Research ArticleSimulation Study on Train-Induced Vibration Control of aLong-Span Steel Truss Girder Bridge by Tuned Mass Dampers

Hao Wang12 Tianyou Tao12 Huaiyu Cheng12 and Xuhui He3

1 Key Laboratory of CampPC Structures of Ministry of Education Southeast University Nanjing 210096 China2 School of Civil Engineering Southeast University Nanjing 210096 China3 School of Civil Engineering Central South University Changsha 410082 China

Correspondence should be addressed to Hao Wang wanghao1980seueducn

Received 15 July 2014 Revised 8 October 2014 Accepted 9 October 2014 Published 16 December 2014

Academic Editor Ting-Hua Yi

Copyright copy 2014 Hao Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Train-induced vibration of steel truss bridges is one of the key issues in bridge engineering This paper talks about the applicationof tunedmass damper (TMD) on the vibration control of a steel truss bridge subjected to dynamic train loadsTheNanjing YangtzeRiver Bridge (NYRB) is taken as the research object and a recorded typical train load is included in this study With dynamic finiteelement (FE) method the real-time dynamic responses of NYRB are analyzed based on a simplified train-bridge time-varyingsystemThereinto two cases including single trainmoving at one side and two trainsmoving oppositely are specifically investigatedAccording to the dynamic characteristics and dynamic responses of NYRB the fourth vertical bending mode is selected as thecontrol target and the parameter sensitivity analysis on vibration control efficiency with TMD is conducted Using the first-orderoptimizationmethod the optimal parameters of TMD are then acquired with the control efficiency of TMD the static displacementofMidspan expenditure of TMDs andmanufacture difficulty of the damper considered Results obtained in this study can providereferences for the vibration control of steel truss bridges

1 Introduction

As one of the most important infrastructures railway bridgeperforms the role that maintains the transportation betweentwo separated locations to overcome the space and terrainlimitations Due to the large gravity of trains and the require-ment for bridge span railway bridges usually demand muchhigher stiffness than highway bridges On account of thesuperior characteristics of steel material and truss structurethe steel truss bridge exhibits the capability to cross widerivers or canyons with light weight and enough stiffness tocope with train loads vehicle loads or both Although theexisting steel truss bridge has been equipped with sufficientcapability for structural safety subjected to train loads fieldtests in China Japan and Europe have indicated obviousdynamic responses exceeding that anticipated on certainsmall andmedium span railway bridges [1ndash3]The theoreticalinvestigations validate the contribution of structural reso-nance arising from the moving of trains with high speed to

this phenomenon [4]When the trainmoves on a steel bridgewith a high speed the train-induced vibrations of the steeltruss bridge will lead to the deterioration of passengersrsquo com-fort the fatigue of steel structural component the decreaseof traffic safety accompanied by damage to the tracks andthe augment of maintenance cost Therefore some effectivemeasures need to be carried out to suppress the vibrations ofsteel truss bridge subjected to dynamic train loads

Since the 1970s the tuned mass damper (TMD) hasbeen introduced to protect the civil structures from suffer-ing vibration-induced major damage because of its lateralmotions [5ndash9] TMD is a classical engineering device whichconsists of a spring a mass and a viscous damper Thesimple structure of TMD brings about its obvious virtuesof economical costing and installation convenience TMDis considered to be sensitive to the frequency variation ofstructures but the viscous damper can broaden its effectivefrequency ranges [10] The existing researches and applica-tions have proved that the TMD is an efficient approach

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 506578 12 pageshttpdxdoiorg1011552014506578

2 Mathematical Problems in Engineering

to attenuate undesirable vibrations of a vibrating systemand it has already been extensively studied and applied tosuppress structural vibrations induced by wind earthquakestrains and so forth As some typical examples the windtunnel and theoretical analyses by Xu et al [11] verifiedthe effectiveness of TMD in suppressing wind-induced tallbuildingmotions Using the perturbation solutions formulasrelated to designing the TMD attached to the single-degree-of-freedom (SDOF) system for various types of loading wereproposed by Fujino and Abe in 1993 [12] Aiming to dissipateearthquake input energy Wong [13] investigated the energytransfer process of TMD in improving the ability of inelasticstructures

Although TMD has been extensively studied or success-fully implemented for suppressing vibrations of structuresdue to environmental loadings such as wind and earthquakesonly a few researches have been conducted on applicationsof TMD in reducing the vibrations of bridges due to trainloads and the existing researches are mainly focused onsimply supported reinforced concrete bridges with moderateor small spans [3 14ndash16] As for the long-span continuoussteel truss girder bridge the studies on vibration controlof bridges subjected to dynamic train loads with TMD arefewer The objective of this study is to apply TMDs withoptimal parameters for suppressing vibrations of a steel trussbridge subjected to moving trains With typical train axleloads the parametric sensitivity analysis is conducted andthe effectiveness of TMD for suppressing dynamic vibrationsis examined based on a simplified train-bridge time-varyingsystem Taking account of the control efficiency expenditureof TMDs TMD-induced static displacement of the bridgeand the manufacturing difficulty of damping of differentmagnitude an objective function for evaluating the controlefficiency is proposed and the optimal parameters of TMDare finally determined with the first optimization methodThe analytical procedure and results can provide referencesfor the vibration control of steel truss bridge subjected todynamic train loads

2 Simplified Train-BridgeTime-Varying System

A bridge is possibly to suffer from different kinds of dynamicloads during its long-term service period such as the vehicleswinds and earthquakes Comparedwith other dynamic loadsacting on the bridge the location of train loads on bridgesis time-varying which makes the dynamic characteristics ofthe train-bridge coupled system dependent on the locationof train loads Therefore the resonance condition can onlyhappen in a shortmoment and it is difficult to establish a clearcorrelation between the moving train loads and the bridgeresponses To clearly identify the train-bridge time-varyingsystem some simplified models have already been proposed[17ndash21]These simplified models provide the basis for furtherinvestigations on the train-bridge coupled system

In this study a broadly used simplified model proposedby Kriloff in 1905 [18] is adopted to simulate the train-bridgetime-varying system The loads induced by train moving at

a constant speed are simplified as a series of concentratedforces moving at the same speed These concentrated forceslocate at the junctions of wheels and the track Consideringthe difficulty in solving differential equations with variablecoefficients the inertia forces of the train are neglected duringthe analysis

3 Engineering Background

31 Bridge Description The Nanjing Yangtze River Bridge(NYRB) as shown in Figure 1 is a double-decked steel trussbridge across the Yangtze River Since 1968 it has operatedas a railway and highway combined bridge for 46 yearsThe upper deck of NYRB is employed as part of NationalHighway 104 in China with four lanes The total width of thecarriage way is 15m accompanied by a 225m wide walkwayon each side The lower deck of NYRB serves for railwaytransportation with a width of 14m Two pairs of parallelseamless railway tracks are installed on the deck so that thetrains can move in opposite directions simultaneously Themain bridge is the combination of three continuous girderbridges and a simply supported bridge with a total length of1568m Thereinto three expansion joints are set at Piers 14 and 7 respectively to separate the three-span continuousgirder bridges and the simply supported bridge Each spanof the continuous bridge is 160m long and the length ofthe simply supported bridge is 128m The main girder iscomposed of the steel truss structure with stiffening chordmembers Each segment for the main girder is 8m with aheight of 16m and connections between each segment aremainly rivet joints In this paper the three-span continuousgirder bridge between Piers 7 and 10 near Nanjing will betaken as the research object

32 Simplified Model of Dynamic Train Loads The structuralhealth monitoring system of NYRB [22] recorded a trainmoving at 70 kmh to cross the Yangtze River The trainis composed of two locomotives and eighteen carriages asshown in Figure 2 There are six axles on each locomotiveand each axle weighs 22 tons which is evenly assigned to twowheels All of the carriages work with four axles and eachaxle weighs 20 tons so that each wheel can transfer 10-tonweight to the bridge structure In order to study the dynamicbehaviour of the train-bridge time-varying system the trainherein is simplified as a series of moving concentrated forceswith fixed intervals The axle distance is detailed in Figure 2and the unit for the distance is meter The overall lengthof the train is 292 meters In regard to the train loads theconcentrated forces are acquired by the axles loadsmultipliedby the gravitational acceleration of 98ms2

4 Finite Element Model and Its Application

41 Finite Element Model Based on ANSYS NYRB is a trussstructure with complicated features due to its various typesof vertical and lateral connection systems and its mainfunctions addressed to both railway and highway The three-span continuous girder bridge contains 7690 elements and 30

Mathematical Problems in Engineering 3

012345678910

128Nanjing Pukou160 times 3 = 480 160 times 3 = 480 160 times 3 = 480

Figure 1 Layout of Nanjing Yangtze River Bridge (unit m)

middot middot middotLocomotive Locomotive Carriage Carriage

2 2 12 2 2 4 2 2 8 2 2 3 2 7 2 14 times 16 + 3 = 227 2 7 2

292

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

980

kN980

kN

980

kN980

kN

980

kN980

kN

980

kN980

kN

Figure 2 Layout of the train loads and axle distance

types of sections With appropriate simplification a spatialFE model of NYRB is established based on ANSYS [23]as shown in Figure 3 The main truss rods track beamsand lateral connection systems are modeled by spatial beamelement (Beam 4) And the pavement on highway surfaceand seamless steel rail which have little contribution tostructural stiffness are simulated by mass element (Mass 21)imposed on the corresponding nodes of the truss bridgegirder To facilitate the exertion of axial forces the trackbeam is partially refined in the mesh generationThe verticallateral vertical-rotational and longitudinal-rotational DOFs(degrees of freedom) are restricted at the four supports[24 25] In addition the bearing between Side-span 1 andMidspan is restricted in longitudinal directionwhile the otherdirections are set free

42 Dynamic Characteristics of NYRB In order to acquirethe natural frequencies and corresponding mode shapes ofNYRB a modal analysis is conducted and the first 200modesare extracted with subspace iterationmethod Table 1 lists thefirst 12 natural frequencies and corresponding characteristicsAmong the first 12 structural modes the lowest modal fre-quency is 10576Hz which corresponds to symmetric verticalbending vibration of the whole bridge The first lateral bend-ingmode appears as the second structuralmode ofNYRBandthis phenomenon reflects a relatively larger lateral stiffnessthan vertical stiffness existing in NYRB All of the first threelateral bendingmodes exhibit the lateral vibration of only onespan (Side-span 1 Midspan or Side-span 2) which revealsthe lateral vibration of each span has little influence on thoseof the other two spans Similar to the lateral bending modethe second and the third vertical bending mode shapes arethe vibration of Side-span 1 and Side-span 2 respectivelyThe torsional vibration modes appear continuously fromthe seventh to the tenth structural mode Specifically thefirst three perform as symmetric or antisymmetric torsionalvibration of the whole bridge accompanied by lateral bendingvibration while the fourth one performs as the torsional

Table 1 The first 12 natural frequencies and mode shapes of NYRB

NumberNaturalfrequency

(Hz)

Modedirection Characteristics of modes

1 10576 V Symmetric vertical bendingvibration of the whole bridge

2 11937 L Lateral bending vibration ofSide-span 1

3 12040 L Lateral bending vibration ofSide-span 2

4 14140 L Lateral bending vibration ofMidspan

5 17649 V Vertical bending vibration ofSide-span 1

6 21638 V Vertical bending vibration ofSide-span 2

7 22736 L + T

Antisymmetric lateral bendingvibration of the whole bridgeaccompanied by antisymmetrictorsional vibration

8 23728 L + T

Symmetric lateral bendingvibration of the whole bridgeaccompanied by antisymmetrictorsional vibration

9 25164 L + T

Symmetric lateral bendingvibration of the whole bridgeaccompanied by symmetrictorsional vibration

10 27352 T Torsional vibration of Side-span 1

11 28726 V Symmetric vertical bendingvibration of the whole bridge

12 32437 V Antisymmetric vertical bendingvibration of the whole bridge

vibration of Side-span 1 The fourth and the fifth verticalbending mode correspond to the eleventh and the twelfth

4 Mathematical Problems in Engineering

Side-span 1

Midspan

Side-span 2

Figure 3 Spatial finite element model of NYRB

0 10 20 30 40 50minus003

minus002

minus001

000

001

002

Vert

ical

disp

lace

men

t (m

)

StaticDynamic

StaticDynamic

15 20 25minus0028minus0026minus0024minus0022minus0020

Amplify

0 10 20 30 40 50minus006

minus004

minus002

000

002

004

Vert

ical

disp

lace

men

t (m

)

Amplify

15 20 25minus0056minus0052minus0048minus0044minus0040

Case 1 single train moving at one side Case 2 two trains moving oppositelyt (s) t (s)

Figure 4 Vertical displacement in the middle of Midspan

structural mode respectively Among them the fourth oneis the symmetric vertical bending vibration and the fifthone is the antisymmetric vertical bending vibration of thewhole bridge In the view of mode shapes only the firstand the fourth vertical bending modes dominate the verticalvibration of the Midspan while others have little influence

43 Dynamic Responses of NYRB Excited by Train LoadsConsidering the functions of NYRB two cases includingsingle train moving at one side and two trains moving oppo-sitely are analyzed in the train-bridge time-varying systemThereinto the second case describes a situation that twotrains set off from the two sides of NYRB simultaneously andthe time phase difference is not included in this study Duringthe transient dynamic analysis by ANSYS the full methodwith intact system matrix is utilized and the Newmark-120573technique is adopted to conduct the time integration Onaccount of the steel material employed in the construction ofNYRB the damping ratio of the whole structure is set as 2

431 Displacement Responses of the Bridge Span With tran-sient analyses of the train-bridge time-varying system thedisplacements in the middle of the spans are the mostprominent Table 2 lists the maximum displacement in themiddle of each span of the two cases Thereinto the torsionaldisplacement has been multiplied by half of the bridge deck

width to unify the unit As shown in Table 2 the verticaldisplacement is the most prominent while the lateral andtorsional displacements aremuch smaller However themax-imum vertical displacement of the two cases does not surpass006 meters and it is much smaller than one-thousandthlength of the main span Hence the small maximum verticaldisplacement of the bridge span under the actions of movingtrains reflects the strong vertical stiffness of NYRB

In order to reveal the difference of vertical displacementbetween static train loads and dynamic train loads thedisplacement response of NYRB under static train loads iscalculated The static and dynamic vertical displacementsin the middle of Midspan of the two cases are shown inFigure 4 Since the trainmoves at a speed of 70 kmh the traincompletely leaves the bridge after 4279 s As for the verticaldisplacements induced by static train loads and dynamictrain loads there is little difference between them during thewhole procedure except for the time duration from 1502 s to2469 s At the moment of 1502 s the train is just completelylanding on the bridge And at the time of 2469 s the trainbegins to leave the bridge and the time of 2469 s correspondsto the last moment that the whole train is on the bridgeDuring the time interval between 1502 s and 2469 s thedynamic vertical displacement is fluctuant However in thisperiod the deviation between the static and dynamic verticaldisplacements is still smallThe detailed difference among the

Mathematical Problems in Engineering 5

Table 2 Maximum displacement in the middle of each span (unit m)

Case Case 1 single train moving at one side Case 2 two trains moving oppositelySide-span 1 Midspan Side-span 2 Side-span 1 Midspan Side-span 2

Vertical 00456 00260 00442 00522 00516 00498Lateral 00006 00015 00016 00008 00013 00019Torsional 00036 00032 00037 00034 0000067 00037Note the displacement in the table has subtracted the gravity-caused static displacement of the structure the values in the table are the absolute values ofmaximum displacement

time interval has been amplified in the lower right cornerof each figure The phenomenon above has similar featureswith the analytical results by Chatterjee et al in 1994 [21]Due to the small vertical displacements induced by dynamictrain loads the displacement cannot be chosen as the targetto conduct vibration control analysis

432 Acceleration Responses of the Bridge Span An obviousdifference between the static and the dynamic train actionsis the excited acceleration response of the bridge structure bythe dynamic train loads The acceleration magnitude reflectsthe intensity of vibration of the whole structure which isclosely related to the deterioration of passengersrsquo comfortthe decrease of traffic safety accompanied by damage tothe tracks and the augment of maintenance cost The timehistories of the acceleration responses in the middle of eachspan of two cases are shown in Figure 5

As shown in Figure 5 the overall trend reveals that theamplitudes of both lateral and torsional acceleration aremuch smaller than that of vertical acceleration in each spanCompared to Case 1 the vertical acceleration of each spanincreases prominently in Case 2 In regard to lateral andtorsional accelerations the differences between the two casesare not consistent In the middle of Side-span 1 the ampli-tudes of both lateral acceleration and torsional accelerationincrease a little inCase 2 comparedwith those of Case 1 In themiddle of Side-span 2 obvious peak accelerations occur inthe lateral and torsional time histories of Case 2 and the peakaccelerations occur near the point of 165 s As for Midspanthe lateral and torsional accelerations are counteracted to alarge extent when two trains move oppositely on the bridgeThe analysis above shows that the concentration should bepaid to vertical acceleration of each span in this study

In Case 1 the maximum vertical accelerations ofSide-span 1 Side-span 2 and Midspan are 00520ms200668ms2 and 01453ms2 respectively In Case 2 themaximum vertical accelerations of Side-span 1 Side-span 2and Midspan are 01027ms2 01509ms2 and 03172ms2respectively From the peak accelerations above it is easy tofind that the acceleration in the middle of Midspan is muchlarger than that of Side-span 1 and Side-span 2 all of thevertical accelerations especially in Case 2 are so prominentthat some measures need to be employed to suppress thedynamic train-induced vibrations of the bridge

The vertical acceleration time histories of each span inFigure 5 are transformed into PSD (power spectral density)with Welch transformation [26] as shown in Figure 6Though the dynamic loads are different in the two cases

the frequency-domain features are similarThe PSDs indicatethat the vibration energy of the two cases concentratesaround an exciting frequency of 2761Hz which is in theneighborhood of the fourth vertical bending mode of NYRBThis phenomenon shows that the fourth vertical bendingmode contributes most to the train-induced vibration ofNYRB Considering that the exciting frequency may be onthe deviation of 2761Hz when the moving speed of thetrain changes the fourth vertical bending mode is selectedas the control mode in this study Consequently the TMDis designed to control the fourth vertical bending mode thatdominates the vertical vibration responses of the train-bridgesystem

5 Analysis of the Vibration ControlEffect of TMD

51 Layout of TMDs onNYRB For structureswhose dynamicresponses are dominated by one structural mode the TMDsshould be deposited near the amplitude of the correspondingmode shape So the TMDs are attached to the center regionof Midspan of NYRB where the mode shape of the fourthvertical bending mode is the largest For the nether deckof NYRB shown in Figure 7 there are four track beamssymmetrically arranged in the inner side for the moving oftrains and two structural beam elements in the outer sideto serve for the whole structure The transverse beams areutilized to strengthen the integrity of the bridge structurewith an adjacent distance of 8 meters In order to guaranteethe simplicity of the loading path each TMD is hung atthe joint of the longitudinal beam and the transverse beamAnd three transverse beams in the central region of Midspanare selected to install the TMDs thus 18 TMDs in total areemployed to conduct the vibration control of NYRB Thedetailed layout of TMDs on NYRB can be seen in Figure 7amongwhich the red blocks represent the TMDs inMidspan

52 Design Parameters for the TMD The TMD is a devicewith a simple structure and explicit working mechanismIt is made up of a mass block and a spring as well asa damper The main contents for designing a TMD areconnected to the mass of the block the stiffness coefficientof the spring and the damping coefficient of the damperTwo mechanical parameters including the natural frequencyand the damping ratio are related with the above threevariables According to the working mechanism of the TMDthe natural frequency of the TMD is determined as the same

6 Mathematical Problems in Engineering

0 10 20 30 40 50minus08minus04

000408

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus15minus08

000815

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

a V (m

middotsminus2)

a L (m

middotsminus2) times10minus2 times10minus2

times10minus2 times10minus2

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s) t (s)

t (s)t (s)

t (s)

(a) Side-span 1

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus16minus08

000816

0 10 20 30 40 50minus30minus15

001530

0 10 20 30 40 50minus20minus10

001020

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus3 times10minus2

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

(b) Side-span 2

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus20minus10

001020

Torsional accelerationLateral accelerationVertical acceleration Torsional acceleration

Lateral accelerationVertical acceleration

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus40minus20

002040

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus2 times10minus2

a L (m

middotsminus2)

a V (m

middotsminus2)

a T (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

a V(m

middotsminus2)

(c) Midspan

Figure 5 Time histories of the acceleration in the middle of each span

Mathematical Problems in Engineering 7

0 5 10 150

1

2

3

4

5

6

7

Side-span 1MidspanSide-span 2

Side-span 1MidspanSide-span 2

0 5 10 1500

05

10

15

20

25

30

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus4 times10minus3S a

(m2 middot

sminus3)

S a (m

2 middotsminus

3)

2761Hz 2761Hz

n (Hz) n (Hz)

Figure 6 Power spectral density of vertical acceleration in the middle of each span

8m 8m

TMDTrack beam 4m

2m

2m

2m

4m

Figure 7 Layout of TMDs on the nether deck of NYRB

with the frequency of the fourth vertical bending mode ofNYRB Meanwhile the mass of a TMD has great influenceson the vibration control efficiency Therefore the designparameters including the mass of the block and the dampingratio should be determined at first while the other parameterscan be determined when the two parameters are certain

The stiffness coefficient of the spring in the TMD can becalculated by [12 27]

119896 = 1198912 sdot 119898 (1)

where 119896 = the stiffness coefficient of the spring119898 = the massof the block and 119891 = the natural frequency of the TMD

The damping coefficient of the damper in the TMD canbe calculated by

119888 = 4120587120585 sdot 119891 sdot 119898 (2)

where 119888 = the damping coefficient of the damper and 120585 = thedamping ratio of the TMD

To reflect the control level of the TMD on the vibrationof NYRB a formula is proposed to quantify the controlefficiency as

120578 =10038161003816100381610038161003816100381610038161003816

120572 minus 120573

120573

10038161003816100381610038161003816100381610038161003816 (3)

where 120578 = the control efficiency and 120572 and 120573 = the peakvertical acceleration in themiddle of the bridge deckwith andwithout TMD respectively

53 Finite ElementModeling of TMD Since there is no specialelement for modeling the TMD in ANSYS the TMD issimulated by the combination of Mass 21 and Combin 14 [2728] in this studyThe mass blocks of the TMDs are simplifiedinto individual masses and simulated by the element Mass21 Mass 21 is a single-node element with six degrees offreedom displacements in the nodal 119909 119910 and 119911 directionsand rotations about the nodal 119909- 119910- and 119911-axes as shownin Figure 8(a) A different mass and rotary inertia can beassigned to each coordinate direction The spring and thedamper in a TMD are simulated by the element Combin 14The element is a uniaxial tension-compression element whichhas three DOFs at each node as shown in Figure 8(b) Afterthe proper stiffness and damping coefficient are selected thefunctions of the spring and the damper in a TMD can beaccurately reflected

54 Initial Static Displacement Induced by the TMDs Theinstallation of TMDs will cause the changes of the overalllayout of initial displacement of NYRB The displacementvariation of the center region of each span induced by thegravities of the TMDs with different masses accompaniedby NYRB itself can be seen in Figure 9 Thereinto thedisplacement is linear to the mass of TMD which manifests

8 Mathematical Problems in Engineering

Z

X

Y

MxMy Mz

Ix Iy Iz

(a) Mass 21

j

k

i

C

(b) Combin 14

Figure 8 Features of Mass 21 and Combin 14 element

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Disp

lace

men

t (m

m)

Mass of TMD (t)

Side-span 1MidspanSide-span 2

Figure 9 Initial static displacement in the middle of each spanversus the mass of TMD

that the whole structure is still in an elastic state evenwhen the mass block of each TMD weighs 45 t The staticdisplacement of each side-span decreases with the augmentof themass of TMD and the displacements of Side-span 1 andSide-span 2 are almost the same which performs consistentlywith the structural features of three-span continuous girderbridge The reduction in the initial static displacement ofeach side-span is beneficial for the whole structure but thedisplacement of Midspan increases sharply with the augmentof the mass of TMD When there is no TMD installed onNYRB the initial static displacement of Midspan is muchsmaller than that of each side-span However the initialstatic displacement of Midspan has exceeded that of eachside-span when the block of each TMD reaches 40 t Hencethe determination of the mass of TMD should consider itsinfluence on the initial static displacement of Midspan

55 Parametric Analysis on the Control Efficiency of TMDThe mass of the block and the damping ratio are the uniquedesign parameters that greatly influence the control efficiencyof TMD on the train-induced vibration of NYRB Thus theparametric analysis is conducted to quantify the control levelof the two influential parameters The control efficiency ofvertical vibration acceleration versus the mass of TMD andthe damping ratio of TMD can be seen in Figure 10 Inaddition the modal mass of the controlled mode is 46235 t

As shown in Figure 10 the control efficiency greatlydepends on themass of TMDThe control efficiency increasessharply with the augment of the mass of TMD when theweight of the block is less than 35 t When the weight ofthe block is larger than 35 t the growth rate slows downgradually Meanwhile the control efficiency will be enhancedby increasing the damping ratio but the gradient is muchsmaller than that of the mass of TMD With regard to thesame parameters of TMD for the two cases the entire controleffect of TMD is slightly more prominent in Case 2 than thatin Case 1

In general the increase of both themass and the dampingratio has positive impacts on the control efficiency of TMDIt seems that a larger mass of TMD will give a bettercontrol performance but the total mass is limited to theavailable budget and the augment of themasswill increase theinitial static displacement of Midspan Though the existenceof viscous damper can suppress the sensitivity of TMDto the frequency variation of structures the augment ofthe damping ratio will inevitably add extra difficulties tothe manufacture of the corresponding damper Thereforethe determination of proper mechanical parameters for aTMD should comprehensively take account of the controlefficiency the economical factor the manufacture difficultyof the damper and so forth

56 Determination of the Optimal Parameters for TMD Theparametric analysis reveals the relationship between thedesign parameters and the control efficiency of TMD andthe selection of design parameters is also closely related tothe overall costs the manufacture difficulty of the TMD andthe static displacement of Midspan Hence it is essential tosearch for an optimal combination of the design parametersto balance all the target factors In order to realize the opti-mization in this analysis the selection of design parametersis transformed into a multiobjective nonlinear optimizationproblem under constraints

There are several techniques available to solve the con-strained optimization problem [29 30] Among them thefirst-order optimization method is utilized to determine theoptimal parameters of the TMD of NYRB Considering thecontrol efficiency expenditure of the mass block and the

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

to attenuate undesirable vibrations of a vibrating systemand it has already been extensively studied and applied tosuppress structural vibrations induced by wind earthquakestrains and so forth As some typical examples the windtunnel and theoretical analyses by Xu et al [11] verifiedthe effectiveness of TMD in suppressing wind-induced tallbuildingmotions Using the perturbation solutions formulasrelated to designing the TMD attached to the single-degree-of-freedom (SDOF) system for various types of loading wereproposed by Fujino and Abe in 1993 [12] Aiming to dissipateearthquake input energy Wong [13] investigated the energytransfer process of TMD in improving the ability of inelasticstructures

Although TMD has been extensively studied or success-fully implemented for suppressing vibrations of structuresdue to environmental loadings such as wind and earthquakesonly a few researches have been conducted on applicationsof TMD in reducing the vibrations of bridges due to trainloads and the existing researches are mainly focused onsimply supported reinforced concrete bridges with moderateor small spans [3 14ndash16] As for the long-span continuoussteel truss girder bridge the studies on vibration controlof bridges subjected to dynamic train loads with TMD arefewer The objective of this study is to apply TMDs withoptimal parameters for suppressing vibrations of a steel trussbridge subjected to moving trains With typical train axleloads the parametric sensitivity analysis is conducted andthe effectiveness of TMD for suppressing dynamic vibrationsis examined based on a simplified train-bridge time-varyingsystem Taking account of the control efficiency expenditureof TMDs TMD-induced static displacement of the bridgeand the manufacturing difficulty of damping of differentmagnitude an objective function for evaluating the controlefficiency is proposed and the optimal parameters of TMDare finally determined with the first optimization methodThe analytical procedure and results can provide referencesfor the vibration control of steel truss bridge subjected todynamic train loads

2 Simplified Train-BridgeTime-Varying System

A bridge is possibly to suffer from different kinds of dynamicloads during its long-term service period such as the vehicleswinds and earthquakes Comparedwith other dynamic loadsacting on the bridge the location of train loads on bridgesis time-varying which makes the dynamic characteristics ofthe train-bridge coupled system dependent on the locationof train loads Therefore the resonance condition can onlyhappen in a shortmoment and it is difficult to establish a clearcorrelation between the moving train loads and the bridgeresponses To clearly identify the train-bridge time-varyingsystem some simplified models have already been proposed[17ndash21]These simplified models provide the basis for furtherinvestigations on the train-bridge coupled system

In this study a broadly used simplified model proposedby Kriloff in 1905 [18] is adopted to simulate the train-bridgetime-varying system The loads induced by train moving at

a constant speed are simplified as a series of concentratedforces moving at the same speed These concentrated forceslocate at the junctions of wheels and the track Consideringthe difficulty in solving differential equations with variablecoefficients the inertia forces of the train are neglected duringthe analysis

3 Engineering Background

31 Bridge Description The Nanjing Yangtze River Bridge(NYRB) as shown in Figure 1 is a double-decked steel trussbridge across the Yangtze River Since 1968 it has operatedas a railway and highway combined bridge for 46 yearsThe upper deck of NYRB is employed as part of NationalHighway 104 in China with four lanes The total width of thecarriage way is 15m accompanied by a 225m wide walkwayon each side The lower deck of NYRB serves for railwaytransportation with a width of 14m Two pairs of parallelseamless railway tracks are installed on the deck so that thetrains can move in opposite directions simultaneously Themain bridge is the combination of three continuous girderbridges and a simply supported bridge with a total length of1568m Thereinto three expansion joints are set at Piers 14 and 7 respectively to separate the three-span continuousgirder bridges and the simply supported bridge Each spanof the continuous bridge is 160m long and the length ofthe simply supported bridge is 128m The main girder iscomposed of the steel truss structure with stiffening chordmembers Each segment for the main girder is 8m with aheight of 16m and connections between each segment aremainly rivet joints In this paper the three-span continuousgirder bridge between Piers 7 and 10 near Nanjing will betaken as the research object

32 Simplified Model of Dynamic Train Loads The structuralhealth monitoring system of NYRB [22] recorded a trainmoving at 70 kmh to cross the Yangtze River The trainis composed of two locomotives and eighteen carriages asshown in Figure 2 There are six axles on each locomotiveand each axle weighs 22 tons which is evenly assigned to twowheels All of the carriages work with four axles and eachaxle weighs 20 tons so that each wheel can transfer 10-tonweight to the bridge structure In order to study the dynamicbehaviour of the train-bridge time-varying system the trainherein is simplified as a series of moving concentrated forceswith fixed intervals The axle distance is detailed in Figure 2and the unit for the distance is meter The overall lengthof the train is 292 meters In regard to the train loads theconcentrated forces are acquired by the axles loadsmultipliedby the gravitational acceleration of 98ms2

4 Finite Element Model and Its Application

41 Finite Element Model Based on ANSYS NYRB is a trussstructure with complicated features due to its various typesof vertical and lateral connection systems and its mainfunctions addressed to both railway and highway The three-span continuous girder bridge contains 7690 elements and 30

Mathematical Problems in Engineering 3

012345678910

128Nanjing Pukou160 times 3 = 480 160 times 3 = 480 160 times 3 = 480

Figure 1 Layout of Nanjing Yangtze River Bridge (unit m)

middot middot middotLocomotive Locomotive Carriage Carriage

2 2 12 2 2 4 2 2 8 2 2 3 2 7 2 14 times 16 + 3 = 227 2 7 2

292

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

980

kN980

kN

980

kN980

kN

980

kN980

kN

980

kN980

kN

Figure 2 Layout of the train loads and axle distance

types of sections With appropriate simplification a spatialFE model of NYRB is established based on ANSYS [23]as shown in Figure 3 The main truss rods track beamsand lateral connection systems are modeled by spatial beamelement (Beam 4) And the pavement on highway surfaceand seamless steel rail which have little contribution tostructural stiffness are simulated by mass element (Mass 21)imposed on the corresponding nodes of the truss bridgegirder To facilitate the exertion of axial forces the trackbeam is partially refined in the mesh generationThe verticallateral vertical-rotational and longitudinal-rotational DOFs(degrees of freedom) are restricted at the four supports[24 25] In addition the bearing between Side-span 1 andMidspan is restricted in longitudinal directionwhile the otherdirections are set free

42 Dynamic Characteristics of NYRB In order to acquirethe natural frequencies and corresponding mode shapes ofNYRB a modal analysis is conducted and the first 200modesare extracted with subspace iterationmethod Table 1 lists thefirst 12 natural frequencies and corresponding characteristicsAmong the first 12 structural modes the lowest modal fre-quency is 10576Hz which corresponds to symmetric verticalbending vibration of the whole bridge The first lateral bend-ingmode appears as the second structuralmode ofNYRBandthis phenomenon reflects a relatively larger lateral stiffnessthan vertical stiffness existing in NYRB All of the first threelateral bendingmodes exhibit the lateral vibration of only onespan (Side-span 1 Midspan or Side-span 2) which revealsthe lateral vibration of each span has little influence on thoseof the other two spans Similar to the lateral bending modethe second and the third vertical bending mode shapes arethe vibration of Side-span 1 and Side-span 2 respectivelyThe torsional vibration modes appear continuously fromthe seventh to the tenth structural mode Specifically thefirst three perform as symmetric or antisymmetric torsionalvibration of the whole bridge accompanied by lateral bendingvibration while the fourth one performs as the torsional

Table 1 The first 12 natural frequencies and mode shapes of NYRB

NumberNaturalfrequency

(Hz)

Modedirection Characteristics of modes

1 10576 V Symmetric vertical bendingvibration of the whole bridge

2 11937 L Lateral bending vibration ofSide-span 1

3 12040 L Lateral bending vibration ofSide-span 2

4 14140 L Lateral bending vibration ofMidspan

5 17649 V Vertical bending vibration ofSide-span 1

6 21638 V Vertical bending vibration ofSide-span 2

7 22736 L + T

Antisymmetric lateral bendingvibration of the whole bridgeaccompanied by antisymmetrictorsional vibration

8 23728 L + T

Symmetric lateral bendingvibration of the whole bridgeaccompanied by antisymmetrictorsional vibration

9 25164 L + T

Symmetric lateral bendingvibration of the whole bridgeaccompanied by symmetrictorsional vibration

10 27352 T Torsional vibration of Side-span 1

11 28726 V Symmetric vertical bendingvibration of the whole bridge

12 32437 V Antisymmetric vertical bendingvibration of the whole bridge

vibration of Side-span 1 The fourth and the fifth verticalbending mode correspond to the eleventh and the twelfth

4 Mathematical Problems in Engineering

Side-span 1

Midspan

Side-span 2

Figure 3 Spatial finite element model of NYRB

0 10 20 30 40 50minus003

minus002

minus001

000

001

002

Vert

ical

disp

lace

men

t (m

)

StaticDynamic

StaticDynamic

15 20 25minus0028minus0026minus0024minus0022minus0020

Amplify

0 10 20 30 40 50minus006

minus004

minus002

000

002

004

Vert

ical

disp

lace

men

t (m

)

Amplify

15 20 25minus0056minus0052minus0048minus0044minus0040

Case 1 single train moving at one side Case 2 two trains moving oppositelyt (s) t (s)

Figure 4 Vertical displacement in the middle of Midspan

structural mode respectively Among them the fourth oneis the symmetric vertical bending vibration and the fifthone is the antisymmetric vertical bending vibration of thewhole bridge In the view of mode shapes only the firstand the fourth vertical bending modes dominate the verticalvibration of the Midspan while others have little influence

43 Dynamic Responses of NYRB Excited by Train LoadsConsidering the functions of NYRB two cases includingsingle train moving at one side and two trains moving oppo-sitely are analyzed in the train-bridge time-varying systemThereinto the second case describes a situation that twotrains set off from the two sides of NYRB simultaneously andthe time phase difference is not included in this study Duringthe transient dynamic analysis by ANSYS the full methodwith intact system matrix is utilized and the Newmark-120573technique is adopted to conduct the time integration Onaccount of the steel material employed in the construction ofNYRB the damping ratio of the whole structure is set as 2

431 Displacement Responses of the Bridge Span With tran-sient analyses of the train-bridge time-varying system thedisplacements in the middle of the spans are the mostprominent Table 2 lists the maximum displacement in themiddle of each span of the two cases Thereinto the torsionaldisplacement has been multiplied by half of the bridge deck

width to unify the unit As shown in Table 2 the verticaldisplacement is the most prominent while the lateral andtorsional displacements aremuch smaller However themax-imum vertical displacement of the two cases does not surpass006 meters and it is much smaller than one-thousandthlength of the main span Hence the small maximum verticaldisplacement of the bridge span under the actions of movingtrains reflects the strong vertical stiffness of NYRB

In order to reveal the difference of vertical displacementbetween static train loads and dynamic train loads thedisplacement response of NYRB under static train loads iscalculated The static and dynamic vertical displacementsin the middle of Midspan of the two cases are shown inFigure 4 Since the trainmoves at a speed of 70 kmh the traincompletely leaves the bridge after 4279 s As for the verticaldisplacements induced by static train loads and dynamictrain loads there is little difference between them during thewhole procedure except for the time duration from 1502 s to2469 s At the moment of 1502 s the train is just completelylanding on the bridge And at the time of 2469 s the trainbegins to leave the bridge and the time of 2469 s correspondsto the last moment that the whole train is on the bridgeDuring the time interval between 1502 s and 2469 s thedynamic vertical displacement is fluctuant However in thisperiod the deviation between the static and dynamic verticaldisplacements is still smallThe detailed difference among the

Mathematical Problems in Engineering 5

Table 2 Maximum displacement in the middle of each span (unit m)

Case Case 1 single train moving at one side Case 2 two trains moving oppositelySide-span 1 Midspan Side-span 2 Side-span 1 Midspan Side-span 2

Vertical 00456 00260 00442 00522 00516 00498Lateral 00006 00015 00016 00008 00013 00019Torsional 00036 00032 00037 00034 0000067 00037Note the displacement in the table has subtracted the gravity-caused static displacement of the structure the values in the table are the absolute values ofmaximum displacement

time interval has been amplified in the lower right cornerof each figure The phenomenon above has similar featureswith the analytical results by Chatterjee et al in 1994 [21]Due to the small vertical displacements induced by dynamictrain loads the displacement cannot be chosen as the targetto conduct vibration control analysis

432 Acceleration Responses of the Bridge Span An obviousdifference between the static and the dynamic train actionsis the excited acceleration response of the bridge structure bythe dynamic train loads The acceleration magnitude reflectsthe intensity of vibration of the whole structure which isclosely related to the deterioration of passengersrsquo comfortthe decrease of traffic safety accompanied by damage tothe tracks and the augment of maintenance cost The timehistories of the acceleration responses in the middle of eachspan of two cases are shown in Figure 5

As shown in Figure 5 the overall trend reveals that theamplitudes of both lateral and torsional acceleration aremuch smaller than that of vertical acceleration in each spanCompared to Case 1 the vertical acceleration of each spanincreases prominently in Case 2 In regard to lateral andtorsional accelerations the differences between the two casesare not consistent In the middle of Side-span 1 the ampli-tudes of both lateral acceleration and torsional accelerationincrease a little inCase 2 comparedwith those of Case 1 In themiddle of Side-span 2 obvious peak accelerations occur inthe lateral and torsional time histories of Case 2 and the peakaccelerations occur near the point of 165 s As for Midspanthe lateral and torsional accelerations are counteracted to alarge extent when two trains move oppositely on the bridgeThe analysis above shows that the concentration should bepaid to vertical acceleration of each span in this study

In Case 1 the maximum vertical accelerations ofSide-span 1 Side-span 2 and Midspan are 00520ms200668ms2 and 01453ms2 respectively In Case 2 themaximum vertical accelerations of Side-span 1 Side-span 2and Midspan are 01027ms2 01509ms2 and 03172ms2respectively From the peak accelerations above it is easy tofind that the acceleration in the middle of Midspan is muchlarger than that of Side-span 1 and Side-span 2 all of thevertical accelerations especially in Case 2 are so prominentthat some measures need to be employed to suppress thedynamic train-induced vibrations of the bridge

The vertical acceleration time histories of each span inFigure 5 are transformed into PSD (power spectral density)with Welch transformation [26] as shown in Figure 6Though the dynamic loads are different in the two cases

the frequency-domain features are similarThe PSDs indicatethat the vibration energy of the two cases concentratesaround an exciting frequency of 2761Hz which is in theneighborhood of the fourth vertical bending mode of NYRBThis phenomenon shows that the fourth vertical bendingmode contributes most to the train-induced vibration ofNYRB Considering that the exciting frequency may be onthe deviation of 2761Hz when the moving speed of thetrain changes the fourth vertical bending mode is selectedas the control mode in this study Consequently the TMDis designed to control the fourth vertical bending mode thatdominates the vertical vibration responses of the train-bridgesystem

5 Analysis of the Vibration ControlEffect of TMD

51 Layout of TMDs onNYRB For structureswhose dynamicresponses are dominated by one structural mode the TMDsshould be deposited near the amplitude of the correspondingmode shape So the TMDs are attached to the center regionof Midspan of NYRB where the mode shape of the fourthvertical bending mode is the largest For the nether deckof NYRB shown in Figure 7 there are four track beamssymmetrically arranged in the inner side for the moving oftrains and two structural beam elements in the outer sideto serve for the whole structure The transverse beams areutilized to strengthen the integrity of the bridge structurewith an adjacent distance of 8 meters In order to guaranteethe simplicity of the loading path each TMD is hung atthe joint of the longitudinal beam and the transverse beamAnd three transverse beams in the central region of Midspanare selected to install the TMDs thus 18 TMDs in total areemployed to conduct the vibration control of NYRB Thedetailed layout of TMDs on NYRB can be seen in Figure 7amongwhich the red blocks represent the TMDs inMidspan

52 Design Parameters for the TMD The TMD is a devicewith a simple structure and explicit working mechanismIt is made up of a mass block and a spring as well asa damper The main contents for designing a TMD areconnected to the mass of the block the stiffness coefficientof the spring and the damping coefficient of the damperTwo mechanical parameters including the natural frequencyand the damping ratio are related with the above threevariables According to the working mechanism of the TMDthe natural frequency of the TMD is determined as the same

6 Mathematical Problems in Engineering

0 10 20 30 40 50minus08minus04

000408

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus15minus08

000815

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

a V (m

middotsminus2)

a L (m

middotsminus2) times10minus2 times10minus2

times10minus2 times10minus2

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s) t (s)

t (s)t (s)

t (s)

(a) Side-span 1

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus16minus08

000816

0 10 20 30 40 50minus30minus15

001530

0 10 20 30 40 50minus20minus10

001020

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus3 times10minus2

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

(b) Side-span 2

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus20minus10

001020

Torsional accelerationLateral accelerationVertical acceleration Torsional acceleration

Lateral accelerationVertical acceleration

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus40minus20

002040

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus2 times10minus2

a L (m

middotsminus2)

a V (m

middotsminus2)

a T (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

a V(m

middotsminus2)

(c) Midspan

Figure 5 Time histories of the acceleration in the middle of each span

Mathematical Problems in Engineering 7

0 5 10 150

1

2

3

4

5

6

7

Side-span 1MidspanSide-span 2

Side-span 1MidspanSide-span 2

0 5 10 1500

05

10

15

20

25

30

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus4 times10minus3S a

(m2 middot

sminus3)

S a (m

2 middotsminus

3)

2761Hz 2761Hz

n (Hz) n (Hz)

Figure 6 Power spectral density of vertical acceleration in the middle of each span

8m 8m

TMDTrack beam 4m

2m

2m

2m

4m

Figure 7 Layout of TMDs on the nether deck of NYRB

with the frequency of the fourth vertical bending mode ofNYRB Meanwhile the mass of a TMD has great influenceson the vibration control efficiency Therefore the designparameters including the mass of the block and the dampingratio should be determined at first while the other parameterscan be determined when the two parameters are certain

The stiffness coefficient of the spring in the TMD can becalculated by [12 27]

119896 = 1198912 sdot 119898 (1)

where 119896 = the stiffness coefficient of the spring119898 = the massof the block and 119891 = the natural frequency of the TMD

The damping coefficient of the damper in the TMD canbe calculated by

119888 = 4120587120585 sdot 119891 sdot 119898 (2)

where 119888 = the damping coefficient of the damper and 120585 = thedamping ratio of the TMD

To reflect the control level of the TMD on the vibrationof NYRB a formula is proposed to quantify the controlefficiency as

120578 =10038161003816100381610038161003816100381610038161003816

120572 minus 120573

120573

10038161003816100381610038161003816100381610038161003816 (3)

where 120578 = the control efficiency and 120572 and 120573 = the peakvertical acceleration in themiddle of the bridge deckwith andwithout TMD respectively

53 Finite ElementModeling of TMD Since there is no specialelement for modeling the TMD in ANSYS the TMD issimulated by the combination of Mass 21 and Combin 14 [2728] in this studyThe mass blocks of the TMDs are simplifiedinto individual masses and simulated by the element Mass21 Mass 21 is a single-node element with six degrees offreedom displacements in the nodal 119909 119910 and 119911 directionsand rotations about the nodal 119909- 119910- and 119911-axes as shownin Figure 8(a) A different mass and rotary inertia can beassigned to each coordinate direction The spring and thedamper in a TMD are simulated by the element Combin 14The element is a uniaxial tension-compression element whichhas three DOFs at each node as shown in Figure 8(b) Afterthe proper stiffness and damping coefficient are selected thefunctions of the spring and the damper in a TMD can beaccurately reflected

54 Initial Static Displacement Induced by the TMDs Theinstallation of TMDs will cause the changes of the overalllayout of initial displacement of NYRB The displacementvariation of the center region of each span induced by thegravities of the TMDs with different masses accompaniedby NYRB itself can be seen in Figure 9 Thereinto thedisplacement is linear to the mass of TMD which manifests

8 Mathematical Problems in Engineering

Z

X

Y

MxMy Mz

Ix Iy Iz

(a) Mass 21

j

k

i

C

(b) Combin 14

Figure 8 Features of Mass 21 and Combin 14 element

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Disp

lace

men

t (m

m)

Mass of TMD (t)

Side-span 1MidspanSide-span 2

Figure 9 Initial static displacement in the middle of each spanversus the mass of TMD

that the whole structure is still in an elastic state evenwhen the mass block of each TMD weighs 45 t The staticdisplacement of each side-span decreases with the augmentof themass of TMD and the displacements of Side-span 1 andSide-span 2 are almost the same which performs consistentlywith the structural features of three-span continuous girderbridge The reduction in the initial static displacement ofeach side-span is beneficial for the whole structure but thedisplacement of Midspan increases sharply with the augmentof the mass of TMD When there is no TMD installed onNYRB the initial static displacement of Midspan is muchsmaller than that of each side-span However the initialstatic displacement of Midspan has exceeded that of eachside-span when the block of each TMD reaches 40 t Hencethe determination of the mass of TMD should consider itsinfluence on the initial static displacement of Midspan

55 Parametric Analysis on the Control Efficiency of TMDThe mass of the block and the damping ratio are the uniquedesign parameters that greatly influence the control efficiencyof TMD on the train-induced vibration of NYRB Thus theparametric analysis is conducted to quantify the control levelof the two influential parameters The control efficiency ofvertical vibration acceleration versus the mass of TMD andthe damping ratio of TMD can be seen in Figure 10 Inaddition the modal mass of the controlled mode is 46235 t

As shown in Figure 10 the control efficiency greatlydepends on themass of TMDThe control efficiency increasessharply with the augment of the mass of TMD when theweight of the block is less than 35 t When the weight ofthe block is larger than 35 t the growth rate slows downgradually Meanwhile the control efficiency will be enhancedby increasing the damping ratio but the gradient is muchsmaller than that of the mass of TMD With regard to thesame parameters of TMD for the two cases the entire controleffect of TMD is slightly more prominent in Case 2 than thatin Case 1

In general the increase of both themass and the dampingratio has positive impacts on the control efficiency of TMDIt seems that a larger mass of TMD will give a bettercontrol performance but the total mass is limited to theavailable budget and the augment of themasswill increase theinitial static displacement of Midspan Though the existenceof viscous damper can suppress the sensitivity of TMDto the frequency variation of structures the augment ofthe damping ratio will inevitably add extra difficulties tothe manufacture of the corresponding damper Thereforethe determination of proper mechanical parameters for aTMD should comprehensively take account of the controlefficiency the economical factor the manufacture difficultyof the damper and so forth

56 Determination of the Optimal Parameters for TMD Theparametric analysis reveals the relationship between thedesign parameters and the control efficiency of TMD andthe selection of design parameters is also closely related tothe overall costs the manufacture difficulty of the TMD andthe static displacement of Midspan Hence it is essential tosearch for an optimal combination of the design parametersto balance all the target factors In order to realize the opti-mization in this analysis the selection of design parametersis transformed into a multiobjective nonlinear optimizationproblem under constraints

There are several techniques available to solve the con-strained optimization problem [29 30] Among them thefirst-order optimization method is utilized to determine theoptimal parameters of the TMD of NYRB Considering thecontrol efficiency expenditure of the mass block and the

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

012345678910

128Nanjing Pukou160 times 3 = 480 160 times 3 = 480 160 times 3 = 480

Figure 1 Layout of Nanjing Yangtze River Bridge (unit m)

middot middot middotLocomotive Locomotive Carriage Carriage

2 2 12 2 2 4 2 2 8 2 2 3 2 7 2 14 times 16 + 3 = 227 2 7 2

292

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

1078

kN1078

kN1078

kN

980

kN980

kN

980

kN980

kN

980

kN980

kN

980

kN980

kN

Figure 2 Layout of the train loads and axle distance

types of sections With appropriate simplification a spatialFE model of NYRB is established based on ANSYS [23]as shown in Figure 3 The main truss rods track beamsand lateral connection systems are modeled by spatial beamelement (Beam 4) And the pavement on highway surfaceand seamless steel rail which have little contribution tostructural stiffness are simulated by mass element (Mass 21)imposed on the corresponding nodes of the truss bridgegirder To facilitate the exertion of axial forces the trackbeam is partially refined in the mesh generationThe verticallateral vertical-rotational and longitudinal-rotational DOFs(degrees of freedom) are restricted at the four supports[24 25] In addition the bearing between Side-span 1 andMidspan is restricted in longitudinal directionwhile the otherdirections are set free

42 Dynamic Characteristics of NYRB In order to acquirethe natural frequencies and corresponding mode shapes ofNYRB a modal analysis is conducted and the first 200modesare extracted with subspace iterationmethod Table 1 lists thefirst 12 natural frequencies and corresponding characteristicsAmong the first 12 structural modes the lowest modal fre-quency is 10576Hz which corresponds to symmetric verticalbending vibration of the whole bridge The first lateral bend-ingmode appears as the second structuralmode ofNYRBandthis phenomenon reflects a relatively larger lateral stiffnessthan vertical stiffness existing in NYRB All of the first threelateral bendingmodes exhibit the lateral vibration of only onespan (Side-span 1 Midspan or Side-span 2) which revealsthe lateral vibration of each span has little influence on thoseof the other two spans Similar to the lateral bending modethe second and the third vertical bending mode shapes arethe vibration of Side-span 1 and Side-span 2 respectivelyThe torsional vibration modes appear continuously fromthe seventh to the tenth structural mode Specifically thefirst three perform as symmetric or antisymmetric torsionalvibration of the whole bridge accompanied by lateral bendingvibration while the fourth one performs as the torsional

Table 1 The first 12 natural frequencies and mode shapes of NYRB

NumberNaturalfrequency

(Hz)

Modedirection Characteristics of modes

1 10576 V Symmetric vertical bendingvibration of the whole bridge

2 11937 L Lateral bending vibration ofSide-span 1

3 12040 L Lateral bending vibration ofSide-span 2

4 14140 L Lateral bending vibration ofMidspan

5 17649 V Vertical bending vibration ofSide-span 1

6 21638 V Vertical bending vibration ofSide-span 2

7 22736 L + T

Antisymmetric lateral bendingvibration of the whole bridgeaccompanied by antisymmetrictorsional vibration

8 23728 L + T

Symmetric lateral bendingvibration of the whole bridgeaccompanied by antisymmetrictorsional vibration

9 25164 L + T

Symmetric lateral bendingvibration of the whole bridgeaccompanied by symmetrictorsional vibration

10 27352 T Torsional vibration of Side-span 1

11 28726 V Symmetric vertical bendingvibration of the whole bridge

12 32437 V Antisymmetric vertical bendingvibration of the whole bridge

vibration of Side-span 1 The fourth and the fifth verticalbending mode correspond to the eleventh and the twelfth

4 Mathematical Problems in Engineering

Side-span 1

Midspan

Side-span 2

Figure 3 Spatial finite element model of NYRB

0 10 20 30 40 50minus003

minus002

minus001

000

001

002

Vert

ical

disp

lace

men

t (m

)

StaticDynamic

StaticDynamic

15 20 25minus0028minus0026minus0024minus0022minus0020

Amplify

0 10 20 30 40 50minus006

minus004

minus002

000

002

004

Vert

ical

disp

lace

men

t (m

)

Amplify

15 20 25minus0056minus0052minus0048minus0044minus0040

Case 1 single train moving at one side Case 2 two trains moving oppositelyt (s) t (s)

Figure 4 Vertical displacement in the middle of Midspan

structural mode respectively Among them the fourth oneis the symmetric vertical bending vibration and the fifthone is the antisymmetric vertical bending vibration of thewhole bridge In the view of mode shapes only the firstand the fourth vertical bending modes dominate the verticalvibration of the Midspan while others have little influence

43 Dynamic Responses of NYRB Excited by Train LoadsConsidering the functions of NYRB two cases includingsingle train moving at one side and two trains moving oppo-sitely are analyzed in the train-bridge time-varying systemThereinto the second case describes a situation that twotrains set off from the two sides of NYRB simultaneously andthe time phase difference is not included in this study Duringthe transient dynamic analysis by ANSYS the full methodwith intact system matrix is utilized and the Newmark-120573technique is adopted to conduct the time integration Onaccount of the steel material employed in the construction ofNYRB the damping ratio of the whole structure is set as 2

431 Displacement Responses of the Bridge Span With tran-sient analyses of the train-bridge time-varying system thedisplacements in the middle of the spans are the mostprominent Table 2 lists the maximum displacement in themiddle of each span of the two cases Thereinto the torsionaldisplacement has been multiplied by half of the bridge deck

width to unify the unit As shown in Table 2 the verticaldisplacement is the most prominent while the lateral andtorsional displacements aremuch smaller However themax-imum vertical displacement of the two cases does not surpass006 meters and it is much smaller than one-thousandthlength of the main span Hence the small maximum verticaldisplacement of the bridge span under the actions of movingtrains reflects the strong vertical stiffness of NYRB

In order to reveal the difference of vertical displacementbetween static train loads and dynamic train loads thedisplacement response of NYRB under static train loads iscalculated The static and dynamic vertical displacementsin the middle of Midspan of the two cases are shown inFigure 4 Since the trainmoves at a speed of 70 kmh the traincompletely leaves the bridge after 4279 s As for the verticaldisplacements induced by static train loads and dynamictrain loads there is little difference between them during thewhole procedure except for the time duration from 1502 s to2469 s At the moment of 1502 s the train is just completelylanding on the bridge And at the time of 2469 s the trainbegins to leave the bridge and the time of 2469 s correspondsto the last moment that the whole train is on the bridgeDuring the time interval between 1502 s and 2469 s thedynamic vertical displacement is fluctuant However in thisperiod the deviation between the static and dynamic verticaldisplacements is still smallThe detailed difference among the

Mathematical Problems in Engineering 5

Table 2 Maximum displacement in the middle of each span (unit m)

Case Case 1 single train moving at one side Case 2 two trains moving oppositelySide-span 1 Midspan Side-span 2 Side-span 1 Midspan Side-span 2

Vertical 00456 00260 00442 00522 00516 00498Lateral 00006 00015 00016 00008 00013 00019Torsional 00036 00032 00037 00034 0000067 00037Note the displacement in the table has subtracted the gravity-caused static displacement of the structure the values in the table are the absolute values ofmaximum displacement

time interval has been amplified in the lower right cornerof each figure The phenomenon above has similar featureswith the analytical results by Chatterjee et al in 1994 [21]Due to the small vertical displacements induced by dynamictrain loads the displacement cannot be chosen as the targetto conduct vibration control analysis

432 Acceleration Responses of the Bridge Span An obviousdifference between the static and the dynamic train actionsis the excited acceleration response of the bridge structure bythe dynamic train loads The acceleration magnitude reflectsthe intensity of vibration of the whole structure which isclosely related to the deterioration of passengersrsquo comfortthe decrease of traffic safety accompanied by damage tothe tracks and the augment of maintenance cost The timehistories of the acceleration responses in the middle of eachspan of two cases are shown in Figure 5

As shown in Figure 5 the overall trend reveals that theamplitudes of both lateral and torsional acceleration aremuch smaller than that of vertical acceleration in each spanCompared to Case 1 the vertical acceleration of each spanincreases prominently in Case 2 In regard to lateral andtorsional accelerations the differences between the two casesare not consistent In the middle of Side-span 1 the ampli-tudes of both lateral acceleration and torsional accelerationincrease a little inCase 2 comparedwith those of Case 1 In themiddle of Side-span 2 obvious peak accelerations occur inthe lateral and torsional time histories of Case 2 and the peakaccelerations occur near the point of 165 s As for Midspanthe lateral and torsional accelerations are counteracted to alarge extent when two trains move oppositely on the bridgeThe analysis above shows that the concentration should bepaid to vertical acceleration of each span in this study

In Case 1 the maximum vertical accelerations ofSide-span 1 Side-span 2 and Midspan are 00520ms200668ms2 and 01453ms2 respectively In Case 2 themaximum vertical accelerations of Side-span 1 Side-span 2and Midspan are 01027ms2 01509ms2 and 03172ms2respectively From the peak accelerations above it is easy tofind that the acceleration in the middle of Midspan is muchlarger than that of Side-span 1 and Side-span 2 all of thevertical accelerations especially in Case 2 are so prominentthat some measures need to be employed to suppress thedynamic train-induced vibrations of the bridge

The vertical acceleration time histories of each span inFigure 5 are transformed into PSD (power spectral density)with Welch transformation [26] as shown in Figure 6Though the dynamic loads are different in the two cases

the frequency-domain features are similarThe PSDs indicatethat the vibration energy of the two cases concentratesaround an exciting frequency of 2761Hz which is in theneighborhood of the fourth vertical bending mode of NYRBThis phenomenon shows that the fourth vertical bendingmode contributes most to the train-induced vibration ofNYRB Considering that the exciting frequency may be onthe deviation of 2761Hz when the moving speed of thetrain changes the fourth vertical bending mode is selectedas the control mode in this study Consequently the TMDis designed to control the fourth vertical bending mode thatdominates the vertical vibration responses of the train-bridgesystem

5 Analysis of the Vibration ControlEffect of TMD

51 Layout of TMDs onNYRB For structureswhose dynamicresponses are dominated by one structural mode the TMDsshould be deposited near the amplitude of the correspondingmode shape So the TMDs are attached to the center regionof Midspan of NYRB where the mode shape of the fourthvertical bending mode is the largest For the nether deckof NYRB shown in Figure 7 there are four track beamssymmetrically arranged in the inner side for the moving oftrains and two structural beam elements in the outer sideto serve for the whole structure The transverse beams areutilized to strengthen the integrity of the bridge structurewith an adjacent distance of 8 meters In order to guaranteethe simplicity of the loading path each TMD is hung atthe joint of the longitudinal beam and the transverse beamAnd three transverse beams in the central region of Midspanare selected to install the TMDs thus 18 TMDs in total areemployed to conduct the vibration control of NYRB Thedetailed layout of TMDs on NYRB can be seen in Figure 7amongwhich the red blocks represent the TMDs inMidspan

52 Design Parameters for the TMD The TMD is a devicewith a simple structure and explicit working mechanismIt is made up of a mass block and a spring as well asa damper The main contents for designing a TMD areconnected to the mass of the block the stiffness coefficientof the spring and the damping coefficient of the damperTwo mechanical parameters including the natural frequencyand the damping ratio are related with the above threevariables According to the working mechanism of the TMDthe natural frequency of the TMD is determined as the same

6 Mathematical Problems in Engineering

0 10 20 30 40 50minus08minus04

000408

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus15minus08

000815

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

a V (m

middotsminus2)

a L (m

middotsminus2) times10minus2 times10minus2

times10minus2 times10minus2

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s) t (s)

t (s)t (s)

t (s)

(a) Side-span 1

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus16minus08

000816

0 10 20 30 40 50minus30minus15

001530

0 10 20 30 40 50minus20minus10

001020

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus3 times10minus2

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

(b) Side-span 2

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus20minus10

001020

Torsional accelerationLateral accelerationVertical acceleration Torsional acceleration

Lateral accelerationVertical acceleration

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus40minus20

002040

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus2 times10minus2

a L (m

middotsminus2)

a V (m

middotsminus2)

a T (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

a V(m

middotsminus2)

(c) Midspan

Figure 5 Time histories of the acceleration in the middle of each span

Mathematical Problems in Engineering 7

0 5 10 150

1

2

3

4

5

6

7

Side-span 1MidspanSide-span 2

Side-span 1MidspanSide-span 2

0 5 10 1500

05

10

15

20

25

30

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus4 times10minus3S a

(m2 middot

sminus3)

S a (m

2 middotsminus

3)

2761Hz 2761Hz

n (Hz) n (Hz)

Figure 6 Power spectral density of vertical acceleration in the middle of each span

8m 8m

TMDTrack beam 4m

2m

2m

2m

4m

Figure 7 Layout of TMDs on the nether deck of NYRB

with the frequency of the fourth vertical bending mode ofNYRB Meanwhile the mass of a TMD has great influenceson the vibration control efficiency Therefore the designparameters including the mass of the block and the dampingratio should be determined at first while the other parameterscan be determined when the two parameters are certain

The stiffness coefficient of the spring in the TMD can becalculated by [12 27]

119896 = 1198912 sdot 119898 (1)

where 119896 = the stiffness coefficient of the spring119898 = the massof the block and 119891 = the natural frequency of the TMD

The damping coefficient of the damper in the TMD canbe calculated by

119888 = 4120587120585 sdot 119891 sdot 119898 (2)

where 119888 = the damping coefficient of the damper and 120585 = thedamping ratio of the TMD

To reflect the control level of the TMD on the vibrationof NYRB a formula is proposed to quantify the controlefficiency as

120578 =10038161003816100381610038161003816100381610038161003816

120572 minus 120573

120573

10038161003816100381610038161003816100381610038161003816 (3)

where 120578 = the control efficiency and 120572 and 120573 = the peakvertical acceleration in themiddle of the bridge deckwith andwithout TMD respectively

53 Finite ElementModeling of TMD Since there is no specialelement for modeling the TMD in ANSYS the TMD issimulated by the combination of Mass 21 and Combin 14 [2728] in this studyThe mass blocks of the TMDs are simplifiedinto individual masses and simulated by the element Mass21 Mass 21 is a single-node element with six degrees offreedom displacements in the nodal 119909 119910 and 119911 directionsand rotations about the nodal 119909- 119910- and 119911-axes as shownin Figure 8(a) A different mass and rotary inertia can beassigned to each coordinate direction The spring and thedamper in a TMD are simulated by the element Combin 14The element is a uniaxial tension-compression element whichhas three DOFs at each node as shown in Figure 8(b) Afterthe proper stiffness and damping coefficient are selected thefunctions of the spring and the damper in a TMD can beaccurately reflected

54 Initial Static Displacement Induced by the TMDs Theinstallation of TMDs will cause the changes of the overalllayout of initial displacement of NYRB The displacementvariation of the center region of each span induced by thegravities of the TMDs with different masses accompaniedby NYRB itself can be seen in Figure 9 Thereinto thedisplacement is linear to the mass of TMD which manifests

8 Mathematical Problems in Engineering

Z

X

Y

MxMy Mz

Ix Iy Iz

(a) Mass 21

j

k

i

C

(b) Combin 14

Figure 8 Features of Mass 21 and Combin 14 element

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Disp

lace

men

t (m

m)

Mass of TMD (t)

Side-span 1MidspanSide-span 2

Figure 9 Initial static displacement in the middle of each spanversus the mass of TMD

that the whole structure is still in an elastic state evenwhen the mass block of each TMD weighs 45 t The staticdisplacement of each side-span decreases with the augmentof themass of TMD and the displacements of Side-span 1 andSide-span 2 are almost the same which performs consistentlywith the structural features of three-span continuous girderbridge The reduction in the initial static displacement ofeach side-span is beneficial for the whole structure but thedisplacement of Midspan increases sharply with the augmentof the mass of TMD When there is no TMD installed onNYRB the initial static displacement of Midspan is muchsmaller than that of each side-span However the initialstatic displacement of Midspan has exceeded that of eachside-span when the block of each TMD reaches 40 t Hencethe determination of the mass of TMD should consider itsinfluence on the initial static displacement of Midspan

55 Parametric Analysis on the Control Efficiency of TMDThe mass of the block and the damping ratio are the uniquedesign parameters that greatly influence the control efficiencyof TMD on the train-induced vibration of NYRB Thus theparametric analysis is conducted to quantify the control levelof the two influential parameters The control efficiency ofvertical vibration acceleration versus the mass of TMD andthe damping ratio of TMD can be seen in Figure 10 Inaddition the modal mass of the controlled mode is 46235 t

As shown in Figure 10 the control efficiency greatlydepends on themass of TMDThe control efficiency increasessharply with the augment of the mass of TMD when theweight of the block is less than 35 t When the weight ofthe block is larger than 35 t the growth rate slows downgradually Meanwhile the control efficiency will be enhancedby increasing the damping ratio but the gradient is muchsmaller than that of the mass of TMD With regard to thesame parameters of TMD for the two cases the entire controleffect of TMD is slightly more prominent in Case 2 than thatin Case 1

In general the increase of both themass and the dampingratio has positive impacts on the control efficiency of TMDIt seems that a larger mass of TMD will give a bettercontrol performance but the total mass is limited to theavailable budget and the augment of themasswill increase theinitial static displacement of Midspan Though the existenceof viscous damper can suppress the sensitivity of TMDto the frequency variation of structures the augment ofthe damping ratio will inevitably add extra difficulties tothe manufacture of the corresponding damper Thereforethe determination of proper mechanical parameters for aTMD should comprehensively take account of the controlefficiency the economical factor the manufacture difficultyof the damper and so forth

56 Determination of the Optimal Parameters for TMD Theparametric analysis reveals the relationship between thedesign parameters and the control efficiency of TMD andthe selection of design parameters is also closely related tothe overall costs the manufacture difficulty of the TMD andthe static displacement of Midspan Hence it is essential tosearch for an optimal combination of the design parametersto balance all the target factors In order to realize the opti-mization in this analysis the selection of design parametersis transformed into a multiobjective nonlinear optimizationproblem under constraints

There are several techniques available to solve the con-strained optimization problem [29 30] Among them thefirst-order optimization method is utilized to determine theoptimal parameters of the TMD of NYRB Considering thecontrol efficiency expenditure of the mass block and the

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Side-span 1

Midspan

Side-span 2

Figure 3 Spatial finite element model of NYRB

0 10 20 30 40 50minus003

minus002

minus001

000

001

002

Vert

ical

disp

lace

men

t (m

)

StaticDynamic

StaticDynamic

15 20 25minus0028minus0026minus0024minus0022minus0020

Amplify

0 10 20 30 40 50minus006

minus004

minus002

000

002

004

Vert

ical

disp

lace

men

t (m

)

Amplify

15 20 25minus0056minus0052minus0048minus0044minus0040

Case 1 single train moving at one side Case 2 two trains moving oppositelyt (s) t (s)

Figure 4 Vertical displacement in the middle of Midspan

structural mode respectively Among them the fourth oneis the symmetric vertical bending vibration and the fifthone is the antisymmetric vertical bending vibration of thewhole bridge In the view of mode shapes only the firstand the fourth vertical bending modes dominate the verticalvibration of the Midspan while others have little influence

43 Dynamic Responses of NYRB Excited by Train LoadsConsidering the functions of NYRB two cases includingsingle train moving at one side and two trains moving oppo-sitely are analyzed in the train-bridge time-varying systemThereinto the second case describes a situation that twotrains set off from the two sides of NYRB simultaneously andthe time phase difference is not included in this study Duringthe transient dynamic analysis by ANSYS the full methodwith intact system matrix is utilized and the Newmark-120573technique is adopted to conduct the time integration Onaccount of the steel material employed in the construction ofNYRB the damping ratio of the whole structure is set as 2

431 Displacement Responses of the Bridge Span With tran-sient analyses of the train-bridge time-varying system thedisplacements in the middle of the spans are the mostprominent Table 2 lists the maximum displacement in themiddle of each span of the two cases Thereinto the torsionaldisplacement has been multiplied by half of the bridge deck

width to unify the unit As shown in Table 2 the verticaldisplacement is the most prominent while the lateral andtorsional displacements aremuch smaller However themax-imum vertical displacement of the two cases does not surpass006 meters and it is much smaller than one-thousandthlength of the main span Hence the small maximum verticaldisplacement of the bridge span under the actions of movingtrains reflects the strong vertical stiffness of NYRB

In order to reveal the difference of vertical displacementbetween static train loads and dynamic train loads thedisplacement response of NYRB under static train loads iscalculated The static and dynamic vertical displacementsin the middle of Midspan of the two cases are shown inFigure 4 Since the trainmoves at a speed of 70 kmh the traincompletely leaves the bridge after 4279 s As for the verticaldisplacements induced by static train loads and dynamictrain loads there is little difference between them during thewhole procedure except for the time duration from 1502 s to2469 s At the moment of 1502 s the train is just completelylanding on the bridge And at the time of 2469 s the trainbegins to leave the bridge and the time of 2469 s correspondsto the last moment that the whole train is on the bridgeDuring the time interval between 1502 s and 2469 s thedynamic vertical displacement is fluctuant However in thisperiod the deviation between the static and dynamic verticaldisplacements is still smallThe detailed difference among the

Mathematical Problems in Engineering 5

Table 2 Maximum displacement in the middle of each span (unit m)

Case Case 1 single train moving at one side Case 2 two trains moving oppositelySide-span 1 Midspan Side-span 2 Side-span 1 Midspan Side-span 2

Vertical 00456 00260 00442 00522 00516 00498Lateral 00006 00015 00016 00008 00013 00019Torsional 00036 00032 00037 00034 0000067 00037Note the displacement in the table has subtracted the gravity-caused static displacement of the structure the values in the table are the absolute values ofmaximum displacement

time interval has been amplified in the lower right cornerof each figure The phenomenon above has similar featureswith the analytical results by Chatterjee et al in 1994 [21]Due to the small vertical displacements induced by dynamictrain loads the displacement cannot be chosen as the targetto conduct vibration control analysis

432 Acceleration Responses of the Bridge Span An obviousdifference between the static and the dynamic train actionsis the excited acceleration response of the bridge structure bythe dynamic train loads The acceleration magnitude reflectsthe intensity of vibration of the whole structure which isclosely related to the deterioration of passengersrsquo comfortthe decrease of traffic safety accompanied by damage tothe tracks and the augment of maintenance cost The timehistories of the acceleration responses in the middle of eachspan of two cases are shown in Figure 5

As shown in Figure 5 the overall trend reveals that theamplitudes of both lateral and torsional acceleration aremuch smaller than that of vertical acceleration in each spanCompared to Case 1 the vertical acceleration of each spanincreases prominently in Case 2 In regard to lateral andtorsional accelerations the differences between the two casesare not consistent In the middle of Side-span 1 the ampli-tudes of both lateral acceleration and torsional accelerationincrease a little inCase 2 comparedwith those of Case 1 In themiddle of Side-span 2 obvious peak accelerations occur inthe lateral and torsional time histories of Case 2 and the peakaccelerations occur near the point of 165 s As for Midspanthe lateral and torsional accelerations are counteracted to alarge extent when two trains move oppositely on the bridgeThe analysis above shows that the concentration should bepaid to vertical acceleration of each span in this study

In Case 1 the maximum vertical accelerations ofSide-span 1 Side-span 2 and Midspan are 00520ms200668ms2 and 01453ms2 respectively In Case 2 themaximum vertical accelerations of Side-span 1 Side-span 2and Midspan are 01027ms2 01509ms2 and 03172ms2respectively From the peak accelerations above it is easy tofind that the acceleration in the middle of Midspan is muchlarger than that of Side-span 1 and Side-span 2 all of thevertical accelerations especially in Case 2 are so prominentthat some measures need to be employed to suppress thedynamic train-induced vibrations of the bridge

The vertical acceleration time histories of each span inFigure 5 are transformed into PSD (power spectral density)with Welch transformation [26] as shown in Figure 6Though the dynamic loads are different in the two cases

the frequency-domain features are similarThe PSDs indicatethat the vibration energy of the two cases concentratesaround an exciting frequency of 2761Hz which is in theneighborhood of the fourth vertical bending mode of NYRBThis phenomenon shows that the fourth vertical bendingmode contributes most to the train-induced vibration ofNYRB Considering that the exciting frequency may be onthe deviation of 2761Hz when the moving speed of thetrain changes the fourth vertical bending mode is selectedas the control mode in this study Consequently the TMDis designed to control the fourth vertical bending mode thatdominates the vertical vibration responses of the train-bridgesystem

5 Analysis of the Vibration ControlEffect of TMD

51 Layout of TMDs onNYRB For structureswhose dynamicresponses are dominated by one structural mode the TMDsshould be deposited near the amplitude of the correspondingmode shape So the TMDs are attached to the center regionof Midspan of NYRB where the mode shape of the fourthvertical bending mode is the largest For the nether deckof NYRB shown in Figure 7 there are four track beamssymmetrically arranged in the inner side for the moving oftrains and two structural beam elements in the outer sideto serve for the whole structure The transverse beams areutilized to strengthen the integrity of the bridge structurewith an adjacent distance of 8 meters In order to guaranteethe simplicity of the loading path each TMD is hung atthe joint of the longitudinal beam and the transverse beamAnd three transverse beams in the central region of Midspanare selected to install the TMDs thus 18 TMDs in total areemployed to conduct the vibration control of NYRB Thedetailed layout of TMDs on NYRB can be seen in Figure 7amongwhich the red blocks represent the TMDs inMidspan

52 Design Parameters for the TMD The TMD is a devicewith a simple structure and explicit working mechanismIt is made up of a mass block and a spring as well asa damper The main contents for designing a TMD areconnected to the mass of the block the stiffness coefficientof the spring and the damping coefficient of the damperTwo mechanical parameters including the natural frequencyand the damping ratio are related with the above threevariables According to the working mechanism of the TMDthe natural frequency of the TMD is determined as the same

6 Mathematical Problems in Engineering

0 10 20 30 40 50minus08minus04

000408

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus15minus08

000815

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

a V (m

middotsminus2)

a L (m

middotsminus2) times10minus2 times10minus2

times10minus2 times10minus2

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s) t (s)

t (s)t (s)

t (s)

(a) Side-span 1

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus16minus08

000816

0 10 20 30 40 50minus30minus15

001530

0 10 20 30 40 50minus20minus10

001020

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus3 times10minus2

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

(b) Side-span 2

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus20minus10

001020

Torsional accelerationLateral accelerationVertical acceleration Torsional acceleration

Lateral accelerationVertical acceleration

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus40minus20

002040

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus2 times10minus2

a L (m

middotsminus2)

a V (m

middotsminus2)

a T (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

a V(m

middotsminus2)

(c) Midspan

Figure 5 Time histories of the acceleration in the middle of each span

Mathematical Problems in Engineering 7

0 5 10 150

1

2

3

4

5

6

7

Side-span 1MidspanSide-span 2

Side-span 1MidspanSide-span 2

0 5 10 1500

05

10

15

20

25

30

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus4 times10minus3S a

(m2 middot

sminus3)

S a (m

2 middotsminus

3)

2761Hz 2761Hz

n (Hz) n (Hz)

Figure 6 Power spectral density of vertical acceleration in the middle of each span

8m 8m

TMDTrack beam 4m

2m

2m

2m

4m

Figure 7 Layout of TMDs on the nether deck of NYRB

with the frequency of the fourth vertical bending mode ofNYRB Meanwhile the mass of a TMD has great influenceson the vibration control efficiency Therefore the designparameters including the mass of the block and the dampingratio should be determined at first while the other parameterscan be determined when the two parameters are certain

The stiffness coefficient of the spring in the TMD can becalculated by [12 27]

119896 = 1198912 sdot 119898 (1)

where 119896 = the stiffness coefficient of the spring119898 = the massof the block and 119891 = the natural frequency of the TMD

The damping coefficient of the damper in the TMD canbe calculated by

119888 = 4120587120585 sdot 119891 sdot 119898 (2)

where 119888 = the damping coefficient of the damper and 120585 = thedamping ratio of the TMD

To reflect the control level of the TMD on the vibrationof NYRB a formula is proposed to quantify the controlefficiency as

120578 =10038161003816100381610038161003816100381610038161003816

120572 minus 120573

120573

10038161003816100381610038161003816100381610038161003816 (3)

where 120578 = the control efficiency and 120572 and 120573 = the peakvertical acceleration in themiddle of the bridge deckwith andwithout TMD respectively

53 Finite ElementModeling of TMD Since there is no specialelement for modeling the TMD in ANSYS the TMD issimulated by the combination of Mass 21 and Combin 14 [2728] in this studyThe mass blocks of the TMDs are simplifiedinto individual masses and simulated by the element Mass21 Mass 21 is a single-node element with six degrees offreedom displacements in the nodal 119909 119910 and 119911 directionsand rotations about the nodal 119909- 119910- and 119911-axes as shownin Figure 8(a) A different mass and rotary inertia can beassigned to each coordinate direction The spring and thedamper in a TMD are simulated by the element Combin 14The element is a uniaxial tension-compression element whichhas three DOFs at each node as shown in Figure 8(b) Afterthe proper stiffness and damping coefficient are selected thefunctions of the spring and the damper in a TMD can beaccurately reflected

54 Initial Static Displacement Induced by the TMDs Theinstallation of TMDs will cause the changes of the overalllayout of initial displacement of NYRB The displacementvariation of the center region of each span induced by thegravities of the TMDs with different masses accompaniedby NYRB itself can be seen in Figure 9 Thereinto thedisplacement is linear to the mass of TMD which manifests

8 Mathematical Problems in Engineering

Z

X

Y

MxMy Mz

Ix Iy Iz

(a) Mass 21

j

k

i

C

(b) Combin 14

Figure 8 Features of Mass 21 and Combin 14 element

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Disp

lace

men

t (m

m)

Mass of TMD (t)

Side-span 1MidspanSide-span 2

Figure 9 Initial static displacement in the middle of each spanversus the mass of TMD

that the whole structure is still in an elastic state evenwhen the mass block of each TMD weighs 45 t The staticdisplacement of each side-span decreases with the augmentof themass of TMD and the displacements of Side-span 1 andSide-span 2 are almost the same which performs consistentlywith the structural features of three-span continuous girderbridge The reduction in the initial static displacement ofeach side-span is beneficial for the whole structure but thedisplacement of Midspan increases sharply with the augmentof the mass of TMD When there is no TMD installed onNYRB the initial static displacement of Midspan is muchsmaller than that of each side-span However the initialstatic displacement of Midspan has exceeded that of eachside-span when the block of each TMD reaches 40 t Hencethe determination of the mass of TMD should consider itsinfluence on the initial static displacement of Midspan

55 Parametric Analysis on the Control Efficiency of TMDThe mass of the block and the damping ratio are the uniquedesign parameters that greatly influence the control efficiencyof TMD on the train-induced vibration of NYRB Thus theparametric analysis is conducted to quantify the control levelof the two influential parameters The control efficiency ofvertical vibration acceleration versus the mass of TMD andthe damping ratio of TMD can be seen in Figure 10 Inaddition the modal mass of the controlled mode is 46235 t

As shown in Figure 10 the control efficiency greatlydepends on themass of TMDThe control efficiency increasessharply with the augment of the mass of TMD when theweight of the block is less than 35 t When the weight ofthe block is larger than 35 t the growth rate slows downgradually Meanwhile the control efficiency will be enhancedby increasing the damping ratio but the gradient is muchsmaller than that of the mass of TMD With regard to thesame parameters of TMD for the two cases the entire controleffect of TMD is slightly more prominent in Case 2 than thatin Case 1

In general the increase of both themass and the dampingratio has positive impacts on the control efficiency of TMDIt seems that a larger mass of TMD will give a bettercontrol performance but the total mass is limited to theavailable budget and the augment of themasswill increase theinitial static displacement of Midspan Though the existenceof viscous damper can suppress the sensitivity of TMDto the frequency variation of structures the augment ofthe damping ratio will inevitably add extra difficulties tothe manufacture of the corresponding damper Thereforethe determination of proper mechanical parameters for aTMD should comprehensively take account of the controlefficiency the economical factor the manufacture difficultyof the damper and so forth

56 Determination of the Optimal Parameters for TMD Theparametric analysis reveals the relationship between thedesign parameters and the control efficiency of TMD andthe selection of design parameters is also closely related tothe overall costs the manufacture difficulty of the TMD andthe static displacement of Midspan Hence it is essential tosearch for an optimal combination of the design parametersto balance all the target factors In order to realize the opti-mization in this analysis the selection of design parametersis transformed into a multiobjective nonlinear optimizationproblem under constraints

There are several techniques available to solve the con-strained optimization problem [29 30] Among them thefirst-order optimization method is utilized to determine theoptimal parameters of the TMD of NYRB Considering thecontrol efficiency expenditure of the mass block and the

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 2 Maximum displacement in the middle of each span (unit m)

Case Case 1 single train moving at one side Case 2 two trains moving oppositelySide-span 1 Midspan Side-span 2 Side-span 1 Midspan Side-span 2

Vertical 00456 00260 00442 00522 00516 00498Lateral 00006 00015 00016 00008 00013 00019Torsional 00036 00032 00037 00034 0000067 00037Note the displacement in the table has subtracted the gravity-caused static displacement of the structure the values in the table are the absolute values ofmaximum displacement

time interval has been amplified in the lower right cornerof each figure The phenomenon above has similar featureswith the analytical results by Chatterjee et al in 1994 [21]Due to the small vertical displacements induced by dynamictrain loads the displacement cannot be chosen as the targetto conduct vibration control analysis

432 Acceleration Responses of the Bridge Span An obviousdifference between the static and the dynamic train actionsis the excited acceleration response of the bridge structure bythe dynamic train loads The acceleration magnitude reflectsthe intensity of vibration of the whole structure which isclosely related to the deterioration of passengersrsquo comfortthe decrease of traffic safety accompanied by damage tothe tracks and the augment of maintenance cost The timehistories of the acceleration responses in the middle of eachspan of two cases are shown in Figure 5

As shown in Figure 5 the overall trend reveals that theamplitudes of both lateral and torsional acceleration aremuch smaller than that of vertical acceleration in each spanCompared to Case 1 the vertical acceleration of each spanincreases prominently in Case 2 In regard to lateral andtorsional accelerations the differences between the two casesare not consistent In the middle of Side-span 1 the ampli-tudes of both lateral acceleration and torsional accelerationincrease a little inCase 2 comparedwith those of Case 1 In themiddle of Side-span 2 obvious peak accelerations occur inthe lateral and torsional time histories of Case 2 and the peakaccelerations occur near the point of 165 s As for Midspanthe lateral and torsional accelerations are counteracted to alarge extent when two trains move oppositely on the bridgeThe analysis above shows that the concentration should bepaid to vertical acceleration of each span in this study

In Case 1 the maximum vertical accelerations ofSide-span 1 Side-span 2 and Midspan are 00520ms200668ms2 and 01453ms2 respectively In Case 2 themaximum vertical accelerations of Side-span 1 Side-span 2and Midspan are 01027ms2 01509ms2 and 03172ms2respectively From the peak accelerations above it is easy tofind that the acceleration in the middle of Midspan is muchlarger than that of Side-span 1 and Side-span 2 all of thevertical accelerations especially in Case 2 are so prominentthat some measures need to be employed to suppress thedynamic train-induced vibrations of the bridge

The vertical acceleration time histories of each span inFigure 5 are transformed into PSD (power spectral density)with Welch transformation [26] as shown in Figure 6Though the dynamic loads are different in the two cases

the frequency-domain features are similarThe PSDs indicatethat the vibration energy of the two cases concentratesaround an exciting frequency of 2761Hz which is in theneighborhood of the fourth vertical bending mode of NYRBThis phenomenon shows that the fourth vertical bendingmode contributes most to the train-induced vibration ofNYRB Considering that the exciting frequency may be onthe deviation of 2761Hz when the moving speed of thetrain changes the fourth vertical bending mode is selectedas the control mode in this study Consequently the TMDis designed to control the fourth vertical bending mode thatdominates the vertical vibration responses of the train-bridgesystem

5 Analysis of the Vibration ControlEffect of TMD

51 Layout of TMDs onNYRB For structureswhose dynamicresponses are dominated by one structural mode the TMDsshould be deposited near the amplitude of the correspondingmode shape So the TMDs are attached to the center regionof Midspan of NYRB where the mode shape of the fourthvertical bending mode is the largest For the nether deckof NYRB shown in Figure 7 there are four track beamssymmetrically arranged in the inner side for the moving oftrains and two structural beam elements in the outer sideto serve for the whole structure The transverse beams areutilized to strengthen the integrity of the bridge structurewith an adjacent distance of 8 meters In order to guaranteethe simplicity of the loading path each TMD is hung atthe joint of the longitudinal beam and the transverse beamAnd three transverse beams in the central region of Midspanare selected to install the TMDs thus 18 TMDs in total areemployed to conduct the vibration control of NYRB Thedetailed layout of TMDs on NYRB can be seen in Figure 7amongwhich the red blocks represent the TMDs inMidspan

52 Design Parameters for the TMD The TMD is a devicewith a simple structure and explicit working mechanismIt is made up of a mass block and a spring as well asa damper The main contents for designing a TMD areconnected to the mass of the block the stiffness coefficientof the spring and the damping coefficient of the damperTwo mechanical parameters including the natural frequencyand the damping ratio are related with the above threevariables According to the working mechanism of the TMDthe natural frequency of the TMD is determined as the same

6 Mathematical Problems in Engineering

0 10 20 30 40 50minus08minus04

000408

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus15minus08

000815

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

a V (m

middotsminus2)

a L (m

middotsminus2) times10minus2 times10minus2

times10minus2 times10minus2

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s) t (s)

t (s)t (s)

t (s)

(a) Side-span 1

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus16minus08

000816

0 10 20 30 40 50minus30minus15

001530

0 10 20 30 40 50minus20minus10

001020

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus3 times10minus2

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

(b) Side-span 2

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus20minus10

001020

Torsional accelerationLateral accelerationVertical acceleration Torsional acceleration

Lateral accelerationVertical acceleration

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus40minus20

002040

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus2 times10minus2

a L (m

middotsminus2)

a V (m

middotsminus2)

a T (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

a V(m

middotsminus2)

(c) Midspan

Figure 5 Time histories of the acceleration in the middle of each span

Mathematical Problems in Engineering 7

0 5 10 150

1

2

3

4

5

6

7

Side-span 1MidspanSide-span 2

Side-span 1MidspanSide-span 2

0 5 10 1500

05

10

15

20

25

30

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus4 times10minus3S a

(m2 middot

sminus3)

S a (m

2 middotsminus

3)

2761Hz 2761Hz

n (Hz) n (Hz)

Figure 6 Power spectral density of vertical acceleration in the middle of each span

8m 8m

TMDTrack beam 4m

2m

2m

2m

4m

Figure 7 Layout of TMDs on the nether deck of NYRB

with the frequency of the fourth vertical bending mode ofNYRB Meanwhile the mass of a TMD has great influenceson the vibration control efficiency Therefore the designparameters including the mass of the block and the dampingratio should be determined at first while the other parameterscan be determined when the two parameters are certain

The stiffness coefficient of the spring in the TMD can becalculated by [12 27]

119896 = 1198912 sdot 119898 (1)

where 119896 = the stiffness coefficient of the spring119898 = the massof the block and 119891 = the natural frequency of the TMD

The damping coefficient of the damper in the TMD canbe calculated by

119888 = 4120587120585 sdot 119891 sdot 119898 (2)

where 119888 = the damping coefficient of the damper and 120585 = thedamping ratio of the TMD

To reflect the control level of the TMD on the vibrationof NYRB a formula is proposed to quantify the controlefficiency as

120578 =10038161003816100381610038161003816100381610038161003816

120572 minus 120573

120573

10038161003816100381610038161003816100381610038161003816 (3)

where 120578 = the control efficiency and 120572 and 120573 = the peakvertical acceleration in themiddle of the bridge deckwith andwithout TMD respectively

53 Finite ElementModeling of TMD Since there is no specialelement for modeling the TMD in ANSYS the TMD issimulated by the combination of Mass 21 and Combin 14 [2728] in this studyThe mass blocks of the TMDs are simplifiedinto individual masses and simulated by the element Mass21 Mass 21 is a single-node element with six degrees offreedom displacements in the nodal 119909 119910 and 119911 directionsand rotations about the nodal 119909- 119910- and 119911-axes as shownin Figure 8(a) A different mass and rotary inertia can beassigned to each coordinate direction The spring and thedamper in a TMD are simulated by the element Combin 14The element is a uniaxial tension-compression element whichhas three DOFs at each node as shown in Figure 8(b) Afterthe proper stiffness and damping coefficient are selected thefunctions of the spring and the damper in a TMD can beaccurately reflected

54 Initial Static Displacement Induced by the TMDs Theinstallation of TMDs will cause the changes of the overalllayout of initial displacement of NYRB The displacementvariation of the center region of each span induced by thegravities of the TMDs with different masses accompaniedby NYRB itself can be seen in Figure 9 Thereinto thedisplacement is linear to the mass of TMD which manifests

8 Mathematical Problems in Engineering

Z

X

Y

MxMy Mz

Ix Iy Iz

(a) Mass 21

j

k

i

C

(b) Combin 14

Figure 8 Features of Mass 21 and Combin 14 element

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Disp

lace

men

t (m

m)

Mass of TMD (t)

Side-span 1MidspanSide-span 2

Figure 9 Initial static displacement in the middle of each spanversus the mass of TMD

that the whole structure is still in an elastic state evenwhen the mass block of each TMD weighs 45 t The staticdisplacement of each side-span decreases with the augmentof themass of TMD and the displacements of Side-span 1 andSide-span 2 are almost the same which performs consistentlywith the structural features of three-span continuous girderbridge The reduction in the initial static displacement ofeach side-span is beneficial for the whole structure but thedisplacement of Midspan increases sharply with the augmentof the mass of TMD When there is no TMD installed onNYRB the initial static displacement of Midspan is muchsmaller than that of each side-span However the initialstatic displacement of Midspan has exceeded that of eachside-span when the block of each TMD reaches 40 t Hencethe determination of the mass of TMD should consider itsinfluence on the initial static displacement of Midspan

55 Parametric Analysis on the Control Efficiency of TMDThe mass of the block and the damping ratio are the uniquedesign parameters that greatly influence the control efficiencyof TMD on the train-induced vibration of NYRB Thus theparametric analysis is conducted to quantify the control levelof the two influential parameters The control efficiency ofvertical vibration acceleration versus the mass of TMD andthe damping ratio of TMD can be seen in Figure 10 Inaddition the modal mass of the controlled mode is 46235 t

As shown in Figure 10 the control efficiency greatlydepends on themass of TMDThe control efficiency increasessharply with the augment of the mass of TMD when theweight of the block is less than 35 t When the weight ofthe block is larger than 35 t the growth rate slows downgradually Meanwhile the control efficiency will be enhancedby increasing the damping ratio but the gradient is muchsmaller than that of the mass of TMD With regard to thesame parameters of TMD for the two cases the entire controleffect of TMD is slightly more prominent in Case 2 than thatin Case 1

In general the increase of both themass and the dampingratio has positive impacts on the control efficiency of TMDIt seems that a larger mass of TMD will give a bettercontrol performance but the total mass is limited to theavailable budget and the augment of themasswill increase theinitial static displacement of Midspan Though the existenceof viscous damper can suppress the sensitivity of TMDto the frequency variation of structures the augment ofthe damping ratio will inevitably add extra difficulties tothe manufacture of the corresponding damper Thereforethe determination of proper mechanical parameters for aTMD should comprehensively take account of the controlefficiency the economical factor the manufacture difficultyof the damper and so forth

56 Determination of the Optimal Parameters for TMD Theparametric analysis reveals the relationship between thedesign parameters and the control efficiency of TMD andthe selection of design parameters is also closely related tothe overall costs the manufacture difficulty of the TMD andthe static displacement of Midspan Hence it is essential tosearch for an optimal combination of the design parametersto balance all the target factors In order to realize the opti-mization in this analysis the selection of design parametersis transformed into a multiobjective nonlinear optimizationproblem under constraints

There are several techniques available to solve the con-strained optimization problem [29 30] Among them thefirst-order optimization method is utilized to determine theoptimal parameters of the TMD of NYRB Considering thecontrol efficiency expenditure of the mass block and the

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 10 20 30 40 50minus08minus04

000408

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus20minus10

001020

0 10 20 30 40 50minus15minus08

000815

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

a V (m

middotsminus2)

a L (m

middotsminus2) times10minus2 times10minus2

times10minus2 times10minus2

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s) t (s)

t (s)t (s)

t (s)

(a) Side-span 1

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus16minus08

000816

0 10 20 30 40 50minus30minus15

001530

0 10 20 30 40 50minus20minus10

001020

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus3 times10minus2

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

a V (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s)

t (s)

t (s)

t (s)

(b) Side-span 2

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus40minus20

002040

0 10 20 30 40 50minus20minus10

001020

Torsional accelerationLateral accelerationVertical acceleration Torsional acceleration

Lateral accelerationVertical acceleration

0 10 20 30 40 50minus06minus03

000306

0 10 20 30 40 50minus10minus05

000510

0 10 20 30 40 50minus40minus20

002040

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus1 times10minus1

times10minus2 times10minus2

times10minus2 times10minus2

a L (m

middotsminus2)

a V (m

middotsminus2)

a T (m

middotsminus2)

a L (m

middotsminus2)

a T (m

middotsminus2)

t (s)

t (s)

t (s) t (s)

t (s)

t (s)

a V(m

middotsminus2)

(c) Midspan

Figure 5 Time histories of the acceleration in the middle of each span

Mathematical Problems in Engineering 7

0 5 10 150

1

2

3

4

5

6

7

Side-span 1MidspanSide-span 2

Side-span 1MidspanSide-span 2

0 5 10 1500

05

10

15

20

25

30

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus4 times10minus3S a

(m2 middot

sminus3)

S a (m

2 middotsminus

3)

2761Hz 2761Hz

n (Hz) n (Hz)

Figure 6 Power spectral density of vertical acceleration in the middle of each span

8m 8m

TMDTrack beam 4m

2m

2m

2m

4m

Figure 7 Layout of TMDs on the nether deck of NYRB

with the frequency of the fourth vertical bending mode ofNYRB Meanwhile the mass of a TMD has great influenceson the vibration control efficiency Therefore the designparameters including the mass of the block and the dampingratio should be determined at first while the other parameterscan be determined when the two parameters are certain

The stiffness coefficient of the spring in the TMD can becalculated by [12 27]

119896 = 1198912 sdot 119898 (1)

where 119896 = the stiffness coefficient of the spring119898 = the massof the block and 119891 = the natural frequency of the TMD

The damping coefficient of the damper in the TMD canbe calculated by

119888 = 4120587120585 sdot 119891 sdot 119898 (2)

where 119888 = the damping coefficient of the damper and 120585 = thedamping ratio of the TMD

To reflect the control level of the TMD on the vibrationof NYRB a formula is proposed to quantify the controlefficiency as

120578 =10038161003816100381610038161003816100381610038161003816

120572 minus 120573

120573

10038161003816100381610038161003816100381610038161003816 (3)

where 120578 = the control efficiency and 120572 and 120573 = the peakvertical acceleration in themiddle of the bridge deckwith andwithout TMD respectively

53 Finite ElementModeling of TMD Since there is no specialelement for modeling the TMD in ANSYS the TMD issimulated by the combination of Mass 21 and Combin 14 [2728] in this studyThe mass blocks of the TMDs are simplifiedinto individual masses and simulated by the element Mass21 Mass 21 is a single-node element with six degrees offreedom displacements in the nodal 119909 119910 and 119911 directionsand rotations about the nodal 119909- 119910- and 119911-axes as shownin Figure 8(a) A different mass and rotary inertia can beassigned to each coordinate direction The spring and thedamper in a TMD are simulated by the element Combin 14The element is a uniaxial tension-compression element whichhas three DOFs at each node as shown in Figure 8(b) Afterthe proper stiffness and damping coefficient are selected thefunctions of the spring and the damper in a TMD can beaccurately reflected

54 Initial Static Displacement Induced by the TMDs Theinstallation of TMDs will cause the changes of the overalllayout of initial displacement of NYRB The displacementvariation of the center region of each span induced by thegravities of the TMDs with different masses accompaniedby NYRB itself can be seen in Figure 9 Thereinto thedisplacement is linear to the mass of TMD which manifests

8 Mathematical Problems in Engineering

Z

X

Y

MxMy Mz

Ix Iy Iz

(a) Mass 21

j

k

i

C

(b) Combin 14

Figure 8 Features of Mass 21 and Combin 14 element

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Disp

lace

men

t (m

m)

Mass of TMD (t)

Side-span 1MidspanSide-span 2

Figure 9 Initial static displacement in the middle of each spanversus the mass of TMD

that the whole structure is still in an elastic state evenwhen the mass block of each TMD weighs 45 t The staticdisplacement of each side-span decreases with the augmentof themass of TMD and the displacements of Side-span 1 andSide-span 2 are almost the same which performs consistentlywith the structural features of three-span continuous girderbridge The reduction in the initial static displacement ofeach side-span is beneficial for the whole structure but thedisplacement of Midspan increases sharply with the augmentof the mass of TMD When there is no TMD installed onNYRB the initial static displacement of Midspan is muchsmaller than that of each side-span However the initialstatic displacement of Midspan has exceeded that of eachside-span when the block of each TMD reaches 40 t Hencethe determination of the mass of TMD should consider itsinfluence on the initial static displacement of Midspan

55 Parametric Analysis on the Control Efficiency of TMDThe mass of the block and the damping ratio are the uniquedesign parameters that greatly influence the control efficiencyof TMD on the train-induced vibration of NYRB Thus theparametric analysis is conducted to quantify the control levelof the two influential parameters The control efficiency ofvertical vibration acceleration versus the mass of TMD andthe damping ratio of TMD can be seen in Figure 10 Inaddition the modal mass of the controlled mode is 46235 t

As shown in Figure 10 the control efficiency greatlydepends on themass of TMDThe control efficiency increasessharply with the augment of the mass of TMD when theweight of the block is less than 35 t When the weight ofthe block is larger than 35 t the growth rate slows downgradually Meanwhile the control efficiency will be enhancedby increasing the damping ratio but the gradient is muchsmaller than that of the mass of TMD With regard to thesame parameters of TMD for the two cases the entire controleffect of TMD is slightly more prominent in Case 2 than thatin Case 1

In general the increase of both themass and the dampingratio has positive impacts on the control efficiency of TMDIt seems that a larger mass of TMD will give a bettercontrol performance but the total mass is limited to theavailable budget and the augment of themasswill increase theinitial static displacement of Midspan Though the existenceof viscous damper can suppress the sensitivity of TMDto the frequency variation of structures the augment ofthe damping ratio will inevitably add extra difficulties tothe manufacture of the corresponding damper Thereforethe determination of proper mechanical parameters for aTMD should comprehensively take account of the controlefficiency the economical factor the manufacture difficultyof the damper and so forth

56 Determination of the Optimal Parameters for TMD Theparametric analysis reveals the relationship between thedesign parameters and the control efficiency of TMD andthe selection of design parameters is also closely related tothe overall costs the manufacture difficulty of the TMD andthe static displacement of Midspan Hence it is essential tosearch for an optimal combination of the design parametersto balance all the target factors In order to realize the opti-mization in this analysis the selection of design parametersis transformed into a multiobjective nonlinear optimizationproblem under constraints

There are several techniques available to solve the con-strained optimization problem [29 30] Among them thefirst-order optimization method is utilized to determine theoptimal parameters of the TMD of NYRB Considering thecontrol efficiency expenditure of the mass block and the

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 5 10 150

1

2

3

4

5

6

7

Side-span 1MidspanSide-span 2

Side-span 1MidspanSide-span 2

0 5 10 1500

05

10

15

20

25

30

Case 1 single train moving at one side Case 2 two trains moving oppositely

times10minus4 times10minus3S a

(m2 middot

sminus3)

S a (m

2 middotsminus

3)

2761Hz 2761Hz

n (Hz) n (Hz)

Figure 6 Power spectral density of vertical acceleration in the middle of each span

8m 8m

TMDTrack beam 4m

2m

2m

2m

4m

Figure 7 Layout of TMDs on the nether deck of NYRB

with the frequency of the fourth vertical bending mode ofNYRB Meanwhile the mass of a TMD has great influenceson the vibration control efficiency Therefore the designparameters including the mass of the block and the dampingratio should be determined at first while the other parameterscan be determined when the two parameters are certain

The stiffness coefficient of the spring in the TMD can becalculated by [12 27]

119896 = 1198912 sdot 119898 (1)

where 119896 = the stiffness coefficient of the spring119898 = the massof the block and 119891 = the natural frequency of the TMD

The damping coefficient of the damper in the TMD canbe calculated by

119888 = 4120587120585 sdot 119891 sdot 119898 (2)

where 119888 = the damping coefficient of the damper and 120585 = thedamping ratio of the TMD

To reflect the control level of the TMD on the vibrationof NYRB a formula is proposed to quantify the controlefficiency as

120578 =10038161003816100381610038161003816100381610038161003816

120572 minus 120573

120573

10038161003816100381610038161003816100381610038161003816 (3)

where 120578 = the control efficiency and 120572 and 120573 = the peakvertical acceleration in themiddle of the bridge deckwith andwithout TMD respectively

53 Finite ElementModeling of TMD Since there is no specialelement for modeling the TMD in ANSYS the TMD issimulated by the combination of Mass 21 and Combin 14 [2728] in this studyThe mass blocks of the TMDs are simplifiedinto individual masses and simulated by the element Mass21 Mass 21 is a single-node element with six degrees offreedom displacements in the nodal 119909 119910 and 119911 directionsand rotations about the nodal 119909- 119910- and 119911-axes as shownin Figure 8(a) A different mass and rotary inertia can beassigned to each coordinate direction The spring and thedamper in a TMD are simulated by the element Combin 14The element is a uniaxial tension-compression element whichhas three DOFs at each node as shown in Figure 8(b) Afterthe proper stiffness and damping coefficient are selected thefunctions of the spring and the damper in a TMD can beaccurately reflected

54 Initial Static Displacement Induced by the TMDs Theinstallation of TMDs will cause the changes of the overalllayout of initial displacement of NYRB The displacementvariation of the center region of each span induced by thegravities of the TMDs with different masses accompaniedby NYRB itself can be seen in Figure 9 Thereinto thedisplacement is linear to the mass of TMD which manifests

8 Mathematical Problems in Engineering

Z

X

Y

MxMy Mz

Ix Iy Iz

(a) Mass 21

j

k

i

C

(b) Combin 14

Figure 8 Features of Mass 21 and Combin 14 element

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Disp

lace

men

t (m

m)

Mass of TMD (t)

Side-span 1MidspanSide-span 2

Figure 9 Initial static displacement in the middle of each spanversus the mass of TMD

that the whole structure is still in an elastic state evenwhen the mass block of each TMD weighs 45 t The staticdisplacement of each side-span decreases with the augmentof themass of TMD and the displacements of Side-span 1 andSide-span 2 are almost the same which performs consistentlywith the structural features of three-span continuous girderbridge The reduction in the initial static displacement ofeach side-span is beneficial for the whole structure but thedisplacement of Midspan increases sharply with the augmentof the mass of TMD When there is no TMD installed onNYRB the initial static displacement of Midspan is muchsmaller than that of each side-span However the initialstatic displacement of Midspan has exceeded that of eachside-span when the block of each TMD reaches 40 t Hencethe determination of the mass of TMD should consider itsinfluence on the initial static displacement of Midspan

55 Parametric Analysis on the Control Efficiency of TMDThe mass of the block and the damping ratio are the uniquedesign parameters that greatly influence the control efficiencyof TMD on the train-induced vibration of NYRB Thus theparametric analysis is conducted to quantify the control levelof the two influential parameters The control efficiency ofvertical vibration acceleration versus the mass of TMD andthe damping ratio of TMD can be seen in Figure 10 Inaddition the modal mass of the controlled mode is 46235 t

As shown in Figure 10 the control efficiency greatlydepends on themass of TMDThe control efficiency increasessharply with the augment of the mass of TMD when theweight of the block is less than 35 t When the weight ofthe block is larger than 35 t the growth rate slows downgradually Meanwhile the control efficiency will be enhancedby increasing the damping ratio but the gradient is muchsmaller than that of the mass of TMD With regard to thesame parameters of TMD for the two cases the entire controleffect of TMD is slightly more prominent in Case 2 than thatin Case 1

In general the increase of both themass and the dampingratio has positive impacts on the control efficiency of TMDIt seems that a larger mass of TMD will give a bettercontrol performance but the total mass is limited to theavailable budget and the augment of themasswill increase theinitial static displacement of Midspan Though the existenceof viscous damper can suppress the sensitivity of TMDto the frequency variation of structures the augment ofthe damping ratio will inevitably add extra difficulties tothe manufacture of the corresponding damper Thereforethe determination of proper mechanical parameters for aTMD should comprehensively take account of the controlefficiency the economical factor the manufacture difficultyof the damper and so forth

56 Determination of the Optimal Parameters for TMD Theparametric analysis reveals the relationship between thedesign parameters and the control efficiency of TMD andthe selection of design parameters is also closely related tothe overall costs the manufacture difficulty of the TMD andthe static displacement of Midspan Hence it is essential tosearch for an optimal combination of the design parametersto balance all the target factors In order to realize the opti-mization in this analysis the selection of design parametersis transformed into a multiobjective nonlinear optimizationproblem under constraints

There are several techniques available to solve the con-strained optimization problem [29 30] Among them thefirst-order optimization method is utilized to determine theoptimal parameters of the TMD of NYRB Considering thecontrol efficiency expenditure of the mass block and the

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Z

X

Y

MxMy Mz

Ix Iy Iz

(a) Mass 21

j

k

i

C

(b) Combin 14

Figure 8 Features of Mass 21 and Combin 14 element

0 5 10 15 20 25 30 35 40 450

20

40

60

80

Disp

lace

men

t (m

m)

Mass of TMD (t)

Side-span 1MidspanSide-span 2

Figure 9 Initial static displacement in the middle of each spanversus the mass of TMD

that the whole structure is still in an elastic state evenwhen the mass block of each TMD weighs 45 t The staticdisplacement of each side-span decreases with the augmentof themass of TMD and the displacements of Side-span 1 andSide-span 2 are almost the same which performs consistentlywith the structural features of three-span continuous girderbridge The reduction in the initial static displacement ofeach side-span is beneficial for the whole structure but thedisplacement of Midspan increases sharply with the augmentof the mass of TMD When there is no TMD installed onNYRB the initial static displacement of Midspan is muchsmaller than that of each side-span However the initialstatic displacement of Midspan has exceeded that of eachside-span when the block of each TMD reaches 40 t Hencethe determination of the mass of TMD should consider itsinfluence on the initial static displacement of Midspan

55 Parametric Analysis on the Control Efficiency of TMDThe mass of the block and the damping ratio are the uniquedesign parameters that greatly influence the control efficiencyof TMD on the train-induced vibration of NYRB Thus theparametric analysis is conducted to quantify the control levelof the two influential parameters The control efficiency ofvertical vibration acceleration versus the mass of TMD andthe damping ratio of TMD can be seen in Figure 10 Inaddition the modal mass of the controlled mode is 46235 t

As shown in Figure 10 the control efficiency greatlydepends on themass of TMDThe control efficiency increasessharply with the augment of the mass of TMD when theweight of the block is less than 35 t When the weight ofthe block is larger than 35 t the growth rate slows downgradually Meanwhile the control efficiency will be enhancedby increasing the damping ratio but the gradient is muchsmaller than that of the mass of TMD With regard to thesame parameters of TMD for the two cases the entire controleffect of TMD is slightly more prominent in Case 2 than thatin Case 1

In general the increase of both themass and the dampingratio has positive impacts on the control efficiency of TMDIt seems that a larger mass of TMD will give a bettercontrol performance but the total mass is limited to theavailable budget and the augment of themasswill increase theinitial static displacement of Midspan Though the existenceof viscous damper can suppress the sensitivity of TMDto the frequency variation of structures the augment ofthe damping ratio will inevitably add extra difficulties tothe manufacture of the corresponding damper Thereforethe determination of proper mechanical parameters for aTMD should comprehensively take account of the controlefficiency the economical factor the manufacture difficultyof the damper and so forth

56 Determination of the Optimal Parameters for TMD Theparametric analysis reveals the relationship between thedesign parameters and the control efficiency of TMD andthe selection of design parameters is also closely related tothe overall costs the manufacture difficulty of the TMD andthe static displacement of Midspan Hence it is essential tosearch for an optimal combination of the design parametersto balance all the target factors In order to realize the opti-mization in this analysis the selection of design parametersis transformed into a multiobjective nonlinear optimizationproblem under constraints

There are several techniques available to solve the con-strained optimization problem [29 30] Among them thefirst-order optimization method is utilized to determine theoptimal parameters of the TMD of NYRB Considering thecontrol efficiency expenditure of the mass block and the

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Mass of T

MD (t) 004900

008500012100157001930022900265003010033700373004090

Damping ratio of TMD

Con

trol e

ffici

ency

()

004200007900011600153001900022700264003010033800375004120

Con

trol e

ffici

ency

()

Mass of T

MD (t)

Damping ratio of TMDCase 1 single train moving at one side

Case 2 two trains moving oppositely

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

45

45

40

40

35

35

30

30

25

25

20

20

15

15

10

10

5

55

010010008

008006006004

004002002000

000

Figure 10 Influence of the mass and the damping ratio of the TMD on the control efficiency

manufacture difficulty of the damper as well as the staticdisplacement ofMidspan an objective function 119869 consideringall the above factors is proposed Then the optimizationproblem can be formed as follows

Minimize 119869 =radic119864 (119898) sdot 119863 (119898) sdot 119862 (120585)

1205782 (119898 120585) 120578 (119898 120585) gt 20

(4)

where 119864(119898) = the expenditure evaluating function propor-tional to the augment of mass of TMD assuming 119864(119898) = 119898119863(119898) = the static displacement evaluating function linear tothe augment of mass of TMD assuming 119863(119898) equals thevalue in Figure 9 119862(120585) = the manufacture difficulty functionassumed to be exponential correlated with the damping ratioof TMD assuming119862(120585) = 1198902120585 120578(119898 120585) = the control efficiencycalculated as (3) 120578

1(119898 120585) = the control efficiency of Case 1

1205782(119898 120585) = the control efficiency of Case 2 Consider

120578 (119898 120585) =1205781(119898 120585) + 120578

2(119898 120585)

2 (5)

The above procedure is carried out by using the optimiza-tion module (OPT) in ANSYS and the first-order optimiza-tion method is applied to obtain the optimal parameters ofTMDThe control efficiency is set beyond 20And the lowerand upper bounds of the design parameters are the same asthose in Figure 10 In order to avoid the local minimum thezero-order optimization method in the optimization moduleof ANSYS is employed to validate the optimization resultsin this study The optimal parameters of the TMD whichcan satisfy the objective function are finally determinedThereinto the mass of TMD is set as 35 t and the dampingratio is quantified as 5 Accordingly the detailed optimaldesign mechanical parameters are calculated as shown inTable 3 Large stiffness coefficient and damping coefficientcan be realized by the combination of multiple springs anddampers with the same parameters

Table 3 Optimal design mechanical parameters of the TMDs onNYRB

Mass of TMDt StiffnesscoefficientkNsdotmminus1

Dampingcoefficienttsdotsminus1

35 1140192 6317

57 The Control Effect of TMD with Optimal ParametersAfter the TMDs with optimal design parameters are installedonNYRB a comparative study on the train-induced vibrationwith and without TMD is conducted The vertical accelera-tions of each span with and without TMD of the two casesare shown in Figure 11

As shown in Figure 11 the vertical acceleration ofMidspan is prominently decreased and the vertical accel-eration of each side-span is reduced to a certain degreeFor Side-span 1 the peak vertical acceleration has beenreduced from 00520ms2 to 00384ms2 in Case 1 and from01027ms2 to 00640ms2 in Case 2 For Side-span 2 thepeak vertical acceleration has been reduced from 00669ms2to 00328ms2 in Case 1 and from 01509ms2 to 00567ms2in Case 2 The control effect is relatively obvious between 15 sand 25 s in Side-span 1 and the vertical acceleration between15 s and 30 s is greatly suppressed by TMD in Side-span2 With regard to Midspan the peak vertical accelerationsare decreased from 01453ms2 to 00931ms2 in Case 1 andfrom03172ms2 to 02049ms2 in Case 2The correspondingcontrol efficiencies are 3593 and 3540 respectively Ingeneral the vertical vibration of Midspan is suppressedeffectively during the whole train-passing procedure

The time histories of the acceleration in the middle ofMidspan are transformed into PSDs with Welch transforma-tion as shown in Figure 12 It is obvious that the energy ofstructural vibrations near the exciting frequency (2761Hz) iseffectively suppressedwith TMD Figure 12 can also verify the

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

0 10 20 30 40 50minus004

minus002

000

002

004

006

Without TMDWith TMD

Side-span 1

0 10 20 30 40 50minus008

minus004

000

004

008Side-span 2

Without TMDWith TMD

0 10 20 30 40 50minus015minus010minus005

000005010015

Midspan

Without TMDWith TMD

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(a) Single train moving at one side

Without TMDWith TMD

0 10 20 30 40 50minus008

minus004

000

004

008

012Side-span 1

Without TMDWith TMD

0 10 20 30 40 50minus016

minus008

000

008

016 Side-span 2

Without TMDWith TMD

0 10 20 30 40 50

minus030minus020minus010

000010020030 Midspan

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

Acce

lera

tion

(mmiddotsminus

2)

t (s)

t (s)

t (s)

(b) Two trains moving oppositely

Figure 11 Train-induced vertical acceleration of NYRB with and without TMD

prominent contribution of the fourth-order vertical bendingmode to the train-induced vibrations In conclusion theTMD with reasonable mechanical parameters can effectivelysuppress the vibration of steel truss bridge subjected todynamic train loads

6 Conclusions

In this paper TMD is utilized to suppress the vibrations ofa steel truss bridge subjected to dynamic train loads Thedynamic performances of the steel truss bridge when a singletrain moves at one side and two trains move oppositelyare investigated An objective function considering all theinfluential factors is proposed and optimal parameters of theTMD are obtained with the first optimization method Thefollowing conclusions can be drawn accordingly

(1) All of the vertical lateral and torsional maximumdisplacements of each span are small in the twocases Relatively the vertical displacement dominatesthe train-induced displacements of NYRB The small

structural displacements of the bridge under theactions of moving trains show the strong stiffness andsufficient capability of NYRB

(2) As for the vertical displacements induced by statictrain loads and dynamic train loads only a smalldifference exists between them during the wholeprocedure Only when the train is wholly on thebridge the dynamic vertical displacement fluctuatesclearly However the deviation between the staticand dynamic vertical displacements is still small inthis period Due to the small vertical displacementsinduced by dynamic train loads the displacementcannot be chosen as the target to conduct vibrationcontrol analysis

(3) The vertical acceleration of each span especially inCase 2 is prominent And the vertical accelerationdominates the train-induced vibration of NYRBAmong them the peak vertical acceleration in themiddle of Midspan is much larger than that of Side-span 1 and Side-span 2

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 5 10 150

1

2

3

4

5

6

7

Without TMDWith TMD

S a (m

2 middotsminus

3)

times10minus4

n (Hz)

(a) Single train moving at one side

Without TMDWith TMD

0 5 10 1500

05

10

15

20

25

30

S a (m

2 middotsminus

3)

times10minus3

n (Hz)

(b) Two trains moving oppositely

Figure 12 Power spectral density of vertical acceleration of Midspan with and without TMD

(4) The control efficiency of TMD increases sharply withthe augment of the mass of TMD when the weightof the mass block is less than 35 t When the weightof the mass block is larger than 35 t the growth rateof the control efficiency begins to become smallerMeanwhile the control efficiency of TMD will beenhanced by increasing the damping ratio but theincreasing gradient is much smaller than that of themass of TMD In general the increase of either themass or the damping ratio has positive influence onthe control efficiency of TMD

(5) Taking the control efficiency expenditure of TMDsTMD-induced static displacement of the bridge andthe manufacturing difficulty of damping of differ-ent magnitude into account the optimal designparameters of the TMD are finally determined withthe first optimization method With the optimaldesign parameters the peak vertical accelerationof each side-span is obviously reduced the peakvertical acceleration of Midspan is decreased from01453ms2 to 00931ms2 in Case 1 and from03172ms2 to 02049ms2 in Case 2 and corre-sponding control efficiencies are 3593 and 3540respectively

(6) The PSDs of vertical acceleration of Midspan revealthat the energy of structural vibrations near theexciting frequency (2761Hz) is effectively suppressedwith TMD which can also reflect the contributionof the fourth-order vertical bending mode to thetrain-induced vibrations In conclusion the TMDwith reasonable mechanical parameters can effec-tively suppress the vibration of long-span steel trussbridge subjected to dynamic train loads

Conflict of Interests

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

Acknowledgments

This work is supported by the Major State Basic ResearchDevelopment Program of China (973 Program) (Grant no2015CB060000) the National Science Foundation of China(Grant nos 51378111 and 51438002) the Fok Ying-TongEducation Foundation for Young Teachers in the HigherEducation Institutions of China (Grant no 142007) theProgram for New Century Excellent Talents in Universityof Ministry of Education of China (Grant no NCET-13-0128) and the Fundamental Research Funds for the CentralUniversities (Grant no 2242012R30002) These supports aregratefully acknowledged

References

[1] Y Fujino and M Abe ldquoDynamic response of concrete railwaybridgesrdquo in International Association for Bridge and StructuralEngineering Proceedings pp 53ndash68 1993

[2] L Fryba ldquoA rough assessment of railway bridges for high speedtrainsrdquo Engineering Structures vol 23 no 5 pp 548ndash556 2001

[3] J Li M Su and L Fan ldquoVibration control of railway bridgesunder high-speed trains using multiple tuned mass dampersrdquoJournal of Bridge Engineering vol 10 no 3 pp 312ndash320 2005

[4] Y-B Yang J-D Yau and L-C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997

[5] R J McNamara ldquoTuned mass dampers for buildingsrdquo Journalof Structural Division vol 103 no 9 pp 1785ndash1798 1977

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[6] RW Luft ldquoOptimal tunedmass dampers for buildingsrdquo Journalof Structural Division vol 105 no 12 pp 2766ndash2772 1979

[7] L A Bergman D M McFarland J K Hall E A Johnsonand A Kareem ldquoOptimal distribution of tuned mass dampersin wind-sensitive structuresrdquo in Proceedings of the 5th Interna-tional Conference on Structural Safety and Reliability (ICOSSARrsquo89) pp 95ndash102 ASCE New York NY USA August 1989

[8] H-N Li and X-L Ni ldquoOptimization of non-uniformly dis-tributed multiple tuned mass damperrdquo Journal of Sound andVibration vol 308 no 1-2 pp 80ndash97 2007

[9] T-H Yi H-N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 no 1 pp 420ndash432 2013

[10] G BWarburton ldquoOptimum absorber parameters for minimiz-ing vibration responserdquo Earthquake Engineering amp StructuralDynamics vol 9 no 3 pp 251ndash262 1981

[11] Y L Xu K C S Kwok and B Samali ldquoControl of wind-inducedtall building vibration by tunedmass dampersrdquo Journal ofWindEngineering and Industrial Aerodynamics vol 40 no 1 pp 1ndash321992

[12] Y Fujino andMAbe ldquoDesign formulas for tunedmass dampersbased on a perturbation techniquerdquo Earthquake Engineering ampStructural Dynamics vol 22 no 10 pp 833ndash854 1993

[13] K K F Wong ldquoSeismic energy dissipation of iInelastic struc-tureswith tunedmass dampersrdquo Journal of EngineeringMechan-ics vol 134 no 2 pp 163ndash172 2008

[14] H-C Kwon M-C Kim and I-W Lee ldquoVibration control ofbridges undermoving loadsrdquoComputers and Structures vol 66no 4 pp 473ndash480 1998

[15] J F Wang C C Lin and B L Chen ldquoVibration suppressionfor high-speed railway bridges using tuned mass dampersrdquoInternational Journal of Solids and Structures vol 40 no 2 pp465ndash491 2003

[16] C C Lin J F Wang and B L Chen ldquoTrain-induced vibrationcontrol of high-speed railway bridges equipped with multipletuned mass dampersrdquo Journal of Bridge Engineering vol 10 no4 pp 398ndash414 2005

[17] RWillis ldquoThe effect produced by causing weights to travel overelastic barsrdquo Report of Commissioners appointed to inquireinto the application of iron to railway structures Appendix HM Stationery Office London UK 1847

[18] A Kriloff ldquoUber die erzwungenen Schwingungen von gle-ichformigen elastischen Stabenrdquo Mathematische Annalen vol61 no 2 pp 211ndash234 1905

[19] S Timoshenko ldquoOn the transverse vibrations of bars of uniformcross sectionrdquo Philosophical Magazine vol 43 pp 125ndash131 1922

[20] J Vellozzi ldquoVibration of suspension bridges under movingloadsrdquo Journal of Structural Engineering vol 93 no 4 pp 123ndash138 1967

[21] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[22] X-H He Z-W Yu and Z-Q Chen ldquoFinite element modelupdating of existing steel bridge based on structural healthmonitoringrdquo Journal of Central South University of Technologyvol 15 no 3 pp 399ndash403 2008

[23] Swanson Analysis Systems (SASI) ANSYS Userrsquos Manual Ver-sion 80 Swanson Analysis Systems (SASI) Houston Pa USA2004

[24] T-H Yi H-N Li andM Gu ldquoA newmethod for optimal selec-tion of sensor location on a high-rise building using simplifiedfinite element modelrdquo Structural Engineering amp Mechanics vol37 no 6 pp 671ndash684 2011

[25] H Wang R Zhou Z Zong C Wang and A Li ldquoStudyon seismic response control of a single-tower self-anchoredsuspension bridge with elastic-plastic steel damperrdquo ScienceChina Technological Sciences vol 55 no 6 pp 1496ndash1502 2012

[26] The MathWorks MATLAB amp SIMULINK Release Notes forR2010b The MathWorks Natick Mass USA 2010

[27] M Gu S R Chen and C C Chang ldquoParametric study onmultiple tuned mass dampers for buffeting control of YangpuBridgerdquo Journal ofWind Engineering and Industrial Aerodynam-ics vol 89 no 11-12 pp 987ndash1000 2001

[28] T H Nguyen I Saidi E F Gad J L Wilson and N HaritosldquoPerformance of distributed multiple viscoelastic tuned massdampers for floor vibration applicationsrdquoAdvances in StructuralEngineering vol 15 no 3 pp 547ndash562 2012

[29] A K Agrawal and J N Yang ldquoOptimal placement of passivedampers on seismic andwind-excited buildings using combina-torial optimizationrdquo Journal of Intelligent Material Systems andStructures vol 10 no 12 pp 997ndash1014 1999

[30] H Wang A Li C Jiao and B F Spencer ldquoDamper placementfor seismic control of super-long-span suspension bridgesbased on the first-order optimization methodrdquo Science ChinaTechnological Sciences vol 53 no 7 pp 2008ndash2014 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of