Hz)1198911
1198912
1198913
1198914
1198915
0 2513 7335 11563 14854 169411 2508 (minus018) 7324 (minus015) 1155 (minus011) 1485 (minus005) 1694 (minus001)2 2504 (minus036) 7313 (minus030) 1154 (minus021) 1484 (minus011) 1694 (minus003)5 2490 (minus094) 7278 (minus078) 1150 (minus053) 1481 (minus026) 1693 (minus007)10 2464 (minus197) 7218 (minus162) 1144 (minus106) 1478 (minus052) 1692 (minus014)20 2407 (minus441) 7088 (minus348) 1132 (minus218) 1471 (minus101) 1690 (minus026)40 2253 (minus116) 6781 (minus816) 1107 (minus450) 1457 (minus192) 1686 (minus046)Note values in brackets are the percentage of change in natural frequency
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 9
0
10
20 1 damageD
amag
e ind
ex
20 damage
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) db2
0
10
20 1 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) db4
Figure 9 Damage detection using different db wavelets under seismic excitation
Dam
age i
ndex
0
100
20020 damage
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
000
50
100
Time (s)
20 damageD
amag
e ind
ex
04 08 12 16 20
(b) Impulse excitation
Figure 10 Damage detection using db4 wavelets
a sudden jump in acceleration responses at damage instantwhichmay commonly introduce high frequency componentsinto the original response signals The crucial procedure indetecting sudden damage is to extract the high frequencycomponents from original acceleration responses using thewavelet transformThe frequency components of accelerationresponses of building subjected to sinusoidal excitation arequite simple and the high frequency signal induced by suddendamage is quite different from other signal componentsAll the three selected wavelets can easily detect the signalsingularity and damage event As far as the seismic exciteddamage building is concerned the acceleration responsescontainmore high frequency components than those inducedby sinusoidal excitations The distinguishing ability in thefrequency domain of the db1 wavelet is coarse due to itsshort supporting length in the time domain which makes itimpossible to capture the sudden damage event under seismicexcitations
The damage events can be captured by using the db2and db4 wavelets due to their finer distinguishing abilitythan db1 wavelet in particular in high frequency rangeThe damage events of the example building under impulseexcitations aremore difficult to be detected because abundanthigh frequency components of acceleration responses mayoverlap the high frequency signal induced by sudden stiffnessreduction If the extent of the damage event is minor
the energy of damage signal is too small to be reflected tothe decomposed wavelet coefficientsThe comparison amongdifferent mother wavelets indicates that only the db4 waveletwith fine frequency distinguishing ability can accuratelycapture the damage event of the building under impulseexcitation
52 Damage Time Instant The first floor of the five-storybuilding is supposed to suffer different levels of suddenstiffness reduction but the sudden reduction occurs at thesame time Two mother wavelets namely db2 and db4are utilized in this section to study their performance fordifferent damage extents as shown in Figure 9 It is clear thatthe db2 wavelet can accurately capture the damage featureswithout noise contamination For small damage cases (1damage) the energy of damage signal is very small andthe detail coefficients of the damage signal are too small toform a distinct spike at damage instant The db4 waveletwith stronger frequency distinguishing ability can detect theminor damage event Therefore the damage detection onsudden stiffness reduction is carried out based on db4waveletin the following sections
The variations of damage index with time under sinu-soidal excitation and impulse excitation using db4 waveletare displayed in Figure 10 The building is subject to
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 The Scientific World Journal
0
200 Floor number 1D
amag
e ind
ex
Floor number 2
Floor number 3
Floor number 4
Floor number 5
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0
200
Dam
age i
ndex
minus200
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(a) Sinusoidal excitation
Floor number 1
Floor number 2
Floor number 3
Floor number 4
Floor number 5
0
100
Dam
age i
ndex
minus100
0
100
minus100
0
100
minus100
0
100
minus100
0
100
minus100
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
(b) Seismic excitation
Floor number 1 Floor number 2
Floor number 3 Floor number 4
Floor number 5
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
00Time (s)
04 08 12 16 20
0
100
minus100Dam
age i
ndex
(c) Impulse excitation
Figure 11 Damage detection for each floor using db4 wavelet
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 11
0
10
20 1 damageD
amag
e ind
ex
0
10
20 2 damage
Dam
age i
ndex
0
20
405 damage
Dam
age i
ndex
0
50
100 10 damage
Dam
age i
ndex
0
50
100
150 20 damage
Dam
age i
ndex
0
50
100
150 40 damage
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Figure 12 Damage detection using different db wavelets under seismic excitation
the same damage severity at the first story only but it occursat time 119905 = 60 seconds for sinusoidal excitation and at time119905 = 02 seconds for impulse excitation Again the sharpdamage index appears only at themoment of sudden stiffnessreduction at the first floor Thus the damage time instantcan be easily captured from the observed occurrence time ofthe sharp damage index While the detection effects reducewith the decreasing damage extent This is because the signalenergy of the minor damage extent holds little informationabout damage event and the projected wavelet coefficients aretoo small to form a distinct spike at damage instant Underthis circumstance the mother wavelet with higher vanishingmoments and longer supporting length also cannot improvethe damage detection efficiency
53 Damage Location Figure 11 shows the variations ofdamage index with time for each floor of the building underthe sinusoidal seismic and impulse excitations It is seenfrom Figure 11(a) that the damage index of the first flooris very large only at time 119905 = 60 seconds which is exactlythe moment when the stiffness of the first story is suddenlyreduced by 20 It is essential to compare the variation ofwavelet coefficients based damage index of the first floorwith those of the second third fourth and fifth floors ofthe building The sharp spike appears clearly only at the firstfloor and no sharp spike emerges in other floors Thereforeby analyzing the distribution of spike along the height of thebuilding the damage location can be easily identified at thefirst story of the building
The variations of damage indices with time for eachfloor of the building are shown in Figure 11(b) for seismicexcitation using the DWT The sharp damage index appears
only at the moment of sudden stiffness reduction at the firstfloor Thus the damage location can be easily captured fromthe observed sharp spikes and its distribution along the heightof the building Similar results are also obtained from thebuilding subject to sinusoidal excitation For the impulseexcited case however the DWT based detection approachmay not give satisfactory results for the building with smalldamage event (1 sudden stiffness reduction)This is becausethe signal fluctuates significantly immediately after the initialvelocity and the energy of damage signal is quite weak
54 Damage Severity The parameter investigation is carriedout in this section to find the sensitivity of damage index todamage severity so as to examine the validity of the proposeddamage index and damage detection approaches The firstfloor of the example building is supposed to suffer differentlevels of sudden stiffness reduction but the damage timeinstants remain unchanged The damage indices of the firstfloor of the building subjected to the seismic excitation areplotted in Figure 12 for the sudden stiffness reduction from1 to 40 It can be seen that even for small damage eventssuch as 1 to 5 sudden stiffness reduction the proposedapproach can easily capture the damage features withoutconsidering noise contaminationThemagnitude of the sharpdamage index also increases with increasing damage severitySimilar observations can be made from the building subjectto sinusoidal excitation as shown in Figure 13(a) For thebuilding under impulse excitation however the proposedapproaches may not provide satisfactory detection effectsfor the building with very small damage event (1 damageseverity) as shown in Figure 13(b)
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 The Scientific World Journal
1 damage
0
20
40D
amag
e ind
ex
2 damage
5 damage
0
100
200
0
50
100
10 damage
0
100
200 20 damage
0
100
200
300 40 damage
0
20
40
Dam
age i
ndex
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
0 2 4 6 8 10
Time (s)
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
Dam
age i
ndex
(a) Sinusoidal excitation
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
1 damage
2 damage
5 damage
10 damage
20 damage
40 damage
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
0
20
40
Dam
age i
ndex
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
00Time (s)
04 08 12 16 20
(b) Impulse excitation
Figure 13 Damage detection using db4 wavelets
The magnitudes of damage index of the example build-ing subjected to different damage severities in the firststory under seismic sinusoidal and impulse excitations arecomputed and listed in Table 2 The relationship betweendamage index and damage severity is also displayed inFigure 14 together with a linear fit in which 119909 representsdamage severity (stiffness reduction) and 119910 represents dam-age index It is observed that there exists a linear rela-tionship between damage index and damage severity forthe building under either seismic or sinusoidal or impulseexcitationThemagnitudes of the damage index increase with
the increasing extents of the stiffness reduction for a givenexternal excitation and the slope of the linear fit is differentfor the building under different excitations The proposeddamage index and detection approach can be used to find thedamage time instant and damage location from themeasuredstructural responses Then one can measure the externalexcitation and input to the structural model with a suddenstiffness reduction at the identified damage location and at theidentified damage time instant to determine the slope of thelinear relationship between the damage severity and damageindexThe linear relationship can be used finally to determine
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 13
Table 2 Variations of damage index with damage severity
Damage severity 1 2 5 10 20 40DI (seismic excitation) 2625 5223 13017 26004 51960 103814DI (sinusoidal excitation) 5172 10344 25857 51704 103369 206579DI (impulse excitation) 1256 2519 6309 12623 25244 50457
0
100
200
300
Dam
age i
ndex
Damage severity ()0 5 10 15 20 25 30 35 40 45
Seismic excitation
Sinusoidal excitationImpulse excitationy = 126156 lowast x minus 000103
y = 2595 lowast x minus 001286
y = 516434 lowast x minus 000668
Figure 14 Relationship between damage index anddamage severity
the damage severity in the actual structure in terms of thedamage index identified from the actual structure
55 Effects of Signal Noise The effect of measurement noiseon the quality of the damage detection is a key issue neededto be addressed in real application Hou et al [10] reportedthat the damage spike identified from the wavelet transformcoefficients could be weakened by measurement noise andstrongmeasurement noise could lead to the failure of damagedetection In reality the sudden damage event may introducea high frequency component to acceleration responses of astructure and the effects of both measurement noise intensityand frequency range on the damage detection are therebyexamined [16]Themeasurement noise in structural responseis assumed to be a random white noise Three frequencyranges are considered (1) white noise with frequency rangefrom 0 to 50Hz (2) white noise with frequency range from 0to 100Hz and (3) white noise with frequency range from 0 to250Hz The measurement noise intensity is defined as
Noise intensity = RMS (noise)RMS (signal)
times 100 (33)
Displayed in Figures 15 16 and 17 are damage detectionresults using the contaminated acceleration responses atthe first floor under the sinusoidal seismic and impulseexcitations respectively The noises are introduced with twonoise intensities and three noise frequency ranges describedas above The sudden stiffness reduction in the first story of
the building is 20 To check the original acceleration timehistories contaminated with noise one can find that withincreasing noise frequency range the acceleration responsesare more fluctuating and the signal discontinuity at the dam-age time instant becomes weak It is clear from Figure 15 thatthe proposed index can still identify the damage time instantfrom the polluted acceleration responses at the first floor forthe designated twonoise intensities and three noise frequencyranges when subjected to sinusoidal excitation Furthermorethe spatial distribution of damage indices along the heightof the building can indicate the damage location from theacceleration responses with noise contamination Similarobservations can be made from the detection observations ofthe example building subjected to the seismic excitation asshown in Figure 16 In the impulse excitation case howeverthe proposed approach fails to identify the damage timeinstant and damage location when the noise frequency rangeis from 0 to 250Hz and the noise intensity is 5 as shown inFigure 17
The effects of measurement noise on the damage detec-tion are assessed and the results indicate that both thenoise intensity and the frequency range have a remarkableinfluence on the results fromDWTbased detection approachIf the noise frequency range is too narrow to overlap thesudden damage signal with high frequency components theinfluence of noise intensity on detection accuracy is smallOtherwise the detection accuracy will decrease with theincrease of noise intensities
6 Concluding Remarks
Thedamage detection on sudden stiffness reduction of build-ing structures has been actively investigated in this studyThe signal jump of the structural acceleration responses of anexample building due to sudden damage is examined and thesignal discontinuity is extracted based on the discrete wavelettransform It is proved that the variation of the first leveldetail coefficients of the wavelet transform at damage instantis proportional to the magnitude of the stiffness reductionIn this regard a new damage index based on the DWT isproposed in terms of the slope of the decomposed detailcoefficients of wavelet transform to detect the damage timeinstant damage location and damage severity
Extensive numerical simulations have been executed on afive-story shear building to assess the performance of DWTbased detection approach The made observations indicatethat the DWT based detection index proposed in this studycan accurately identify the damage time instant and damagelocation due to a sudden stiffness reduction in terms of theoccurrence time and spatial distribution of damage indexspikesThe proposed damage index is linearly proportional to
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 The Scientific World Journal
0
100
200 2 noise
Dam
age i
ndex
5 noise
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
100
200
Dam
age i
ndex
0
100
200
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 15 Detection results from contaminated acceleration responses (sinusoidal excitation)
2 noise
0
50
100
150 5 noise
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(a) Noise frequency range 0sim50Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(b) Noise frequency range 0sim100Hz
2 noise 5 noise
0
50
100
150
Dam
age i
ndex
0
50
100
150
Dam
age i
ndex
0 2 4 6 8 10
Time (s)0 2 4 6 8 10
Time (s)
(c) Noise frequency range 0sim250Hz
Figure 16 Detection results from contaminated acceleration responses (seismic excitation)
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 15
0
50
1002 noise
Dam
age i
ndex
0
50
100
Dam
age i
ndex 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(a) Noise frequency range 0sim50Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(b) Noise frequency range 0sim100Hz
0
50
100
Dam
age i
ndex
0
50
100
Dam
age i
ndex
2 noise 5 noise
00Time (s)
04 08 12 16 20 00Time (s)
04 08 12 16 20
(c) Noise frequency range 0sim250Hz
Figure 17 Detection results from contaminated acceleration responses (impulse excitation)
damage severity but the slope of the linear function dependson external excitation and damage time instant A possibleway of determining damage severity has been suggested usingthe calibrated structural model and the measured excitationThe proposed damage index can identify the damage eventsfrom the contaminated acceleration responses if the noisefrequency range is limited If the noise frequency range iswide enough the reliability of damage detection using theproposed approach remarkably deteriorates with the increaseof noise intensity
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The writers are grateful for the financial support from theNational Natural Science Foundation of China (51178366 and51108395) the Technological Project of the Chinese SouthernPower Grid Co Ltd (K-GD2013-0222) and the FundamentalResearch Funds for the Central Universities (WUT 2013-II-015)
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] Y Lei Y Jiang and Z Xu ldquoStructural damage detection withlimited input and output measurement signalsrdquo MechanicalSystems and Signal Processing vol 28 pp 229ndash243 2012
[3] K Gurley and A Kareem ldquoApplications of wavelet transformsin earthquake wind and ocean engineeringrdquo Engineering Struc-tures vol 21 no 2 pp 149ndash167 1999
[4] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013
[5] T H Yi H N Li and H M Sun ldquoMulti-stage structuraldamage diagnosis method based on energy-damage theoryrdquoSmart Structures and Systems vol 12 no 3-4 pp 345ndash361 2013
[6] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010
[7] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[8] A Masuda A Nakaoka A Sone and S Yamamoto ldquoHealthmonitoring system of structures based on orthonormal wavelettransformrdquo Seismic Engineering vol 312 no 1 pp 161ndash167 1995
[9] A Sone S Yamamoto A Masuda A Nakaoka and R AshinoldquoEstimation of cumulative damage of a building with hystereticrestoring force by using wavelet analysis of strong motionrecordsrdquo Japan Structural Construction Engineering vol 476pp 67ndash74 1995
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
16 The Scientific World Journal
[10] Z Hou M Noori and R S Amand ldquoWavelet-based approachfor structural damage detectionrdquo Journal of EngineeringMechanics vol 126 no 7 pp 677ndash683 2000
[11] H Sohn A N Robertson and C R Farrar ldquoHolder exponentanalysis for discontinuity detectionrdquo Structural Engineering andMechanics vol 17 no 3-4 pp 409ndash428 2004
[12] H T Vincent S L J Hu and Z Hou ldquoDamage detection usingempirical mode decomposition method and a comparisonwith wavelet analysisrdquo in Proceedings of the 2nd InternationalWorkshop on Structural HealthMonitoring Stanford UniveristyStandford Calif USA 1999
[13] J N Yang Y Lei and N E Huang ldquoDamage identificationof civil engineering structures using Hilbert-Huang transformrdquoin Proceedings of the 3rd International Workshop on StructuralHealth Monitoring pp 544ndash553 Stanford Calif USA 2001
[14] J N Yang Y Lei S Lin and N Huang ldquoHilbert-Huangbased approach for structural damage detectionrdquo Journal ofEngineering Mechanics vol 130 no 1 pp 85ndash95 2004
[15] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A Mathematical Physical and Engineering Sciences vol454 no 1971 pp 903ndash995 1998
[16] N E Huang Z Shen and S R Long ldquoA new view of nonlinearwater waves the Hilbert spectrumrdquo Annual Review of FluidMechanics vol 31 pp 417ndash457 1999
[17] Y L Xu and J Chen ldquoStructural damage detection usingempirical mode decomposition experimental investigationrdquoJournal of Engineering Mechanics vol 130 no 11 pp 1279ndash12882004
[18] B Chen and Y L Xu ldquoA new damage index for detectingsudden change of structural stiffnessrdquo Structural Engineeringand Mechanics vol 26 no 3 pp 315ndash341 2007
[19] H Li T Yi M Gu and L Huo ldquoEvaluation of earthquake-induced structural damages by wavelet transformrdquo Progress inNatural Science vol 19 no 4 pp 461ndash470 2009
[20] I Daubechies ldquoWavelet transform time-frequency localizationand signal analysisrdquo IEEE Transactions on Information Theoryvol 36 no 5 pp 961ndash1005 1990
[21] S Mallat A Wavelet Four of Signal Processing Academic PressWaltham Mass USA 1998
[22] B Chen and Y L Xu ldquoIntegrated vibration control and healthmonitoring of building structures using semi-active frictiondampersmdashpart IImdashnumerical investigationrdquo Engineering Struc-tures vol 30 no 3 pp 573ndash587 2008
[23] B Chen Y L Xu and X Zhao ldquoIntegrated vibration controland health monitoring of building structures a time-domainapproachrdquo Smart Structures and Systems vol 6 no 7 pp 811ndash833 2010
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
