review article a summary review of correlations between...

Review ArticleA Summary Review of Correlations between Temperatures andVibration Properties of Long-Span Bridges

Guang-Dong Zhou1 and Ting-Hua Yi23

1 College of Civil and Transportation Engineering Hohai University Nanjing 210098 China2 School of Civil Engineering Dalian University of Technology Dalian 116023 China3 State Key Laboratory for Disaster Reduction in Civil Engineering Tongji University Shanghai 200092 China

Correspondence should be addressed to Guang-Dong Zhou zhougdhhueducn

Received 8 January 2014 Accepted 6 March 2014 Published 13 April 2014

Academic Editor Hua-Peng Chen

Copyright copy 2014 G-D Zhou and T-H Yi 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

The shift ofmodal parameters induced by temperature fluctuationmaymask the changes of vibration properties caused by structuraldamage and result in false structural condition identification Thoroughly understanding the temperature effects on vibrationproperties of long-span bridges becomes an especially important issue before vibration-based damage detection methodologies areapplied in real bridges This paper presents an overview of current research activities and developments in the field of correlationsbetween temperatures and vibration properties of long-span bridges The theoretical derivation methods using classical structuraldynamics and closed-form formulations are first briefly introduced Then the trend analysis methods that are intended to extractthe degree of variability in vibration property under temperature variation for different bridges by numerical analysis laboratorytest or field monitoring are reviewed in detail Following that the development of quantitative models to quantify the temperatureinfluence on vibration properties is discussed including the linear model nonlinear model and learning model Finally somepromising research efforts for promoting the study of correlations between temperatures and vibration properties of long-spanbridges are suggested

1 Introduction

The long-span bridges which represent the key elements interms of the safety and functionality of the entire trans-portation system are susceptible to the combined action ofenvironmental conditions (eg temperature winds rainsand humidity) service loads (eg railway loads highwayloads) and catastrophic events (eg earthquakes hurricanesfloods and vehicle collision) [1ndash3] In the United States 15of 595000 bridges were rated as being deficient for structuralreasons by the US Federal Highway Agency (FHWA) in2005 and more than 134 bridges were reported to havepartially or totally collapsed between 1989 and 2000 [4ndash6] Therefore how to maintain structural safety becomesa challenging task Structural health monitoring (SHM)which aims at monitoring structural behavior in real-timeevaluating structural performance under various loads andidentifying structural damage or deterioration provides a

better solution for this problem [7ndash10] Recent advances insensing data acquisition computing communication anddata and informationmanagement have greatly promoted theapplications of SHM technology in bridge structures [11] Atpresent SHM has been extensively implemented on existingor newly built bridges in Europe USA Canada Japan KoreaChina and other countries [12 13]

For a sophisticated bridge SHM system one of theextremely important functions is damage detection which isalso the original intention of developing SHM technology Inorder to fulfill this requirement a variety of damage detectionmethods have been proposed for different types of long-spanbridges Among them the vibration-based methodology isdominant [14ndash16] The basic concept underlying the use ofvibration-based damage detection is that global vibrationproperties notably natural frequencies mode shapes andmodal damping are functions of the physical parameters

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 638209 19 pageshttpdxdoiorg1011552014638209

2 Mathematical Problems in Engineering

of the structural mass damping and stiffness Changes inthe physical parameters will cause detectable changes inthe vibration properties [17] Up to now various damagedetection methods that are based on the vibration propertiesor their derivatives have been developed such as coordinatemodal assurance criterion (COMAC) enhanced coordinatemodal assurance criterion (ECOMAC) damage index (DI)method mode shape curvature (MSC) method modal flexi-bility index (MFI) changes in stiffness matrix obtained frommeasured mode shapes and changes in the curvature ofuniform load surface [18ndash21] However the application ofvibration-based damage detection methods for real long-span bridges is complicated because vibration properties ofa real structure are strongly affected by factors other thanstructural damages [22] Environmental changes especiallytemperature also cause changes in vibration propertiessuch as frequency mode shapes and damping ratio whichmay mask the changes caused by structural damage [23]To avoid false condition assessments the variability of thevibration properties due to temperature must be thoroughlyunderstood and quantified so that it can be discriminatedfrom changes caused by damage [17] As a matter of fact thevariation of temperatures affects vibration properties of long-span bridges in a complicated manner Firstly the tempera-ture distribution and its variation in a bridge are generallynonuniform and time dependent which induces that thechanges of physical parameters among different structuralparts are asynchronous [24] Secondly all members of abridge change their size under different temperatures for theeffect of thermal expansion and contraction [25]Thirdly theYoungrsquos modulus and the shear modulus of material whichare the main parameters determining vibration propertiesof bridges vary because of the fluctuation of temperatureFourthly daily or seasonal variations in temperature maychange the stiffness of bridge expansion joints or supportsand thus strengthen or weaken the constraints resultingin an apparent shift of vibration properties [26] Lastlytemperature variations result in considerable thermal stressesor stress redistribution in structure which may change thevibration properties of bridges Numerous investigationsindicate that temperature is the critical source causing thevariability of vibration properties of long-span bridges andthe variations of modal frequencies caused by temperaturemay reach 5 to 10 for highway bridges which inmost casesexceed the changes of frequencies due to structural damage ordeterioration [12]

During the past decades considerable effort has beendevoted to investigating the correlations between tempera-tures and vibration properties of long-span bridges Accord-ing to the approach of researching the existing achievementscan be categorized into three types termed as theoreticalderivation trend analysis and quantitative model The the-oretical derivation is used to establish closed-form formu-lations of the correlations between temperatures and vibra-tion properties of long-span bridges by classical structuraldynamic methods The trend analysis is applied to analyzethe variation of vibration properties of long-span bridgescaused by temperature changes through numerical simula-tion laboratory test and field monitoring The quantitative

model is intended to develop no physical blindmodels whichcan quantificationally forecast changes in vibration propertiesbased on the input of time series of new temperaturesthrough a large amount of measurement data This paperaims to present a summary review of correlations betweentemperatures and vibration properties of long-span bridgesAlthough Xia et al [30] gave a literature review concerningvariations in vibration properties of civil structures underchanging temperature conditions before the content is toosketchy to give reference for later research Furthermoresome important contributions were omitted In the fol-lowing sections research progress of correlations betweentemperatures and vibration properties of long-span bridgesis extensively described from theoretical derivation methodtrend analysis approach and quantitative model In additionsome existing problems and promising research efforts aboutthis topic are expected to be extracted through pertinentassessment This paper is not intended to list all the literatureabout the effects of temperature on vibration properties ofbridges but exhibit some representative achievements whichare expected to provide reference for future research

2 Progress of Theoretical Derivation Method

A long-span bridge is a complex multi-degree-of-freedomnonlinear system This makes it is extremely difficult toformulate the vibration properties by closed-form equa-tions Furthermore temperature change affects a bridge in acomplicated way It not only affects mechanical propertiesbut also geometrical features constraint conditions andboundary conditions As a result only a few researchershave investigated the correlations between temperatures andvibration properties of long-span bridges through the theo-retical derivation methods

In order to simplify the problem Xia et al [34] assumedthat mass and the boundary conditions remain unchangedand temperature affects only the geometry and mechanicalproperties A single-span beammade of an isotropic materialwas used as an example to model the effect of temperatureon natural frequencies For this beam its undamped flexuralvibration frequency of order n is [35]

119891119899=1198992120587ℎ

21198972radic

119864

12120588 (1)

where119891119899denotes the 119899th frequency ℎ and 119897 are the height and

length of the beam respectively 119864 represents the modulusof elasticity and 120588 denotes the density of the material Thevariation of the above equation can be expressed as

120575119891119899

119891119899

=120575ℎ

ℎminus 2

120575119897

119897+1

2

120575119864

119864minus1

2

120575120588

120588 (2)

Mathematical Problems in Engineering 3

where 120575 represents an increment in the correspondingparameter Assuming the thermal coefficient of linear expan-sion of the material is 120579

119905and the temperature coefficient of

modulus is 120579119864 one obtains

120575ℎ

ℎ= 120579119905120575119905

120575119897

119897= 120579119905120575119905

120575120588

120588= minus3120579

119905120575119905

120575119864

119864= 120579119864120575119905

(3)

where 119905 is the temperature of the beam Here it is assumedthat the Youngrsquos modulus variation with temperature is linearfor small temperature changes Consequently (2) yields

120575119891119899

119891119899

=1

2(120579119905+ 120579119864) 120575119905 (4)

For prismatic multispan beams the natural frequencies canbe expressed in a similar form to those of a simply supportedbeam as [30 34]

119891119899=1205822119899120587ℎ

21198972radic

119864

12120588 (5)

where 120582119899is a dimensionless parameter and is a function

of the boundary conditions Consequently (2)ndash(4) are stillapplicable for multispan beam structures

The thermal axial force that arises because of changesin the support bearings alters the natural frequencies bychanging the longitudinal bending stiffness The frequenciesof the beam under constant axial force are based on theanalysis of Clough and Penzien [35]

1198911015840119899= 119891119899

radic1 +1198651198972

11989921205872119864119868 (6)

where 1198911015840119899denotes the modal frequency including axial force

F represents the constant axial force and I is the moment ofinertia

Equation (4) estimates the dimensionless rate of thefrequency variation with the temperature change For steelor concrete 120579

119864is obviously much larger than 120579

119905 which

indicates that variations in natural frequency subjected totemperature change are controlled by 120579

119864 It also shows that

the variation percentage of natural frequency is a function ofthe modulus thermal coefficients only regardless of modesspans and beam types (simply supported multispan orcantilever beam)

3 Developments of Trend Analysis Approach

For a real bridge it is impossible to describe its vibrationproperties by theoretical formulations exactly except byintroducingmany rigorous hypotheses Trend analysis whichintends to assess the trend and degree that the vibrationproperties of long-span bridges change with the fluctuationof temperature becomes an indispensable alternative Threeapproaches can be used to conduct the trend analysis the

numerical simulation approach the laboratory test approachand the field monitoring approach The SHM system obtainslong-term bridge vibration data and corresponding tempera-tures of the structure which provides a wonderful opportu-nity to extracting the correlations between temperatures andvibration properties of long-span bridges

31 Numerical Simulation Approach Fu and DeWolf [36]developed a nonlinear dynamic finite element (FE) analysisto assess the changes of frequencies in a two-span skewedcomposite bridge due to the partially restrained bearingsTheFE model was based on use of beam elements for the girdersand diaphragms and shell elements for the slabThe field datain the unrestrained state was used to calibrate the FE modelto account for the nonlinear cracking behavior in the slab anddiaphragm interaction The FE model was then modified toaccount for changes in the vibrational behavior due to theeccentrically applied bearing forces which occurred whenthe bearings were partially restrained in colder weatherWiththis analysis it was confirmed that the variations of boundaryconditions due to temperature fluctuation cause a change inthe natural frequencies

Macdonald and Daniell [37] established a detailed FEmodel for a cable-stayed bridge with a main span of 456mThe reinforced concrete (RC) slab of the deck was modeledwith four shell elements and the longitudinal girder wasmodeled with beam elements The transverse trusses andcable anchorage beams were represented by beam elementsThe bearings were modeled as fixed in all translationaldirections whilst being free to rotate Modal analyses wereperformed on the FE model for two different temperatureconditions The first condition was a uniform temperaturechange of 5∘C to the whole structure and the second wasa temperature differential of +10∘C for the top of the deckcompared to the rest of the structure It was found thatfor both conditions the natural frequencies of about onethird of the modes shifted by up to plusmn02 while there wasno noticeable change in the others This analysis indicatedthat the modal parameters of the bridge are not influencedsignificantly by normal temperature changes which is a littlebit different with the following outcomes

Xu and Wu [38] implemented a systemic investigationabout the frequency and mode shape curvature variations ofa cable-stayed concrete box girder bridge caused by uniformtemperature changes and nonuniform temperature changesthrough three-dimensional FE analysis The FE modelemploys beam elements link elements contact elementsspring elements and mass elements Three factors includingthe size thermal effect induced bymaterial thermal expansioncoefficient the change of elastic modulus of concrete andthe cables sag induced by cable gravity were considered Theresults showed that the variation behavior of frequency andmode shape curvature caused by nonuniform temperaturechange is similar to those caused by uniform temperaturechange The maximum variation ratio of frequency could be2 as uniform temperature increases from minus20 to +40∘CThe effect of temperature on frequencies corresponding tobending mode type is more obvious especially for the


vertical bending mode Changes of mode shape curvaturedue to ambient temperature variations usually happen atthe midspan and one-fourth span of the central span themidspan of the side span and in supports in the cable-stayedbridgeThe corresponding variation ratio is approximately 1ndash8

In order to outline the temperature effect an FE modelof bridge deck was developed by Balmes et al [39] Thewhole bridge span is modeled using 9600 standard 8-node isoparametric volume elements and 13668 nodesThe temperature variations were modeled using a uniformtemperature elevation It was found that the first frequencyincreases while the temperature decreases Miao et al [40]analyzed the dynamic characteristics of a three-tower andtwo-span suspension steel-box girder bridge at differentglobal temperatures by spatial FE method This bridge isthe first multitower suspension bridge whose span exceeds1000 m in the world An equivalent cable force method wasproposed to simulate temperature effects and the effects ofenvironmental temperature on dynamic characteristics ofthe bridge were studied The result revealed that the effectsof temperature cannot be neglected in dynamic analysis ofsuspension bridges The relationship between temperatureand frequency is negative

As the temperature of an entire structure is gener-ally nonuniformly distributed using the air temperatureor surface temperatures alone may not sufficiently capturethe relation between the bridge vibration properties andtemperatures Xia et al [28] proposed a new approach toinvestigate the variation of bridge vibration characteristicswith respect to structural temperatures In contrast to theconventional studies that use air temperature or temperaturesat a few points of the structure this study aims to obtain thethermal distribution of the entire structure from the ther-modynamic models As the material mechanical propertiesare temperature dependent the structure was regarded asa composite structure consisting of elements with differentYoungrsquos module At first an FE model was established toconduct a transient thermal analysis and estimate the tem-perature distribution of the slab And then the temperaturedata at all components and thermal properties of the materialwere inputted to the model to calculate the frequenciesConsequently the relation between the frequencies and thetemperature at the whole structure was established A simplysupported RC slab model was constructed and used asa proof-of-concept example The outcomes of this studyverified that the frequencies are global properties and areassociated with the temperature distribution of the entirestructure Therefore consideration of the temperature distri-bution of a whole structure will lead to more accurate resultsof the temperature effect on the vibration properties of thestructure

32 Laboratory Test Approach When the numericalapproach is adopted in trend analysis of the temperatureeffect on structural vibration properties many hypothesesand simplifications must be employed and the outcomesdepend on the input parameters Therefore the results

obtained by the numerical approach may deviate fromactual occasions Contrarily laboratory test can reveal thevariations of structural vibration properties under differentclimates and provides more rational conclusions

Xia et al [34] carried out periodical vibration tests on areinforced concrete slab for nearly two years The slab whichmeasures 6400mm times 800mm times 100mm with 3000mmspans and 200mm overhang on each end was constructedand placed outside the laboratory exposed to weather condi-tions Its vibration properties namely vibration frequenciesmode shapes and damping were measured together withthe temperature during each measurement It was found thatthe frequencies of first four modes have a strong negativecorrelation with temperature damping ratios have a positivecorrelation but no clear correlation of mode shapes withtemperature change can be observed The three bendingmodes gave similar results namely that the modal frequen-cies decrease 013ndash023 when temperature increases byone degree and the torsional mode that is affected by theshear rigidity of the structure has different sensitivity totemperature than bending rigidity does The damping ratioincreases slightly with the increase of temperature

Kim et al [27] performed a series of forced vibration testson a model bridge for about 7 months from December 1999to July 2000 The test model bridge is a single-span stainlesssteel plate girder bridgeThe bridge spans 20m and is simplysupported with a set of pin supports at the left edge and aset of roller supports at the other edge The superstructureof the bridge consists of the deck the deck supportingsystems and piers The substructure consists of four steelblocks and a system of steel frame bases Environmentalfactors other than temperature were not considered duringthe tests Experimental setups for locations and arrangementsof accelerometers and excitation sources on the test structurewere designed as shown in Figure 1 When these tests wereperformed temperatures varied between a low of minus3∘C anda high of 23∘C The results demonstrated that in all modesnatural frequencies go down as the temperature goes upBending modes are more sensitive than torsional modes Asone progresses to lowermodes the temperature has relativelygreater effects on the change in natural frequency Naturalfrequencies at 6 different temperatures are listed in Table 1The first four frequencies decreased by about 064 033 044and 022 respectively when temperature increased per unitdegree Two levels of damage were inflicted on a girder nearthe center of the structure the bottom flange was cut halfwayin from outside and the bottom flange was cut completelyin from either side The variations of vibration propertiesinduced by damages are almost the same level as that causedby temperature changes

Balmes et al [41] carried out an aluminum beam experi-ment in a climatic chamberThe results demonstrated that theaxial prestress of this beam because of thermal change variedsignificantly and led to changes in the first four frequencies by16 8 5 and 3 respectively The variations in mode shapeswere invisible when temperature decreased by about 17∘CThe effect of temperature on damping was also concernedbut no consistent conclusion was drawn They stated thatone possible reason is that the damping of structures is more


Table 1 Predamage natural frequencies of model bridge at varioustemperatures (Kim et al [27])

Modenumber

Frequency (Hz) of undamaged structure at varioustemperatures

0∘C 10∘C 20∘C 30∘C 40∘C 50∘C1 76150 71822 68034 66194 65328 631642 101455 98320 95185 94244 93617 920503 212223 203694 195165 192606 190901 1866364 295347 289123 282899 281031 279787 276674

difficult tomeasure accurately comparedwith the frequenciesand the mode shapes A large uncertainty in modal dampingmight mask the temperature effects

Xia et al [28] constructed a simply supported RC slabwhich measures 3200mm long 800mm wide and 120mmthick with 100mm overhang at each end as shown inFigure 2 This study aims to investigate the variation of thestructural vibration characteristics with respect to differenttemperatures Sets of modal testing were carried out hourlyfrom 800 am to 2200 pm on February 11 2009 Thelocations of the acceleration sensors are shown in Figure 3At the same time structural temperatures were automaticallyrecorded from the embedded thermocouples every 2minFigure 4 illustrates the variation of the measured first naturalfrequency with respect to temperatures measured from thethermocouples It clearly shows that the frequency of thestructure decreases when temperature goes up before noonwhile the frequency increases as temperature drops down inthe afternoon as observed by many researchers Howeverthe temperature effect on damping ratios is not so obviousFigure 5 illustrates the variation of the first two modaldamping ratios extracted from measurement with respect totime It was found that damping ratios increase slightly whenthe temperature rises Since the variation of temperature hereis not significant and the uncertainty level of measurementof damping is relatively high the change of damping may bemasked by the measurement noise

Xia et al [30] examined the effect of temperature onvibration properties of different structures made of differentmaterials in particular steel aluminum and RC throughlaboratory studies An RC slab a steel beam and an alu-minum beam were constructed outside the laboratory andexposed to solar irradiationThe RC slab is simply supportedwhile the support conditions of the two metallic beamsare cantilever Temperature variations were recorded fromthe installed thermocouples Modal testing was carried outusing an instrumented hammerThe outcomes indicated thatall natural frequencies of the three beams decreased withan increase in surface temperature and increased with adecrease in surface temperatureThe variation trendmatchedthe surface temperature very well although the frequencyvariations were very small The frequency variations of theRC slab are much larger than those of the metallic beams Noclear correlation between damping ratios of the three beamsand temperature can be found indicating that temperaturehas little effect on damping ratios

In addition Sun et al [42 43] performed an experi-mental study on correlation between dynamic properties ofa cable-stayed bridge model and its structural temperatureThe results indicate a tendency that the frequency of thecable-stayed bridge model decreases with the increase oftemperature

33 Field Monitoring Approach The temperature variationsin bridges are affected by air temperature wind humidityintensity of solar radiation material type and so forth [44]The vibration properties of bridges are governed not onlyby material characteristics and geometrical configurationsbut also by boundary conditions and load cases The abovetwo reasons make the correlations between temperatures andvibration properties of long-span bridges difficult to extractby theoretical derivation numerical simulation and labora-tory test Fieldmeasurement thatmonitored the temperaturesand vibration responses of a bridge by sensors installedon it can give the most objective data subjected to realservice circumstanceThe correlations between temperaturesand vibration properties deduced from those data may bein turn more convincing With the development of sensorand computer technology continuous field measurements oftemperatures and vibration responses in structures have beencarried out extensively in various bridges

In the 1980s being limited to the measurement instru-ments long-termdynamic tests of bridges are seldom Turner[45] tested a three-span concrete foot-bridge which consistsof one central 16m and two 8m side spans Results showedthat the natural frequencies of this small bridge appearedunaffected by temperature changes Then he further studieda highway bridge and stated that there is ldquosome evidencerdquoto show that the natural frequencies of the bridge structurethey tested did not change more than 05 as a result ofenvironmental effects [46] In 1988 Askegaard and Mossing[47] tested a three-spanRC footbridge for a three-year periodand about 10 seasonal changes in the frequencies wererepeatedly observed for each year The authors concludedthat the changes were partially attributed to the variation ofambient temperature

Wood [48] reported that bridge responses were closelyrelated to the structural temperature based on vibrationtests of five bridges in the United Kingdom Here the testresults are different from those obtained by Turner This isbecause the span of the bridge tested by Turner is smalland the effect of temperature on the dynamic characteristicsof the bridge is slight while the bridges tested by Woodare long-span bridges and the effect of temperature on thedynamic characteristics of these bridges is obvious Analysesbased on the data compiled suggested that the variability ofthe asphalt elastic modulus due to temperature effects is amajor contributor to the changes in the structural stiffnessTemperature variation not only changes the material stiffnessbut also alters the boundary conditions of a system

Rohrmann et al [49 50] presented the results of testsperformed on a 7-span highway bridge in Berlin over a six-month time period Those studies showed that the temper-ature effects on the dynamics bridge cannot be neglected


Station numberActuator

North girder

South girder

Power supply

Signal

Sensor

Model analysis

1 2 3 4 5 6 7 8 9

1110 12 13 14 15 16

R A

17 18

R reference channelA active channel

generator

Figure 1 Test setup for model plate-girder bridge [27]

Figure 2 The RC slab [28]

Accelerometers

Support Support

Thermocouples Thermocouples(T1 sim T7)

1 2 3 4 5 6 7

8 9 10 11 12 13 14

100mm 100mm

(T9 sim T11)

50

mm

50

mm

700

mm

375 times 8 = 3000mm

Figure 3 Locations of the acceleration sensors [28]

The natural frequencies increase as the mean temperaturedecreases On a time scale of days these tests indicated thefirst mode frequency varying from 23 to 28Hz DeWolfet al [51 52] studied the interaction between the variabletemperature and the governing natural frequencies of a twospan skewed composite steel-girder bridge It was found thatthe maximum changes in the natural frequencies based oncomparison with the baseline frequencies at 55∘F (128∘C)were 116 for the first frequency 154 for the second fre-quency and 94 for the third frequency respectively Theseoccurred at the temperatures of 4∘F (minus156∘C) 4∘F (minus156∘C)and 15∘F (minus94∘C) respectively As noted previously thenatural frequency increased with decreasing temperatureBetween the temperature at 55∘F (128∘C) and 10∘F (minus122∘C)the maximum percent difference of modal displacements forthe first bending mode the first torsional mode and thesecond bendingmode are 221 100 and 471 and the percent

3700

3500

3300

3100

2900

2700

2500

2300

2100

1900

1700

Tem

pera

ture

(∘C)

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2015

2010

2005

2000

1995

1990

1985

1980

1975

1970

1965

Temperature-topTemperature-bottom

FrequencyTime (hhmm)

Figure 4 Variation of the first frequency versus temperatures [28]

difference in the diagonal terms of the modal flexibilityis 208 341 and 223 for the first second and thirdmodeThemodal flexibility for the two different temperaturelevels produces significantly larger changes than the naturalfrequencies

Researchers from Los Alamos National Laboratory [53ndash55] performed several tests on the Alamosa Canyon Bridgelocated in southern New Mexico This bridge has sevenindependent spans and each span consists of a concreteddeck supported by six W30 times 116 steel girders The resultsrevealed that the first three natural frequencies of the bridgevaried about 47 66 and 50 during a 24 h period asthe temperature of the bridge deck changed by approximately22∘C The temperature gradient largely influences the vari-ation of the fundamental frequency They also found thata significant artificial cut in the I-40 Bridge over the Rio


2

18

16

14

12

1

08

06

02

04

0

Time

Dam

ping

ratio

()

1st mode2nd mode

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

Figure 5 Variation of the damping ratios [28]

Grande caused only insignificant frequency changes [56]Wahab and De Roeck [57] investigated the seasonal effect oftemperature variations on natural frequencies of a concretebridge and found that changes in natural frequencies of thebridges can be up to 5-6 during a year Roberts and Pearson[58] described a monitoring program on a 9-span 840mlong bridge It was found that normal environmental changescould account for changes in frequencies of as much as 3-4during the year

Alampalli [59] conducted several tests over nine monthson a steel-stringer bridge with a concrete deck to examinethe sensitivity of measured modal parameters to variationsresulting from test and in-service environmental conditionsThe results confirmed that for the bridge with a span of676m and a width of 526m the natural frequency changescaused by the freezing of the bridge supports were an orderof magnitude larger than the variation introduced by an arti-ficial saw cut across the bottom flanges of both girders It wassuggested that bridges should be monitored for at least onecomplete cycle of in-service conditions before establishingwarning triggers or using damage assessment algorithms sothat the influence of temperature can be well understoodRohrmann et al [60] further studied the thermal effecton modal frequencies of an eight-span prestressed concretebridge by using three-year continuous monitoring data inan attempt to establish functional proportionality betweenthe temperature variations and changing natural frequenciesThey concluded that vibration frequencies decrease with thetemperature increase

Peeters et al [61ndash63] conducted a study devoted tomonitoring the effect of changing environmental conditionson structural vibration properties on the Z24 Bridge inSwitzerland It was continuously monitored for nearly a yearIt was reported that the first four vibration frequencies variedby 14ndash18 during the 10 months It was also found thatthe frequencies of all the modes analyzed except the secondmode decreased with the temperature increase The second

mode frequency however increased with increasing temper-atures (for positive temperatures) In particular a bilinearrelationship was observed between the measured frequenciesand temperature There was a clear linear relationship abovethe freezing temperature (0∘C) and a different linear corre-lation below the freezing temperature It was demonstratedthat this bilinear behavior was attributed to the asphalt layeron the deck Although the asphalt layer did not contribute tothe overall stiffness atwarm temperatures it added significantstiffness to the bridge at cold temperatures Similar outcomeswere also found in a ballasted railway bridge in Sweden byGonzales et al [29] The natural frequencies of the bridgeincrease by 15 and 35 for the first vertical bending andfirst torsional modes during the cold period of the yearrespectively Furthermore the increase in natural frequenciestook place only after the temperatures had dropped below0∘Cfor a number of days as shown in Figure 6

Ni et al [64] investigated the influence of temperatureon modal frequencies for the cable-stayed Ting Kau Bridge(Hong Kong) which has been instrumented with a long-termSHM system Based on one-year measurement data obtainedfrom 45 accelerometers and 83 temperature sensors themodal frequencies and temperatures of the bridge at variouslocations were obtained at one-hour intervals to cover a fullcycle of in-serviceoperating conditions It was found thatthe environmental temperature changes from 283 to 5346∘Caccount for variation in modal frequencies with variancefrom 020 to 152 for the first ten modes which may maskthe modal changes caused by structural damage Desjardinset al [65] examined the variations in frequencies of theConfederation Bridge (made of prestressed concrete) over a6-month period They reported a clear trend of reductionin the modal frequencies by about 4 when the averagetemperature of the concrete of the bridge varied from minus20 to+25∘C

Liu and DeWolf [17 66] explored how temperaturevariations influence themodal properties based on frequencyand temperature measurements obtained over the period ofone year from a long-term SHM system that was installed onan in-service curved bridgeThe curved bridge is 235m longwith three unequal-length continuums of 58 88 and 88mrespectivelyThe superstructure is a five-cell curved prismaticbox girderThe results indicated that the long-term variationsof modal frequencies have a reciprocal relation with respectto the variation in the in situ concrete temperature for allthree measured modes The natural frequencies varied bya maximum of 6 in a peak to peak temperature rangeof 70∘F during one full year The higher modal frequencyis less sensitive to temperature changes and the bendingmodes are more easily affected by temperature than torsionmodes which is similar to that obtained by Kim et al [67]in an experimental study of a model plate-girder bridge Inaddition the results reported in this study argued that useof the natural frequencies alone is not sufficient for damagedetection because of their sensitivity to temperature and theinfluence of temperature on the natural frequency is likelystructure specific

Ding et al [68 69] observed the daily and seasonalcorrelations of frequency-temperature using 215 days of


Transition from the warm to the cold period

46

44

42

4

38

minus30 minus20 minus10 0 10 20 30

Temperature (∘C)

Transition from the

Freq

uenc

y (H

z)

cold to the warm period

Figure 6The variation in the frequency of the first vertical bendingmode as function of temperature [29]

health monitoring data obtained on the Runyang SuspensionBridge The bridge is a single-span steel suspension bridgewith the main span 1490m The aerodynamically shapedclosed box steel girder is 363m wide and 30m highFrom the daily correlations of temperature-frequency it wasreported that an overall decrease in modal frequency wasobserved with the increase in temperature of the bridgeand measured modal frequencies of higher modes are moresensitive to ambient temperatures About 2 variation in thefirst six modal frequencies during a period of 10 monthswas found as the ambient temperature of the steel bridgevaried from minus5 to +50∘C Moreover the conclusion thatthe daily averaged modal frequencies of vibration modeshave remarkable seasonal correlationswith the daily averagedtemperature was also deduced

Moser and Moaveni [32] presented results from a con-tinuous monitoring system installed on the Dowling HallFootbridge on the campus of Tufts University The bridge isa two-span continuous steel frame bridge of 44m long and37m wide It was observed that (1) the identified naturalfrequencies increase as temperatures decrease (2) this effectis more significant when the temperatures are below thefreezing point resulting in a nonlinear relationship and(3) Modes 1 3 and 4 show more clear correlation withtemperature while Modes 2 5 and 6 show more scatterOne explanation for the third observation is that Modes 25 and 6 are identified with larger estimation uncertaintiesThe natural frequencies of the first six identified modesvaried by 4ndash8 while temperatures ranged from minus14 to 39∘Cand the relationship between identified natural frequenciesand measured temperatures was nonlinear However noobvious variation in damping ratios and mode shapes wasobserved One possible reason is that those parameters arenot identified with enough precision

Xia et al [30] investigated the correlations betweentemperatures and vibration properties of the Tsing Ma Sus-pension Bridge which has a total span of 2132m and carrieshighway and railway tracks Figure 7 shows the variations inthe first four vertical frequencies and temperature on January17 2005 It was found that all frequencies generally decreasewhen the temperature goes up before noon whereas theyincrease as the temperature drops in the afternoon althoughthe variations are quite small The results are consistentwith that obtained from the laboratory model [28] The

minimum frequencies and the maximum temperatures donot occur at the same time and the time difference is about3 h Two reasons were deduced The first one is that thetemperature is nonuniformly distributed across the bridgeand the frequencies are global properties and are associatedwith the temperature distribution of the entire structure Andthe second one is that the variations in temperature causedinternal stresses of the components changes for example thecables The variation trends of the vertical modal frequenciesare very similar and the maximum variations are about1ndash15 No clear correlation between damping ratios andtemperature was observed

Mosavi et al [70] monitored the response of a skewedtwo-span steel-concrete composite bridge in North Carolinaduring a 24 h period to address possible reasons for the dailyvariations inmodal frequencies of the bridge An almost 1-2change in natural frequencies was investigated in all modes ofthe bridge from night to noon while the natural frequenciesdid not change significantly from night to morning Theshifts in the natural frequencies from night to noon werecoincident with an inversion of the temperature gradients inthe composite cross section of the bridge from night to noonIt was argued that this inversion of the temperature gradientsfrom night to noon is the main reason for a relatively largeupward deformation of the steel girders and variations of themeasured natural frequencies during this time

Besides other research groups like Lloyd et al [71 72]Kullaa et al [73] Breccolotti et al [74] Song and Dyke [75]Siddique et al [26] Yang et al [76] Di et al [77] Zabelet al [78] Wang et al [79] Cury et al [80] and Koo et al[81] also carried out structural vibrations and temperaturestests on different types of bridgesThe similar conclusion thatvibration properties of long-span bridges are significantlyimpacted by varying temperatures was drawn

4 Progress of Quantitative Model

The trend analysis can reveal the degree that the vibra-tion properties are affected by temperature variations Moststudies in the above literature indicate that the naturalfrequencies decrease as the temperature rises Those resultsthat are directly deduced from simulated or measurementdata are bridge-dependent and time-dependent For differentbridges the changes of natural frequencies per temperatureunit are different Even using the data obtained from anidentical bridge at different times the extracted outcomesmay be different To eliminate the environmental effects invibration-based damage detection the quantitative modelthat is used to quantify the variations of vibration propertiesunder temperature changes becomes necessary The SHMsystems installed on large-scale bridges permanently obtainlong-term vibration data and corresponding temperaturedata which provides a great opportunity for quantitativeunderstanding and modeling of the effect of temperatureon modal properties Three types of quantitative modelsincluding linearmodel nonlinearmodel and learningmodelwere developed in the past few years

Mathematical Problems in Engineering 9Te

mpe

ratu

re

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

115

1145

114

1135

113

Freq

uenc

y (0001

Hz)

(a)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

138

1375

137

1365

136

1355

(b)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

184

1835

183

1825

182

1815

TemperatureFrequency

(c)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)


242

241

240

239

238

237

236

(d)

Figure 7 Variations in frequencies with respect to the temperature of the TsingMaBridge on January 17 2005 ((a) 1st vertical (b) 2nd vertical(c) 3rd vertical and (d) 4th vertical) [30]

41 Linear Model The linear model assumes that the vibra-tion properties change with temperature linearly The coeffi-cients of the linear model are usually estimated by regressionmethods This model is consistent with simplified theoreticalformulation and very easy to implement

Sohn et al [31 82] proposed a linear adaptive filter thataccommodates the changes in temperature to the damagedetection system of a large-scale bridge by introducing theassumption that changes in themodal parameters are linearlyproportional to changes in temperatureThe linear filtermod-els the relationship between the selected bridge temperatureinputs 119909 and its measured fundamental frequency 119910 as alinear function

119910 = 119909119908 + 120576 (7)

where 119908 is a column vector of coefficients that weighs eachtemperature input and 120576 is the filter errorWith 119899 observations

119910 =[[[[

[

119910 (1)119910 (2)

119910 (119899)

]]]]

]

119883 =[[[[

[

1 1199091(1) 119909

2(1) sdot sdot sdot 119909

119903(1)

1 1199091(2) 119909


119903(2)

sdot sdot sdot

1 1199091(119899) 119909

2(119899) sdot sdot sdot 119909

119903(119899)

]]]]

]

120576 =[[[[

[

120576 (1)120576 (2)

120576 (119899)

]]]]

]

(8)

where 119903 represents the number of inputs


78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)

Estimated from the regression modelMeasured (first data set)

Figure 8 Reproduction of the first mode frequency using a linearfilter [31]

The Least-Mean-Squares (LMS) error minimization wasemployed to estimate the filter coefficients Data from theAlamosaCanyonBridge in the state ofNewMexicowere usedto demonstrate the effectiveness of the adaptive filterThe firstdataset from 1996 was used to train the adaptive filter whilethe second dataset from 1997 was used to test the predictionperformance A linear filter with two spatially separated andtwo temporally separated temperature measurements wasthen developedThe reproduction of the firstmode frequencyis shown in Figure 8 It indicates that a linear adaptive filterwith four temperature inputs could reproduce the naturalvariability of the frequencies with respect to time of dayreasonably well Then the regression model can be used toestablish confidence intervals of the frequencies for a newtemperature profile

Peeters et al [62 63] proposed a dynamic linearregression model called ARX to filter out the temperatureeffects from the measured frequencies using monitoring dataobtained from the healthy Z24-Bridge in Switzerland Single-input and single-output (SISO) ARX models were con-structed for the first four fundamental frequencies describingthe relation between one input temperature and one outputfrequency The loss function value and the final predictionerror were used as the quality criteria to find a good modelThe measurement data of the Z24-Bridge were split in twoparts an estimation part and a validation partThe estimationdata were used to estimate the ARX model whereas thevalidation data were applied to validate the model It wasvalidated that an ARX model that includes the thermaldynamics of the bridge is superior to a ldquostaticrdquo regressionmodel Also it turned out that a temperature measurementat one location was sufficient to find an accurate model TheARXmodel can be used for simulating the frequencies whennew temperaturemeasurements are fed to themodel If a newmeasured frequency lies outside the estimated confidenceintervals it is likely that the bridge is damaged In case of theZ24-Bridge and the applied damage scenarios the damagewas successfully detected

It is interesting that the regression approaches performedby Sohn et al [31] and Peeters et al [62] were comparedby Sohn [83] It was pointed out that the Peteersrsquo work

emphasizes the thermal dynamics of the bridge by using asingle temperature measurement with multiple time lags (thetemporal variation of temperature) while the work by Sohnet al uses temperature readings frommultiple thermocouplesto take into account the temperature gradient across thebridge (the spatial variation of temperature) as well as thetemporal variation The comparison of these two approachesclearly demonstrated that the two models are problem-specific Because the temperature of the Alamos CanyonBridge can easily go above 45∘C the bridge experiences a largetemperature gradient across the bridge throughout the dayTherefore it is feasible that the spatial variation of the bridgetemperature might have been the main driving factor for thefrequency variation On the other hand the magnitude of theZ24 Bridge which had a 30m long main span and two 20mlong side spans with 86m width is certainly larger than theAlamos Canyon Bridge Therefore it is possible that it takesa longer time before the temperature affected the dynamicproperties of the bridge and the time-lag information of thetemperature is more important for the Z24 Bridge than theAlamos Canyon Bridge

A reinforced concrete slab which was constructed andplaced outside the laboratory has been periodically vibrationtested by Xia et al [34] for nearly two years Linear regressionmodels between modal properties and temperature wereproposedThe corresponding linear regressionmodel is takenas

119902 = 1205730+ 120573119905119905 + 120576119891 (9)

where 119902 represents the dynamic parameter such as frequencyand percentage damping ratio 120573

0(intercept) and 120573

119905(slope)

are regression coefficients and 120576119891is the error 120573

1199051205730indicates

the percentage of the frequency change when the structuraltemperature increases by unit degree Celsius With least-squares fitting the regression coefficients and confidencebounds are obtained The correlation coefficient was used toexamine the goodness of fit of the linear relation betweenfrequencies and temperature The results implied that thereis a good linear correlation between the two variables It isworth noting that the estimated linear regression functionagrees well with the theoretical model of (4) which demon-strates the reliability of the theoretical derivation Then thislinear regression model was utilized to examine the relationbetween the frequencies and the structural temperature ofthree beam models that are made of steel aluminum andreinforced concrete (RC) [30] The correlation coefficients ofthe first four natural frequencies of the steel beam are minus093minus099 minus096 and minus097 respectively implying a very goodlinear correlation between the average temperature and thenatural frequencies The regression coefficients of the threestructures were also obtained and listed in Table 2 for eachmode The variations of modal frequencies per degree arevery close to half of the modulus thermal coefficients ofthe material Subsequently the linear regression model wasapplied to the first four vertical modal frequencies of theTsing Ma Suspension Bridge in 4 days as shown in Figure 9Good linear correlation can be observed from the figureTheaveraged value (120573

1199051205730) for the four modes is very close to


Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)

(a) 1st vertical mode

15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)

(b) 2nd vertical mode

1845

184

1835

183

1825

182

1815

181

ObservationFitted data95 bound

Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)

(c) 3rd vertical mode

242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)

(d) 4th vertical mode

Figure 9 Relation of natural frequencies to temperature of the Tsing Ma Bridge [30]

that of the laboratory tested steel beam and also to half of themodulus thermal coefficients of steel Both laboratory modelresults and real bridge results demonstrated that variations inthe bending frequencies of the structures are mainly causedby the change in the modulus of the material although thetwo structures are quite different in terms of size type andboundary conditions Besides applying the linear regressionmodel to the damping data showed that their correlationcoefficients are 010 007 minus011 and minus033 respectively

Similarly Liu and DeWolf [17 84] developed a linearregression model with a 5 estimate error to representthe three natural frequencies of a three-span curved post-tensioned box concrete bridge located in Connecticut asa function of temperature The model is mathematicallydescribed as

119865119894= 119891119894+ 119886119879 (10)

where 119891119894is mean natural frequency 119879 denotes concrete

temperature and 119865119894represents natural frequency at temper-

ature 119879 The parameter 119886 is determined from the measuredmonitoring data Based on a full year data beginning withJanuary 2002 recorded by a long-term health monitoringsystem the functions of the first three natural frequencies are

1198651

= 15588 minus 0007119879 (first natural frequency 1519Hz)

1198652

= 21584 minus 0008119879 (second natural frequency 211Hz)

1198653

= 36391 minus 0007119879 (third natural frequency 359Hz) (11)


Table 2 Regression coefficients of the models (Xia et al [30])

Mode number Steel beam Aluminum beam Concrete beam1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1 369 minus00014 minus378 times 10minus4 328 minus00024 minus731 times 10minus4 2065 minus0029 minus14 times 10minus3


3 6664 minus00132 minus198 times 10minus4 6001 minus00163 minus272 times 10minus4


The proposed models were used for comparison with datacollected in 2005 Results of the hypothesis test showed thatthey both conform to the normality distribution but they donot always have identical medians

42 NonlinearModel Although linearmodels are simple andintuitionistic the responses of long-span bridges particularlycable-supported bridges are nonlinear The error that gen-erates from linear models may mask the shift of vibrationproperties caused by structural damages From this point ofview the nonlinear models are more reasonable

Ding andLi [68] proposed a polynomial regressionmodelto describe the frequency-temperature seasonal correlationsof the Runyang Suspension Bridge which can be mathemat-ically described as

119891 (119879 119899) = 119901119899119879119899 + 119901

119899minus1119879119899minus1 + sdot sdot sdot + 119901

21198792 + 119901

11198791 + 119901

0 (12)

where 119879 is the daily averaged value of temperature 119891 denotesthe daily averaged values of the frequency 119899 means theorder of polynomial regression model and 119901

119894(119894 = 0simn)

represents the coefficient of the regression model The 175-day monitoring data were discontinuously extracted fromthe total 215-day data for training the model and the other40-day monitoring data were used for testing the predictioncapability By increasing the number of 119899 from 1 to 10 theregression models were formulated and the relative errors ofmeasured and predicted frequencies were obtained The bestorder n of the regression model was selected as 6 for thisbridgeThe 40-day temperature data were fed into the trainedregression model to generate the prediction values of modalfrequency which were compared with the measured valuesto evaluate the prediction capabilityThe result demonstratedthat the developed polynomial regression model exhibitsgood capabilities for mapping between the temperature andmeasured modal frequency

Using results from a vibration-based continuous mon-itoring system deployed on the Dowling Hall Footbridgeat Tufts University in Medford Moaveni et al [32 85]proposed four models including a static linear model anARX model a bilinear model and polynomials of variousorders to represent the relationship between the identifiednatural frequencies and measured temperatures Throughcorrelation coefficient three records were used to model therelationship between the identified natural frequencies andthe measured temperatures The estimation error varianceand two objective criteria including the Akaikersquos InformationCriterion (AIC) and the Bayesian Information Criterion

(BIC) were employed as the model quality metric After eval-uating those candidate models the fourth-order polynomialwithout cross terms was selected as the final model whilethe bilinear model also performs well and shows promisefor environmental modeling near the freezing point Thevalidity of the selected fourth-ordermodelwas assessed basedon separate sets of validation data from two different data-splitting methods Figure 10 displays one of the comparisonsIt was demonstrated that the model can be applied to thefuture data with almost the same accuracy that it fits themodeling data as long as future temperatures are not outsidethe range of modeling temperatures

43 LearningModel Thetemperature distribution in a bridgeis governed by air temperature solar radiation wind speedand so on all of which vary in a random manner As aresult the temperature distribution in a bridge is randomThe measurement error caused by uncertainties in actualtests are stochastic which induces the tested vibration prop-erties of a bridge notably natural frequency are scatteredunder the same temperature For these two reasons thereliability is questionable when the relationship betweentemperatures and vibration properties of long-span bridgesis described by simple linear regression model or polynomialregression model To discriminate the variations of vibrationproperties due to temperature changes from those causedby structural damage the sophisticated models should bedeveloped which are called learning models in this paperThe learning models integrated intelligent algorithm andshow a better robustness and performance to predict thevibration properties of bridges under different temperatures

Ni et al [64] applied the support vector machine (SVM)technique to formulate regression models which quantify theeffect of temperature on modal frequencies for the cable-stayed Ting Kau Bridge (Hong Kong) based on long-termmonitoring data In order to achieve a trade-off betweensimulation performance and generalization only a portionof the measurement data was employed to train the SVMmodels and the remaining measurement data was usedfor model validation A squared correlation coefficient wasdefined for optimizing the SVM coefficients to obtain goodgeneralization performance The results obtained by theSVM models were compared with those produced by amultivariate linear regression model and showed that theSVM models exhibit good capabilities for mapping betweenthe temperature and modal frequencies In this model theaverage temperature was used but several studies showedthat the measured temperatures from different locations of


48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)

Figure 10 Time-variation of identified and fourth-order model simulated natural frequencies of all six modes [32]

a structure are highly correlated The correlated featurescould seriously deteriorate the generalization performanceof regression model Subsequently the principal componentanalysis (PCA) was added before conducting SVM algorithm[86]The PCA is first applied to extract principal componentsfrom the measured temperatures for dimensionality reduc-tion The predominant feature vectors in conjunction withthe measured modal frequencies are then fed into an SVMalgorithm to formulate regression models that may take intoaccount thermal inertia effect The proper hyperparametersto formulate SVR models with good generalization perfor-mance were chosen The proposed method was comparedwith the method directly using measurement data to trainSVM models through the use of long-term measurementdata It was shown that PCA-compressed features make

the training and validation of SVM models more efficientin both model accuracy and computational costs Whencontinuously measured data is available the dynamic SVRmodel which is trained using the augmented temperaturevector to account for temporal correlation provides moreaccurate frequency prediction than the static SVR modelwithout considering temporal correlation

With one year of measurement data from the instru-mented Ting Kau Bridge a comparative study of evaluatingthe effectiveness of four statistical regressionlearning meth-ods including linear regression nonlinear regression neuralnetwork and SVM for modeling the effect of temperatureon modal frequencies was conducted by Ko and Ni [12] Thepredicted results are plotted in Figure 11 It was observedthat the linear model reproduces the training data well but


0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)

Modal frequency by dataModal frequency by model

(a) Frequency sequences generated by the linear regression model

501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160

(b) Frequency sequences generated by the nonlinear regression model

0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured

(c) Frequency sequences generated by the neural network model

0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured

(d) Frequency sequences generated by the support vector machine model

Figure 11 The comparison of prediction performance of different models [12]

is poor in predicting fresh validation data The nonlinearregression model shows stronger generalization capabilitythan the linearmodel but is still short of accurately predictingthe frequency variations The neural network and SVMmodels exhibited good capabilities in both reproduction andprediction It was found that a perception neural networkwith a single hidden layer is sufficient for modeling thecorrelation and an appropriate number of hidden nodes iscrucial to achieve superior prediction performance of thetrained model (excessive hidden nodes may cause seriousattenuation of the prediction capability) When an SVM isused for modeling the correlation the prediction capabilityof the trained model is heavily dependent on the selection ofthe SVM coefficients An advisable way is to determine themodel parameters of the SVM model by means of trainingdata while optimizing the SVM coefficients with the use ofindependent validation data

Considering that the variations of frequencies of realbridges may be induced by the combined action of tempera-ture andwind Li et al [22] investigated the effects of tempera-ture andwind velocity onmodal parameters includingmodalfrequencies modal shapes and the associated damping ratiosof cable-stayed bridges based on nonlinear principal compo-nent analysis (NLPCA) and artificial neural network (ANN)

The NLPCA was first employed as a signal preprocessingtool to distinguish temperature and wind effects on struc-tural modal parameters from other environmental factorsThen the ANN technique was introduced to model therelationship between the preprocessedmodal parameters andenvironmental factors Sixteen-day continuousmeasurementof acceleration temperature and wind velocity which wasrecorded by a sophisticated long-term SHM system wereused in this study Numerical results indicated that modalparameters preprocessed by NLPCA can retain the mostfeatures of original signalsThe ANN regression models havegood capacities for mapping the relationship of environmen-tal factors and modal frequency The proposed regressionmodels showed that the preprocessed modal frequency anddamping ratios are dramatically affected by temperatureHowever temperature effects on the entire modal shapes areinsignificant A similar method was then adopted by Loh andChen [87]

Ni et al [33 88] studied the prediction capability of neuralnetwork models established using long-term monitoringdata for correlation between themodal frequencies and envi-ronmental temperatures A total of 770 h modal frequencyand temperature data obtained from an instrumented bridgeare available for this study which are further divided into


0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order

(a) The baseline BPNNmodel

0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)

(b) The BPNNmodel with the early stopping technique

0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)

(c) The BPNNmodel with the Bayesian regularization technique

Figure 12 Comparison between BPNN-predicted and measured frequencies for testing data [33]

three sets training data validation data and testing dataTwo new back-propagation neural networks (BPNNs) wereconfigured with the same data by applying the early stop-ping technique and the Bayesian regularization techniquerespectively The reproduction and prediction capabilities ofthe two new BPNNs were examined with respect to thetraining data and the unseen testing data and comparedwith the performance of the baseline BPNN model Theresults are shown in Figure 12 It validated that both the earlystopping and Bayesian regularization techniques can signif-icantly ameliorate the generalization capability of BPNN-based correlation models and the BPNN model formulatedusing the early stopping technique outperforms that usingthe Bayesian regularization technique in both reproductionand prediction capabilities Then they addressed the con-struction of appropriate input to neural networks with intentto enhance the reproduction and prediction capabilities ofthe formulated correlation models [89] Making use of thetemperature data measured from different portions of the

bridge three kinds of input that is mean temperatureseffective temperatures and principal components (PCs) oftemperatures were constructed It was revealed that the tem-perature profile characterized by the effective temperaturesis insufficient for formulating a good correlation modelbetween the modal frequencies and temperatures When asufficient number of PCs are used the BPNN with input ofthe PCs of temperatures performs better than the BPNNwithinput of the mean temperatures in both reproduction andprediction capabilities

5 Conclusions and Recommendations

The variations in vibration properties of long-span bridgesare not only caused by structural damage or deteriorationbut also caused by environmental conditions especially tem-perature Inappropriately estimating the temperature effectson the changes of vibration characteristics of a bridge may


result in false-positive or false-negative damage detection Tosolve this problem considerable efforts have been devotedto investigating the correlations between temperatures andvibration properties in different types of bridges with theaim of eliminating the temperature effect in vibration-basedstructural damage detectionHowever it is far from sufficientThe temperature distribution in a bridge that is affected byair temperature wind humidity intensity of solar radiationmaterial type and so forth changes in a complex way Themore and more complicated configuration of bridges makesthe vibration properties multivariate So at least the followingexisting problems and promising efforts need particularattention

(1) The changes of vibration properties in long-spanbridges induced by structural damage or temperature areonly several percentages So the extraction of temperature-caused vibration property variability and damage-causedvibration property variability relies heavily on the accuracyof model parameters How to identify bridge model param-eters notably mode shape and damping ratio with a highresolution in noise environment is the fundamental problemof studying the temperature effects on vibration proper-ties of long-span bridges and executing damage detectionalgorithms Improving the precision of structural vibrationproperties identification is a significant work not only forstudies of the temperature effect but also for the research ofdamage detection

(2) Many new bridges with complicated geometric con-figuration and novel materials are constructed The thermaldynamic characteristics such as conduction emissivity andabsorptivity of those new structures are different fromgeneralcases Besides modern transport system has extended toextraordinary climate regionsThe temperature in those areasis particularly high or particularly low It was already foundthat two different relationships between frequencies andtemperatures existed below and above freezing Thereforestudying the correlations between temperatures and vibrationproperties of various bridges in broad regions is helpfulin understanding the activity of temperature on vibrationproperty variations

(3) Although numerical simulation approach has insuf-ficient accuracy for quantifying the effect of temperaturevariations on the vibration properties of long-span bridges itis characterized by high efficiency and easy implementationThis method is an advisable choice when field measurementdata are unavailable The development of the thermody-namics simulation makes the calculation of temperaturedistribution in bridges by the FE method more reliableTherefore studying the correlations between temperaturesand vibration properties by the coupled thermomechanicssimulation is a promising work

(4) Successful implementation and operation of long-term SHM systems on bridges have been widely reportedwhich provide a large number of structural vibration dataand corresponding temperature records But the observa-tion of the correlations between temperatures and vibrationproperties focuses on only a few bridges Manymeasurementresults are unused which induces the potentiality of SHMis considerably reduced Carrying out an intensive study of

temperature influence on bridge vibration properties fromSHM data should be encouraged and particular emphasisshould be placed on extracting quantitative models fordamage detection

(5) Fromfieldmeasurement data of acceleration and tem-perature many promising models were trained to describethe temperature effects on the variation of vibration proper-ties However previous prediction models have been effec-tive when they were used for a specific bridge type Theyhave not been shown to be applicable to different bridgetypes Understanding the potential mechanism of vibrationproperty variability because of the temperature variations canhelp in developing more rational mathematical relationshipsbetween the measured vibration properties and temperaturesthat are suitable for a wide range of bridge types In additionestablishing parametric stochastic models are beneficial forpredicting temperature-induced variations in the vibrationproperties of new bridges when temperature data are avail-able

(6) Advanced statistical methods such as the Bayesianprobabilistic framework and the Markov chain Monte Carlosimulation provide effective approaches for establishing sta-tistical models of temperatures and vibration propertiesModern signal processing technologies like the wavelettransform and the Hilbert-Huang transform can extractsignificant information from complicated and unsystematicmeasurement data So developing the correlations betweentemperatures and vibration properties by advanced statisticalmethods and modern signal processing technologies may bemore robust against uncertainties

Conflict of Interests

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

Acknowledgments

This research work was jointly supported by the NationalNatural Science Foundation of China (Grant nos 5112100551308186 and 51222806) the Natural Science Foundationof Jiangsu Province of China (Grant no BK20130850) andthe Research Fund of State Key Laboratory for DisasterReduction in Civil Engineering (Grant no SLDRCE12-MB-03)

References

[1] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering vol 138 no 3 pp 299ndash307 2011

[2] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011

[3] T H Yi H N Li and M Gu ldquoWavelet based multi-stepfiltering method for bridge health monitoring using GPS andaccelerometerrdquo Smart Structures and Systems vol 11 no 4 pp331ndash348 2013


[4] K Wardhana and F C Hadipriono ldquoAnalysis of recent bridgefailures in the United Statesrdquo Journal of Performance of Con-structed Facilities vol 17 no 3 pp 144ndash150 2003

[5] B F Spencer Jr and S Cho ldquoWireless smart sensor technologyfor monitoring civil infrastructure technological developmentsand full-scale applicationsrdquo in Proceedings of theWorld Congresson Advances in Structural Engineering and Mechanics (ASEMrsquo11) Seoul Republic of Korea 2011

[6] T H Yi H N Li and M Gu ldquoExperimental assessment ofhigh-rate GPS receivers for deformation monitoring of bridgerdquoMeasurement vol 46 no 1 pp 420ndash432 2013

[7] T Yi H Li and M Gu ldquoFull-scale measurements of dynamicresponse of suspension bridge subjected to environmental loadsusing GPS technologyrdquo Science China Technological Sciencesvol 53 no 2 pp 469ndash479 2010

[8] T H Yi H N Li andM Gu ldquoRecent research and applicationsof GPS based technology for bridge health monitoringrdquo ScienceChina Technological Sciences vol 53 no 10 pp 2597ndash2610 2010

[9] G D Zhou and T H Yi ldquoRecent developments on wirelesssensor networks technology for bridge health monitoringrdquoMathematical Problems in Engineering vol 2013 Article ID947867 33 pages 2013

[10] G D Zhou and T H Yi ldquoThe nonuniform node configurationof wireless sensor networks for long-span bridge health mon-itoringrdquo International Journal of Distributed Sensor Networksvol 2013 Article ID 797650 9 pages 2013

[11] Y Q Ni X W Ye and J M Ko ldquoMonitoring-based fatiguereliability assessment of steel bridges analytical model andapplicationrdquo Journal of Structural Engineering vol 136 no 12pp 1563ndash1573 2010

[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005

[13] Y Q Ni X W Ye and J M Ko ldquoModeling of stress spectrumusing long-term monitoring data and finite mixture distribu-tionsrdquo Journal of Engineering Mechanics vol 138 no 2 pp 175ndash183 2012

[14] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998

[15] P J S Cruz and R Salgado ldquoPerformance of vibration-baseddamage detection methods in bridgesrdquo Computer-Aided Civiland Infrastructure Engineering vol 24 no 1 pp 62ndash79 2009

[16] G D Zhou and T H Yi ldquoThe node arrangement methodologyof wireless sensor networks for long-span bridge health mon-itoringrdquo International Journal of Distributed Sensor Networksvol 2013 Article ID 865324 8 pages 2013

[17] C Liu and J T DeWolf ldquoEffect of temperature on modalvariability of a curved concrete bridge under ambient loadsrdquoJournal of Structural Engineering vol 133 no 12 pp 1742ndash17512007

[18] Y Zou L Tong and G P Steven ldquoVibration-based model-dependent damage (delamination) identification and healthmonitoring for composite structuresmdasha reviewrdquo Journal ofSound and Vibration vol 230 no 2 pp 357ndash378 2000

[19] J-T Kim Y-S Ryu H-M Cho and N Stubbs ldquoDamage iden-tification in beam-type structures frequency-based method vsmode-shape-based methodrdquo Engineering Structures vol 25 no1 pp 57ndash67 2003

[20] A Alvandi and C Cremona ldquoAssessment of vibration-baseddamage identification techniquesrdquo Journal of Sound and Vibra-tion vol 292 no 1-2 pp 179ndash202 2006

[21] I Talebinejad C Fischer and F Ansari ldquoNumerical evaluationof vibration-based methods for damage assessment of cable-stayed bridgesrdquo Computer-Aided Civil and Infrastructure Engi-neering vol 26 no 3 pp 239ndash251 2011

[22] H Li S Li J Ou and H Li ldquoModal identification of bridgesunder varying environmental conditions temperature andwind effectsrdquo Structural Control and Health Monitoring vol 17no 5 pp 495ndash512 2010

[23] P Cawley ldquoLong range inspection of structures using lowfrequency ultrasoundrdquo in Structural Damage Assessment UsingAdvanced Signal Processing Procedures University of SheffieldSheffield UK 1997

[24] G D Zhou and T H Yi ldquoThermal load in large-scale bridgesa state-of-the-art reviewrdquo International Journal of DistributedSensor Networks vol 2013 Article ID 217983 17 pages 2013

[25] Y L Xu B Chen C L Ng K Y Wong and W Y ChanldquoMonitoring temperature effect on a long suspension bridgerdquoStructural Control and HealthMonitoring vol 17 no 6 pp 632ndash653 2010

[26] A B Siddique B F Sparling and L D Wegner ldquoAssessmentof vibration-based damage detection for an integral abutmentbridgerdquo Canadian Journal of Civil Engineering vol 34 no 3 pp438ndash452 2007

[27] J-T Kim J-H Park and B-J Lee ldquoVibration-based damagemonitoring in model plate-girder bridges under uncertaintemperature conditionsrdquo Engineering Structures vol 29 no 7pp 1354ndash1365 2007

[28] Y Xia Y-L Xu Z-L Wei H-P Zhu and X-Q ZhouldquoVariation of structural vibration characteristics versus non-uniform temperature distributionrdquo Engineering Structures vol33 no 1 pp 146ndash153 2011

[29] I Gonzales M Ulker-Kaustell and R Karoumi ldquoSeasonaleffects on the stiffness properties of a ballasted railway bridgerdquoEngineering Structures vol 57 pp 63ndash72 2013

[30] Y Xia B Chen S Weng Y Q Ni and Y L Xu ldquoTemperatureeffect on vibration properties of civil structures a literaturereview and case studiesrdquo Journal of Civil Structural HealthMonitoring vol 2 no 1 pp 29ndash46 2012

[31] H Sohn M Dzwonczyk E G Straser A S Kiremidjian KLaw and T Meng ldquoAn experimental study of temperatureeffect on modal parameters of the Alamosa Canyon BridgerdquoEarthquake Engineering and Structural Dynamics vol 28 no7-8 pp 879ndash897 1999

[32] P Moser and B Moaveni ldquoEnvironmental effects on theidentified natural frequencies of the Dowling Hall FootbridgerdquoMechanical Systems and Signal Processing vol 25 no 7 pp2336ndash2357 2011

[33] Y Q Ni H F Zhou and J M Ko ldquoGeneralization capability ofneural network models for temperaturefrequency correlationusing monitoring datardquo Journal of Structural Engineering vol135 no 10 pp 1290ndash1300 2009

[34] Y Xia H Hao G Zanardo and A Deeks ldquoLong term vibrationmonitoring of an RC slab temperature and humidity effectrdquoEngineering Structures vol 28 no 3 pp 441ndash452 2006

[35] R W Clough and J Penzien Dynamics of Structure McGraw-Hill New York NY USA 2nd edition 1993

[36] Y Fu and J TDeWolf ldquoMonitoring and analysis of a bridgewithpartially restrained bearingsrdquo Journal of Bridge Engineering vol6 no 1 pp 23ndash29 2001

[37] J H G Macdonald and W E Daniell ldquoVariation of modalparameters of a cable-stayed bridge identified from ambient


vibration measurements and FE modellingrdquo Engineering Struc-tures vol 27 no 13 pp 1916ndash1930 2005

[38] Z-D Xu and Z Wu ldquoSimulation of the effect of temperaturevariation on damage detection in a long-span cable-stayedbridgerdquo Structural Health Monitoring vol 6 no 3 pp 177ndash1892007

[39] E BalmesM Basseville LMevel andHNasser ldquoHandling thetemperature effect in vibration monitoring of civil structuresa combined subspace-based and nuisance rejection approachrdquoControl Engineering Practice vol 17 no 1 pp 80ndash87 2009

[40] CMiao L Chen and Z Feng ldquoStudy on effects of environmen-tal temperature on dynamic characteristics of Taizhou YangtzeRiver Bridgerdquo Engineering Science vol 9 no 2 pp 79ndash84 2011

[41] E Balmes M Basseville F Bourquin L Mevel H Nasser andF Treyssede ldquoMerging sensor data from multiple temperaturescenarios for vibration monitoring of civil structuresrdquo Struc-tural Health Monitoring vol 7 no 2 pp 129ndash142 2008

[42] L M Sun and Z H Min ldquoEvaluations on dynamic behaviorsand its environmental influence factors of bridge based onlong term monitoringrdquo in Proceedings of the 4th InternationalSymposium on Lifetime Engineering of Civil InfrastructureChangsha China 2009

[43] L Sun Y Zhou and X Li ldquoCorrelation study on modalfrequency and temperature effects of a cable-stayed bridgemodelrdquo Advanced Materials Research vol 446ndash449 pp 3264ndash3272 2012

[44] Y Xia B Chen X-Q Zhou andY-L Xu ldquoFieldmonitoring andnumerical analysis of Tsing Ma Suspension Bridge temperaturebehaviorrdquo Structural Control and HealthMonitoring vol 20 no4 pp 560ndash575 2013

[45] J D Turner An experimental and theoretical study of dynamicmethods of bridge conditioning monitoring [PhD thesis] Uni-versity of Reading Berkshire UK 1984

[46] J D Turner and A J Pretlove ldquoA study of the spectrumof traffic-induced bridge vibrationrdquo Journal of Sound andVibration vol 122 no 1 pp 31ndash42 1988

[47] V Askegaard and P Mossing ldquoLong term observation of RC-bridge using changes in natural frequencyrdquo Nordic ConcreteResearch no 7 pp 20ndash27 1988

[48] M G Wood Damage analysis of bridge structures using vibra-tional techniques [PhD thesis] Aston University 1992

[49] R G Rohrmann and W F Rucker ldquoSurveillance of structuralproperties of large bridges using dynamic methodsrdquo in Pro-ceedings of 6th International Conference on Structural Safety andReliability pp 977ndash980 1994

[50] W F Rucker S Said R G Rohrmann and W Schmid ldquoLoadand condition monitoring of a highway bridge in a continuousmannerrdquo in Proceedings of the IABSE Symposium on Extendingthe Lifespan of Structures vol 73 San Francisco Calif USA1995

[51] J T DeWolf P E Conn and P N Orsquoleary ldquoContinuous mon-itoring of bridge structuresrdquo in Proceedings of the InternationalAssociation for Bridge and Structural Engineering Symposium(IABSE rsquo95) pp 934ndash940 1995

[52] J Zhao and J T Dewolf ldquoDynamic monitoring of steel girderhighway bridgerdquo Journal of Bridge Engineering vol 7 no 6 pp350ndash356 2002

[53] C R Farrar S W Doebling P J Cornwell and E G StraserldquoVariability of modal parameters measured on the AlamosaCanyon Bridgerdquo in Proceedings of the 15th International ModalAnalysis Conference (IMAC rsquo97) pp 257ndash263 February 1997

[54] S W Doebling and C R Farrar ldquoUsing statistical analysis toenhance modal-based damage identificationrdquo in Proceedingsof the Structural Damage Assessment Using Advanced SignalProcessing Procedures pp 199ndash210 1997

[55] P Cornwell C R Farrar S W Doebling and H SohnldquoEnvironmental variability of modal propertiesrdquo ExperimentalTechniques vol 23 no 6 pp 45ndash48 1999

[56] C R Farrar and D A Jauregui ldquoComparative study of damageidentification algorithms applied to a bridge I ExperimentrdquoSmart Materials and Structures vol 7 no 5 pp 704ndash719 1998

[57] M A Wahab and G De Roeck ldquoEffect of temperature ondynamic system parameters of a highway bridgerdquo StructuralEngineering International vol 7 no 4 pp 266ndash270 1997

[58] G P Roberts and A J Pearson ldquoHealth monitoring ofstructures-towards a stethoscope for bridgesrdquo in Proceedings ofthe International Seminar onModal Analysis vol 2 pp 947ndash9521999

[59] S Alampalli ldquoInfluence of in-service environment on modalparametersrdquo in Proceedings of the 16th International ModalAnalysis Conference (IMAC rsquo98) pp 111ndash116 February 1998

[60] R G Rohrmann M Baessler S Said W Schmid and W FRuecker ldquoStructural causes of temperature affected modal dataof civil structures obtained by long time monitoringrdquo in Societyof Photo-Optical Instrumentation Engineers SPIE ProceedingsSeries pp 1ndash7 February 2000

[61] J Maeck B Peeters and G De Roeck ldquoDamage identifica-tion on the Z24-bridge using vibration monitoring analysisrdquoin Proceedings of European COST F3 Conference on SystemIdentification and Structural Health Monitoring pp 233ndash2422000

[62] B Peeters and G De Roeck ldquoOne-year monitoring of the Z24-Bridge environmental effects versus damage eventsrdquo Earth-quake EngineeringampStructuralDynamics vol 30 no 2 pp 149ndash171 2001

[63] B Peeters J Maeck and G De Roeck ldquoVibration-baseddamage detection in civil engineering excitation sources andtemperature effectsrdquo Smart Materials and Structures vol 10 no3 pp 518ndash527 2001

[64] Y Q Ni X G Hua K Q Fan and J M Ko ldquoCorrelating modalproperties with temperature using long-term monitoring dataand support vector machine techniquerdquo Engineering Structuresvol 27 no 12 pp 1762ndash1773 2005

[65] S L Desjardins N A Londono D T Lau and H Khoo ldquoReal-time data processing analysis and visualization for structuralmonitoring of the confederation bridgerdquoAdvances in StructuralEngineering vol 9 no 1 pp 141ndash157 2006

[66] C Liu and J T DeWolf ldquoEffect of temperature on modalvariability for a curved concrete bridgerdquo in Smart Structuresand Materials 2006 Sensors and Smart Structures Technologiesfor Civil Mechanical and Aerospace Systems vol 6174 ofProceedings of SPIE March 2006

[67] J-T Kim W-B Na J-H Park and J-S Lee ldquoStructuralhealth monitoring and risk alarming in plate-girder bridgesunder uncertain temperature conditionrdquo in Smart Structuresand Materials 2005 Sensors and Smart Structures Technologiesfor Civil Mechanical and Aerospace Systems vol 5765 ofProceedings of SPIE pp 1002ndash1011 SanDiego Calif USAMarch2005

[68] Y L Ding and A Q Li ldquoTemperature-induced variations ofmeasured modal frequencies of steel box girder for a long-spansuspension bridgerdquo International Journal of Steel Structures vol11 no 2 pp 145ndash155 2011


[69] Y L Ding A Q Li and F F Geng ldquoVariability of mea-sured modal frequencies of a suspension bridge under actualenvironmental effectsrdquo in Proceedings of the 5th InternationalConference on Bridge Maintenance Safety and Management(IABMAS rsquo10) pp 1392ndash1398 Lehigh University July 2010

[70] AAMosavi R Seracino and S Rizkalla ldquoEffect of temperatureon daily modal variability of a steel-concrete composite bridgerdquoJournal of Bridge Engineering vol 17 no 6 pp 979ndash983 2012

[71] G M Lloyd M L Wang and V Singh ldquoObserved variationsof mode frequencies of a prestressed concrete bridge withtemperaturerdquo in Proceedings of the 14th Engineering MechanicsConference pp 179ndash189 American Society of Civil EngineersMay 2000

[72] G M Lloyd M L Wang and V Singh ldquoObserved variations ofmode frequencies of a prestressed concrete bridge with temper-aturerdquo in Condition Monitoring of Materials and Structures pp179ndash189 2000

[73] J Kullaa ldquoEliminating environmental influences in structuralhealth monitoring using spatiotemporal correlation modelsrdquo inProceedings of the 1st European Workshop on Structural HealthMonitoring pp 742ndash749 2002

[74] M Breccolotti G Franceschini and A L Materazzi ldquoSensi-tivity of dynamic methods for damage detection in structuralconcrete bridgesrdquo Shock and Vibration vol 11 no 3-4 pp 383ndash394 2004

[75] W Song and S J Dyke ldquoAmbient vibration based modalidentification of the Emerson bridge considering temperatureeffectsrdquo in Proceedings of the 4thWorld Conference on StructuralControl and Monitoring pp 11ndash13 2006

[76] Z J Yang U Dutta D Zhu E Marx and N Biswas ldquoSea-sonal frost effects on the soil-foundation-structure interactionsystemrdquo Journal of Cold Regions Engineering vol 21 no 4 pp108ndash120 2007

[77] S K Di L X Wang H Li and Z S Wu ldquoStochasticsubspace identification research of concrete girders consideringtemperature influencerdquo in Proceedings of the 4th InternationalSymposium on Environment Vibrations-Prediction MonitoringMitigation and Evaluation Beijing China 2009

[78] V Zabel M Brehm and S Nikulla ldquoThe influence of tem-perature varying material parameters on the dynamic behaviorof short span railway bridgesrdquo in Proceedings of the 24thInternational Conference on Noise and Vibration EngineeringLeuven Belgium 2010

[79] L Wang J L Hou and J P Ou ldquoTemperature effect on modalfrequencies for a rigid continuous bridge based on long termmonitoringrdquo in Conference on Nondestructive Characterizationfor CompositeMaterials Aerospace Engineering Civil Infrastruc-ture and Homeland Proceedings of SPIE 2011

[80] A Cury C Cremona and J Dumoulin ldquoLong-term monitor-ing of a PSC box girder bridge operational modal analysisdata normalization and structural modification assessmentrdquoMechanical Systems and Signal Processing vol 33 pp 13ndash372012

[81] K Y Koo JMW BrownjohnD I List andRCole ldquoStructuralhealth monitoring of the Tamar suspension bridgerdquo StructuralControl andHealthMonitoring vol 20 no 4 pp 609ndash625 2013

[82] H Sohn M Dzwonczyk E G Straser K H Law T Meng andA S Kiremidjian ldquoAdaptive modeling of environmental effectsinmodal parameters for damage detection in civil structuresrdquo inProceedings of the 5thAnnual International Symposiumon SmartStructures and Materials pp 127ndash138 International Society forOptics and Photonics March 1998

[83] H Sohn ldquoEffects of environmental and operational variabilityon structural health monitoringrdquo Philosophical Transactions ofthe Royal Society A vol 365 no 1851 pp 539ndash560 2007

[84] C Liu J T DeWolf and J-H Kim ldquoDevelopment of a baselinefor structural health monitoring for a curved post-tensionedconcrete box-girder bridgerdquo Engineering Structures vol 31 no12 pp 3107ndash3115 2009

[85] B Moaveni and I Behmanesh ldquoEffects of changing ambienttemperature on finite element model updating of the DowlingHall Footbridgerdquo Engineering Structures vol 43 pp 58ndash682012

[86] X G Hua Y Q Ni J M Ko and K Y Wong ldquoModeling oftemperature-frequency correlation using combined principalcomponent analysis and support vector regression techniquerdquoJournal of Computing in Civil Engineering vol 21 no 2 pp 122ndash135 2007

[87] C H Loh and M C Chen ldquoModeling of environmental effectsfor vibration-based shm using recursive stochastic subspaceidentification analysisrdquo in Proceedings of the 4th Asia-PacificWorkshop on Structural Health Monitoring Melbourne Aus-tralia 2013

[88] H F Zhou Y Q Ni and J M Ko ldquoEliminating temperatureeffect in vibration-based structural damage detectionrdquo Journalof Engineering Mechanics vol 137 no 12 pp 785ndash796 2012

[89] H F Zhou Y Q Ni and J M Ko ldquoConstructing inputto neural networks for modeling temperature-caused modalvariability mean temperatures effective temperatures andprincipal components of temperaturesrdquo Engineering Structuresvol 32 no 6 pp 1747ndash1759 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

Journal of


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


of the structural mass damping and stiffness Changes inthe physical parameters will cause detectable changes inthe vibration properties [17] Up to now various damagedetection methods that are based on the vibration propertiesor their derivatives have been developed such as coordinatemodal assurance criterion (COMAC) enhanced coordinatemodal assurance criterion (ECOMAC) damage index (DI)method mode shape curvature (MSC) method modal flexi-bility index (MFI) changes in stiffness matrix obtained frommeasured mode shapes and changes in the curvature ofuniform load surface [18ndash21] However the application ofvibration-based damage detection methods for real long-span bridges is complicated because vibration properties ofa real structure are strongly affected by factors other thanstructural damages [22] Environmental changes especiallytemperature also cause changes in vibration propertiessuch as frequency mode shapes and damping ratio whichmay mask the changes caused by structural damage [23]To avoid false condition assessments the variability of thevibration properties due to temperature must be thoroughlyunderstood and quantified so that it can be discriminatedfrom changes caused by damage [17] As a matter of fact thevariation of temperatures affects vibration properties of long-span bridges in a complicated manner Firstly the tempera-ture distribution and its variation in a bridge are generallynonuniform and time dependent which induces that thechanges of physical parameters among different structuralparts are asynchronous [24] Secondly all members of abridge change their size under different temperatures for theeffect of thermal expansion and contraction [25]Thirdly theYoungrsquos modulus and the shear modulus of material whichare the main parameters determining vibration propertiesof bridges vary because of the fluctuation of temperatureFourthly daily or seasonal variations in temperature maychange the stiffness of bridge expansion joints or supportsand thus strengthen or weaken the constraints resultingin an apparent shift of vibration properties [26] Lastlytemperature variations result in considerable thermal stressesor stress redistribution in structure which may change thevibration properties of bridges Numerous investigationsindicate that temperature is the critical source causing thevariability of vibration properties of long-span bridges andthe variations of modal frequencies caused by temperaturemay reach 5 to 10 for highway bridges which inmost casesexceed the changes of frequencies due to structural damage ordeterioration [12]

During the past decades considerable effort has beendevoted to investigating the correlations between tempera-tures and vibration properties of long-span bridges Accord-ing to the approach of researching the existing achievementscan be categorized into three types termed as theoreticalderivation trend analysis and quantitative model The the-oretical derivation is used to establish closed-form formu-lations of the correlations between temperatures and vibra-tion properties of long-span bridges by classical structuraldynamic methods The trend analysis is applied to analyzethe variation of vibration properties of long-span bridgescaused by temperature changes through numerical simula-tion laboratory test and field monitoring The quantitative

model is intended to develop no physical blindmodels whichcan quantificationally forecast changes in vibration propertiesbased on the input of time series of new temperaturesthrough a large amount of measurement data This paperaims to present a summary review of correlations betweentemperatures and vibration properties of long-span bridgesAlthough Xia et al [30] gave a literature review concerningvariations in vibration properties of civil structures underchanging temperature conditions before the content is toosketchy to give reference for later research Furthermoresome important contributions were omitted In the fol-lowing sections research progress of correlations betweentemperatures and vibration properties of long-span bridgesis extensively described from theoretical derivation methodtrend analysis approach and quantitative model In additionsome existing problems and promising research efforts aboutthis topic are expected to be extracted through pertinentassessment This paper is not intended to list all the literatureabout the effects of temperature on vibration properties ofbridges but exhibit some representative achievements whichare expected to provide reference for future research

2 Progress of Theoretical Derivation Method

A long-span bridge is a complex multi-degree-of-freedomnonlinear system This makes it is extremely difficult toformulate the vibration properties by closed-form equa-tions Furthermore temperature change affects a bridge in acomplicated way It not only affects mechanical propertiesbut also geometrical features constraint conditions andboundary conditions As a result only a few researchershave investigated the correlations between temperatures andvibration properties of long-span bridges through the theo-retical derivation methods

In order to simplify the problem Xia et al [34] assumedthat mass and the boundary conditions remain unchangedand temperature affects only the geometry and mechanicalproperties A single-span beammade of an isotropic materialwas used as an example to model the effect of temperatureon natural frequencies For this beam its undamped flexuralvibration frequency of order n is [35]

119891119899=1198992120587ℎ

21198972radic

119864

12120588 (1)

where119891119899denotes the 119899th frequency ℎ and 119897 are the height and

length of the beam respectively 119864 represents the modulusof elasticity and 120588 denotes the density of the material Thevariation of the above equation can be expressed as

120575119891119899

119891119899

=120575ℎ

ℎminus 2

120575119897

119897+1

2

120575119864

119864minus1

2

120575120588

120588 (2)





120575ℎ

ℎ= 120579119905120575119905

120575119897

119897= 120579119905120575119905

120575120588

120588= minus3120579

119905120575119905

120575119864

119864= 120579119864120575119905

(3)


120575119891119899

119891119899

=1

2(120579119905+ 120579119864) 120575119905 (4)


119891119899=1205822119899120587ℎ

21198972radic

119864

12120588 (5)




1198911015840119899= 119891119899

radic1 +1198651198972

11989921205872119864119868 (6)





119905 which





















Modenumber


0∘C 10∘C 20∘C 30∘C 40∘C 50∘C1 76150 71822 68034 66194 65328 631642 101455 98320 95185 94244 93617 920503 212223 203694 195165 192606 190901 1866364 295347 289123 282899 281031 279787 276674











North girder

South girder

Power supply

Signal

Sensor

Model analysis

1 2 3 4 5 6 7 8 9

1110 12 13 14 15 16

R A

17 18


generator



Accelerometers

Support Support


1 2 3 4 5 6 7

8 9 10 11 12 13 14

100mm 100mm

(T9 sim T11)

50

mm

50

mm

700

mm

375 times 8 = 3000mm



3700

3500

3300

3100

2900

2700

2500

2300

2100

1900

1700

Tem

pera

ture

(∘C)

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2015

2010

2005

2000

1995

1990

1985

1980

1975

1970

1965







2

18

16

14

12

1

08

06

02

04

0

Time

Dam

ping

ratio

()

1st mode2nd mode

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200











46

44

42

4

38


Temperature (∘C)

Transition from the

Freq

uenc

y (H

z)












mpe

ratu

re

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

115

1145

114

1135

113

Freq

uenc

y (0001

Hz)

(a)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

138

1375

137

1365

136

1355

(b)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

184

1835

183

1825

182

1815


(c)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)


242

241

240

239

238

237

236

(d)




119910 = 119909119908 + 120576 (7)


119910 =[[[[

[

119910 (1)119910 (2)

119910 (119899)

]]]]

]

119883 =[[[[

[

1 1199091(1) 119909


119903(1)

1 1199091(2) 119909


119903(2)

sdot sdot sdot

1 1199091(119899) 119909

2(119899) sdot sdot sdot 119909

119903(119899)

]]]]

]

120576 =[[[[

[

120576 (1)120576 (2)

120576 (119899)

]]]]

]

(8)



78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)








119902 = 1205730+ 120573119905119905 + 120576119891 (9)



119905(slope)






Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120575ℎ

ℎ= 120579119905120575119905

120575119897

119897= 120579119905120575119905

120575120588

120588= minus3120579

119905120575119905

120575119864

119864= 120579119864120575119905

(3)


120575119891119899

119891119899

=1

2(120579119905+ 120579119864) 120575119905 (4)


119891119899=1205822119899120587ℎ

21198972radic

119864

12120588 (5)




1198911015840119899= 119891119899

radic1 +1198651198972

11989921205872119864119868 (6)





119905 which





















Modenumber


0∘C 10∘C 20∘C 30∘C 40∘C 50∘C1 76150 71822 68034 66194 65328 631642 101455 98320 95185 94244 93617 920503 212223 203694 195165 192606 190901 1866364 295347 289123 282899 281031 279787 276674











North girder

South girder

Power supply

Signal

Sensor

Model analysis

1 2 3 4 5 6 7 8 9

1110 12 13 14 15 16

R A

17 18


generator



Accelerometers

Support Support


1 2 3 4 5 6 7

8 9 10 11 12 13 14

100mm 100mm

(T9 sim T11)

50

mm

50

mm

700

mm

375 times 8 = 3000mm



3700

3500

3300

3100

2900

2700

2500

2300

2100

1900

1700

Tem

pera

ture

(∘C)

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2015

2010

2005

2000

1995

1990

1985

1980

1975

1970

1965







2

18

16

14

12

1

08

06

02

04

0

Time

Dam

ping

ratio

()

1st mode2nd mode

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200











46

44

42

4

38


Temperature (∘C)

Transition from the

Freq

uenc

y (H

z)












mpe

ratu

re

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

115

1145

114

1135

113

Freq

uenc

y (0001

Hz)

(a)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

138

1375

137

1365

136

1355

(b)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

184

1835

183

1825

182

1815


(c)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)


242

241

240

239

238

237

236

(d)




119910 = 119909119908 + 120576 (7)


119910 =[[[[

[

119910 (1)119910 (2)

119910 (119899)

]]]]

]

119883 =[[[[

[

1 1199091(1) 119909


119903(1)

1 1199091(2) 119909


119903(2)

sdot sdot sdot

1 1199091(119899) 119909

2(119899) sdot sdot sdot 119909

119903(119899)

]]]]

]

120576 =[[[[

[

120576 (1)120576 (2)

120576 (119899)

]]]]

]

(8)



78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)








119902 = 1205730+ 120573119905119905 + 120576119891 (9)



119905(slope)






Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Modenumber


0∘C 10∘C 20∘C 30∘C 40∘C 50∘C1 76150 71822 68034 66194 65328 631642 101455 98320 95185 94244 93617 920503 212223 203694 195165 192606 190901 1866364 295347 289123 282899 281031 279787 276674











North girder

South girder

Power supply

Signal

Sensor

Model analysis

1 2 3 4 5 6 7 8 9

1110 12 13 14 15 16

R A

17 18


generator



Accelerometers

Support Support


1 2 3 4 5 6 7

8 9 10 11 12 13 14

100mm 100mm

(T9 sim T11)

50

mm

50

mm

700

mm

375 times 8 = 3000mm



3700

3500

3300

3100

2900

2700

2500

2300

2100

1900

1700

Tem

pera

ture

(∘C)

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2015

2010

2005

2000

1995

1990

1985

1980

1975

1970

1965







2

18

16

14

12

1

08

06

02

04

0

Time

Dam

ping

ratio

()

1st mode2nd mode

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200











46

44

42

4

38


Temperature (∘C)

Transition from the

Freq

uenc

y (H

z)












mpe

ratu

re

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

115

1145

114

1135

113

Freq

uenc

y (0001

Hz)

(a)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

138

1375

137

1365

136

1355

(b)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

184

1835

183

1825

182

1815


(c)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)


242

241

240

239

238

237

236

(d)




119910 = 119909119908 + 120576 (7)


119910 =[[[[

[

119910 (1)119910 (2)

119910 (119899)

]]]]

]

119883 =[[[[

[

1 1199091(1) 119909


119903(1)

1 1199091(2) 119909


119903(2)

sdot sdot sdot

1 1199091(119899) 119909

2(119899) sdot sdot sdot 119909

119903(119899)

]]]]

]

120576 =[[[[

[

120576 (1)120576 (2)

120576 (119899)

]]]]

]

(8)



78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)








119902 = 1205730+ 120573119905119905 + 120576119891 (9)



119905(slope)






Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











North girder

South girder

Power supply

Signal

Sensor

Model analysis

1 2 3 4 5 6 7 8 9

1110 12 13 14 15 16

R A

17 18


generator



Accelerometers

Support Support


1 2 3 4 5 6 7

8 9 10 11 12 13 14

100mm 100mm

(T9 sim T11)

50

mm

50

mm

700

mm

375 times 8 = 3000mm



3700

3500

3300

3100

2900

2700

2500

2300

2100

1900

1700

Tem

pera

ture

(∘C)

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200

2015

2010

2005

2000

1995

1990

1985

1980

1975

1970

1965







2

18

16

14

12

1

08

06

02

04

0

Time

Dam

ping

ratio

()

1st mode2nd mode

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200











46

44

42

4

38


Temperature (∘C)

Transition from the

Freq

uenc

y (H

z)












mpe

ratu

re

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

115

1145

114

1135

113

Freq

uenc

y (0001

Hz)

(a)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

138

1375

137

1365

136

1355

(b)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

184

1835

183

1825

182

1815


(c)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)


242

241

240

239

238

237

236

(d)




119910 = 119909119908 + 120576 (7)


119910 =[[[[

[

119910 (1)119910 (2)

119910 (119899)

]]]]

]

119883 =[[[[

[

1 1199091(1) 119909


119903(1)

1 1199091(2) 119909


119903(2)

sdot sdot sdot

1 1199091(119899) 119909

2(119899) sdot sdot sdot 119909

119903(119899)

]]]]

]

120576 =[[[[

[

120576 (1)120576 (2)

120576 (119899)

]]]]

]

(8)



78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)








119902 = 1205730+ 120573119905119905 + 120576119891 (9)



119905(slope)






Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

18

16

14

12

1

08

06

02

04

0

Time

Dam

ping

ratio

()

1st mode2nd mode

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2000

2100

2200











46

44

42

4

38


Temperature (∘C)

Transition from the

Freq

uenc

y (H

z)












mpe

ratu

re

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

115

1145

114

1135

113

Freq

uenc

y (0001

Hz)

(a)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

138

1375

137

1365

136

1355

(b)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

184

1835

183

1825

182

1815


(c)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)


242

241

240

239

238

237

236

(d)




119910 = 119909119908 + 120576 (7)


119910 =[[[[

[

119910 (1)119910 (2)

119910 (119899)

]]]]

]

119883 =[[[[

[

1 1199091(1) 119909


119903(1)

1 1199091(2) 119909


119903(2)

sdot sdot sdot

1 1199091(119899) 119909

2(119899) sdot sdot sdot 119909

119903(119899)

]]]]

]

120576 =[[[[

[

120576 (1)120576 (2)

120576 (119899)

]]]]

]

(8)



78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)








119902 = 1205730+ 120573119905119905 + 120576119891 (9)



119905(slope)






Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











46

44

42

4

38


Temperature (∘C)

Transition from the

Freq

uenc

y (H

z)












mpe

ratu

re

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

115

1145

114

1135

113

Freq

uenc

y (0001

Hz)

(a)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

138

1375

137

1365

136

1355

(b)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

184

1835

183

1825

182

1815


(c)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)


242

241

240

239

238

237

236

(d)




119910 = 119909119908 + 120576 (7)


119910 =[[[[

[

119910 (1)119910 (2)

119910 (119899)

]]]]

]

119883 =[[[[

[

1 1199091(1) 119909


119903(1)

1 1199091(2) 119909


119903(2)

sdot sdot sdot

1 1199091(119899) 119909

2(119899) sdot sdot sdot 119909

119903(119899)

]]]]

]

120576 =[[[[

[

120576 (1)120576 (2)

120576 (119899)

]]]]

]

(8)



78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)








119902 = 1205730+ 120573119905119905 + 120576119891 (9)



119905(slope)






Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










mpe

ratu

re

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

115

1145

114

1135

113

Freq

uenc

y (0001

Hz)

(a)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

138

1375

137

1365

136

1355

(b)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)

184

1835

183

1825

182

1815


(c)

Tem

pera

ture

Time

24

22

20

18

16

14

12

0000 0600 1200 1800 0000

Freq

uenc

y (0001

Hz)


242

241

240

239

238

237

236

(d)




119910 = 119909119908 + 120576 (7)


119910 =[[[[

[

119910 (1)119910 (2)

119910 (119899)

]]]]

]

119883 =[[[[

[

1 1199091(1) 119909


119903(1)

1 1199091(2) 119909


119903(2)

sdot sdot sdot

1 1199091(119899) 119909

2(119899) sdot sdot sdot 119909

119903(119899)

]]]]

]

120576 =[[[[

[

120576 (1)120576 (2)

120576 (119899)

]]]]

]

(8)



78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)








119902 = 1205730+ 120573119905119905 + 120576119891 (9)



119905(slope)






Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










78

77

76

75

74

730 5 10 15 20 25

Time (hr)

Firs

t fre

quen

cy (H

z)








119902 = 1205730+ 120573119905119905 + 120576119891 (9)



119905(slope)






Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Temperature

115

1145

114

1135

113

112515 20 25 30 35 40

Freq

uenc

y (0001

Hz)


15 20 25 30 35 40

138

1375

137

1365

136

1355

Temperature

Freq

uenc

y (0001

Hz)


1845

184

1835

183

1825

182

1815

181


Temperature15 20 25 30 35 40

Freq

uenc

y (0001

Hz)


242

240

238

236

23415 20 25 30 35 40

Temperature


Freq

uenc

y (0001

Hz)





119865119894= 119891119894+ 119886119879 (10)




1198651


1198652


1198653





120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120573119905

1205731199051205730

1205730

120573119905

1205731199051205730

1205730

120573119905

1205731199051205730








119891 (119879 119899) = 119901119899119879119899 + 119901


21198792 + 119901

11198791 + 119901

0 (12)


119894(119894 = 0simn)







48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










48

475

47

465

0126 0130 0203 0207Date

f(H

z)

(a)

62

6

58

0126 0130 0203 0207Date

f(H

z)

(b)

0126 0130 0203 0207Date

74

72

7

f(H

z)

(c)

0126 0130 0203 0207Date

91

9

89

f(H

z)

(d)

0126 0130 0203 0207Date

136

134

132

13

SimulatedIdentified

f(H

z)

(e)

142

14

138

136

134

0126 0130 0203 0207Date

SimulatedIdentified

f(H

z)

(f)






0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0175

0170

0165

0160501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)



501 520 540 560 680 600

Freq

uenc

y (H

z)

Time (hour)


0175

0170

0165

0160


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured


0174

0172

0170

0168

0166

0164

0162

0160

Freq

uenc

y (H

z)

0 50 100 150 200 250 300 350 400

Sample order

PredictedMeasured








0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0174

0172

0170

0166

0168

0164

0162

0160

01580 20 40 60 80 100 120 140 160 180 200

Freq

uenc

y (H

z)

Sample order


0 20 40 60 80 100 120 140 160 180 200

Sample order

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


0 20 40 60 80 100 120 140 160 180 200

Sample order

PredictedMeasured

0174

0172

0170

0166

0168

0164

0162

0160

0158

Freq

uenc

y (H

z)


















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Acknowledgments


References





































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










































































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









review article a summary review of correlations between...

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