Research ArticleThe Tunnel Structural Mode Frequency CharacteristicsIdentification and Analysis Based on a Modified StochasticSubspace Identification Method
Biao Zhou 12 Xiongyao Xie12 and Xiaojian Wang12
1Key Laboratory of Geotechnical amp Underground Engineering Ministry of Education Tongji University Shanghai 200092 China2Department of Geotechnical Engineering Tongji University Shanghai 200092 China
Correspondence should be addressed to Biao Zhou zhoubiaotongjieducn
Received 2 August 2018 Accepted 22 October 2018 Published 2 December 2018
Academic Editor Filippo Ubertini
Copyright copy 2018 Biao Zhou et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
With the rapid development of underground engineering in China the heavy structural maintenance work followed is expected tobe a great challenge in the future (e development also provides a promising application prospect for the newly developedvibration-based health assessment and monitoring methods However the fact that tunnels are embedded in soil makes collectingand identifying the vibration characteristics more difficult especially for the online monitoring In this paper a new identificationmethod that combines the natural excitation technique (NExT) and stochastic subspace identification (SSI) method is developed(e newmethod is compared with the traditional SSI method and mode frequency analysis is made based on a series of field testscarried out at the subway and power tunnel It is found that both stability and efficiency of the mode frequency identification havebeen greatly improved and it more suitable for online monitoring Meanwhile a mathematical model is used to analyze theoriginal mode characteristics and the influence of soil coupling (e results are also compared with the field tests results by usingthe NExT-SSI method and some recommendations are also made for how to choose the vibration modals for vibration-basedmonitoring in the tunnel
1 Introduction
At present the underground engineering is graduallymoving from the large-scale construction stage into thefollowing heavy maintenance stage in China (e safe op-eration of underground facilities is threatened by the in-creasing servicing time and the increasing number of thesurrounding construction activities such as the large ex-cavation (erefore the health monitoring system is greatlyneeded to determine the service state of underground fa-cilities in real time(e static level gauge and crack meter arefrequently used to monitor tunnel settlement and jointopening [1 2] which can effectively prevent large de-formation and leakage accidents caused by improper con-struction behavior such as deep excavation around subwaytunnels However the related static health monitoringmethod can only characterize the structural behavior near
the monitoring points and the dense layout is thus neededand difficult to obtain the structural elastic modulus andother indicators directly It can be used instead of somenondestructive testing methods but the detection efficien-cies of these methods are difficult to meet the real-timerequirements for large-scale applications (erefore there isa strong need of developing health monitoring methodsbased on vibration or wave propagation characteristicswhich have sensing capability of the overall undergroundstructure and the surrounding soil
(e present wave or vibration-based methods can utilizecharacteristics of wave propagation within the structures orthe normal vibration modes to identify structural perfor-mance Examples of current structural healthy monitoring(SHM) techniques include using fiber optics ultrasonicspiezoelectrics and acoustic emissions thermography andembedded thin film [4ndash6] For these methods the effective
HindawiShock and VibrationVolume 2018 Article ID 6595841 12 pageshttpsdoiorg10115520186595841
identification and extraction of vibration characteristicsbecome very important For applications in bridges andhigh-rise buildings the vibration modes and relatedmethods are more commonly used eg the natural fre-quency and frequency response function (FRF) spectrum-based methods [7] the mode shape-based methods [8] themodel and related matrix-based methods [9] and the datadriven methods [10] Some signal processing techniqueshave also been proposed for vibration characteristics ex-traction [11]
It is well known that the soil-structure interaction willchange the wave propagation characteristics [12 13] sincethe underground structures are strongly coupled with thesurrounding soil and the constraining and damping effectsand the dynamic behaviors of surrounding soil will causemajor challenges for the identification of vibration char-acteristics [14] In previous studies the difference betweenfrequency and dispersion characteristics of the subwaytunnel with or without surrounding soil is studied by nu-merical calculation [15ndash17] It is found that the mode fre-quency is more difficult to identify from the frequencyspectrum under the condition of soil coupling Healthmonitoring of underground pipelines is mostly based onwave propagation characteristics and related methods[18ndash20] Leinov et al [21] Vogt et al [22] and Eybpooshet al [23] found that the propagation modes and theirdispersion characteristics of the propagating ultrasonic-guided waves changed from the pipeline to the surround-ing medium Meanwhile a series of signal processing andmodal parameter identification methods have been pro-posed such as the peak-picking method the random dec-rement technique (RAMA) [24] the natural excitationtechnique (NExT) [25] and the stochastic subspace iden-tification (SSI) method [26] (e NExT is a method of modetesting that allows structures to be tested in their ambientenvironments where the auto- or cross-correlation functionis used for mode identification (e SSI method is originallypresented by van Overschee and de Moor [27] and anextension of the original SSI method that does not requireoutput covariance was proposed by Peeters and de Roeck[28 29] as the reference-based SSI (e stochastic realizationalgorithm mainly focused on the data-driven method (SSI-DATA) that considers the problem of identifying a sto-chastic state-space model from output-only data As op-posed to SSI-DATA the covariance-driven stochasticsubspace identification (SSI-COV) algorithm is also de-veloped to avoid the computation of orthogonal projectionFurthermore to reduce the effect of noise on the results ofidentification some filtering techniques need to be used toenhance the early emergence of a stable diagram for theidentifiable modes For instance SSI is combined withmultivariate singular spectrum analysis (MSSA) for vibra-tion monitoring of the rotating turbine system [30] (esemethods are widely used in bridges and high-rise buildingsbut their applicability in the underground engineering re-mains to be explored
An equally important consideration in developing themonitoring system is minimizing the amount of data pro-cessing and improving the stability and robust of the
monitoring network Automatic processing technology isalso used to reduce the amount of data transmission fromthe terminal nodes to the server(erefore the following keyproblems need to be considered in the application of thevibration-based monitoring method to tunnels and otherunderground structures
(1) Understanding of the tunnel-soil coupling effect onthe tunnel dynamic characteristics and selection ofthe appropriate vibration characteristics for de-termining the current structural state
(2) Proper signal processing and vibration characteris-tics identification methods with high stability andaccuracy
(3) Development of the self-processing and automaticidentification capability of terminal nodes to meetthe low energy cost and robust needs of the wirelessnetwork transmission
In this paper to explore the vibration characteristics ofthe tunnel and its identification method a series of field testsare carried out at the subway and power tunnels with dif-ferent diameters of 64m and 32m(e SSI-COVmethod isused and the NExT is also combined as the filtering tech-niques to identify tunnel mode natural frequency and somecomparisons on the difference between using SSI-COV andNExT-SSI-COV method on the result of identification arecarried out through the stability diagram At last a theo-retical pipe-in-pipe (PiP) model [31] is employed to studythe distribution of the vibration modes and some scientificsuggestions for the tunnel mode parameters identificationare provided at last
2 The NExT-SSI-COV Method
(e SSI-COV method is a well-known multivariate iden-tification technique However the uncertainty of parameterselection such as the Toeplitz matrix row number [32] andthe long input data will result in the sharp drop of thecomputational efficiency and stability In this study theNExTmethod is employed and combined with the SSI-COVmethod to reduce the singularity of covariance matrix andimprove the calculation efficiency and stability (e detailsand process of this method are shown in Figure 1
Step 1 Form the input matrix N by the NExT method
(e autocorrelation function amm of every measure-ment points and cross-correlation function amn for anytwo measurement points were calculated by the NExTmethod [33] and the formedmatrix is denoted asN isin Rltimesnwhere l is the number of autocorrelation and cross-correlation functions and n is the sampling number ofthe NExTmethod (e n value is always less than 2048 foridentifying the low-frequency modes (erefore comparedwith the conventional SSI-COV method the size andsingularity of the input matrix will be greatly reduced andthe computational efficiency and stability can be obviouslyimproved
2 Shock and Vibration
Step 2 Form the output Hankel matrix Ys from inputmatrix N
Firstly as Equation (1) shows a series of block matrix yiare obtained with a time lag i from the input matrix N andevery block matrix yi has l lines same as YN And the dataHankel data matrix is established by the future measure-ments matrix Yf and past measurements Yp as Equation (1)shows
Step 3 Obtain covariance-driven Hankel matrix P by theconventional matrix and then form the block Toeplitz matrixby a multiplication between future measurements matrix Yf
and transpose of past measurements Yp based on Equation(2)
P
Ri Riminus1 middot middot middot R1
Ri+1 Ri middot middot middot R2
⋮ ⋮ ⋱ ⋮
R2iminus1 R2iminus2 middot middot middot Ri
C
CA
middot middot middot
CAiminus1
Aiminus1G middot middot middot AG G( ) Yf Yp( )T
Ri E yk+i yTk[ ] lim
j⟶infin
1jsumjminus1
k0yk+iy
Tk
(2)
Step 4 Perform singular value decomposition (SVD) on theHankel covariance matrix P and obtain eigenmatrix S[28 29] en separate the matrix S into two submatrices S1and S2 by reducing the eigenvalues e smallest singularvalues in the matrix S are grouped as S2 and will be neglectedin the following steps by parameters Csvd e Csvd can becalculated by the following equations
P USVT U1 U2( )S1 0
0 S2
VT1
VT2
U1S1VT1
(3)
Csvd sumVSnsumVS
(4)
wheresumVSn is the cumulative value of the rst n eigenvaluesfor the S matrix and sumVS is the cumulative value of alleigenvalues
Steps 5sim7 Based on the conventional SSI-COV method theextended observability matrix ΓL is rstly calculatedaccording to Equation (5) and then the system parametermatrices A and C and mode frequency ωn and dampingcoecient ξn are obtained separately by Equations (6) and(7) [34 35]
ΓL S121 VT1 (5)
A ΓLprimeΓLC the first L rows of ΓL
(6)
where ΓL
and ΓL denote ΓL without the last L rows and therst L rows respectively
Calculate the correlation function matrix N by NExT methodN = [a11 aij] isin Rntimesl aij is the correlation function ofevery two measurement points
(i)
Calculate the covariance matrix P(P = YpYfT)(iii)
Calculate SVD of P and determine the order n by neglectingthe smaller singular values in S2
(iv)
00S1 V1
TP = USVT = (U1 U2) V2
TS2
(ii) Form the output Hankel matrix Ys from matrix N
Ys = isin R2nLtimesk k = l ndash L + 1 is the user-defined indexYpYf
Calculate the extended observability matrix ΓL = U1 S1frac12(v)
Calculate the system parameter matrices A and CA = ΓprimeLΓL ΓL denotes ΓL without the last l rows while ΓL withoutthe first l rows C = the first l rows of ΓL
(vi)
Determine the model frequency ωn and damping coefficient ξn(vii)
3 Field Test and Mode FrequencyIdentification of a Power Tunnel
31Measurement Site andArrangement As Figure 2 showsthe measured power tunnel is a pipe jacking tunnel whichconsists of concrete pipes with a length of 25m (e outerdiameter is 32m and inner diameter is 27m Nine ac-celerometers (LANCE 130 series) are installed at the walland the bottom of the tunnel (ree of them are equipped ateach measured point to collect the response from theradical longitudinal and tangential directions and theletter R means the sensor is installed in the radial directionL means the longitudinal direction while T is the tangentialdirection
32 Ambient Vibration Test To compare the responsecharacteristics of the tunnel in daytime and nighttime theambient vibration test is continued for seven days Accel-eration response records of 30 minutes in the nighttime anddaytime respectively at the measurement point B-1 areshown in Figure 3 For the measured power tunnel close to asubway the vibration from the operational trains makes thevibration amplitude at daytime about 4 times than that atnight Based on the above data the tunnel mode frequencycharacteristics are studied and a comparison on the dif-ference between using SSI-COV and NExT-SSI-COV on theresult of identification is carried out through the con-struction of a stability diagram (e stability diagram hasbeen proved very helpful to identify the dominant fre-quencies of the vibration mode while using the SSI-COVmethod [32]
321 2e Tunnel Response Characteristics at Night Toanalyze the tunnel response characteristics at night with the30-minute records collected from nine accelerometers theinput matrix dimension will be 3600000 times 9 using thetraditional SSI method (e stability diagram can be ob-tained and is shown in Figure 4 after running for three hourson a microcomputer with an Intel i5 processor While theNExTmethod is employed based on the procedures of Step1 and Step 2 introduced in Section 2 the auto- and cross-correlation functions of every degree of freedom were cal-culated with a sampling window of 2048 points and the inputmatrix N is formed with the matrix dimension of 2048 times 28
(e operation time of the stability diagram on the samemicrocomputer then can be reduced to less than 2 minutes(e calculation results are shown in Figure 5
By comparing the stability diagrams of Figures 4 and 5 itwas observed that several modes between 10 and 15Hz canbe found in Figure 4 by using the SSI-COV method directlybut the data fluctuation along the number of rows makes itdifficult to identify the natural frequency automatically andaccurately It thus will limit its application of SHM at theunderground structure However as the NExT method iscombined there is only one mode left with the frequency of105Hz In addition in the implementation of SSI-COV theselecting Csvd in Step 4 at Section 2 can lead to a change inthe number of the vibration modes and their identificationaccuracy From Figure 5 it is observed that there is almostno change on the identified model frequencies as the Csvdvalue changes from 0815 to 0915 by using the NExT-SSI-COV method (erefore the NExTmethod can develop themode frequency identification stability and accurately andmake it much suitable for application in the undergroundstructure
322 2e Tunnel Response Characteristics at Daytime(e tunnel response characteristics are analyzed during thedaytime especially for considering those vibration modesexcited by the train-induced vibration from the nearbysubway By employing the method both of traditional SSI-COV method and NExT-SSI-COV method a 30-minuterecord is used and the stability diagrams are shown inFigure 6 By comparing Figures 6(a) and 6(b) the modefrequency identification accuracy and stability are im-proved by using the NExT-SSI-COV method Meanwhileit also can be observed that besides 105Hz there are muchmore recognition results that can be found from Figure 6and several modes of which are around 48Hz of which isclose to the subway track-rail resonant frequency
From the ambient test at the power tunnel both atdaytime and nighttime and by comparing the stability di-agrams of using the traditional SSI method and modifiedmethod it is found that the NExT-SSI-COV method hasbetter mode natural frequency recognition accuracy andstability and several mode frequencies can be clearlyidentified around 105Hz 48Hz and 71Hz from the am-bient test Especially for the frequency of 105 stability canbe recognized both at daytime and nighttime
33 Hammering Test (e mode hammering test is acommonly used method to study the mode characteristicsand has a wider test frequency band than the ambient test(us it was also carried out at the power tunnel along theradial longitudinal and tangential directions (e impulseload is applied at the connecting steel ring of the tunnelsegment (Figure 7(a)) the time history of the pulse load isshown in Figure 7(b) and the frequency test band is around300Hz as shown in Figure 7(c)
(e NExT-SSI-COV method is also used here to analyzethe acceleration response and the stability diagrams areshown in Figure 8 From Figure 8(a) the modes at 53Hz
4 Shock and Vibration
203Hz and 223Hz can be identied from the radical re-sponses as hammered along the radial direction of thetunnel And much more modes can be found from thelongitudinal direction response as hammered along thelongitudinal direction and there are stable recognition re-sults at 105Hz 15Hz 108Hz 123Hz 203Hz and 223HzWhen the excitation is applied along the tangential directionalong the top pipe from the stability diagram of longitudinaldirectional response four modes at 105Hz 123Hz 203Hzand 223Hz can be clearly identied
By comparing with the results of the ambient test be-sides the rst mode of 105Hz there are much more modesat 53Hz 123Hz 203Hz and 223Hz that can be found byhammering excitation Among them the stability de-termination of the mode at 53Hz means that the modesaround 48Hz that are found from the ambient test in
Figure 6 are false modes and it is generated by the trainvibration of the nearby subway
4 Field Test and Mode FrequencyIdentification of a Subway Tunnel
For analysis of the tunnel mode characteristics with dierentsizes a mode hammer test is also applied in the nearbysubway tunnel As shown in Figure 9(b) the internal di-ameter of the tunnel is 55m and the outer diameter is 62me structure of each ring is assembled from six segmentsand the thickness of the segment is 35 cm ere is onebidirectional (1-3) and three radial acceleration sensors (1-11-2 and 2-1) placed on the tunnel structure to collect theresponse from dierent directions as shown in Figure 9 ehammer pulsersquos frequency spectrum is shown in Figure 10
T
R
L
Tangential
Longitudinal
Radial
B-1 B-2
W-1
TR
W-1
RT
B-1(2)
L
L
RLT
TLR
RLT
3000
250
250
2500 1
1
1350
1600
1ndash1
Figure 2 Arrangement map for the site measurement at the Xizang Road power tunnel
Figure 3 e time history of the acceleration for the measurement points B-1 (a) at nighttime (b) at daytime
Shock and Vibration 5
(e SSI-NExT method is employed to the accelerationresponse collected in the tunnel and the stability diagramsare shown in Figure 11 From Figure 11(a) two modes canbe found from the radial response at 1425Hz and 233HzAnd relatively stable recognition results can be obtained at
235 Hz and 1285Hz in the longitudinal response(Figure 11(b)) and 119Hz and 1285Hz in the tangentialdirection (Figure 11(c)) In addition it is also observed thatthere are two peaks at 1075Hz and 176Hz which appear atthe spectrum function from Figure 11 and 1075Hz of
100 101 102ndash150
ndash100
ndash50
0
50
100
150
200
Frequency (Hz)
Num
ber o
f row
sSSIndashCOV Csvd = 0815
(a)
10 11 12 13 14 15 16
40
60
80
100
120
140
160
180
200
Frequency (Hz)
Num
ber o
f row
s
Csvd = 0815
(b)
Figure 4 (e stability diagram of the data collected at night by using the traditional SSI COV method
1 10 100
40
0
80
120
160
200
Num
ber o
f row
s
Frequency (Hz)
Csvd = 0815Csvd = 0915
Figure 5 (e stability diagram of the data collected at night by using the NExT-SSI-COV method
SSI-COV Csvd = 0915200
Num
ber o
f row
s
160
120
80
40
0 25 50 75 100 125 150Frequency (Hz)
(a)
SSI-COV Csvd = 0915200
Num
ber o
f row
s
Frequency (Hz)
160
120
80
40
0 25 50 75 100 125 150
(b)
Figure 6 (e comparison of the stability diagram of the data collected at daytime (a) NExT-SSI-COV method (b) traditional SSI-COVmethod
6 Shock and Vibration
which is close to one of the power tunnelrsquos natural fre-quency at 105Hz
From the tests and analysis introduced at above it canbe seen that the modes of the tunnel can be stably identiedfrom the hammer test by using the NExT-SSI-COVmethod
5 Numerical Verification and Analysis
As the SSI method may generate false mode the dispersionanalysis is thus carried out in this section to explore thevibration modes distribution both of the tunnel mentionedabove based on the pipe-in-pipe (PiP) model [31] By
40
1
60
80
100
120
140
160
180
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(a)
40
1
80
120
160
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(b)
01
50
100
150
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(c)
Figure 8 e stability diagrams of the response by applying impulse along dierent directions at the power tunnel (a) radical (b)longitudinal (c) tangential directions
30
25a
(ms
2 )
000 002 004Time (s)
006 008
20
15
10
05
00
ndash05
(a)
20
PSD
(1endash6
)
f (Hz)
15
1 10 100
10
5
0
(b)
Site photo
(c)
Figure 7 e hammer mode test at the power tunnel
Shock and Vibration 7
03
02
01
00
ndash01
Forc
e (kN
)
01 02 03Time (s)
(a)
15
10
05
00
Forc
e (N
Hz)
f (Hz)0 100 200 300 400
(b)
Figure 10 e hammer impulse load applied in the subway tunnel (a) time spectrum (b) frequency spectrum
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
R direction
1425Hz
1075Hz
233Hz
(a)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
L direction
1285Hz235Hz
137Hz
176Hz
(b)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
T direction
1285Hz
11Hz
119Hz
(c)
Figure 11 e stability diagrams of the response by applying impulse along dierent directions at the subway tunnel (a) radical(b) longitudinal (c) tangential directions
6m (5 rings)
T
L
Tangential
Longitudinal
R Radial
1-1 Measure point number
1-1
1-2
1-3
2-1
TL
R
R
R RT
L
RR
T
L
12m
(a) (b) (c) (d)
Figure 9 Arrangement map for the site measurement at the subway tunnel
8 Shock and Vibration
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
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Hindawi
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Advances in
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Submit your manuscripts atwwwhindawicom
identification and extraction of vibration characteristicsbecome very important For applications in bridges andhigh-rise buildings the vibration modes and relatedmethods are more commonly used eg the natural fre-quency and frequency response function (FRF) spectrum-based methods [7] the mode shape-based methods [8] themodel and related matrix-based methods [9] and the datadriven methods [10] Some signal processing techniqueshave also been proposed for vibration characteristics ex-traction [11]
It is well known that the soil-structure interaction willchange the wave propagation characteristics [12 13] sincethe underground structures are strongly coupled with thesurrounding soil and the constraining and damping effectsand the dynamic behaviors of surrounding soil will causemajor challenges for the identification of vibration char-acteristics [14] In previous studies the difference betweenfrequency and dispersion characteristics of the subwaytunnel with or without surrounding soil is studied by nu-merical calculation [15ndash17] It is found that the mode fre-quency is more difficult to identify from the frequencyspectrum under the condition of soil coupling Healthmonitoring of underground pipelines is mostly based onwave propagation characteristics and related methods[18ndash20] Leinov et al [21] Vogt et al [22] and Eybpooshet al [23] found that the propagation modes and theirdispersion characteristics of the propagating ultrasonic-guided waves changed from the pipeline to the surround-ing medium Meanwhile a series of signal processing andmodal parameter identification methods have been pro-posed such as the peak-picking method the random dec-rement technique (RAMA) [24] the natural excitationtechnique (NExT) [25] and the stochastic subspace iden-tification (SSI) method [26] (e NExT is a method of modetesting that allows structures to be tested in their ambientenvironments where the auto- or cross-correlation functionis used for mode identification (e SSI method is originallypresented by van Overschee and de Moor [27] and anextension of the original SSI method that does not requireoutput covariance was proposed by Peeters and de Roeck[28 29] as the reference-based SSI (e stochastic realizationalgorithm mainly focused on the data-driven method (SSI-DATA) that considers the problem of identifying a sto-chastic state-space model from output-only data As op-posed to SSI-DATA the covariance-driven stochasticsubspace identification (SSI-COV) algorithm is also de-veloped to avoid the computation of orthogonal projectionFurthermore to reduce the effect of noise on the results ofidentification some filtering techniques need to be used toenhance the early emergence of a stable diagram for theidentifiable modes For instance SSI is combined withmultivariate singular spectrum analysis (MSSA) for vibra-tion monitoring of the rotating turbine system [30] (esemethods are widely used in bridges and high-rise buildingsbut their applicability in the underground engineering re-mains to be explored
An equally important consideration in developing themonitoring system is minimizing the amount of data pro-cessing and improving the stability and robust of the
monitoring network Automatic processing technology isalso used to reduce the amount of data transmission fromthe terminal nodes to the server(erefore the following keyproblems need to be considered in the application of thevibration-based monitoring method to tunnels and otherunderground structures
(1) Understanding of the tunnel-soil coupling effect onthe tunnel dynamic characteristics and selection ofthe appropriate vibration characteristics for de-termining the current structural state
(2) Proper signal processing and vibration characteris-tics identification methods with high stability andaccuracy
(3) Development of the self-processing and automaticidentification capability of terminal nodes to meetthe low energy cost and robust needs of the wirelessnetwork transmission
In this paper to explore the vibration characteristics ofthe tunnel and its identification method a series of field testsare carried out at the subway and power tunnels with dif-ferent diameters of 64m and 32m(e SSI-COVmethod isused and the NExT is also combined as the filtering tech-niques to identify tunnel mode natural frequency and somecomparisons on the difference between using SSI-COV andNExT-SSI-COV method on the result of identification arecarried out through the stability diagram At last a theo-retical pipe-in-pipe (PiP) model [31] is employed to studythe distribution of the vibration modes and some scientificsuggestions for the tunnel mode parameters identificationare provided at last
2 The NExT-SSI-COV Method
(e SSI-COV method is a well-known multivariate iden-tification technique However the uncertainty of parameterselection such as the Toeplitz matrix row number [32] andthe long input data will result in the sharp drop of thecomputational efficiency and stability In this study theNExTmethod is employed and combined with the SSI-COVmethod to reduce the singularity of covariance matrix andimprove the calculation efficiency and stability (e detailsand process of this method are shown in Figure 1
Step 1 Form the input matrix N by the NExT method
(e autocorrelation function amm of every measure-ment points and cross-correlation function amn for anytwo measurement points were calculated by the NExTmethod [33] and the formedmatrix is denoted asN isin Rltimesnwhere l is the number of autocorrelation and cross-correlation functions and n is the sampling number ofthe NExTmethod (e n value is always less than 2048 foridentifying the low-frequency modes (erefore comparedwith the conventional SSI-COV method the size andsingularity of the input matrix will be greatly reduced andthe computational efficiency and stability can be obviouslyimproved
2 Shock and Vibration
Step 2 Form the output Hankel matrix Ys from inputmatrix N
Firstly as Equation (1) shows a series of block matrix yiare obtained with a time lag i from the input matrix N andevery block matrix yi has l lines same as YN And the dataHankel data matrix is established by the future measure-ments matrix Yf and past measurements Yp as Equation (1)shows
Step 3 Obtain covariance-driven Hankel matrix P by theconventional matrix and then form the block Toeplitz matrixby a multiplication between future measurements matrix Yf
and transpose of past measurements Yp based on Equation(2)
P
Ri Riminus1 middot middot middot R1
Ri+1 Ri middot middot middot R2
⋮ ⋮ ⋱ ⋮
R2iminus1 R2iminus2 middot middot middot Ri
C
CA
middot middot middot
CAiminus1
Aiminus1G middot middot middot AG G( ) Yf Yp( )T
Ri E yk+i yTk[ ] lim
j⟶infin
1jsumjminus1
k0yk+iy
Tk
(2)
Step 4 Perform singular value decomposition (SVD) on theHankel covariance matrix P and obtain eigenmatrix S[28 29] en separate the matrix S into two submatrices S1and S2 by reducing the eigenvalues e smallest singularvalues in the matrix S are grouped as S2 and will be neglectedin the following steps by parameters Csvd e Csvd can becalculated by the following equations
P USVT U1 U2( )S1 0
0 S2
VT1
VT2
U1S1VT1
(3)
Csvd sumVSnsumVS
(4)
wheresumVSn is the cumulative value of the rst n eigenvaluesfor the S matrix and sumVS is the cumulative value of alleigenvalues
Steps 5sim7 Based on the conventional SSI-COV method theextended observability matrix ΓL is rstly calculatedaccording to Equation (5) and then the system parametermatrices A and C and mode frequency ωn and dampingcoecient ξn are obtained separately by Equations (6) and(7) [34 35]
ΓL S121 VT1 (5)
A ΓLprimeΓLC the first L rows of ΓL
(6)
where ΓL
and ΓL denote ΓL without the last L rows and therst L rows respectively
Calculate the correlation function matrix N by NExT methodN = [a11 aij] isin Rntimesl aij is the correlation function ofevery two measurement points
(i)
Calculate the covariance matrix P(P = YpYfT)(iii)
Calculate SVD of P and determine the order n by neglectingthe smaller singular values in S2
(iv)
00S1 V1
TP = USVT = (U1 U2) V2
TS2
(ii) Form the output Hankel matrix Ys from matrix N
Ys = isin R2nLtimesk k = l ndash L + 1 is the user-defined indexYpYf
Calculate the extended observability matrix ΓL = U1 S1frac12(v)
Calculate the system parameter matrices A and CA = ΓprimeLΓL ΓL denotes ΓL without the last l rows while ΓL withoutthe first l rows C = the first l rows of ΓL
(vi)
Determine the model frequency ωn and damping coefficient ξn(vii)
3 Field Test and Mode FrequencyIdentification of a Power Tunnel
31Measurement Site andArrangement As Figure 2 showsthe measured power tunnel is a pipe jacking tunnel whichconsists of concrete pipes with a length of 25m (e outerdiameter is 32m and inner diameter is 27m Nine ac-celerometers (LANCE 130 series) are installed at the walland the bottom of the tunnel (ree of them are equipped ateach measured point to collect the response from theradical longitudinal and tangential directions and theletter R means the sensor is installed in the radial directionL means the longitudinal direction while T is the tangentialdirection
32 Ambient Vibration Test To compare the responsecharacteristics of the tunnel in daytime and nighttime theambient vibration test is continued for seven days Accel-eration response records of 30 minutes in the nighttime anddaytime respectively at the measurement point B-1 areshown in Figure 3 For the measured power tunnel close to asubway the vibration from the operational trains makes thevibration amplitude at daytime about 4 times than that atnight Based on the above data the tunnel mode frequencycharacteristics are studied and a comparison on the dif-ference between using SSI-COV and NExT-SSI-COV on theresult of identification is carried out through the con-struction of a stability diagram (e stability diagram hasbeen proved very helpful to identify the dominant fre-quencies of the vibration mode while using the SSI-COVmethod [32]
321 2e Tunnel Response Characteristics at Night Toanalyze the tunnel response characteristics at night with the30-minute records collected from nine accelerometers theinput matrix dimension will be 3600000 times 9 using thetraditional SSI method (e stability diagram can be ob-tained and is shown in Figure 4 after running for three hourson a microcomputer with an Intel i5 processor While theNExTmethod is employed based on the procedures of Step1 and Step 2 introduced in Section 2 the auto- and cross-correlation functions of every degree of freedom were cal-culated with a sampling window of 2048 points and the inputmatrix N is formed with the matrix dimension of 2048 times 28
(e operation time of the stability diagram on the samemicrocomputer then can be reduced to less than 2 minutes(e calculation results are shown in Figure 5
By comparing the stability diagrams of Figures 4 and 5 itwas observed that several modes between 10 and 15Hz canbe found in Figure 4 by using the SSI-COV method directlybut the data fluctuation along the number of rows makes itdifficult to identify the natural frequency automatically andaccurately It thus will limit its application of SHM at theunderground structure However as the NExT method iscombined there is only one mode left with the frequency of105Hz In addition in the implementation of SSI-COV theselecting Csvd in Step 4 at Section 2 can lead to a change inthe number of the vibration modes and their identificationaccuracy From Figure 5 it is observed that there is almostno change on the identified model frequencies as the Csvdvalue changes from 0815 to 0915 by using the NExT-SSI-COV method (erefore the NExTmethod can develop themode frequency identification stability and accurately andmake it much suitable for application in the undergroundstructure
322 2e Tunnel Response Characteristics at Daytime(e tunnel response characteristics are analyzed during thedaytime especially for considering those vibration modesexcited by the train-induced vibration from the nearbysubway By employing the method both of traditional SSI-COV method and NExT-SSI-COV method a 30-minuterecord is used and the stability diagrams are shown inFigure 6 By comparing Figures 6(a) and 6(b) the modefrequency identification accuracy and stability are im-proved by using the NExT-SSI-COV method Meanwhileit also can be observed that besides 105Hz there are muchmore recognition results that can be found from Figure 6and several modes of which are around 48Hz of which isclose to the subway track-rail resonant frequency
From the ambient test at the power tunnel both atdaytime and nighttime and by comparing the stability di-agrams of using the traditional SSI method and modifiedmethod it is found that the NExT-SSI-COV method hasbetter mode natural frequency recognition accuracy andstability and several mode frequencies can be clearlyidentified around 105Hz 48Hz and 71Hz from the am-bient test Especially for the frequency of 105 stability canbe recognized both at daytime and nighttime
33 Hammering Test (e mode hammering test is acommonly used method to study the mode characteristicsand has a wider test frequency band than the ambient test(us it was also carried out at the power tunnel along theradial longitudinal and tangential directions (e impulseload is applied at the connecting steel ring of the tunnelsegment (Figure 7(a)) the time history of the pulse load isshown in Figure 7(b) and the frequency test band is around300Hz as shown in Figure 7(c)
(e NExT-SSI-COV method is also used here to analyzethe acceleration response and the stability diagrams areshown in Figure 8 From Figure 8(a) the modes at 53Hz
4 Shock and Vibration
203Hz and 223Hz can be identied from the radical re-sponses as hammered along the radial direction of thetunnel And much more modes can be found from thelongitudinal direction response as hammered along thelongitudinal direction and there are stable recognition re-sults at 105Hz 15Hz 108Hz 123Hz 203Hz and 223HzWhen the excitation is applied along the tangential directionalong the top pipe from the stability diagram of longitudinaldirectional response four modes at 105Hz 123Hz 203Hzand 223Hz can be clearly identied
By comparing with the results of the ambient test be-sides the rst mode of 105Hz there are much more modesat 53Hz 123Hz 203Hz and 223Hz that can be found byhammering excitation Among them the stability de-termination of the mode at 53Hz means that the modesaround 48Hz that are found from the ambient test in
Figure 6 are false modes and it is generated by the trainvibration of the nearby subway
4 Field Test and Mode FrequencyIdentification of a Subway Tunnel
For analysis of the tunnel mode characteristics with dierentsizes a mode hammer test is also applied in the nearbysubway tunnel As shown in Figure 9(b) the internal di-ameter of the tunnel is 55m and the outer diameter is 62me structure of each ring is assembled from six segmentsand the thickness of the segment is 35 cm ere is onebidirectional (1-3) and three radial acceleration sensors (1-11-2 and 2-1) placed on the tunnel structure to collect theresponse from dierent directions as shown in Figure 9 ehammer pulsersquos frequency spectrum is shown in Figure 10
T
R
L
Tangential
Longitudinal
Radial
B-1 B-2
W-1
TR
W-1
RT
B-1(2)
L
L
RLT
TLR
RLT
3000
250
250
2500 1
1
1350
1600
1ndash1
Figure 2 Arrangement map for the site measurement at the Xizang Road power tunnel
Figure 3 e time history of the acceleration for the measurement points B-1 (a) at nighttime (b) at daytime
Shock and Vibration 5
(e SSI-NExT method is employed to the accelerationresponse collected in the tunnel and the stability diagramsare shown in Figure 11 From Figure 11(a) two modes canbe found from the radial response at 1425Hz and 233HzAnd relatively stable recognition results can be obtained at
235 Hz and 1285Hz in the longitudinal response(Figure 11(b)) and 119Hz and 1285Hz in the tangentialdirection (Figure 11(c)) In addition it is also observed thatthere are two peaks at 1075Hz and 176Hz which appear atthe spectrum function from Figure 11 and 1075Hz of
100 101 102ndash150
ndash100
ndash50
0
50
100
150
200
Frequency (Hz)
Num
ber o
f row
sSSIndashCOV Csvd = 0815
(a)
10 11 12 13 14 15 16
40
60
80
100
120
140
160
180
200
Frequency (Hz)
Num
ber o
f row
s
Csvd = 0815
(b)
Figure 4 (e stability diagram of the data collected at night by using the traditional SSI COV method
1 10 100
40
0
80
120
160
200
Num
ber o
f row
s
Frequency (Hz)
Csvd = 0815Csvd = 0915
Figure 5 (e stability diagram of the data collected at night by using the NExT-SSI-COV method
SSI-COV Csvd = 0915200
Num
ber o
f row
s
160
120
80
40
0 25 50 75 100 125 150Frequency (Hz)
(a)
SSI-COV Csvd = 0915200
Num
ber o
f row
s
Frequency (Hz)
160
120
80
40
0 25 50 75 100 125 150
(b)
Figure 6 (e comparison of the stability diagram of the data collected at daytime (a) NExT-SSI-COV method (b) traditional SSI-COVmethod
6 Shock and Vibration
which is close to one of the power tunnelrsquos natural fre-quency at 105Hz
From the tests and analysis introduced at above it canbe seen that the modes of the tunnel can be stably identiedfrom the hammer test by using the NExT-SSI-COVmethod
5 Numerical Verification and Analysis
As the SSI method may generate false mode the dispersionanalysis is thus carried out in this section to explore thevibration modes distribution both of the tunnel mentionedabove based on the pipe-in-pipe (PiP) model [31] By
40
1
60
80
100
120
140
160
180
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(a)
40
1
80
120
160
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(b)
01
50
100
150
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(c)
Figure 8 e stability diagrams of the response by applying impulse along dierent directions at the power tunnel (a) radical (b)longitudinal (c) tangential directions
30
25a
(ms
2 )
000 002 004Time (s)
006 008
20
15
10
05
00
ndash05
(a)
20
PSD
(1endash6
)
f (Hz)
15
1 10 100
10
5
0
(b)
Site photo
(c)
Figure 7 e hammer mode test at the power tunnel
Shock and Vibration 7
03
02
01
00
ndash01
Forc
e (kN
)
01 02 03Time (s)
(a)
15
10
05
00
Forc
e (N
Hz)
f (Hz)0 100 200 300 400
(b)
Figure 10 e hammer impulse load applied in the subway tunnel (a) time spectrum (b) frequency spectrum
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
R direction
1425Hz
1075Hz
233Hz
(a)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
L direction
1285Hz235Hz
137Hz
176Hz
(b)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
T direction
1285Hz
11Hz
119Hz
(c)
Figure 11 e stability diagrams of the response by applying impulse along dierent directions at the subway tunnel (a) radical(b) longitudinal (c) tangential directions
6m (5 rings)
T
L
Tangential
Longitudinal
R Radial
1-1 Measure point number
1-1
1-2
1-3
2-1
TL
R
R
R RT
L
RR
T
L
12m
(a) (b) (c) (d)
Figure 9 Arrangement map for the site measurement at the subway tunnel
8 Shock and Vibration
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
Step 2 Form the output Hankel matrix Ys from inputmatrix N
Firstly as Equation (1) shows a series of block matrix yiare obtained with a time lag i from the input matrix N andevery block matrix yi has l lines same as YN And the dataHankel data matrix is established by the future measure-ments matrix Yf and past measurements Yp as Equation (1)shows
Step 3 Obtain covariance-driven Hankel matrix P by theconventional matrix and then form the block Toeplitz matrixby a multiplication between future measurements matrix Yf
and transpose of past measurements Yp based on Equation(2)
P
Ri Riminus1 middot middot middot R1
Ri+1 Ri middot middot middot R2
⋮ ⋮ ⋱ ⋮
R2iminus1 R2iminus2 middot middot middot Ri
C
CA
middot middot middot
CAiminus1
Aiminus1G middot middot middot AG G( ) Yf Yp( )T
Ri E yk+i yTk[ ] lim
j⟶infin
1jsumjminus1
k0yk+iy
Tk
(2)
Step 4 Perform singular value decomposition (SVD) on theHankel covariance matrix P and obtain eigenmatrix S[28 29] en separate the matrix S into two submatrices S1and S2 by reducing the eigenvalues e smallest singularvalues in the matrix S are grouped as S2 and will be neglectedin the following steps by parameters Csvd e Csvd can becalculated by the following equations
P USVT U1 U2( )S1 0
0 S2
VT1
VT2
U1S1VT1
(3)
Csvd sumVSnsumVS
(4)
wheresumVSn is the cumulative value of the rst n eigenvaluesfor the S matrix and sumVS is the cumulative value of alleigenvalues
Steps 5sim7 Based on the conventional SSI-COV method theextended observability matrix ΓL is rstly calculatedaccording to Equation (5) and then the system parametermatrices A and C and mode frequency ωn and dampingcoecient ξn are obtained separately by Equations (6) and(7) [34 35]
ΓL S121 VT1 (5)
A ΓLprimeΓLC the first L rows of ΓL
(6)
where ΓL
and ΓL denote ΓL without the last L rows and therst L rows respectively
Calculate the correlation function matrix N by NExT methodN = [a11 aij] isin Rntimesl aij is the correlation function ofevery two measurement points
(i)
Calculate the covariance matrix P(P = YpYfT)(iii)
Calculate SVD of P and determine the order n by neglectingthe smaller singular values in S2
(iv)
00S1 V1
TP = USVT = (U1 U2) V2
TS2
(ii) Form the output Hankel matrix Ys from matrix N
Ys = isin R2nLtimesk k = l ndash L + 1 is the user-defined indexYpYf
Calculate the extended observability matrix ΓL = U1 S1frac12(v)
Calculate the system parameter matrices A and CA = ΓprimeLΓL ΓL denotes ΓL without the last l rows while ΓL withoutthe first l rows C = the first l rows of ΓL
(vi)
Determine the model frequency ωn and damping coefficient ξn(vii)
3 Field Test and Mode FrequencyIdentification of a Power Tunnel
31Measurement Site andArrangement As Figure 2 showsthe measured power tunnel is a pipe jacking tunnel whichconsists of concrete pipes with a length of 25m (e outerdiameter is 32m and inner diameter is 27m Nine ac-celerometers (LANCE 130 series) are installed at the walland the bottom of the tunnel (ree of them are equipped ateach measured point to collect the response from theradical longitudinal and tangential directions and theletter R means the sensor is installed in the radial directionL means the longitudinal direction while T is the tangentialdirection
32 Ambient Vibration Test To compare the responsecharacteristics of the tunnel in daytime and nighttime theambient vibration test is continued for seven days Accel-eration response records of 30 minutes in the nighttime anddaytime respectively at the measurement point B-1 areshown in Figure 3 For the measured power tunnel close to asubway the vibration from the operational trains makes thevibration amplitude at daytime about 4 times than that atnight Based on the above data the tunnel mode frequencycharacteristics are studied and a comparison on the dif-ference between using SSI-COV and NExT-SSI-COV on theresult of identification is carried out through the con-struction of a stability diagram (e stability diagram hasbeen proved very helpful to identify the dominant fre-quencies of the vibration mode while using the SSI-COVmethod [32]
321 2e Tunnel Response Characteristics at Night Toanalyze the tunnel response characteristics at night with the30-minute records collected from nine accelerometers theinput matrix dimension will be 3600000 times 9 using thetraditional SSI method (e stability diagram can be ob-tained and is shown in Figure 4 after running for three hourson a microcomputer with an Intel i5 processor While theNExTmethod is employed based on the procedures of Step1 and Step 2 introduced in Section 2 the auto- and cross-correlation functions of every degree of freedom were cal-culated with a sampling window of 2048 points and the inputmatrix N is formed with the matrix dimension of 2048 times 28
(e operation time of the stability diagram on the samemicrocomputer then can be reduced to less than 2 minutes(e calculation results are shown in Figure 5
By comparing the stability diagrams of Figures 4 and 5 itwas observed that several modes between 10 and 15Hz canbe found in Figure 4 by using the SSI-COV method directlybut the data fluctuation along the number of rows makes itdifficult to identify the natural frequency automatically andaccurately It thus will limit its application of SHM at theunderground structure However as the NExT method iscombined there is only one mode left with the frequency of105Hz In addition in the implementation of SSI-COV theselecting Csvd in Step 4 at Section 2 can lead to a change inthe number of the vibration modes and their identificationaccuracy From Figure 5 it is observed that there is almostno change on the identified model frequencies as the Csvdvalue changes from 0815 to 0915 by using the NExT-SSI-COV method (erefore the NExTmethod can develop themode frequency identification stability and accurately andmake it much suitable for application in the undergroundstructure
322 2e Tunnel Response Characteristics at Daytime(e tunnel response characteristics are analyzed during thedaytime especially for considering those vibration modesexcited by the train-induced vibration from the nearbysubway By employing the method both of traditional SSI-COV method and NExT-SSI-COV method a 30-minuterecord is used and the stability diagrams are shown inFigure 6 By comparing Figures 6(a) and 6(b) the modefrequency identification accuracy and stability are im-proved by using the NExT-SSI-COV method Meanwhileit also can be observed that besides 105Hz there are muchmore recognition results that can be found from Figure 6and several modes of which are around 48Hz of which isclose to the subway track-rail resonant frequency
From the ambient test at the power tunnel both atdaytime and nighttime and by comparing the stability di-agrams of using the traditional SSI method and modifiedmethod it is found that the NExT-SSI-COV method hasbetter mode natural frequency recognition accuracy andstability and several mode frequencies can be clearlyidentified around 105Hz 48Hz and 71Hz from the am-bient test Especially for the frequency of 105 stability canbe recognized both at daytime and nighttime
33 Hammering Test (e mode hammering test is acommonly used method to study the mode characteristicsand has a wider test frequency band than the ambient test(us it was also carried out at the power tunnel along theradial longitudinal and tangential directions (e impulseload is applied at the connecting steel ring of the tunnelsegment (Figure 7(a)) the time history of the pulse load isshown in Figure 7(b) and the frequency test band is around300Hz as shown in Figure 7(c)
(e NExT-SSI-COV method is also used here to analyzethe acceleration response and the stability diagrams areshown in Figure 8 From Figure 8(a) the modes at 53Hz
4 Shock and Vibration
203Hz and 223Hz can be identied from the radical re-sponses as hammered along the radial direction of thetunnel And much more modes can be found from thelongitudinal direction response as hammered along thelongitudinal direction and there are stable recognition re-sults at 105Hz 15Hz 108Hz 123Hz 203Hz and 223HzWhen the excitation is applied along the tangential directionalong the top pipe from the stability diagram of longitudinaldirectional response four modes at 105Hz 123Hz 203Hzand 223Hz can be clearly identied
By comparing with the results of the ambient test be-sides the rst mode of 105Hz there are much more modesat 53Hz 123Hz 203Hz and 223Hz that can be found byhammering excitation Among them the stability de-termination of the mode at 53Hz means that the modesaround 48Hz that are found from the ambient test in
Figure 6 are false modes and it is generated by the trainvibration of the nearby subway
4 Field Test and Mode FrequencyIdentification of a Subway Tunnel
For analysis of the tunnel mode characteristics with dierentsizes a mode hammer test is also applied in the nearbysubway tunnel As shown in Figure 9(b) the internal di-ameter of the tunnel is 55m and the outer diameter is 62me structure of each ring is assembled from six segmentsand the thickness of the segment is 35 cm ere is onebidirectional (1-3) and three radial acceleration sensors (1-11-2 and 2-1) placed on the tunnel structure to collect theresponse from dierent directions as shown in Figure 9 ehammer pulsersquos frequency spectrum is shown in Figure 10
T
R
L
Tangential
Longitudinal
Radial
B-1 B-2
W-1
TR
W-1
RT
B-1(2)
L
L
RLT
TLR
RLT
3000
250
250
2500 1
1
1350
1600
1ndash1
Figure 2 Arrangement map for the site measurement at the Xizang Road power tunnel
Figure 3 e time history of the acceleration for the measurement points B-1 (a) at nighttime (b) at daytime
Shock and Vibration 5
(e SSI-NExT method is employed to the accelerationresponse collected in the tunnel and the stability diagramsare shown in Figure 11 From Figure 11(a) two modes canbe found from the radial response at 1425Hz and 233HzAnd relatively stable recognition results can be obtained at
235 Hz and 1285Hz in the longitudinal response(Figure 11(b)) and 119Hz and 1285Hz in the tangentialdirection (Figure 11(c)) In addition it is also observed thatthere are two peaks at 1075Hz and 176Hz which appear atthe spectrum function from Figure 11 and 1075Hz of
100 101 102ndash150
ndash100
ndash50
0
50
100
150
200
Frequency (Hz)
Num
ber o
f row
sSSIndashCOV Csvd = 0815
(a)
10 11 12 13 14 15 16
40
60
80
100
120
140
160
180
200
Frequency (Hz)
Num
ber o
f row
s
Csvd = 0815
(b)
Figure 4 (e stability diagram of the data collected at night by using the traditional SSI COV method
1 10 100
40
0
80
120
160
200
Num
ber o
f row
s
Frequency (Hz)
Csvd = 0815Csvd = 0915
Figure 5 (e stability diagram of the data collected at night by using the NExT-SSI-COV method
SSI-COV Csvd = 0915200
Num
ber o
f row
s
160
120
80
40
0 25 50 75 100 125 150Frequency (Hz)
(a)
SSI-COV Csvd = 0915200
Num
ber o
f row
s
Frequency (Hz)
160
120
80
40
0 25 50 75 100 125 150
(b)
Figure 6 (e comparison of the stability diagram of the data collected at daytime (a) NExT-SSI-COV method (b) traditional SSI-COVmethod
6 Shock and Vibration
which is close to one of the power tunnelrsquos natural fre-quency at 105Hz
From the tests and analysis introduced at above it canbe seen that the modes of the tunnel can be stably identiedfrom the hammer test by using the NExT-SSI-COVmethod
5 Numerical Verification and Analysis
As the SSI method may generate false mode the dispersionanalysis is thus carried out in this section to explore thevibration modes distribution both of the tunnel mentionedabove based on the pipe-in-pipe (PiP) model [31] By
40
1
60
80
100
120
140
160
180
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(a)
40
1
80
120
160
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(b)
01
50
100
150
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(c)
Figure 8 e stability diagrams of the response by applying impulse along dierent directions at the power tunnel (a) radical (b)longitudinal (c) tangential directions
30
25a
(ms
2 )
000 002 004Time (s)
006 008
20
15
10
05
00
ndash05
(a)
20
PSD
(1endash6
)
f (Hz)
15
1 10 100
10
5
0
(b)
Site photo
(c)
Figure 7 e hammer mode test at the power tunnel
Shock and Vibration 7
03
02
01
00
ndash01
Forc
e (kN
)
01 02 03Time (s)
(a)
15
10
05
00
Forc
e (N
Hz)
f (Hz)0 100 200 300 400
(b)
Figure 10 e hammer impulse load applied in the subway tunnel (a) time spectrum (b) frequency spectrum
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
R direction
1425Hz
1075Hz
233Hz
(a)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
L direction
1285Hz235Hz
137Hz
176Hz
(b)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
T direction
1285Hz
11Hz
119Hz
(c)
Figure 11 e stability diagrams of the response by applying impulse along dierent directions at the subway tunnel (a) radical(b) longitudinal (c) tangential directions
6m (5 rings)
T
L
Tangential
Longitudinal
R Radial
1-1 Measure point number
1-1
1-2
1-3
2-1
TL
R
R
R RT
L
RR
T
L
12m
(a) (b) (c) (d)
Figure 9 Arrangement map for the site measurement at the subway tunnel
8 Shock and Vibration
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
3 Field Test and Mode FrequencyIdentification of a Power Tunnel
31Measurement Site andArrangement As Figure 2 showsthe measured power tunnel is a pipe jacking tunnel whichconsists of concrete pipes with a length of 25m (e outerdiameter is 32m and inner diameter is 27m Nine ac-celerometers (LANCE 130 series) are installed at the walland the bottom of the tunnel (ree of them are equipped ateach measured point to collect the response from theradical longitudinal and tangential directions and theletter R means the sensor is installed in the radial directionL means the longitudinal direction while T is the tangentialdirection
32 Ambient Vibration Test To compare the responsecharacteristics of the tunnel in daytime and nighttime theambient vibration test is continued for seven days Accel-eration response records of 30 minutes in the nighttime anddaytime respectively at the measurement point B-1 areshown in Figure 3 For the measured power tunnel close to asubway the vibration from the operational trains makes thevibration amplitude at daytime about 4 times than that atnight Based on the above data the tunnel mode frequencycharacteristics are studied and a comparison on the dif-ference between using SSI-COV and NExT-SSI-COV on theresult of identification is carried out through the con-struction of a stability diagram (e stability diagram hasbeen proved very helpful to identify the dominant fre-quencies of the vibration mode while using the SSI-COVmethod [32]
321 2e Tunnel Response Characteristics at Night Toanalyze the tunnel response characteristics at night with the30-minute records collected from nine accelerometers theinput matrix dimension will be 3600000 times 9 using thetraditional SSI method (e stability diagram can be ob-tained and is shown in Figure 4 after running for three hourson a microcomputer with an Intel i5 processor While theNExTmethod is employed based on the procedures of Step1 and Step 2 introduced in Section 2 the auto- and cross-correlation functions of every degree of freedom were cal-culated with a sampling window of 2048 points and the inputmatrix N is formed with the matrix dimension of 2048 times 28
(e operation time of the stability diagram on the samemicrocomputer then can be reduced to less than 2 minutes(e calculation results are shown in Figure 5
By comparing the stability diagrams of Figures 4 and 5 itwas observed that several modes between 10 and 15Hz canbe found in Figure 4 by using the SSI-COV method directlybut the data fluctuation along the number of rows makes itdifficult to identify the natural frequency automatically andaccurately It thus will limit its application of SHM at theunderground structure However as the NExT method iscombined there is only one mode left with the frequency of105Hz In addition in the implementation of SSI-COV theselecting Csvd in Step 4 at Section 2 can lead to a change inthe number of the vibration modes and their identificationaccuracy From Figure 5 it is observed that there is almostno change on the identified model frequencies as the Csvdvalue changes from 0815 to 0915 by using the NExT-SSI-COV method (erefore the NExTmethod can develop themode frequency identification stability and accurately andmake it much suitable for application in the undergroundstructure
322 2e Tunnel Response Characteristics at Daytime(e tunnel response characteristics are analyzed during thedaytime especially for considering those vibration modesexcited by the train-induced vibration from the nearbysubway By employing the method both of traditional SSI-COV method and NExT-SSI-COV method a 30-minuterecord is used and the stability diagrams are shown inFigure 6 By comparing Figures 6(a) and 6(b) the modefrequency identification accuracy and stability are im-proved by using the NExT-SSI-COV method Meanwhileit also can be observed that besides 105Hz there are muchmore recognition results that can be found from Figure 6and several modes of which are around 48Hz of which isclose to the subway track-rail resonant frequency
From the ambient test at the power tunnel both atdaytime and nighttime and by comparing the stability di-agrams of using the traditional SSI method and modifiedmethod it is found that the NExT-SSI-COV method hasbetter mode natural frequency recognition accuracy andstability and several mode frequencies can be clearlyidentified around 105Hz 48Hz and 71Hz from the am-bient test Especially for the frequency of 105 stability canbe recognized both at daytime and nighttime
33 Hammering Test (e mode hammering test is acommonly used method to study the mode characteristicsand has a wider test frequency band than the ambient test(us it was also carried out at the power tunnel along theradial longitudinal and tangential directions (e impulseload is applied at the connecting steel ring of the tunnelsegment (Figure 7(a)) the time history of the pulse load isshown in Figure 7(b) and the frequency test band is around300Hz as shown in Figure 7(c)
(e NExT-SSI-COV method is also used here to analyzethe acceleration response and the stability diagrams areshown in Figure 8 From Figure 8(a) the modes at 53Hz
4 Shock and Vibration
203Hz and 223Hz can be identied from the radical re-sponses as hammered along the radial direction of thetunnel And much more modes can be found from thelongitudinal direction response as hammered along thelongitudinal direction and there are stable recognition re-sults at 105Hz 15Hz 108Hz 123Hz 203Hz and 223HzWhen the excitation is applied along the tangential directionalong the top pipe from the stability diagram of longitudinaldirectional response four modes at 105Hz 123Hz 203Hzand 223Hz can be clearly identied
By comparing with the results of the ambient test be-sides the rst mode of 105Hz there are much more modesat 53Hz 123Hz 203Hz and 223Hz that can be found byhammering excitation Among them the stability de-termination of the mode at 53Hz means that the modesaround 48Hz that are found from the ambient test in
Figure 6 are false modes and it is generated by the trainvibration of the nearby subway
4 Field Test and Mode FrequencyIdentification of a Subway Tunnel
For analysis of the tunnel mode characteristics with dierentsizes a mode hammer test is also applied in the nearbysubway tunnel As shown in Figure 9(b) the internal di-ameter of the tunnel is 55m and the outer diameter is 62me structure of each ring is assembled from six segmentsand the thickness of the segment is 35 cm ere is onebidirectional (1-3) and three radial acceleration sensors (1-11-2 and 2-1) placed on the tunnel structure to collect theresponse from dierent directions as shown in Figure 9 ehammer pulsersquos frequency spectrum is shown in Figure 10
T
R
L
Tangential
Longitudinal
Radial
B-1 B-2
W-1
TR
W-1
RT
B-1(2)
L
L
RLT
TLR
RLT
3000
250
250
2500 1
1
1350
1600
1ndash1
Figure 2 Arrangement map for the site measurement at the Xizang Road power tunnel
Figure 3 e time history of the acceleration for the measurement points B-1 (a) at nighttime (b) at daytime
Shock and Vibration 5
(e SSI-NExT method is employed to the accelerationresponse collected in the tunnel and the stability diagramsare shown in Figure 11 From Figure 11(a) two modes canbe found from the radial response at 1425Hz and 233HzAnd relatively stable recognition results can be obtained at
235 Hz and 1285Hz in the longitudinal response(Figure 11(b)) and 119Hz and 1285Hz in the tangentialdirection (Figure 11(c)) In addition it is also observed thatthere are two peaks at 1075Hz and 176Hz which appear atthe spectrum function from Figure 11 and 1075Hz of
100 101 102ndash150
ndash100
ndash50
0
50
100
150
200
Frequency (Hz)
Num
ber o
f row
sSSIndashCOV Csvd = 0815
(a)
10 11 12 13 14 15 16
40
60
80
100
120
140
160
180
200
Frequency (Hz)
Num
ber o
f row
s
Csvd = 0815
(b)
Figure 4 (e stability diagram of the data collected at night by using the traditional SSI COV method
1 10 100
40
0
80
120
160
200
Num
ber o
f row
s
Frequency (Hz)
Csvd = 0815Csvd = 0915
Figure 5 (e stability diagram of the data collected at night by using the NExT-SSI-COV method
SSI-COV Csvd = 0915200
Num
ber o
f row
s
160
120
80
40
0 25 50 75 100 125 150Frequency (Hz)
(a)
SSI-COV Csvd = 0915200
Num
ber o
f row
s
Frequency (Hz)
160
120
80
40
0 25 50 75 100 125 150
(b)
Figure 6 (e comparison of the stability diagram of the data collected at daytime (a) NExT-SSI-COV method (b) traditional SSI-COVmethod
6 Shock and Vibration
which is close to one of the power tunnelrsquos natural fre-quency at 105Hz
From the tests and analysis introduced at above it canbe seen that the modes of the tunnel can be stably identiedfrom the hammer test by using the NExT-SSI-COVmethod
5 Numerical Verification and Analysis
As the SSI method may generate false mode the dispersionanalysis is thus carried out in this section to explore thevibration modes distribution both of the tunnel mentionedabove based on the pipe-in-pipe (PiP) model [31] By
40
1
60
80
100
120
140
160
180
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(a)
40
1
80
120
160
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(b)
01
50
100
150
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(c)
Figure 8 e stability diagrams of the response by applying impulse along dierent directions at the power tunnel (a) radical (b)longitudinal (c) tangential directions
30
25a
(ms
2 )
000 002 004Time (s)
006 008
20
15
10
05
00
ndash05
(a)
20
PSD
(1endash6
)
f (Hz)
15
1 10 100
10
5
0
(b)
Site photo
(c)
Figure 7 e hammer mode test at the power tunnel
Shock and Vibration 7
03
02
01
00
ndash01
Forc
e (kN
)
01 02 03Time (s)
(a)
15
10
05
00
Forc
e (N
Hz)
f (Hz)0 100 200 300 400
(b)
Figure 10 e hammer impulse load applied in the subway tunnel (a) time spectrum (b) frequency spectrum
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
R direction
1425Hz
1075Hz
233Hz
(a)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
L direction
1285Hz235Hz
137Hz
176Hz
(b)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
T direction
1285Hz
11Hz
119Hz
(c)
Figure 11 e stability diagrams of the response by applying impulse along dierent directions at the subway tunnel (a) radical(b) longitudinal (c) tangential directions
6m (5 rings)
T
L
Tangential
Longitudinal
R Radial
1-1 Measure point number
1-1
1-2
1-3
2-1
TL
R
R
R RT
L
RR
T
L
12m
(a) (b) (c) (d)
Figure 9 Arrangement map for the site measurement at the subway tunnel
8 Shock and Vibration
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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Navigation and Observation
International Journal of
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Advances in
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Submit your manuscripts atwwwhindawicom
203Hz and 223Hz can be identied from the radical re-sponses as hammered along the radial direction of thetunnel And much more modes can be found from thelongitudinal direction response as hammered along thelongitudinal direction and there are stable recognition re-sults at 105Hz 15Hz 108Hz 123Hz 203Hz and 223HzWhen the excitation is applied along the tangential directionalong the top pipe from the stability diagram of longitudinaldirectional response four modes at 105Hz 123Hz 203Hzand 223Hz can be clearly identied
By comparing with the results of the ambient test be-sides the rst mode of 105Hz there are much more modesat 53Hz 123Hz 203Hz and 223Hz that can be found byhammering excitation Among them the stability de-termination of the mode at 53Hz means that the modesaround 48Hz that are found from the ambient test in
Figure 6 are false modes and it is generated by the trainvibration of the nearby subway
4 Field Test and Mode FrequencyIdentification of a Subway Tunnel
For analysis of the tunnel mode characteristics with dierentsizes a mode hammer test is also applied in the nearbysubway tunnel As shown in Figure 9(b) the internal di-ameter of the tunnel is 55m and the outer diameter is 62me structure of each ring is assembled from six segmentsand the thickness of the segment is 35 cm ere is onebidirectional (1-3) and three radial acceleration sensors (1-11-2 and 2-1) placed on the tunnel structure to collect theresponse from dierent directions as shown in Figure 9 ehammer pulsersquos frequency spectrum is shown in Figure 10
T
R
L
Tangential
Longitudinal
Radial
B-1 B-2
W-1
TR
W-1
RT
B-1(2)
L
L
RLT
TLR
RLT
3000
250
250
2500 1
1
1350
1600
1ndash1
Figure 2 Arrangement map for the site measurement at the Xizang Road power tunnel
Figure 3 e time history of the acceleration for the measurement points B-1 (a) at nighttime (b) at daytime
Shock and Vibration 5
(e SSI-NExT method is employed to the accelerationresponse collected in the tunnel and the stability diagramsare shown in Figure 11 From Figure 11(a) two modes canbe found from the radial response at 1425Hz and 233HzAnd relatively stable recognition results can be obtained at
235 Hz and 1285Hz in the longitudinal response(Figure 11(b)) and 119Hz and 1285Hz in the tangentialdirection (Figure 11(c)) In addition it is also observed thatthere are two peaks at 1075Hz and 176Hz which appear atthe spectrum function from Figure 11 and 1075Hz of
100 101 102ndash150
ndash100
ndash50
0
50
100
150
200
Frequency (Hz)
Num
ber o
f row
sSSIndashCOV Csvd = 0815
(a)
10 11 12 13 14 15 16
40
60
80
100
120
140
160
180
200
Frequency (Hz)
Num
ber o
f row
s
Csvd = 0815
(b)
Figure 4 (e stability diagram of the data collected at night by using the traditional SSI COV method
1 10 100
40
0
80
120
160
200
Num
ber o
f row
s
Frequency (Hz)
Csvd = 0815Csvd = 0915
Figure 5 (e stability diagram of the data collected at night by using the NExT-SSI-COV method
SSI-COV Csvd = 0915200
Num
ber o
f row
s
160
120
80
40
0 25 50 75 100 125 150Frequency (Hz)
(a)
SSI-COV Csvd = 0915200
Num
ber o
f row
s
Frequency (Hz)
160
120
80
40
0 25 50 75 100 125 150
(b)
Figure 6 (e comparison of the stability diagram of the data collected at daytime (a) NExT-SSI-COV method (b) traditional SSI-COVmethod
6 Shock and Vibration
which is close to one of the power tunnelrsquos natural fre-quency at 105Hz
From the tests and analysis introduced at above it canbe seen that the modes of the tunnel can be stably identiedfrom the hammer test by using the NExT-SSI-COVmethod
5 Numerical Verification and Analysis
As the SSI method may generate false mode the dispersionanalysis is thus carried out in this section to explore thevibration modes distribution both of the tunnel mentionedabove based on the pipe-in-pipe (PiP) model [31] By
40
1
60
80
100
120
140
160
180
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(a)
40
1
80
120
160
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(b)
01
50
100
150
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(c)
Figure 8 e stability diagrams of the response by applying impulse along dierent directions at the power tunnel (a) radical (b)longitudinal (c) tangential directions
30
25a
(ms
2 )
000 002 004Time (s)
006 008
20
15
10
05
00
ndash05
(a)
20
PSD
(1endash6
)
f (Hz)
15
1 10 100
10
5
0
(b)
Site photo
(c)
Figure 7 e hammer mode test at the power tunnel
Shock and Vibration 7
03
02
01
00
ndash01
Forc
e (kN
)
01 02 03Time (s)
(a)
15
10
05
00
Forc
e (N
Hz)
f (Hz)0 100 200 300 400
(b)
Figure 10 e hammer impulse load applied in the subway tunnel (a) time spectrum (b) frequency spectrum
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
R direction
1425Hz
1075Hz
233Hz
(a)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
L direction
1285Hz235Hz
137Hz
176Hz
(b)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
T direction
1285Hz
11Hz
119Hz
(c)
Figure 11 e stability diagrams of the response by applying impulse along dierent directions at the subway tunnel (a) radical(b) longitudinal (c) tangential directions
6m (5 rings)
T
L
Tangential
Longitudinal
R Radial
1-1 Measure point number
1-1
1-2
1-3
2-1
TL
R
R
R RT
L
RR
T
L
12m
(a) (b) (c) (d)
Figure 9 Arrangement map for the site measurement at the subway tunnel
8 Shock and Vibration
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
(e SSI-NExT method is employed to the accelerationresponse collected in the tunnel and the stability diagramsare shown in Figure 11 From Figure 11(a) two modes canbe found from the radial response at 1425Hz and 233HzAnd relatively stable recognition results can be obtained at
235 Hz and 1285Hz in the longitudinal response(Figure 11(b)) and 119Hz and 1285Hz in the tangentialdirection (Figure 11(c)) In addition it is also observed thatthere are two peaks at 1075Hz and 176Hz which appear atthe spectrum function from Figure 11 and 1075Hz of
100 101 102ndash150
ndash100
ndash50
0
50
100
150
200
Frequency (Hz)
Num
ber o
f row
sSSIndashCOV Csvd = 0815
(a)
10 11 12 13 14 15 16
40
60
80
100
120
140
160
180
200
Frequency (Hz)
Num
ber o
f row
s
Csvd = 0815
(b)
Figure 4 (e stability diagram of the data collected at night by using the traditional SSI COV method
1 10 100
40
0
80
120
160
200
Num
ber o
f row
s
Frequency (Hz)
Csvd = 0815Csvd = 0915
Figure 5 (e stability diagram of the data collected at night by using the NExT-SSI-COV method
SSI-COV Csvd = 0915200
Num
ber o
f row
s
160
120
80
40
0 25 50 75 100 125 150Frequency (Hz)
(a)
SSI-COV Csvd = 0915200
Num
ber o
f row
s
Frequency (Hz)
160
120
80
40
0 25 50 75 100 125 150
(b)
Figure 6 (e comparison of the stability diagram of the data collected at daytime (a) NExT-SSI-COV method (b) traditional SSI-COVmethod
6 Shock and Vibration
which is close to one of the power tunnelrsquos natural fre-quency at 105Hz
From the tests and analysis introduced at above it canbe seen that the modes of the tunnel can be stably identiedfrom the hammer test by using the NExT-SSI-COVmethod
5 Numerical Verification and Analysis
As the SSI method may generate false mode the dispersionanalysis is thus carried out in this section to explore thevibration modes distribution both of the tunnel mentionedabove based on the pipe-in-pipe (PiP) model [31] By
40
1
60
80
100
120
140
160
180
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(a)
40
1
80
120
160
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(b)
01
50
100
150
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(c)
Figure 8 e stability diagrams of the response by applying impulse along dierent directions at the power tunnel (a) radical (b)longitudinal (c) tangential directions
30
25a
(ms
2 )
000 002 004Time (s)
006 008
20
15
10
05
00
ndash05
(a)
20
PSD
(1endash6
)
f (Hz)
15
1 10 100
10
5
0
(b)
Site photo
(c)
Figure 7 e hammer mode test at the power tunnel
Shock and Vibration 7
03
02
01
00
ndash01
Forc
e (kN
)
01 02 03Time (s)
(a)
15
10
05
00
Forc
e (N
Hz)
f (Hz)0 100 200 300 400
(b)
Figure 10 e hammer impulse load applied in the subway tunnel (a) time spectrum (b) frequency spectrum
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
R direction
1425Hz
1075Hz
233Hz
(a)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
L direction
1285Hz235Hz
137Hz
176Hz
(b)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
T direction
1285Hz
11Hz
119Hz
(c)
Figure 11 e stability diagrams of the response by applying impulse along dierent directions at the subway tunnel (a) radical(b) longitudinal (c) tangential directions
6m (5 rings)
T
L
Tangential
Longitudinal
R Radial
1-1 Measure point number
1-1
1-2
1-3
2-1
TL
R
R
R RT
L
RR
T
L
12m
(a) (b) (c) (d)
Figure 9 Arrangement map for the site measurement at the subway tunnel
8 Shock and Vibration
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
which is close to one of the power tunnelrsquos natural fre-quency at 105Hz
From the tests and analysis introduced at above it canbe seen that the modes of the tunnel can be stably identiedfrom the hammer test by using the NExT-SSI-COVmethod
5 Numerical Verification and Analysis
As the SSI method may generate false mode the dispersionanalysis is thus carried out in this section to explore thevibration modes distribution both of the tunnel mentionedabove based on the pipe-in-pipe (PiP) model [31] By
40
1
60
80
100
120
140
160
180
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(a)
40
1
80
120
160
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(b)
01
50
100
150
200
Num
ber o
f row
s
10Frequency (Hz)
100 1000
Csvd = 097
(c)
Figure 8 e stability diagrams of the response by applying impulse along dierent directions at the power tunnel (a) radical (b)longitudinal (c) tangential directions
30
25a
(ms
2 )
000 002 004Time (s)
006 008
20
15
10
05
00
ndash05
(a)
20
PSD
(1endash6
)
f (Hz)
15
1 10 100
10
5
0
(b)
Site photo
(c)
Figure 7 e hammer mode test at the power tunnel
Shock and Vibration 7
03
02
01
00
ndash01
Forc
e (kN
)
01 02 03Time (s)
(a)
15
10
05
00
Forc
e (N
Hz)
f (Hz)0 100 200 300 400
(b)
Figure 10 e hammer impulse load applied in the subway tunnel (a) time spectrum (b) frequency spectrum
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
R direction
1425Hz
1075Hz
233Hz
(a)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
L direction
1285Hz235Hz
137Hz
176Hz
(b)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
T direction
1285Hz
11Hz
119Hz
(c)
Figure 11 e stability diagrams of the response by applying impulse along dierent directions at the subway tunnel (a) radical(b) longitudinal (c) tangential directions
6m (5 rings)
T
L
Tangential
Longitudinal
R Radial
1-1 Measure point number
1-1
1-2
1-3
2-1
TL
R
R
R RT
L
RR
T
L
12m
(a) (b) (c) (d)
Figure 9 Arrangement map for the site measurement at the subway tunnel
8 Shock and Vibration
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
03
02
01
00
ndash01
Forc
e (kN
)
01 02 03Time (s)
(a)
15
10
05
00
Forc
e (N
Hz)
f (Hz)0 100 200 300 400
(b)
Figure 10 e hammer impulse load applied in the subway tunnel (a) time spectrum (b) frequency spectrum
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
R direction
1425Hz
1075Hz
233Hz
(a)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
L direction
1285Hz235Hz
137Hz
176Hz
(b)
200
160
120
80
40
Num
ber o
f row
s
0 50 100 150 200 250 300Frequency (Hz)
T direction
1285Hz
11Hz
119Hz
(c)
Figure 11 e stability diagrams of the response by applying impulse along dierent directions at the subway tunnel (a) radical(b) longitudinal (c) tangential directions
6m (5 rings)
T
L
Tangential
Longitudinal
R Radial
1-1 Measure point number
1-1
1-2
1-3
2-1
TL
R
R
R RT
L
RR
T
L
12m
(a) (b) (c) (d)
Figure 9 Arrangement map for the site measurement at the subway tunnel
8 Shock and Vibration
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
comparing the calculated cut-off frequencies with the resultsfrom the field test the structural and soil-body couplingvibration characteristics will be analyzed
51 2e PiP Model and Dispersion Analysis (e shield orpipe-jacking tunnel is an assembled structure composedof segments As Figure 12(a) shows when only the lowfrequency response is concerned it can be approximatelyanalogous to an infinite continuous concrete hollowcylinder due to the wavelength greater than the tunnelsegment size (erefore the PiP model is introduced hereand the details are given in [17] based on dispersion andwave propagation theories In this model as shown inFigure 12(a) the tunnel structure is analogous to aninfinite continuous concrete hollow cylinder and thehomogeneous surrounding soil is considered by couplinga concentric 3D thick-walled cylinder outside the tunnelin the PiP model with its inner diameter set equal tothe diameter of the tunnel and outer diameter set toinfinity
(e PiP model is established based on the 25D periodicapproach [31] By assuming constant material and geometricproperties along the infinite extended direction x and fromFigures 12(b) and 12(c) the loads supplied in the tunnelinvert can be treated as a sum of sequence of unit harmonicloads along the direction of x and θ (e function can beexpressed as shown in Equation (8) in the frequency-wavenumber domain
1113957F(r θ x t) 1113944infin
n0
1113957Qrn cos nθ1113957Qθn sin nθ1113957Qxn cos nθ
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦e
i(kx+ωt) (8)
where 1113957Qxn 1113957Qθn and 1113957Qrn are the force components whoseelements are given in [31]
And for a steady response system it is known that thedisplacement response can be expressed as the same stylewith the applied load Finally for a given wavenumber nwhen applying an impulse load at the tunnel invert alongdifferent directions the displacement components of 1113957Un1113957Vn and 1113957Wn for the tunnel structure only is shown inEquation (9) and for the tunnel-soil coupled system thedisplacement components of 1113957Urn 1113957Uθn and 1113957Uzn can becalculated by Equation (10) in the frequency-wavenumberdomain
[A]
1113957Un
1113957Vn
1113957Wn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
minusr 1minus υ2( 1113857
Eh
1113957Qrn
1113957Qθn
1113957Qxn
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
⎫⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎭
(9)
wherematrixA is the coefficients matrix whose elements aregiven in [31]
where matrix Tinfin and Uinfin are the 3 times 3 complex matrixwhose elements are given in [31]
It is known that the dispersion curves are useful ininvestigating the mechanism of wave propagation in amedium which are plotted as the wavenumber k of prop-agating modes versus the frequency ω And the mode fre-quency can be obtained by searching for the cut-onfrequency of each propagation mode (ere are differentmethods to calculate the dispersion curves [36] (e mostcommonly used method is to solve the determinant of thecoefficient matrix of the equilibrium equation As Equation(11) shows the dispersion relation of the tunnel structurecan be obtained by defining the coefficient matrix A ofEquation (9) to zero
det(A) 0 (11)
where A is a 3 times 3 matrix with the parameters of wave-number k tunnel density (ρ) Poisson ratio (]) segmentthickness (h) and tunnel radius which is equal to the averageof the inner and outer radii (e detailed description of thematrix A can be found in [31]
52 2e Power Tunnel Dispersion Characteristics and Com-parison with the Field Test Based on the PiP model thedispersion characteristics of the power tunnel will be ob-tained According to Equation (4) the parameters areYoungrsquos modulus Et 225 times 104MPa Poissonrsquos ratio υ
02 density ρ 2500 kgm3 the outer diameter 32m andinner diameter 27m (e tunnel dispersion curves withinthe range of 0ndash300Hz are obtained and plotted in Figure 13By dispersion analysis theory every curve in Figure 13corresponds to one vibration mode and the starting fre-quency at the x-axis is the cut-on frequency of the corre-sponding vibration mode (e vibration modes arise whenthe excited frequency is higher than the corresponding cut-on frequency (e cut-on frequencies of the 2ndndash4thpropagation modes and the comparison with the field testresults are listed in Table 1
Shock and Vibration 9
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
From Table 1 it is found that most mode frequencyplotted in the eld test can be found by dispersion analysisHowever 105Hz has not matched which can be clearlyidentied in the ambient vibration test is may be causedby the coupling eect of the tunnel and surrounding soil andthe discussion will be continued in the following section
53 e Subway Tunnel Dispersion Characteristics andComparison with Field Test Similarly the dispersion of thesubway tunnel is studied and plotted in Figure 14 HereYoungrsquos modulus Et 435 times 104MPa Poissonrsquos ratio υ 02 density ρ 2500 kgm3 the outer diameter 62m andinner diameter 55m e dispersion curves within therange of 0ndash250Hz are obtained and plotted in Figure 14And the cut-on frequencies of the propagation modes arecompared with the eld test results in Table 2
From Table 2 the most cut-on frequencies from dis-persion analysis can well match with the eld test resultsSimilarly the rst mode frequency 1075Hz cannot beobtained by dispersion analysis which is close 105Hz foundin the eld test of the power tunnel It is further conrmedthat the rst mode comes from the surrounding soil or thecoupled mode of the tunnel and surrounding soil
From the analysis above it can be found that except therst mode most of the modes natural frequency identiedby the SSI-COV and NExTmethod are well consistent withthe dispersion analysis results It is proved that the methodproposed in this paper has a good accuracy in mode fre-quency identication And these modes can be used to trackstructural changes in tunnels by combing some inversionalgorithms
F~ = 1ei(ωt ndash kx)
r θO
O3
O2
O1
Tunnel
L (m)
x S (m)
(a)
n = 0
n = 2 n = 3
n = 1
(b)
x
+ + +ndash ndash
ei(ωt ndash kx)
λ =k
2π
(c)
Figure 12 Decomposition of a radial line load and the resulting tunnel response (a) the rst four Fourier components in cross section (b)schematic map of the steady response system of the tunnel structure (c) spatial distribution along the x direction [31]
20
15
10
05
00
f (Hz)
k (1
m)
n = 2
n = 1 n = 3
n = 0
n = 0
n = 1
n = 4
0 50 100 150 200 250 300
Figure 13 Dispersion curves of the power tunnel modeled as ahollow cylinder shell by the PiP model
Table 1 e dispersion analysis of the power tunnel mode fre-quency distribution and comparison with the eld test results
Number 1 2 3 4 5Dispersion analysis mdash 48 121 203 231Field test 105 53 123 203 223
15
10
05
00
k (r
adm
)
n = 1n = 2
n = 3 n = 4
n = 0
n = 0
n = 1n = 5
f (Hz)0 50 100 150 200 250
Figure 14 Dispersion curves of the subway tunnel modeled as ahollow cylinder shell by the PiP model
Table 2 e dispersion analysis of the subway tunnel mode fre-quency distribution and comparison with the eld test results
Field test 1075 235 mdash 119 1285 142 176 203 233
10 Shock and Vibration
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
And the mismatch of the first mode at 105Hz of thepower tunnel and 1075Hz of the subway tunnel means thatit may come from the surrounding soil or generate from thecoupling of the tunnel and surrounding soil and furtheranalysis on the mechanism needs to be discussed in futureand is not include here What is more meaningful is that thefirst mode can be stability recognized and it is the only onemode from the ambient test at night and therefore it verysuitable for monitoring the natural frequency changescaused by soil excavation around the tunnel
6 Conclusions
In this paper in order to explore the mode frequency dis-tribution of the tunnel and its automatic recognitionmethod the SSI and NExT methods are combined andapplied to analyze the recorded response from ambient andhammer tests and have been proved very suitable for themode frequency identification in the underground structure(e recognition results are also verified by dispersionanalysis based on the PiP model and some conclusions andsuggestion for vibration-based monitoring are obtained asfollows
(1) (e first-order mode frequencies can be clearly andstably identified by the ambient test which has greatapplication potential for monitoring the naturalfrequency changes caused by soil excavation aroundthe tunnel
(2) Traffic environment excitation in the daytime willinterfere with the recognition of structural modecharacteristics and result in some false modes
(3) Most of the higher-order vibration modes can befound by the hammer test and are well consistentwith the dispersion analysis results It providesconditions for the analysis of the structural servicecondition more accurately by some inversionalgorithms
Data Availability
(e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
(e authors declare that they have no conflicts of interest
Acknowledgments
(is research was supported by the National Key RampDProgram of China under the grant no 2018YFC0808702National Natural Science Foundation of China under thegrant nos 51608379 and 51778476 and Shanghai Scienceand Technology Innovation Plan Funds under the grant nos17DZ1204203 and 18DZ1205200 (ese supports are greatlyappreciated
References
[1] H Zhang J Chen F Fan and J Wang ldquoDeformationmonitoring and performance analysis on the shield tunnelinfluenced by adjacent deep excavationsrdquo Journal of Aero-space Engineering vol 30 no 2 article B4015002 2015
[2] J Y Han J Guo and Y S Jiang ldquoMonitoring tunnel de-formations by means of multi-epoch dispersed 3D LiDARpoint clouds an improved approachrdquo Tunnelling and Un-derground Space Technology vol 38 pp 385ndash389 2013
[3] T Hao and C D F Rogers ldquoCondition assessment of theburied utility service infrastructurerdquo Tunnelling and Un-derground Space Technology vol 28 pp 331ndash344 2012
[4] X Y Xie and L Feng ldquoReal-time health monitoring systemfor power tunnelrdquo in Proceedings of Geo Congresspp 3099ndash3108 Oakland CA USA March 2012
[5] D Delaloye M S Diederichs and G Walton ldquoSensitivitytesting of the newly developed elliptical fitting method for themeasurement of convergence in tunnels and shaftsrdquo RockMechanics and Rock Engineering vol 48 no 2 pp 651ndash6672015
[6] M Wilcock K Soga and P Wright ldquoMonitoring mecha-nisms of tunnel lining settlement using instrumented boltsand conventional survey method assessing neutral axis oflongitudinal flexurerdquo Unspecified pp 398ndash407 2012
[7] D Bindi and B Petrovic ldquoSeismic response of an 8-story RC-building from ambient vibration analysisrdquo Bulletin ofEarthquake Engineering vol 13 no 7 pp 2095ndash2120 2015
[8] D Daniele and C Gabriele ldquoDamage identification tech-niques via mode curvature analysis overview and compari-sonrdquo Mechanical Systems and Signal Processing vol 52-53pp 181ndash205 2016
[9] F Musafere A Sadhu and K Liu ldquoTowards damage de-tection using blind source separation integrated with timevarying auto-regressive modelingrdquo Smart Materials andStructures vol 25 no 1 article 015013 2016
[10] J A Goulet C Michel Kiureghian and A D KiureghianldquoData-driven post-earthquake rapid structural safety assess-mentrdquo Earthquake Engineering and Structural Dynamicsvol 44 no 4 pp 549ndash562 2015
[11] J P Amezquita-Sanchez and H Adeli ldquoSignal processingtechniques for vibration-based health monitoring of smartstructuresrdquo Archives of Computational Methods in Engi-neering vol 23 no 1 pp 1ndash15 2016
[12] Z N Ba and X Gao ldquoSoil-structure interaction in transverselyisotropic layered media subjected to incident plane SHwavesrdquo Shock and Vibration vol 2017 Article ID 283427413 pages 2017
[13] A Anvarsamarin F R Rofooei and M Nekooei ldquoSoil-structure interaction effect on fragility curve of 3D modelsof concrete moment-resisting buildingsrdquo Shock and Vibra-tion vol 2018 Article ID 7270137 13 pages 2018
[14] B Zhou F S Zhang and X Y Xie ldquoVibration characteristicsof underground structure and surrounding soil underneathhigh speed railway based on field vibration testsrdquo Shock andVibration vol 2018 Article ID 3526952 13 pages 2018
[15] S Gupta M F M Hussein G Degrande H E M Hunt andD Clouteau ldquoA comparison of two numerical models for theprediction of vibrations from underground railway trafficrdquoSoil Dynamics and Earthquake Engineering vol 27 no 7pp 608ndash624 2007
[16] B Zhou X Y Xie Y B Yang and J C Jiang ldquoA novelvibration-based structure health monitoring approach for the
Shock and Vibration 11
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Navigation and Observation
International Journal of
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
Submit your manuscripts atwwwhindawicom
shallow buried tunnelrdquo Computer Modeling in Engineeringand Sciences vol 86 no 4 pp 321ndash348 2012
[17] B Zhou X Y Xie and Y S Li ldquoA structural health as-sessment method for shield tunnels based on torsional wavespeedrdquo Science China Technological Sciences vol 57 no 6pp 1109ndash1120 2014
[18] A Galvagni and P Cawley ldquo(e reflection of guided wavesfrom simple supports in pipesrdquo Journal of Acoustical Society ofAmerica vol 129 no 4 pp 1869ndash1880 2011
[19] A B (ien H C Chiamori J T Ching J R Wait andG Park ldquoModel-based SHM the use of macro-fibre com-posites for pipeline structural health assessmentrdquo StructuralControl and Health Monitoring vol 15 no 1 pp 43ndash63 2008
[20] B K RaghuPrasad N Lakshmanan N GopalakrishnanK Sathishkumar and R Sreekala ldquoDamage identification ofbeam-like structures with contiguous and distributed dam-agerdquo Structural Control and Health Monitoring vol 20 no 4pp 496ndash519 2013
[21] E Leinov M J S Lowe and P Cawley ldquoInvestigation ofguided wave propagation and attenuation in pipe buried insandrdquo Journal of Sound and Vibration vol 347 no 7pp 96ndash114 2015
[22] T Vogt M Lowe and P Cawley ldquo(e scattering of ultrasonicguided waves in partly embedded cylindrical structuresrdquoAcoustical Society of America vol 113 no 3 pp 1259ndash12722003
[23] M Eybpoosh M Berges and H Y Noh ldquoSparse represen-tation of ultrasonic guided-waves for robust damage detectionin pipelines under varying environmental and operationalconditionsrdquo Structural Control and Health Monitoringvol 23 no 2 pp 369ndash391 2016
[24] P Andersen Identification of civil engineering structures usingvector ARMA models PhD thesis Department of BuildingTechnology and Structural Engineering Aalborg UniversityAalborg Denmark 1997
[25] G H James T G Carne and J P Lauffer ldquo(e naturalexcitation technique (NExT) for modal parameter extractionfrom operating structuresrdquo International of Analytical andExperimental Modal Analysis vol 10 no 4 pp 260ndash2771995
[26] Y C Liu C H Loh and Y Q Ni ldquoStochastic subspaceidentification for output-only mode analysis application tosuper high-rise tower under abnormal loading conditionrdquoEarthquake Engineering Structure Dynamics vol 42 no 4pp 477ndash498 2013
[27] P van Overschee and B L R deMoor Subspace Identificationfor Linear Systems 2eory-Implementation-ApplicationsKluwer Academic Publishers Dordrecht Netherlands 1996
[28] I Goethals L Mevel A Benveniste and B D Moor ldquoRe-cursive output only subspace identification for in-flight fluttermonitoringrdquo in Proceedings of 22nd International ModeAnalysis Conference Dearborn MI USA 2004
[29] B Peeters and G de Roeck ldquoReference-based stochasticsubspace identification for output-only mode analysisrdquo Me-chanical Systems and Signal Processing vol 13 no 6pp 855ndash878 1999
[30] B Peeters and G D Roeck ldquoStochastic system identificationfor operational mode analysis a reviewrdquo Journal of DynamicSystems Measurement and Control vol 123 no 4pp 659ndash667 2001
[31] J A Forrest and H E M Hunt ldquoA three-dimensional modelfor calculation of train-induced ground vibrationrdquo Journal ofSound and Vibration vol 294 no 4-5 pp 678ndash705 2006
[32] Y Wang X C Hang D Jiang X L Han and Q G FeildquoSelection method of Toeplitz matrix row number based oncovariance driven stochastic subspace identificationrdquo Journalof Vibration and Shock vol 34 no 7 pp 71ndash75 2015
[33] R E Akins ldquoCross-spectral measurements in the testing ofwind turbinesrdquo in Proceedings of 9th ASME Wind EnergySymposium New Orleans LA USA 1990
[34] J H Weng C H Loh J P Lynch K C Lu P Y Lin andY Wang ldquoOutput-only mode identification of a cable-stayedbridge using wireless monitoring systemsrdquo EngineeringStructures vol 30 no 7 pp 1820ndash1830 2008
[35] C H Loh K J Loh Y S Yang W Y Hsiung andY T Huang ldquoVibration-based system identification of windturbine systemrdquo Structural Control and Health Monitoringvol 24 no 3 article e187 2016
[36] X Sheng C J C Jones and D J (ompson ldquoA theoreticalstudy on the influence of the track on train-induced groundvibrationrdquo Journal of Sound and Vibration vol 272 no 3ndash5pp 909ndash936 2004
