Research Article Mathematical Modeling for Lateral ...downloads.hindawi.com/journals/mpe/2014/723152.pdf · from Main Girder of Sutong Cable-Stayed Bridge ... According to aerostatic

Research ArticleMathematical Modeling for Lateral DisplacementInduced by Wind Velocity Using Monitoring Data Obtainedfrom Main Girder of Sutong Cable-Stayed Bridge

Gao-Xin Wang and You-Liang Ding

The Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of EducationSoutheast University Nanjing 210096 China

Correspondence should be addressed to You-Liang Ding civilchinahotmailcom

Received 2 April 2014 Accepted 22 May 2014 Published 19 June 2014

Academic Editor Giuseppe Rega

Copyright copy 2014 G-X Wang and Y-L Ding This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Based on the health monitoring system installed on the main span of Sutong Cable-Stayed Bridge GPS displacement andwind field are real-time monitored and analyzed According to analytical results apparent nonlinear correlation with certaindiscreteness exists between lateral static girder displacement and lateral static wind velocity thus time series of lateral staticgirder displacement are decomposed into nonlinear correlation term and discreteness term nonlinear correlation term of whichis mathematically modeled by third-order Fourier series with intervention of lateral static wind velocity and discreteness term ofwhich ismathematicallymodeled by the combinedmodels of ARMA(7 4) and EGARCH(2 1) Additionally stable power spectrumdensity exists in time series of lateral dynamic girder displacement which can be well described by the fourth-order Gaussianseries thus time series of lateral dynamic girder displacement are mathematically modeled by harmonic superposition functionBy comparison and verification between simulative and monitoring lateral girder displacements from September 1 to September 3the presented mathematical models are effective to simulate time series of lateral girder displacement from main girder of SutongCable-Stayed Bridge

1 Introduction

Nowadays long span cable-stayed and suspension bridgestructures are commonly constructed at home and abroadOn account of their flexible structural characteristics dis-placement response from main girder of long-span bridgestructure swings obviously impacted by strong aerostatic andfluctuating wind actions According to aerostatic responseanalysis on Sutong Cable-Stayed Bridge by Xu et al thelateral displacement response frommain girder can approach12m under strong wind velocity 40ms with attack angle 0∘[1] and research results from buffeting response analysis onGolden Gate Bridge by Vincent showed that extreme buffet-ing amplitude frommain girder can reach 17m under strongwind velocity 31ms [2] Such large amplitude can definitelythreaten comfort and safety of the whole bridge structure Forexample severe wind vibration from main girder of Tacoma

Suspension Bridge in Washington state eventually broughtabout collapse of the whole bridge structure under windvelocity 19ms [3] Therefore it is of great significance toresearch displacement response impacted bywind loads frommain girder of long span bridge structures and especially thelateral displacement response as one fairly important part formain girder should be specifically valued

Theoretical exploration numerical simulation and windtunnel tests for lateral displacement response have beencarried out to some extent Cheng andXiao improved the cal-culationmethod for aerostatic stability and further concludedthat instable lateral displacement was 424m under criticalstatic wind [4] Long et al analyzed lateral displacementresponse from Sidu Suspension Bridge through ANSYS finiteelement simulation and concluded that maximum lateraldisplacement at middle span was 3226 cm which is closeto 11000 length of main span [5] Yu et al researched

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 723152 16 pageshttpdxdoiorg1011552014723152

2 Mathematical Problems in Engineering

2088100 100 1001003001088

3D ultrasonic anemometer

GPS monitoring station 300

Figure 1 Longitudinal layout of monitoring equipment on the Sutong Bridge (unit m)

41000

354002

4000

90009000 23000

GPSMS GPSMS

3DUA 3DUAUpstream Downstream

354002

Figure 2 Transverse layout of monitoring equipment on the flat steel box girder (unit mm) (1)GPSMS GPSmonitoring station (2) 3DUA3D ultrasonic anemometer

lateral displacement response from Xihoumen SuspensionBridge through wind tunnel tests and concluded that lateraldisplacement at horizontal angle 10∘ was larger than that ofother angles [6]

However for mechanism complexity of lateral dis-placement response impacted by aerostatic and fluctuatingwind actions traditional methods of theoretical deductionnumerical simulation and wind tunnel tests are difficult toaccurately reflect the actual lateral displacement responseof bridge structure on account of uncertain boundarycondition imprecise assignment of initial parameters andinappropriate ignorance of subordinate factors In recentyears with development of structural healthmonitoring tech-nology it is feasible to installmonitoring sensors on long spanbridge structures monitoring data of which can authenticallyreflect bridge structural behaviors under actual environmentand load actions Although wind field of long span bridgestructures has been widely monitored in recent years [7ndash9]lateral displacement response is rarely monitored andresearched thus real correlation regularity between lateraldisplacement response and wind action is still covered Addi-tionally lateral displacement response under actual operationenvironment is also affected by other random factors whichhave never been taken into account by researchers beforeTherefore lateral displacement response from main girder isnecessarily researched upon monitoring data to reveal realstructural behavior of long span bridges

In this paper based on the health monitoring systeminstalled on the main span of Sutong Cable-Stayed BridgeGPS displacement and wind field are real-time monitoredand analyzed According to analytical results apparent non-linear correlation with certain discreteness exists between

lateral static girder displacement and lateral static windvelocity thus time series of lateral static girder displace-ment are decomposed into nonlinear correlation term anddiscreteness term nonlinear correlation term of which ismathematically modeled by 119899th-order Fourier series withintervention of lateral static wind velocity and discretenessterm of which is mathematically modeled by the combinedmodels of ARMA(119901 119902) and EGARCH(119898 119899) Additionallystable power spectrum density exists in time series of lateraldynamic girder displacement thus time series of lateraldynamic girder displacement are mathematically modeledby harmonic superposition function By comparison andverification between simulative and monitoring lateral dis-placements from September 1 to September 3 mathematicalmodels are feasible and effective to simulate time series oflateral girder displacement frommain girder of SutongCable-Stayed Bridge

2 Bridge Monitoring and Sample Analysis

The bridge monitoring object for this research is theworldwide famous Sutong Cable-Stayed Bridge (in JiangsuProvince China) Its whole structure form is single-spannedand double-hinged with the main span reaching 1088m asshown in Figure 1 and the main girder employs flat steelbox type with 363m wide and 40m high as shown inFigure 2 3D ultrasonic anemometers and GPS monitoringstation are installed on two flanks of midspan cross-sectionfrom main girder (resp shown in Figures 1 and 2) tocontinuously acquire wind data and displacement data withsample frequency of 1Hz

Mathematical Problems in Engineering 3

x

y

X (longitude orientation)

Y (latitude orientation)

z or Z

Along the bridge Across the bridge

Geographical north 106∘

106∘

Figure 3 Description of the defined local andWGS-84 coordinate system Notes (1) local coordinate system 119909-119910-119911 (2)WGS-84 coordinatesystem119883-119884-119885 (3) 90∘ of wind horizontal angle denotes the axis angle from 119909 to 119910 (within range of [0∘ 360∘]) (4) 90∘ of wind vertical angledenotes axis angle from 119910 to 119911 (within range of [minus90∘ 90∘]) (5) 106∘ axis angle exists between axes 119884 and 119910 of across the bridge

0 5 10 15 20 25 30

0

10

20

30

Time (day)

Win

d ve

loci

ty al

ong y

-axi

s (m

s)

minus10

minus20

(a) Time series of original wind velocity

0 5 10 15 20 25 30

0

10

20

30

Time (day)

Stat

ic w

ind

velo

city

(ms

)

minus10

minus20

(b) Time series of static wind velocity

0 5 10 15 20 25 30

0

5

10

15

Time (day)

Dyn

amic

win

d ve

loci

ty (m

s)

minus10

minus10

minus15

(c) Time series of dynamic wind velocity

Figure 4 Time series of wind velocity along 119910-axis in the whole August


0 5 10 15 20 25 30

005

015

025

035

Time (day)

Gird

er d

ispla

cem

ent a

long

y-a

xis (

m)

minus005

minus015

(a) Time series of girder displacement

0 5 10 15 20 25 30

0

01

02

03

Time (day)

Stat

ic g

irder

disp

lace

men

t (m

)

minus01

(b) Time series of static girder displacement

0 5 10 15 20 25 30

0

004

008

012

Time (day)

Dyn

amic

gird

er d

ispla

cem

ent (

m)

minus012

minus008

minus004

(c) Time series of dynamic girder displacement

Figure 5 Time series of girder displacements along 119910-axis in the whole August

5 15 250

005

01

015

02

025

03

Static wind velocity (ms)

Stat

ic g

irder

disp

lace

men

t (m

)

minus15 minus5

Figure 6 Correlation scatter plots between static wind velocity and static girder displacement


Frequency (Hz)

Day 1 to day 8Day 9 to day 16 Day 17 to day 24

Day 25 to day 31

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7

Figure 7 Power spectrum densities of dynamic girder displacement in different periods

0 10 20 300

006

012

018

024

03


Fitte

d lat

eral

gird

er d

ispla

cem

ent (

m)

minus20 minus10

(a) The fitting Fourier series of nonlinear correlation term

0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01

(b) Time series of fitting static displacement

Figure 8 The fitting Fourier series and time series of fitting static displacement

Specifically the wind data from 3D ultrasonic anemome-ters embrace such three types as wind velocity horizontalangle and vertical angle in local coordinate system (Figure 3)and the girder displacement data from GPS monitoringstation contains absolute locations in WGS-84 coordinatesystem (Figure 3) which are supposed to deduct referencelocations for analysis Until now the storage amount ofmoni-toring data has increased to 93 million for each measurementpoint Such considerable monitoring data cannot be totallyapplied to actual analysis thus the monitoring data fromupstream flank in the year 2012 are specially chosen

Taking the monitoring data in the whole August forexample considering that lateral displacement effect of maingirder is primarily reflected by wind load across the SutongBridge hence time series of wind velocity and girder dis-placement are decomposed into the119910-axis in local coordinate

system as shown in Figures 4(a) and 5(a) respectively whichcan be obviously observed that either of two time seriescontain static variant trend in whole and dynamic stochasticfluctuation in part Such two kinds of variation characteristicscan be furthermore separated by 10-minute average process asshown in Figures 4(b) 4(c) 5(b) and 5(c) respectively

By comparison between Figures 4(b) and 5(b) similarvariation characteristics exist between static wind veloc-ity and static girder displacement which can be visu-ally described by correlation scatter plots as shown inFigure 6 indicating apparent nonlinear correlation similar toquadratic parabolic curve Therefore static girder displace-ment can bemathematically expressed by static wind velocitywith consideration of definite discreteness affected by otherrandom factors Moreover time series of dynamic girderdisplacement depict obvious steady stochastic fluctuation


0 6 12 18 24 30

003

007

011

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus005

minus001

(a) Time series of original discreteness

0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s

Autocorrelation valueUpper limit with 95 confidence intervalsLower limit with 95 confidence intervals

minus02

Auto

corr

elat

ion

valu

e120588(b) The autocorrelation function 120588(119904)

Partial correlation valueUpper limit with 95 confidence intervalsLower limit with 95 confidence intervals

0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s

minus02

Part

ial c

orre

latio

n va

lue 120588

(c) The partial correlation function 120573(119904)

Figure 9 Time series of original discreteness with its 120588(119904) 120573(119904)

as well as no variation of its power spectrum densities bytime shown in Figure 7 thus dynamic girder displacementcan be mathematically simulated by harmonic superpositionmethod

3 Modeling Theory and Procedure

31 Modeling for Static Girder Displacement

311 Modeling Theory and Method Correlation scatter plotsin Figure 6 can be acquired by combination of nonlinear cor-relation term and discreteness term Furthermore nonlinear

correlation term can be mathematically modeled by the 119899-order Fourier series that is

1199061(119905) = 119886

0+

119899

sum119896=1

(119886119896cos (119896120596V (119905)) + 119887

119896sin (119896120596V (119905))) (1)

where 1199061(119905) denotes time series of fitting static displacement

for nonlinear correlation term V(119905) denotes time series ofstatic wind velocity (Figure 4(b)) and 119886

119896 119887119896(119896 = 0 1 119899)

and 120596 are fitting parameters of 119899th-order Fourier seriesWith denotation of original static girder displacement as

119906(119905) (Figure 5(b)) 119906(119905) minus 1199061(119905) acquires time series of

discreteness 1199062(119905) which contains three types of stochastic

Mathematical Problems in Engineering 7St

atic

disp

lace

men

t of d

iscre

tene

ss (m

)

0 6 12 18 24 30

001

003

005

Time (day)

minus005

minus003

minus001

(a) Time series of processed discreteness


Lag phase s 0 10 20 30 40 50

0

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

Part

ial c

orre

latio

n va

lue 120588

minus02

minus04


Figure 10 Time series of processed discreteness with its 120588(119904) 120573(119904)

characteristics (autoregression moving average and het-eroscedasticity) and can be mathematically described by thecombined models of ARMA(119901 119902) and EGARCH(119898 119899) Indetail the ARMA(119901 119902) model defines the stochastic charac-teristics of autoregression and moving average as follows

1199062(119905) = 119888 +

119901

sum119895=1

1198871198951199062(119905 minus 119895) +

119902

sum119895=0

119888119895120576119905minus119895 (2)

where 119888 is the constant term 119901 and 119902 respectively denotethe orders of autoregression or moving average of 119906

2(119905) 119887119895

and 119888119895 respectively denote the coefficients of autoregression

or moving average of 1199062(119905) with 119888

0= 1 and 120576

119905minus119895denotes

the innovations process with time delay of 119895 Meanwhile theother EGARCH(119898 119899)model defines the stochastic character-istic of heteroscedasticity as follows [9 10]

log1205902119905

= 120581 +

119898

sum119896=1

119892119896log1205902119905minus119896

+

119899

sum119896=1

V119896[

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

minus 119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816]

+

119899

sum119896=1

119897119896(120576119905minus119896

120590119905minus119896

)

(3)


1 2 3 4 5 6 7 8minus33615

minus33605

minus33595

minus33585

minus33575

33565

The value of m

The s

tatis

tical

val

ue o

f AIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

(a) The statistical values of AIC

The s

tatis

tical

val

ue o

f BIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

1 2 3 4 5 6 7 8minus33530

minus33496

minus33458

minus33422

minus33386

minus33350

The value of m

(b) The statistical values of BIC

Figure 11 The statistical values of AIC and BIC

where 120590119905denotes the conditional variance of the innovations

process 120576119905 120581 is the constant term119898 and 119899 respectively denote

the orders of the EGARCH(119898 119899)model119892119896 V119896 and 119897

119896 respec-

tively denote the coefficients of the EGARCH(119898 119899) model119911119905is a standard independent and identically distributed

random draw from some specified probability distributionsuch as Gaussian or Studentrsquos 119905 and

119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816 = 119864(

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

)

=

radic2

120587Gaussian

radic] minus 2120587

sdotΓ ((] minus 1) 2)Γ (]2)

Studentrsquos 119905

(4)

with degrees of freedom V gt 2 Considering that theEGARCH(119898 119899) model is treated as ARMA(119901 119902) models forlog1205902119905

thus the stationarity constraint for the EGARCH(119898 119899)model is included by ensuring that the eigenvalues of thecharacteristic polynomial

120582119898

minus 1198661120582119898minus1

minus 1198662120582119898minus2

minus sdot sdot sdot minus 119866119898 (5)

are inside the unit circle where 120582 and 119866119895are the variable and

coefficients of the characteristic polynomial respectivelyDuring modeling process for time series of discreteness

1199062(119905) the autocorrelation function 120588(119904) and partial corre-

lation function 120573(119904) with their lag phase 119904 are introduced

for stationary test of 1199062(119905) Specifically the autocorrelation

function 120588(119904) can be calculated as follows [11]

120588 (119904) =(1119873)sum

119873minus119904

119905=1

(1199062(119905) minus 119906

2) (1199062(119905 + 119904) minus 119906

2)

1205902120588

119904 = 0 1 2 119873 minus 1

(6)

where

1199062=1

119873

119873

sum119905=1

1199062(119905) 120590

2

120588

=1

119873

119873

sum119905=1

(1199062(119905) minus 119906

2)2 (7)

with 119873 denoting the amount of 1199062(119905) And the other partial

correlation function 120573(119904) can be calculated through fittingsuccessive autoregressive models of orders by ordinary leastsquares retaining the last coefficient of each regression [12]Besides the AIC and BIC delimitation criteria are applied todetermine model orders with their statistical values aic andbic being respectively calculated as follows [13 14]

119886119894119888 = minus2 ln (119871) + 21198701

bic = minus2 ln (119871) + 1198701sdot ln (119870

2)

(8)

where 119871 denotes the optimized log-likelihood objectivefunction (LLF) values associated with parameter estimates ofthe combined models 119870

1denotes the number of estimated

parameters associated with each value in LLF and1198702denotes

the sample size of the observed 1199062(119905) associatedwith each LLF

value

312 Detailed Procedure Based on the theory and methodabove the detailed procedure for mathematically modeling


0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588

(a) The autocorrelation function 120588(119904)

0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588

(b) The partial correlation function 120573(119904)

Figure 12 The correlation functions 120588(119904) and 120573(119904) of residuals

1 2 3 4 5 6 7590

591

592

593

594

595

596

The value of order p

q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3

(a) The standard errors when119898 = 1 119899 = 1

1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3

(b) The standard errors when119898 = 2 119899 = 1

Figure 13 The standard errors between processed discreteness and its models with lower orders

time series of static girder displacement is illustrated takingthe correlation scatter plots in the whole August in Figure 6for example as follows

Step 1 Fitting Fourier series for nonlinear correlation termBy means of the MATLAB fitting tools (utilizing the third-order Fourier series (1) to fit the correlation scatter plots)[10] the mathematical model of nonlinear correlation termis straightforward and acquired as shown in Figure 8(a)together with the estimated values of Fourier parameterspresented in Table 1 By substitution of estimated valuestogether with V(119905) into formula (1) time series of fitting static

displacement1199061(119905) in thewholeAugust are acquired as shown

in Figure 8(b)

Step 2 Stationary test for time series of discreteness Timeseries of discreteness 119906

2(119905) can be acquired by 119906(119905) minus

1199061(119905) as shown in Figure 9(a) Autocorrelation function 120588(119904)

and partial correlation function 120573(119904) of 1199062(119905) are calcu-

lated with 50 lag phases shown in Figures 9(b) and 9(c)respectively presenting that 120588(119904) is slowly converging intothe 95 confidence intervals as the lag phase 119904 increaseswhich indicates bad stationarity of 119906

2(119905) for mathematical

modeling Due to this process of first-order difference for


0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005

(a) Time series of simulative processed discreteness

minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005

(b) Time series of simulative original discreteness 1199062(119905)

0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01

(c) Time series of simulative static displacement 119906(119905)

Figure 14 Simulation for time series of discreteness 1199062

(119905) and static displacement 119906(119905)

Table 1 Estimated values of fitting parameters (minus12 lt V(119905) lt 21)

Fitting parameters 1198860

1198861

1198871

1198862

1198872

1198863

1198873

119908

Estimated values 01102 minus01026 00218 001214 0006845 minus000618 minus0003763 01374

1199062(119905) is carried out as shown in Figure 10(a) with its 120588(119904) and

120573(119904) shown in Figures 10(b) and 10(c) both presenting rapidconvergence into the 95 confidence intervals and verifyinggood stationarity for processed discreteness

Step 3 Order determination of ARMA(119901 119902) andEGARCH(119898 119899) The orders of 119901 and 119902 are relative tothe convergent forms of 120588(119904) and 120573(119904) That is the 120588(119904)

and 120573(119904) clearly present the trailing property in Figures10(b) and 10(c) with 4th and 7th of lag phase 119904 initiallyconverging into the 95 confidence intervals inferring that7 and 4 are appropriately assigned to the orders of 119901 and 119902respectively [11 12] Furthermore the orders of 119898 and 119899 aredetermined utilizing the AIC and BIC delimitation criterion

In detail the statistical values of AIC and BIC from timeseries of processed discreteness (Figure 10(a)) are calculatedrespectively under integer assignment of119898 and 119899 between 0and 8 as shown in Figure 11 presenting that the most suitableorders of 119898 and 119899 are 2 and 1 respectively corresponding tothe minimum statistical values [13 14]

Step 4 Parameter estimation and residual test of orderedmodels Based on the specified orders above model param-eters (119887

119895 119888119895 119892119896 V119896 119888 and 120581) are further estimated to fit

time series of processed discreteness as shown in Tables2 and 3 respectively For testing the fitting effectiveness ofestimated parameters residuals between processed discrete-ness (Figure 10(a)) and its defined models with estimated


August 1 to August 8

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7

(a) Original power spectrum density


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

(b) Processed power spectrum density

Figure 15 Original and processed power spectrum densities

Table 2 Estimating model parameters of the ARMA(7 4) model

The combined models The ARMA(7 4) modelThe fitting parameters 119888 119887

1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884

The values 000 minus049 094 075 minus032 minus001 004 002 006 minus126 minus048 068

Table 3 Estimatingmodel parameters of the EGARCH(2 1) model

The combined models The EGARCH(2 1) modelThe fitting parameters 120581 119892

1

1198922

V1

1198971

The values minus076 027 065 028 006

parameters are analyzed using 120588(119904) and 120573(119904) as shownin Figure 12 which presents that both 120588(119904) and 120573(119904) areconsistent in the 95 confidence intervals (except 119904 is 0)and thus verifies good availability of estimated parametersfor ordered models Moreover the standard errors betweenprocessed discreteness (Figure 10(a)) and its models withlower orders are shown in Figure 13

Step 5 Simulation for time series of discreteness 1199062(119905)

and static displacement 119906(119905) Based on the mathematicalmodels of ARMA(7 4) and EGARCH(2 1) with estimatedparameters time series of simulative processed discretenessare shown in Figure 14(a) and through inverse calculationof first-order difference time series of simulative originaldiscreteness 119906

2(119905) are obtained as shown in Figure 14(b)

Together with time series of fitting static displacement 1199061(119905)

in Figure 8(b) time series of simulative static displacement119906(119905) are ultimately shown in Figure 14(c) definitely similar totime series of monitoring static displacement in Figure 5(b)

32 Modeling for Dynamic Girder Displacement

321 Modeling Theory and Method Time series of dynamicgirder displacement show consistent power spectrum den-sity (Figure 7) which can be mathematically simulated byharmonic superposition method [15ndash17] Primarily functionexpression of power spectrum density should be confirmedto lay foundation for harmonic superposition Consideringthat straightforward function fitting will ignore local featherof acute peak at 10minus1Hz around of frequency thus the powerspectrum density are decomposed into two parts part one tofit the whole trend with ignorance of acute peak part two tospecifically fit the acute peak Furthermore each part can beexpressed by the 119899-order Gaussian series in logarithmic formthat is

lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)

where 1199011(119891) denotes function expression of part one 119901

2(119891)

denotes function expression of part two and 119886119896 119887119896 119888119896 119889119896 ℎ119896

and 119903119896

are fitting parameters of 1199011(119891) and 119901

2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)

(a) The first part and its fitting curve 1199011(119891)

Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101

10minus110minus210minus310minus410minus5100

Fitting curve p2(f)

(b) The second part and its fitting curve 1199012(119891)

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)

(c) The combination 119901(119891) of 1199011(119891) and 119901

2(119891)

Figure 16 Two parts of processed power spectrum densities and their fitting curves

119896 = 0 1 119899 Accordingly the function expression ofpower spectrum density 119901(119891) can be formulized as follows

119901 (119891) = 1199011(119891) + 119901

2(119891) (10)

Based on power spectrum density119901(119891) above variance 1205902within frequency band of [119891

11198912] can be calculated utilizing

the expanding characteristics of average power spectrum forstationary and stochastic process as follows

1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)

Then the frequency band of [11989111198912] is divided into 119899 subin-

tervals and the whole values 119901(119891) in the 119894th subintervalcan be represented by the corresponding value 119901(119891mid119894) at

the middle frequency 119891mid119894 thus formula (11) is furthermorediscretized as follows

1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)

As for the 119894th subinterval its sinusoidal function 119908(119905 119894)

containing middle frequency 119891mid119894 and standard deviationradic119901(119891mid119894) sdot Δ119891 can be expressed as follows

119908 (119905 119894) = radic2119901 (119891mid119894) sdot Δ119891 sin (2120587119891mid119894119905 + 120579119894) (14)


0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002

(a) Time series of simulative dynamic displacement 119908(119905)

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

(b) Power spectrum density of 119908(119905)

Figure 17 Time series of simulative dynamic displacement 119908(119905) and its power spectrum density

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01

(a) Time series of simulative lateral displacement

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01

(b) Time series of monitoring lateral displacement

Figure 18 Time series of simulative and monitoring lateral girder displacement

where 120579119894is the random variable of uniform distribution

within [0 2120587] With superposition of119908(119905 119894) from each subin-terval the time series of dynamic girder displacement 119908(119905)can be mathematically modeled by harmonic superpositionfunction as follows

119908 (119905) =

119899

sum119894=1

radic2119901 (119891mid119894) sdot Δ119891 sin (2120587119891mid119894119905 + 120579119894) (15)

322 Detailed Procedure Based on the theory and methodabove the detailed procedure for mathematically modeledtime series of dynamic girder displacement is illustrated

taking the correlation scatter plots from August 1 to August8 in Figure 15(a) for example as follows

Step 1 Decreasing discreteness for original power spectrumdensity Considering that certain discreteness existing inoriginal power spectrum density can cover the acute peakat 10minus1Hz adverse to fitting the 119899-order Gaussian seriestherefore average process with double frequency scales iscarried out to decrease discreteness specifically by 10minus3Hzaverage process within frequency band [10minus15Hz 10minus05Hz]and 10minus2Hz average process within other frequency bandsProcess result is shown in Figure 15(b) showing distinct acutepeak for fitting the 119899-order Gaussian series


0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

Simulative lateral displacement Monitoring lateral displacement

minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)

(a) Simulative and monitoring results

0 005 01 015

0

005

01

015

Simulative lateral displacement (m)M

onito

ring

later

al d

ispla

cem

ent (

m)

Scatter plotsFitting curve

minus005minus005

ds(t) = 1008dm(t) + 00016

(b) Correlation scatter plots with fitting curve

Figure 19 Simulative and monitoring results and their correlation scatter plots with fitting curve

Simulative resultMonitoring result

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7

Figure 20 Power spectrum densities of simulative and monitoring results

Step 2 Fitting Gaussian series for two parts of processedpower spectrum density The processed power spectrumdensity can be divided into two parts the whole trendwith ignorance of acute peak as shown in Figure 16(a) thespecific acute peak as shown in Figure 16(b) By means of theMATLAB fitting tools (utilizing the fourth-order Gaussianseries (9) to fit two parts) [10] themathematicalmodels119901

1(119891)

and 1199012(119891) of two parts are respectively shown in Figures

15(a) and 15(b) together with estimated parameters presentedin Table 4The combination119901(119891) of119901

1(119891) and119901

2(119891) is shown

in Figure 16(c)

Step 3 Harmonic superposition for fitting power spectrumdensity 119901(119891) sdot 119901(119891) is divided into 25000 subintervals withinfrequency bands [10minus5Hz 10minus05Hz] In each subintervalone middle frequency 119891mid119894 and its corresponding value119901(119891mid119894) are existent and then substituted into (15) forsummation thus time series of dynamic girder displacement119908(119905) can be acquired as shown in Figure 17(a) (simulated for86400 s) By comparing its power spectrum density shown inFigure 17(b) with the monitoring one shown in Figure 15(a)good similarity of the whole trend and the acute peak verifies


Table 4 Estimated parameter values of the fourth-order Gaussian series

Fitting curves Estimated parameter values of the fourth-order Gaussian Series1198861

1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011

(119891) minus536 045 minus486 minus073 minus498 minus298 minus075 minus022 169 038 173 0361199012

(119891) 058 085 014 070 minus096 minus096 minus120 minus110 001 005 004 015

appropriateness of mathematical models for dynamic girderdisplacement

4 Model Test and Evaluation

According to mathematical modeling process above timeseries of lateral girder displacement are expressed by com-bination of third-order Fourier series 119906

1(119905) ARMA(74)

EGARCH(2 1) and harmonic superposition function 119908(119905)For verifying feasibility and effectiveness of the whole math-ematical models time series from September 1 to September3 are simulated with intervention of monitoring static wind(Figure 18(a)) and then compared with monitoring onesduring same period (Figure 18(b))

According to mathematical modeling theory and proce-dure above comparison is divided into two parts (1) timeseries of static girder displacement (2) time series of dynamicgirder displacement As for the first part its simulative andmonitoring results are shown in Figure 19(a) and linear fittingcurve of correlation scatter plots is shown in Figure 19(b)presenting consistent variation tendency in Figure 19(a) and119889119904(119905) approximating to 119889

119898(119905) in Figure 19(b) (where 119889

119904(119905)

and 119889119898(119905) respectively denote simulative and monitoring

results) which verifies good feasibility and effectiveness ofmathematical models for static girder displacement As forthe second part power spectrum densities of simulativeand monitoring results are shown in Figure 20 present-ing uniform variation tendency of both whole trends andacute peaks which verifies good feasibility and effectivenessof mathematical models for dynamic girder displacementTherefore mathematical models above can be reasonablyutilized to simulate time series of lateral girder displacementfrom main girder of Sutong Cable-Stayed Bridge

5 Conclusions

Based onmonitoring data frommain girder of Sutong Cable-Stayed Bridge time series of lateral girder displacement effectare mathematically modeled by methods of fitting Fourierseries and Gaussian series combined models of ARMA(74)and EGARCH(21) and harmonic superposition functionAnd conclusions can be drawn as follows

(1) Scatter plots between lateral static wind velocity andlateral static displacement present apparent nonlinearcorrelation which is similar to quadratic paraboliccurve and time series of lateral dynamic displacementcontain obvious stable power spectrum density withno variation by time

(2) Time series of lateral static displacement can bedecomposed into nonlinear correlation term and

discreteness term Moreover nonlinear correlationterm can be mathematically modeled by third-orderFourier series with intervention of lateral static windvelocity and discreteness term can bemathematicallymodeled by the combinedmodels of ARMA(7 4) andEGARCH(2 1)

(3) Through decreasing discreteness in double frequencyscales and division in double frequency bands powerspectrum density of lateral dynamic displacementcan be mathematically modeled by the fourth-orderGaussian series and time series of lateral dynamicdisplacement can be further mathematically modeledby harmonic superposition function

(4) By the comparison between simulative and moni-toring lateral displacement effect from September 1to September 3 mathematical models are feasibleand effective to simulate time series of lateral girderdisplacement from main girder of Sutong Cable-Stayed Bridge

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) and KeyProgram of Ministry of Transport (no 2011318223190 and no2013319223120)

References

[1] F Xu A Chen and Z Zhang ldquoAerostatic wind effects on theSutong Bridgerdquo in Proceedings of the 3rd International Confer-ence on Intelligent System Design and Engineering Applications(ISDEA 2013) pp 247ndash256 January 2013

[2] M Gu and H F Xiang ldquoAnalysis of buffeting response and itscontrol of Yangpu Bridgerdquo Journal of Tongji University vol 21no 3 pp 307ndash314 1993 (Chinese)

[3] D Green andW G Unruh ldquoThe failure of the Tacoma bridge aphysical modelrdquo American Journal of Physics vol 74 no 8 pp706ndash716 2006

[4] J Cheng and R-C Xiao ldquoA simplified method for lateralresponse analysis of suspension bridges under wind loadsrdquoCommunications in Numerical Methods in Engineering vol 22no 8 pp 861ndash874 2006


[5] X-H Long L Li and LHu ldquoTime domain analysis of buffetingresponses of sidu river suspension bridgerdquoEngineeringMechan-ics vol 27 no 1 pp 113ndash117 2010 (Chinese)

[6] M Yu H-L Liao M-S Li C-M Ma and M Liu ldquoFieldmeasurement and wind tunnel test of buffeting response oflong-span bridge under skew windrdquo Journal of Experiments inFluid Mechanics vol 27 no 3 pp 51ndash55 2013 (Chinese)

[7] H Wang A Li J Niu Z Zong and J Li ldquoLong-term moni-toring of wind characteristics at Sutong Bridge siterdquo Journal ofWind Engineering and Industrial Aerodynamics vol 115 pp 39ndash47 2013

[8] F T Lombardo J AMain and E Simiu ldquoAutomated extractionand classification of thunderstorm and non-thunderstormwinddata for extreme-value analysisrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 97 no 3-4 pp 120ndash131 2009

[9] H Wang A-Q Li C-K Jiao and X-P Li ldquoCharacteristics ofstrong winds at the Runyang Suspension Bridge based on fieldtests from 2005 to 2008rdquo Journal of Zhejiang University ScienceA vol 11 no 7 pp 465ndash476 2010

[10] MATLAB MATLAB online manual Natick Mass USA 2010[11] E Castano and J Martınez ldquoUse of the crosscorrelation func-

tion in the identification ofARMAmodelsrdquoRevistaColombianade Estadistica vol 31 no 2 pp 293ndash310 2008

[12] T I Lin and H J Ho ldquoA simplified approach to invertingthe autocovariance matrix of a general ARMA(p q) processrdquoStatistics and Probability Letters vol 78 no 1 pp 36ndash41 2008

[13] B Clarke J Clarke and C W Yu ldquoStatistical problem classesand their links to information theoryrdquoEconometric Reviews vol33 no 1ndash4 pp 337ndash371 2014

[14] J C Escanciano I N Lobato and L Zhu ldquoAutomatic spec-ification testing for vector autoregressions and multivariatenonlinear time series modelsrdquo Journal of Business amp EconomicStatistics vol 31 no 4 pp 426ndash437 2013

[15] J A Munoz J R Espinoza and C R Baier ldquoDecoupled andmodular harmonic compensation formultilevel statcomsrdquo IEEETransactions on Industrial Electronics vol 61 no 6 pp 2743ndash2753 2014

[16] S Yan and W Zheng ldquoWind load simulation by superpositionof harmonicrdquo Journal of Shenyang Architectural and CivilEngineering Institute vol 21 no 1 pp 1ndash4 2005 (Chinese)

[17] C-X Li and C-Z Liu ldquoRBF-neural-network-based harmonysuperposition methodrdquo Journal of Vibration and Shock vol 29no 1 pp 112ndash116 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2088100 100 1001003001088

3D ultrasonic anemometer

GPS monitoring station 300

Figure 1 Longitudinal layout of monitoring equipment on the Sutong Bridge (unit m)

41000

354002

4000

90009000 23000

GPSMS GPSMS

3DUA 3DUAUpstream Downstream

354002

Figure 2 Transverse layout of monitoring equipment on the flat steel box girder (unit mm) (1)GPSMS GPSmonitoring station (2) 3DUA3D ultrasonic anemometer

lateral displacement response from Xihoumen SuspensionBridge through wind tunnel tests and concluded that lateraldisplacement at horizontal angle 10∘ was larger than that ofother angles [6]

However for mechanism complexity of lateral dis-placement response impacted by aerostatic and fluctuatingwind actions traditional methods of theoretical deductionnumerical simulation and wind tunnel tests are difficult toaccurately reflect the actual lateral displacement responseof bridge structure on account of uncertain boundarycondition imprecise assignment of initial parameters andinappropriate ignorance of subordinate factors In recentyears with development of structural healthmonitoring tech-nology it is feasible to installmonitoring sensors on long spanbridge structures monitoring data of which can authenticallyreflect bridge structural behaviors under actual environmentand load actions Although wind field of long span bridgestructures has been widely monitored in recent years [7ndash9]lateral displacement response is rarely monitored andresearched thus real correlation regularity between lateraldisplacement response and wind action is still covered Addi-tionally lateral displacement response under actual operationenvironment is also affected by other random factors whichhave never been taken into account by researchers beforeTherefore lateral displacement response from main girder isnecessarily researched upon monitoring data to reveal realstructural behavior of long span bridges

In this paper based on the health monitoring systeminstalled on the main span of Sutong Cable-Stayed BridgeGPS displacement and wind field are real-time monitoredand analyzed According to analytical results apparent non-linear correlation with certain discreteness exists between

lateral static girder displacement and lateral static windvelocity thus time series of lateral static girder displace-ment are decomposed into nonlinear correlation term anddiscreteness term nonlinear correlation term of which ismathematically modeled by 119899th-order Fourier series withintervention of lateral static wind velocity and discretenessterm of which is mathematically modeled by the combinedmodels of ARMA(119901 119902) and EGARCH(119898 119899) Additionallystable power spectrum density exists in time series of lateraldynamic girder displacement thus time series of lateraldynamic girder displacement are mathematically modeledby harmonic superposition function By comparison andverification between simulative and monitoring lateral dis-placements from September 1 to September 3 mathematicalmodels are feasible and effective to simulate time series oflateral girder displacement frommain girder of SutongCable-Stayed Bridge

2 Bridge Monitoring and Sample Analysis

The bridge monitoring object for this research is theworldwide famous Sutong Cable-Stayed Bridge (in JiangsuProvince China) Its whole structure form is single-spannedand double-hinged with the main span reaching 1088m asshown in Figure 1 and the main girder employs flat steelbox type with 363m wide and 40m high as shown inFigure 2 3D ultrasonic anemometers and GPS monitoringstation are installed on two flanks of midspan cross-sectionfrom main girder (resp shown in Figures 1 and 2) tocontinuously acquire wind data and displacement data withsample frequency of 1Hz


x

y



z or Z



106∘


0 5 10 15 20 25 30

0

10

20

30

Time (day)

Win

d ve

loci

ty al

ong y

-axi

s (m

s)

minus10

minus20


0 5 10 15 20 25 30

0

10

20

30

Time (day)

Stat

ic w

ind

velo

city

(ms

)

minus10

minus20


0 5 10 15 20 25 30

0

5

10

15

Time (day)

Dyn

amic

win

d ve

loci

ty (m

s)

minus10

minus10

minus15




0 5 10 15 20 25 30

005

015

025

035

Time (day)

Gird

er d

ispla

cem

ent a

long

y-a

xis (

m)

minus005

minus015


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Stat

ic g

irder

disp

lace

men

t (m

)

minus01


0 5 10 15 20 25 30

0

004

008

012

Time (day)

Dyn

amic

gird

er d

ispla

cem

ent (

m)

minus012

minus008

minus004



5 15 250

005

01

015

02

025

03


Stat

ic g

irder

disp

lace

men

t (m

)

minus15 minus5



Frequency (Hz)


Day 25 to day 31

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7


0 10 20 300

006

012

018

024

03


Fitte

d lat

eral

gird

er d

ispla

cem

ent (

m)

minus20 minus10


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01








0 6 12 18 24 30

003

007

011

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus005

minus001


0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s


minus02

Auto

corr

elat

ion

valu



0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s

minus02

Part

ial c

orre

latio

n va

lue 120588








1199061(119905) = 119886

0+

119899

sum119896=1

(119886119896cos (119896120596V (119905)) + 119887

119896sin (119896120596V (119905))) (1)



119896 119887119896(119896 = 0 1 119899)





atic

disp

lace

men

t of d

iscre

tene

ss (m

)

0 6 12 18 24 30

001

003

005

Time (day)

minus005

minus003

minus001



Lag phase s 0 10 20 30 40 50

0

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

Part

ial c

orre

latio

n va

lue 120588

minus02

minus04




1199062(119905) = 119888 +

119901

sum119895=1

1198871198951199062(119905 minus 119895) +

119902

sum119895=0

119888119895120576119905minus119895 (2)


2(119905) 119887119895



0= 1 and 120576



log1205902119905

= 120581 +

119898

sum119896=1

119892119896log1205902119905minus119896

+

119899

sum119896=1

V119896[

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

minus 119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816]

+

119899

sum119896=1

119897119896(120576119905minus119896

120590119905minus119896

)

(3)


1 2 3 4 5 6 7 8minus33615

minus33605

minus33595

minus33585

minus33575

33565

The value of m

The s

tatis

tical

val

ue o

f AIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8


The s

tatis

tical

val

ue o

f BIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

1 2 3 4 5 6 7 8minus33530

minus33496

minus33458

minus33422

minus33386

minus33350

The value of m






119896 respec-



119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816 = 119864(

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

)

=

radic2

120587Gaussian




(4)



120582119898

minus 1198661120582119898minus1

minus 1198662120582119898minus2








120588 (119904) =(1119873)sum

119873minus119904

119905=1

(1199062(119905) minus 119906

2) (1199062(119905 + 119904) minus 119906

2)

1205902120588

119904 = 0 1 2 119873 minus 1

(6)

where

1199062=1

119873

119873

sum119905=1

1199062(119905) 120590

2

120588

=1

119873

119873

sum119905=1

(1199062(119905) minus 119906

2)2 (7)



119886119894119888 = minus2 ln (119871) + 21198701


2)

(8)





value



0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588



1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3


1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3






in Figure 8(b)









0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

y



z or Z



106∘


0 5 10 15 20 25 30

0

10

20

30

Time (day)

Win

d ve

loci

ty al

ong y

-axi

s (m

s)

minus10

minus20


0 5 10 15 20 25 30

0

10

20

30

Time (day)

Stat

ic w

ind

velo

city

(ms

)

minus10

minus20


0 5 10 15 20 25 30

0

5

10

15

Time (day)

Dyn

amic

win

d ve

loci

ty (m

s)

minus10

minus10

minus15




0 5 10 15 20 25 30

005

015

025

035

Time (day)

Gird

er d

ispla

cem

ent a

long

y-a

xis (

m)

minus005

minus015


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Stat

ic g

irder

disp

lace

men

t (m

)

minus01


0 5 10 15 20 25 30

0

004

008

012

Time (day)

Dyn

amic

gird

er d

ispla

cem

ent (

m)

minus012

minus008

minus004



5 15 250

005

01

015

02

025

03


Stat

ic g

irder

disp

lace

men

t (m

)

minus15 minus5



Frequency (Hz)


Day 25 to day 31

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7


0 10 20 300

006

012

018

024

03


Fitte

d lat

eral

gird

er d

ispla

cem

ent (

m)

minus20 minus10


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01








0 6 12 18 24 30

003

007

011

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus005

minus001


0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s


minus02

Auto

corr

elat

ion

valu



0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s

minus02

Part

ial c

orre

latio

n va

lue 120588








1199061(119905) = 119886

0+

119899

sum119896=1

(119886119896cos (119896120596V (119905)) + 119887

119896sin (119896120596V (119905))) (1)



119896 119887119896(119896 = 0 1 119899)





atic

disp

lace

men

t of d

iscre

tene

ss (m

)

0 6 12 18 24 30

001

003

005

Time (day)

minus005

minus003

minus001



Lag phase s 0 10 20 30 40 50

0

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

Part

ial c

orre

latio

n va

lue 120588

minus02

minus04




1199062(119905) = 119888 +

119901

sum119895=1

1198871198951199062(119905 minus 119895) +

119902

sum119895=0

119888119895120576119905minus119895 (2)


2(119905) 119887119895



0= 1 and 120576



log1205902119905

= 120581 +

119898

sum119896=1

119892119896log1205902119905minus119896

+

119899

sum119896=1

V119896[

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

minus 119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816]

+

119899

sum119896=1

119897119896(120576119905minus119896

120590119905minus119896

)

(3)


1 2 3 4 5 6 7 8minus33615

minus33605

minus33595

minus33585

minus33575

33565

The value of m

The s

tatis

tical

val

ue o

f AIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8


The s

tatis

tical

val

ue o

f BIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

1 2 3 4 5 6 7 8minus33530

minus33496

minus33458

minus33422

minus33386

minus33350

The value of m






119896 respec-



119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816 = 119864(

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

)

=

radic2

120587Gaussian




(4)



120582119898

minus 1198661120582119898minus1

minus 1198662120582119898minus2








120588 (119904) =(1119873)sum

119873minus119904

119905=1

(1199062(119905) minus 119906

2) (1199062(119905 + 119904) minus 119906

2)

1205902120588

119904 = 0 1 2 119873 minus 1

(6)

where

1199062=1

119873

119873

sum119905=1

1199062(119905) 120590

2

120588

=1

119873

119873

sum119905=1

(1199062(119905) minus 119906

2)2 (7)



119886119894119888 = minus2 ln (119871) + 21198701


2)

(8)





value



0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588



1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3


1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3






in Figure 8(b)









0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30

005

015

025

035

Time (day)

Gird

er d

ispla

cem

ent a

long

y-a

xis (

m)

minus005

minus015


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Stat

ic g

irder

disp

lace

men

t (m

)

minus01


0 5 10 15 20 25 30

0

004

008

012

Time (day)

Dyn

amic

gird

er d

ispla

cem

ent (

m)

minus012

minus008

minus004



5 15 250

005

01

015

02

025

03


Stat

ic g

irder

disp

lace

men

t (m

)

minus15 minus5



Frequency (Hz)


Day 25 to day 31

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7


0 10 20 300

006

012

018

024

03


Fitte

d lat

eral

gird

er d

ispla

cem

ent (

m)

minus20 minus10


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01








0 6 12 18 24 30

003

007

011

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus005

minus001


0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s


minus02

Auto

corr

elat

ion

valu



0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s

minus02

Part

ial c

orre

latio

n va

lue 120588








1199061(119905) = 119886

0+

119899

sum119896=1

(119886119896cos (119896120596V (119905)) + 119887

119896sin (119896120596V (119905))) (1)



119896 119887119896(119896 = 0 1 119899)





atic

disp

lace

men

t of d

iscre

tene

ss (m

)

0 6 12 18 24 30

001

003

005

Time (day)

minus005

minus003

minus001



Lag phase s 0 10 20 30 40 50

0

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

Part

ial c

orre

latio

n va

lue 120588

minus02

minus04




1199062(119905) = 119888 +

119901

sum119895=1

1198871198951199062(119905 minus 119895) +

119902

sum119895=0

119888119895120576119905minus119895 (2)


2(119905) 119887119895



0= 1 and 120576



log1205902119905

= 120581 +

119898

sum119896=1

119892119896log1205902119905minus119896

+

119899

sum119896=1

V119896[

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

minus 119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816]

+

119899

sum119896=1

119897119896(120576119905minus119896

120590119905minus119896

)

(3)


1 2 3 4 5 6 7 8minus33615

minus33605

minus33595

minus33585

minus33575

33565

The value of m

The s

tatis

tical

val

ue o

f AIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8


The s

tatis

tical

val

ue o

f BIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

1 2 3 4 5 6 7 8minus33530

minus33496

minus33458

minus33422

minus33386

minus33350

The value of m






119896 respec-



119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816 = 119864(

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

)

=

radic2

120587Gaussian




(4)



120582119898

minus 1198661120582119898minus1

minus 1198662120582119898minus2








120588 (119904) =(1119873)sum

119873minus119904

119905=1

(1199062(119905) minus 119906

2) (1199062(119905 + 119904) minus 119906

2)

1205902120588

119904 = 0 1 2 119873 minus 1

(6)

where

1199062=1

119873

119873

sum119905=1

1199062(119905) 120590

2

120588

=1

119873

119873

sum119905=1

(1199062(119905) minus 119906

2)2 (7)



119886119894119888 = minus2 ln (119871) + 21198701


2)

(8)





value



0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588



1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3


1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3






in Figure 8(b)









0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Frequency (Hz)


Day 25 to day 31

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7


0 10 20 300

006

012

018

024

03


Fitte

d lat

eral

gird

er d

ispla

cem

ent (

m)

minus20 minus10


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01








0 6 12 18 24 30

003

007

011

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus005

minus001


0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s


minus02

Auto

corr

elat

ion

valu



0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s

minus02

Part

ial c

orre

latio

n va

lue 120588








1199061(119905) = 119886

0+

119899

sum119896=1

(119886119896cos (119896120596V (119905)) + 119887

119896sin (119896120596V (119905))) (1)



119896 119887119896(119896 = 0 1 119899)





atic

disp

lace

men

t of d

iscre

tene

ss (m

)

0 6 12 18 24 30

001

003

005

Time (day)

minus005

minus003

minus001



Lag phase s 0 10 20 30 40 50

0

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

Part

ial c

orre

latio

n va

lue 120588

minus02

minus04




1199062(119905) = 119888 +

119901

sum119895=1

1198871198951199062(119905 minus 119895) +

119902

sum119895=0

119888119895120576119905minus119895 (2)


2(119905) 119887119895



0= 1 and 120576



log1205902119905

= 120581 +

119898

sum119896=1

119892119896log1205902119905minus119896

+

119899

sum119896=1

V119896[

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

minus 119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816]

+

119899

sum119896=1

119897119896(120576119905minus119896

120590119905minus119896

)

(3)


1 2 3 4 5 6 7 8minus33615

minus33605

minus33595

minus33585

minus33575

33565

The value of m

The s

tatis

tical

val

ue o

f AIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8


The s

tatis

tical

val

ue o

f BIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

1 2 3 4 5 6 7 8minus33530

minus33496

minus33458

minus33422

minus33386

minus33350

The value of m






119896 respec-



119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816 = 119864(

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

)

=

radic2

120587Gaussian




(4)



120582119898

minus 1198661120582119898minus1

minus 1198662120582119898minus2








120588 (119904) =(1119873)sum

119873minus119904

119905=1

(1199062(119905) minus 119906

2) (1199062(119905 + 119904) minus 119906

2)

1205902120588

119904 = 0 1 2 119873 minus 1

(6)

where

1199062=1

119873

119873

sum119905=1

1199062(119905) 120590

2

120588

=1

119873

119873

sum119905=1

(1199062(119905) minus 119906

2)2 (7)



119886119894119888 = minus2 ln (119871) + 21198701


2)

(8)





value



0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588



1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3


1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3






in Figure 8(b)









0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 6 12 18 24 30

003

007

011

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus005

minus001


0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s


minus02

Auto

corr

elat

ion

valu



0 10 20 30 40 50

0

02

04

06

08

1

12

Lag phase s

minus02

Part

ial c

orre

latio

n va

lue 120588








1199061(119905) = 119886

0+

119899

sum119896=1

(119886119896cos (119896120596V (119905)) + 119887

119896sin (119896120596V (119905))) (1)



119896 119887119896(119896 = 0 1 119899)





atic

disp

lace

men

t of d

iscre

tene

ss (m

)

0 6 12 18 24 30

001

003

005

Time (day)

minus005

minus003

minus001



Lag phase s 0 10 20 30 40 50

0

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

Part

ial c

orre

latio

n va

lue 120588

minus02

minus04




1199062(119905) = 119888 +

119901

sum119895=1

1198871198951199062(119905 minus 119895) +

119902

sum119895=0

119888119895120576119905minus119895 (2)


2(119905) 119887119895



0= 1 and 120576



log1205902119905

= 120581 +

119898

sum119896=1

119892119896log1205902119905minus119896

+

119899

sum119896=1

V119896[

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

minus 119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816]

+

119899

sum119896=1

119897119896(120576119905minus119896

120590119905minus119896

)

(3)


1 2 3 4 5 6 7 8minus33615

minus33605

minus33595

minus33585

minus33575

33565

The value of m

The s

tatis

tical

val

ue o

f AIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8


The s

tatis

tical

val

ue o

f BIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

1 2 3 4 5 6 7 8minus33530

minus33496

minus33458

minus33422

minus33386

minus33350

The value of m






119896 respec-



119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816 = 119864(

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

)

=

radic2

120587Gaussian




(4)



120582119898

minus 1198661120582119898minus1

minus 1198662120582119898minus2








120588 (119904) =(1119873)sum

119873minus119904

119905=1

(1199062(119905) minus 119906

2) (1199062(119905 + 119904) minus 119906

2)

1205902120588

119904 = 0 1 2 119873 minus 1

(6)

where

1199062=1

119873

119873

sum119905=1

1199062(119905) 120590

2

120588

=1

119873

119873

sum119905=1

(1199062(119905) minus 119906

2)2 (7)



119886119894119888 = minus2 ln (119871) + 21198701


2)

(8)





value



0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588



1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3


1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3






in Figure 8(b)









0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










atic

disp

lace

men

t of d

iscre

tene

ss (m

)

0 6 12 18 24 30

001

003

005

Time (day)

minus005

minus003

minus001



Lag phase s 0 10 20 30 40 50

0

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

Part

ial c

orre

latio

n va

lue 120588

minus02

minus04




1199062(119905) = 119888 +

119901

sum119895=1

1198871198951199062(119905 minus 119895) +

119902

sum119895=0

119888119895120576119905minus119895 (2)


2(119905) 119887119895



0= 1 and 120576



log1205902119905

= 120581 +

119898

sum119896=1

119892119896log1205902119905minus119896

+

119899

sum119896=1

V119896[

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

minus 119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816]

+

119899

sum119896=1

119897119896(120576119905minus119896

120590119905minus119896

)

(3)


1 2 3 4 5 6 7 8minus33615

minus33605

minus33595

minus33585

minus33575

33565

The value of m

The s

tatis

tical

val

ue o

f AIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8


The s

tatis

tical

val

ue o

f BIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

1 2 3 4 5 6 7 8minus33530

minus33496

minus33458

minus33422

minus33386

minus33350

The value of m






119896 respec-



119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816 = 119864(

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

)

=

radic2

120587Gaussian




(4)



120582119898

minus 1198661120582119898minus1

minus 1198662120582119898minus2








120588 (119904) =(1119873)sum

119873minus119904

119905=1

(1199062(119905) minus 119906

2) (1199062(119905 + 119904) minus 119906

2)

1205902120588

119904 = 0 1 2 119873 minus 1

(6)

where

1199062=1

119873

119873

sum119905=1

1199062(119905) 120590

2

120588

=1

119873

119873

sum119905=1

(1199062(119905) minus 119906

2)2 (7)



119886119894119888 = minus2 ln (119871) + 21198701


2)

(8)





value



0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588



1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3


1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3






in Figure 8(b)









0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 2 3 4 5 6 7 8minus33615

minus33605

minus33595

minus33585

minus33575

33565

The value of m

The s

tatis

tical

val

ue o

f AIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8


The s

tatis

tical

val

ue o

f BIC

n = 1n = 2n = 3n = 4

n = 5n = 6n = 7n = 8

1 2 3 4 5 6 7 8minus33530

minus33496

minus33458

minus33422

minus33386

minus33350

The value of m






119896 respec-



119864 1003816100381610038161003816119911119905minus119896

1003816100381610038161003816 = 119864(

1003816100381610038161003816120576119905minus1198961003816100381610038161003816

120590119905minus119896

)

=

radic2

120587Gaussian




(4)



120582119898

minus 1198661120582119898minus1

minus 1198662120582119898minus2








120588 (119904) =(1119873)sum

119873minus119904

119905=1

(1199062(119905) minus 119906

2) (1199062(119905 + 119904) minus 119906

2)

1205902120588

119904 = 0 1 2 119873 minus 1

(6)

where

1199062=1

119873

119873

sum119905=1

1199062(119905) 120590

2

120588

=1

119873

119873

sum119905=1

(1199062(119905) minus 119906

2)2 (7)



119886119894119888 = minus2 ln (119871) + 21198701


2)

(8)





value



0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588



1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3


1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3






in Figure 8(b)









0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

0

10 20 30 40 50


Lag phase s

02

04

06

08

1

minus02

minus04

Auto

corr

elat

ion

valu

e120588


0 10 20 30 40 50

0

02

04

06

08

1


Lag phase s

minus02

minus04

Part

ial c

orre

latio

n va

lue 120588



1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

r times10

minus3


1 2 3 4 5 6 7590

591

592

593

594

595

596


q = 1q = 2

q = 3q = 4

The s

tand

ard

erro

rtimes10

minus3






in Figure 8(b)









0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 6 12 18 24 30

minus001

001

003

005

Time (day)

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

minus003

minus005


minus001

Stat

ic d

ispla

cem

ent o

f disc

rete

ness

(m)

0 6 12 18 24 30

003

007

011

Time (day)

minus005


0 5 10 15 20 25 30

0

01

02

03

Time (day)

Fitti

ng st

atic

disp

lace

men

t (m

)

minus01






1198861

1198871

1198862

1198872

1198863

1198873

119908












Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus5

10minus6

10minus610minus7



Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7





1

1198872

1198873

1198874

1198875

1198876

1198877

1198881

1198882

1198883

1198884




1

1198922

V1

1198971










lg1199011(119891) =

119899

sum119896=1

119886119896119890minus((lg119891minus119887119896)119888119896)

2

lg1199012(119891) =

119899

sum119896=1

119889119896119890minus((lg119891minus119903119896)ℎ119896)

2

(9)


2(119891)


and 119903119896


2(119891)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p1(f)


Monitoring curve

Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

102

100

101


Fitting curve p2(f)


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7

Fitting curve p(f)


2(119891)



119901 (119891) = 1199011(119891) + 119901

2(119891) (10)




1205902

= int1198912

1198911

119901 (119891) 119889119891 (11)




1205902

asymp

119899

sum119894=1

119901 (119891mid119894) sdot Δ119891 (12)

where

Δ119891 =1198912minus 1198911

119899 (13)





0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 14400 28800 43200 57600 72000 86400

0

002

004

Time (s)

Sim

ulat

ive d

ynam

ic d

ispla

cem

ent (

m)

minus004

minus002


Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

100

100

10minus1

10minus1

10minus2

10minus2

10minus3

10minus3

10minus4

10minus4

10minus5

10minus6

10minus510minus7



Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)

minus005

minus01


Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

0

005

01

015

02

Time (day)

Mon

itorin

g lat

eral

disp

lace

men

t (m

)

minus005

minus01





119908 (119905) =

119899

sum119894=1






0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

005

01

015

02

Sim

ulat

ive l

ater

al d

ispla

cem

ent (

m)


minus005

minus01

Sept

embe

r 1

Sept

embe

r 2

Sept

embe

r 3

Sept

embe

r 4

Time (day)


0 005 01 015

0

005

01

015


onito

ring

later

al d

ispla

cem

ent (

m)


minus005minus005

ds(t) = 1008dm(t) + 00016




Frequency (Hz)

Pow

er sp

ectr

um d

ensit

y (m

2s)

10minus1

10minus110minus2

10minus3

10minus3

10minus5

10minus410minus9

10minus7



1(119891)



1(119891) and119901

2(119891) is shown

in Figure 16(c)





1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1198862

1198863

1198864

1198871

1198872

1198873

1198874

1198881

1198882

1198883

1198884

1199011






1(119905) ARMA(74)



119898(119905) in Figure 19(b) (where 119889

119904(119905)



5 Conclusions









Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Mathematical Modeling for Lateral ...downloads.hindawi.com/journals/mpe/2014/723152.pdf · from Main Girder of Sutong Cable-Stayed Bridge ... According to aerostatic