Research ArticleLateral Vibrations of a Cable-Stayed Bridge underCrowd Excitation

Lijun Ouyang1 Caihong Wang2 Bin Zhen1 and Jian Xu3

1School of Environment and Architecture University of Shanghai for Science and Technology Shanghai 200093 China2Shanghai Publishing and Printing College Shanghai Publication and Media Research Institute Shanghai 200092 China3School of Aerospace Engineering and Mechanics Tongji University Shanghai 200092 China

Correspondence should be addressed to Bin Zhen zhenbin80163com

Received 13 June 2015 Revised 29 August 2015 Accepted 30 August 2015

Academic Editor Stefano Lenci

Copyright copy 2015 Lijun Ouyang 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

A cable-stayed bridge model under crowd excitation is established in this paper by considering the geometric nonlinear propertyof the cables Lateral vibrations of the model are investigated by employing the center manifold theory and the first-orderapproximation solution of the periodic vibration of the bridge is derived by using the energy method Numerical simulationsare carried out to verify the validity of our analytical expressions Our research shows that the existence of the cables can reducethe amplitude and frequency of the bridge especially for the large amplitude case This might explain why measured data of acable-stayed bridge (T-bridge in Japan) vibrating under crowd excitation are much less than the theoretical results reported inprevious studies in which the cable-stayed bridge is viewed as a single-degree-of-freedom system Our analysis results suggest thatthe structure types of footbridges should not be easily ignored in the study of pedestrian-footbridge interaction

1 Introduction

In the last decade attention has been focused on the lateralvibrations of footbridges induced by pedestrians since theLondon Millennium Bridge was opened on 10 June 2000[1 2] After that more andmore excessive lateral vibrations offootbridges in various structures have been reported includ-ing suspension bridges [3 4] arch bridges truss bridges [1]and cable-stayed bridges [5] It seems that the occurrence ofexcessive lateral vibrations of a footbridge is independent ofits structure type

The mechanisms of the lateral vibrations of footbridgesexerted by pedestrians have been paid lots of attention inrecent ten years Plentiful of experiments suggested that thelateral vibrations of footbridges are caused by the zigzagmovements of pedestriansWhen pedestrians on a footbridgestep with their right and left feet in turn the gravity centerof their bodies has a lateral oscillation which induces lateraldynamic time-varying forces on the surface of the bridgeAs a consequence the bridge begins to sway laterally Thepedestrians instinctively feel more comfortable to walk insynchronization with the lateral motion of the bridge which

further increases its response The frequency of the lateralforce induced by pedestrians is about between 07 and 12Hz[6] If the natural frequency of the bridge is within therange of the lateral walking frequency direct resonancebetween pedestrians and the footbridge may be achieved [5]If there exists a 2 1 ratio between the vertical and lateralmode frequencies of the bridge internal resonance [3 7] ispossible The energy may flow from the vertical mode tothe lateral mode due to structural nonlinearities Dynamicinteraction mechanisms have attracted much interests inpast decade because they allow the research for the criticalnumber of pedestrians A lot of dynamic models describingthe interaction between pedestrians and a footbridge havebeen proposed to understand the occurrence of excessivelateral vibrations of the footbridge [1 4 7ndash11] To explainthe occurrence of lateral vibrations with 048Hz in theMillennium Bridge Piccardo and Tubino [12] proposed theparametric excitationmechanismby assuming that the lateralpedestrian-induced force is proportional to the lateral bridgedisplacement The above research for dynamic interactionsuggested that the key to lateral vibration problem mainlylies in the measurement andor modeling of the lateral

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 309645 11 pageshttpdxdoiorg1011552015309645

2 Mathematical Problems in Engineering

force exerted by pedestrians on footbridges However theforce exerted by pedestrians is generally tackled by using anempirical approach strict and exactmathematicalmodels arenot available yet

On the other side in almost all dynamic models pro-posed to describe the interaction between pedestrians anda footbridge the footbridge always is simplified as a single-degree-freedom system On an intuitive level such simpli-fication may not be rational to all types of footbridges forexample suspension bridges and cable-stayed bridges In factMcRobie et al [13] experimentally investigated the ldquolock-inrdquophenomenonby constructing a sectionmodelwith cables andbridge deck Zhou and Ji [14] theoretically and experimentallyanalyzed a generalized suspension system developed fromthe section model Their research showed that cables havegreat influence on free vibrations of the suspension systemwhereas the dynamic behaviour of the suspension systemunder crowd excitation has not been considered Cable-stayed bridges are important engineering structures widelyapplied all over the world A cable-stayed bridge consists ofa deck suspended from stay cables and supported on piersand towers Geurts et al [15] reported that Erasmus bridgein Netherlands opened to the public for just two monthsin 1996 had to be closed because of the large amplitudevibrations of the cables Lilien and Pinto Da Costa [16]found that a dynamic instability of the cables may occurdue to small periodic movements of the girder and themasts Fujino et al [5] argued that cables may have intensivevibrations induced by deck when the natural frequency ofthe cables approximates to half natural frequency of thedeck Nevertheless seldom theoretical analysis for cable-stayed bridges with nonlinear couples between cables anddeck under crowd excitation has been carried out

The paper focuses on the lateral vibrations of a cable-stayed bridge under crowd excitationThe cable-stayed bridgemodel consists of cables and deck which are nonlinearlycoupled Nakamurarsquos assumption [4] is adopted to model thelateral force induced by crowd excitation We will pull all ouremphasis on whether and how the cables affect the lateralvibrations of the deck under crowd excitation The rest ofthe paper is organized as follows the governing equationof the pedestrian-bridge system is established in Section 2the qualitative and quantitative analysis for the governingequation are presented in Sections 3 and 4 respectivelynumerical verifications and discussion are demonstrated inSection 5 conclusions are drawn in Section 6

2 The Governing Equation

The cable-stayed bridge model investigated in this paper isillustrated in Figure 1 in which line 119860119861 represents cablesand the rigid block of the deck Line 119860119861 has a length 119871mass119898

1 stiffness 119864119860 tension 119879

0 and damping coefficient 119888

1

For simplicity all the matters in line 119860119861 are assumed to betogether in the center of mass (point 119862) The rigid block hasa mass 119898

2 stiffness 119896 and damping coefficient 119888

2 Assume

that the lateral displacements of point 119862 and the rigid blockare 1199091and 119909

2 respectively

12 12

represent corresponding

A

C

Cables

x1

F

x2

B

m1

m2

k

Deck

c1

c2

Figure 1 The cable-stayed bridge diagram

velocities and accelerations respectively The lateral forceinduced by pedestrians on the bridge is denoted by 119865

According to Nakamurarsquos assumption [4] 119865 is given by

119865 = 11989611198962119872119901119892119866 (119891)119867 (

2) (1)

where coefficient 1198961is the ratio of lateral force to pedestriansrsquo

weight coefficient 1198962is the percentage of pedestrians who

synchronize to bridge vibration119872119901119892 is modal self-weight of

pedestrians 119866(119891) is a function describing how pedestrianssynchronizewith the natural frequency of a bridge and119867(

2)

is a function that describes the pedestriansrsquo synchronizationnature Nakamura assumed that the pedestrians synchronizeproportionally with the deck velocity

2at low velocities

When the deck velocity becomes large pedestrians feeluncomfortable and then they decrease their walking pacewhich limits the deck response at a certain level rather thaninfinite one Based on the analysis of Zhen et al [17] 119867(

2)

can be expressed as

119867(2) = tanh (119896

32) (2)

in which the coefficient 1198963indicates the saturation rate of the

pedestrian-induced forceBy applying Newtonrsquos second law of motion and consid-

ering the geometric nonlinearity of the cables the governingequation of the cable-stayed bridge under crowd excitation isderived as follows

1+ 12057211+ 12057311199091+ 1205741[1199093

1+ (1199091minus 1199092)3

] = 0

2+ 12057222+ 12057321199092+ 1205742(1199091minus 1199092)3

= 120590 tanh (11989632)

(3)

where 1205721= 11988811198981 1205722= 11988821198982 1205731= 211987901198981119871 1205732= 119896119898

2

1205741= 1198641198602119898

11198713 1205742= 1198641198602119898

21198713 and 120590 = (119896

11198962119866(119891)119872

119901119892)

1198982According to the physics meanings 120572

12 12057312 12057412gt 0

Furthermore we assume that 1205732gt 1205742in the following

analysis

Mathematical Problems in Engineering 3

3 Qualitative Analysis for the Cable-StayedBridge Model

By letting

1199091= 1199101

1= 1199102

1199092= 1199103

2= 1199104

(4)

(3) can be rewritten as

1199101= 1199102

1199102= minus12057311199101minus 12057211199102minus 1205741[1199103

1+ (1199101minus 1199103)3

]

1199103= 1199104

1199104= minus12057321199103minus 12057221199104+ 120590 tanh (119896

31199104) minus 1205742(1199101minus 1199103)3

(5)

Obviously (0 0 0 0) is an equilibrium of (5)The charac-teristic equation of (5) near the origin is

1205824+ (1205721minus 120575) 120582

3+ (1205731+ 1205732minus 1205721120575) 1205822

+ (12057211205732minus 1205751205731) 120582 + 120573

11205732= 0

(6)

where 120575 = 1205901198963minus 1205722 Since 120572

119894 120573119894 120574119894gt 0 119894 = 1 2 it is easy to

verify that (5) has a stable origin if 120575 lt 0 and an unstable oneif 120575 gt 0 When 120575 = 0 (6) has a pair of purely imaginary roots12058212= plusmn119894radic120573

2 Regarding 120582 as a function of 120575 in (6) and taking

the derivative of 120582 with respect to 120575 one has

119889120582

119889120575

=

120582 (1205822+ 1205721120582 + 1205731)

41205823 minus 3 (120575 minus 1205721) 1205822 minus 2 (120575120572

1minus 1205731minus 1205732) 120582 minus 120573

1120575 + 12057211205732

(7)

Substituting 120582 = plusmn119894radic1205732and 120575 = 0 into (7) yields

(119889120582

119889120575)

120575=0

=1

2gt 0 (8)

According to Hopf bifurcation theory system (5) under-goes a Hopf bifurcation near 120575 = 0 and a limit cycle appearswhen 120575 gt 0 This means that periodic vibrations occur in thecable-stayed bridgemodel under crowd excitation To furtherdetermine the stability of the limit cycle we employ centermanifold theory to analyze system (5)

If 120575 = 0 (6) has a pair of purely imaginary roots and twonegative real roots which satisfies the condition for centermanifold theory Assume that a local center manifold can beexpressed by

1199101= ℎ1(1199103 1199104)

1199102= ℎ2(1199103 1199104)

(9)

where ℎ1and ℎ2are vector functions of nonlinear terms of 119910

3

and 1199104 Assume that 119910

1and 1199102can be transformed into power

series

1199101= ℎ101199103

3+ ℎ111199102

31199104+ ℎ1211991031199102

4+ ℎ131199103

4

+ 119874 (1199103 1199104)

1199102= ℎ201199103

3+ ℎ211199102

31199104+ ℎ2211991031199102

4+ ℎ231199103

4

+ 119874 (1199103 1199104)

(10)

where ℎ1119894and ℎ

2119894(119894 = 0 1 2 3 ) are coefficients to be

determined 119874(sdot) represents the higher orders of 11991034 Substi-

tuting (10) into (5) and setting the coefficients of each powerof11991034

equal to zerowould yield the following set of equations

ℎ10=1

119860[(712057311205732minus 91205732

2) 12057411205722

1

+ 1205741(1205733

1minus 17120573

2

11205732+ 79120573

11205732

2minus 63120573

3

2)]

ℎ11=1

119860(312057321205722

1+ 1205732

1minus 612057311205732+ 21120573

2

2)

ℎ12=61205741

119860(12057311205722

1minus 1205732

1+ 10120573

11205732minus 91205732

2)

ℎ13= minus

612057411205721

119860(1205722

1minus 21205731+ 10120573

2)

(11)

where

119860 = 91205732

21205724

1+ 2 (minus18120573

11205732

2+ 10120573

2

11205732+ 90120573

3

2) 1205722

11205734

minus 201205733

11205732+ 118120573

2

11205732

2minus 180120573

11205733

2+ 81120573

4

2

(12)

Then the control equations on the center manifold are givenby

1199103= 1199104

1199104= minus12057321199103minus 12057221199104+ 120590 tanh 119896

31199104

minus 1205742[ℎ1(1199103 1199104) minus 1199103]3

(13)

By letting

119906 = 1199103

V = minus1

radic1205732

1199104

(14)

(13) can be rewritten as

= minusradic1205732V

V = radic1205732119906 + 119876

(15)

where

119876 = minus1205742

radic1205732

1199063minus1

312057321205903V3 + 119874 (1199063 V3) (16)

4 Mathematical Problems in Engineering

When 120575 = 1205901198963minus 1205722= 0 the first Lyapunov coefficient of

system (13) can be calculated

1198971=1

16(1205973119865

1205971199062120597V+1205973119865

120597V3)

minus1

16radic1205732

[1205972119865

120597119906120597V(1205972119865

1205971199062+1205972119865

120597V2)]

= minus1

81205732(1205722

1198963

)

3

lt 0

(17)

The first Lyapunov coefficient is less than zero which meansthat original system (5) undergoes a Hopf bifurcation near120575 = 0 and a stable limit cycle occurs To discuss the dynamicresponse of cables and its influence on the deck in greaterdetail we will analytically calculate the periodic solution insystem (3) in next section

4 Quantitative Analysis for the Cables-StayedBridge Model

In this section we use the energy method [18] to calculate theperiodic solution of (3)

(1) Denote

1198921(1199091) = 12057311199091+ 212057411199093

1

1198922(1199092) = 12057321199092minus 12057421199093

2

1198911(119909 ) = 120572

11+ 1205741(311990911199092

2minus 31199092

11199092minus 1199093

2)

1198912(119909 ) = 120572

22minus 120590 tanh (119896

32)

+ 1205742(1199093

1minus 31199092

11199092+ 311990911199092

2)

(18)

Considering the physics meanings of parameters in (3) andassuming that the vibration amplitudes are not too large(|11990912| lt 1119898) one has

11990911198921(1199091)1199091=0= 12057311199092

1+ 212057411199094

1gt 0

11990921198922(1199092)1199092=0= 12057321199092

2minus 12057421199094

2gt 0

(19)

Therefore the energy method can be applied to (3) to cal-culate its approximation periodic solution

(2) The potential energy functions of system (3) can beexpressed by

1198811(1199091) = int

1199091

0

(12057311199091+ 212057411199093

1) 1198891199091

=1

2(12057311199092

1+ 12057411199094

1)

1198812(1199092) = int

1199092

0

(12057321199092minus 12057421199093

2) 1198891199092

=1

4(212057321199092

2minus 12057421199094

2)

(20)

119881119894(119909) 119894 = 1 2 are even functions thus 119887

12= 0 hold [18] The

coordinate change formulas can be written as

119909119894= 119886119894cos (120579

119894) (119894 = 1 2)

1= plusmnradic2 (119881

1(1198861) minus 1198811(1198861cos (120579

1)))

= plusmn1198861

1003816100381610038161003816sin (1205791)1003816100381610038161003816radic1205731+3

212057411198862

1+1

212057411198862

1cos (2120579

1)

2= plusmnradic2 (119881

2(1198862) minus 1198812(1198862cos (120579

2)))

= plusmn1198862

1003816100381610038161003816sin (1205792)1003816100381610038161003816radic1205732minus3

412057421198862

2minus1

412057421198862

2cos (2120579

2)

(21)

Ignoring the terms of the harmonic expansions higher thanthe second the coordinate change formulas can be simplifiedas

1= minus1198861sin (1205791) 1198601(1 + 119860

2cos (2120579

1))

2= minus1198862sin (1205792) 1198611(1 + 119861

2cos (2120579

2))

(22)

where

1198601= radic1205731+3

212057411198862

1gt 0

0 lt 1198602=

12057411198862

1

41205731+ 612057411198862

1

lt 1

1198611= radic1205732minus3

412057421198862

2gt 0

minus1 lt 1198612= minus

12057421198862

2

81205732minus 612057421198862

2

lt 0

(23)

(3) Denote the energy of 1198981and 119898

2by 1198641and 119864

2

respectively 1198891198641119889119905 and 119889119864

2119889119905 can be calculated in which

the terms of the harmonic expansions higher than the secondwill be ignored

1198891198641

119889119905= minus1198911(119909 )

1

= 1198621+ 1198622cos (2120579

1)

+ 1198623[sin (120579

1minus 1205792) + sin (120579

1+ 1205792)]

+ 1198624sin (2120579

1minus 21205792) + 1198625sin (3120579

1minus 31205792)

+ 1198626sin (2120579

1) + 1198627sin (3120579

1minus 1205792)

+ 1198628sin (1205791minus 31205792) + 1198629sin (4120579

1minus 21205792)

≜ 1198651(119864 120579)

Mathematical Problems in Engineering 5

1198891198642

119889119905= minus1198912(119909 )

2

= 1198631+ 1198632cos (2120579

2)

+ 1198633[sin (120579

1minus 1205792) minus sin (120579

1+ 1205792)]

+ 1198634sin (2120579

1minus 21205792) + 1198635sin (3120579

1minus 31205792)

+ 1198636sin (2120579

2) + 1198637sin (3120579

1minus 1205792)

+ 1198638sin (1205791minus 31205792) + 1198639sin (2120579

1minus 41205792)

≜ 1198652(119864 120579)

(24)

where

1198621=1

4(21198602minus 2 minus 119860

2

2)1198602

11198862

11205721

1198622=1

8(4 + 3119860

2

2minus 81198602)1198602

112057211198862

1

1198623= minus

3

812057411198863

111986011198862+3

161205741(1198602minus 2)119860

11198863

21198861

1198624=3

812057411198862

111986011198862

2

1198625= minus

1

161205741119886111986011198863

21198602

1198626=3

412057411198862

111986011198862

2

1198627= minus

3

161205741(1198602+ 2)119860

111988621198863

1minus3

161205741119886111986011198863

21198602

1198628=1

161205741(1198602minus 2)119860

11198863

21198861

1198629=3

1612057411198862

111986011198862

21198602

1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

1198632=1

12(2 minus 7119861

2) 1198963

31198614

11198864

2120590 +

1

8(81198612minus 4) 119886

2

21198612

1(1198963120590

minus 1205722)

1198633=1

16((minus6 + 3119861

2) 11988621198863

1minus 61198863

21198861) 12057421198611

1198634=3

812057421198862

211986111198862

1

1198635= minus

1

161205742119886211986111198863

11198612

1198636= minus

3

412057421198862

211986111198862

1

1198637=1

16(1198612minus 2) 119886

21198863

112057421198611

1198638= minus

1

16(311988621198863

11198612+ (6 + 3119861

2) 1198863

21198861) 12057421198611

1198639=3

1612057421198862

211986111198862

11198612

(25)

(4) 1198891205791119889119905 and 119889120579

2119889119905 are given by

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)[1198671(1 + cos (2120579

2))

+ 1198672sin (2120579

1) + 1198673cos (2120579

1)

+ 1198674(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198675cos (2120579

1minus 21205792) + 1198676cos (3120579

1minus 31205792)

+ 1198677cos (120579

1minus 31205792) + 1198678cos (3120579

1minus 1205792)

+ 1198679cos (4120579

1minus 21205792)] + 119860

1(1 + 119860

2cos (2120579

1))

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)[1198691(1 + cos (2120579

1))

+ 1198692sin (2120579

2) + 1198693cos (2120579

2)

+ 1198694(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198695cos (2120579

1minus 21205792) + 1198696cos (3120579

1minus 31205792)

+ 1198697cos (120579

1minus 31205792) + 1198698cos (3120579

1minus 1205792)

+ 1198699cos (2120579

1minus 41205792)] + 119861

1(1 + 119861

2cos (2120579

2))

(26)

where

1198671=3

8(2 + 119860

2) 119860112057411198862

21198862

1

1198672= minus

1

8(4 + 119860

2

2)1198602

11198862

11205721

1198673= minus

3

4(1 + 119860

2) 11986011198862

11198862

21205741

1198674= minus

3

16((2 + 119860

2) 11988611198863

2+ (6 + 4119860

2) 1198863

11198862)11986011205741

1198675=3

8(1 + 119860

2) 11986011198862

11198862

21205741

1198676= minus

1

161205741119886111986011198863

21198602

1198677= minus

1

16(2 + 119860

2) 11988611198863

211986011205741

1198678= minus

3

16(11988611198863

21198602+ (2 + 3119860

2) 1198863

11198862)11986011205741

1198679=3

1612057411198862

111986011198862

21198602

6 Mathematical Problems in Engineering

1198691= minus

3

8(2 + 119861

2) 1198862

11198862

211986111205742

1198692=1

12(1198612minus 1) 119886

4

21198611

41198963

3+ (1

21198963120590 minus 48120572

2) 1198862

21198612

1

1198693= minus

3

4(1198612+ 1) 119861

11198862

21198862

11205742

1198694=3

16((2 + 119861

2) 11988621198863

1+ (4119861

2+ 6) 119886

3

21198861) 12057421198611

1198695= minus

3

8(1198612+ 1) 119861

11198862

21198862

11205742

1198696=1

161205742119886211986111198863

11198612

1198697=3

16(11988621198863

11198612+ (3119861

2+ 2) 119886

3

21198861) 12057421198611

1198698=1

16(2 + 119861

2) 11988621198863

112057421198611

1198699= minus

3

1612057421198862

211986111198862

11198612

(27)

(5) Letting

1205791= 120596119905 + 120601

1

1205792= 120596119905 + 120601

2

(28)

and substituting them into (24) (26) yields

1198891198641

119889119905= 1198621+ 1198623sin (120601

1minus 1206012) + 1198624sin (2120601

1minus 21206012)

+ 1198625sin (3120601

1minus 31206012) + [119862

2cos (2120601

1) + 1198623

sdot sin (1206011+ 1206012) + 1198626sin (2120601

1) + 1198627sin (3120601

1minus 1206012)

+ 1198628sin (120601

1minus 31206012) + 1198629sin (4120601

1minus 21206012)] cos (2120596119905)

+ [minus1198622sin (2120601

1) + 1198623cos (120601

1+ 1206012) + 1198626cos (2120601

1)

+ 1198627cos (3120601

1minus 1206012) minus 1198628cos (120601

1minus 31206012) + 1198629

sdot cos (41206011minus 21206012)] sin (2120596119905)

1198891198642

119889119905= 1198631+ 1198633sin (120601

1minus 1206012) + 1198634sin (2120601

1minus 21206012)

+ 1198635sin (3120601

1minus 31206012) + [119863

2cos (2120601

1) minus 1198633

sdot sin (1206011+ 1206012) + 1198636sin (2120601

2) + 1198637sin (3120601

1minus 1206012)

+ 1198638sin (120601

1minus 31206012) + 1198639sin (2120601

1minus 41206012)]

sdot cos (2120596119905) + [minus1198632sin (2120601

1) minus 1198633cos (120601

1+ 1206012)

+ 1198636cos (2120601

2) + 1198637cos (3120601

1minus 1206012) minus 1198638

sdot cos (1206011minus 31206012) minus 1198639cos (2120601

1minus 41206012)] sin (2120596119905)

(29)

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)1198862

1(1205731+ 212057411198862

1)1198601+ 1198671

+ 1198674cos (120601

1minus 1206012) + 1198675cos (2120601

1minus 21206012) + 1198676

sdot cos (31206011minus 31206012) + [119867

1cos (2120601

2) + 1198672sin (2120601

1)

+ (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) cos (2120601

1)

+ 1198674cos (120601

1+ 1206012) + 1198677cos (120601

1minus 31206012)

+ 1198678cos (3120601

1minus 1206012) + 1198679cos (4120601

1minus 21206012)]

sdot cos (2120596119905) + [minus1198671sin (2120601

2) + 1198672cos (2120601

1)

minus (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) sin (2120601

1)

minus 1198674sin (120601

1+ 1206012) + 1198677sin (120601

1minus 31206012)

minus 1198678sin (3120601

1minus 1206012) minus 1198679sin (4120601

1minus 21206012)]

sdot sin (2120596119905)

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)1198862

2(1205732minus 12057421198862

2) 1198611+ 1198691+ 1198694

sdot cos (1206011minus 1206012) + 1198695cos (2120601

1minus 21206012) + 1198696

sdot cos (31206011minus 31206012) + [1198691cos (2120601

1) + 1198692sin (2120601

2)

+ (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) cos (2120601

2)

+ 1198694cos (120601

1+ 1206012) + 1198697cos (120601

1minus 31206012)

+ 1198698cos (3120601

1minus 1206012) + 1198699cos (2120601

1minus 41206012)] cos (2120596119905)

+ [minus1198691sin (2120601

1) + 1198692cos (2120601

2)

minus (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) sin (2120601

2)

minus 1198694sin (120601

1+ 1206012) + 1198697sin (120601

1minus 31206012)

minus 1198698sin (3120601

1minus 1206012) minus 1198699sin (2120601

1minus 41206012)]

sdot sin (2120596119905)

(30)

If there exists a periodic solution with period 2120587120596 in (3)then the energy of the system should remain the same at ini-tial and finial positions in one period Integrating both sidesof (29) in one period (from 119905 = Δ to 119905 = Δ + 2120587120596) leads to

(1198641)2120587120596+Δ

minus (1198641)Δ= 1198621= minus

1

4[(1198602minus 1)2

minus 1]

sdot 1198602

11198862

11205721lt 0

(1198642)2120587120596+Δ

minus (1198642)Δ= 1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

(31)

Mathematical Problems in Engineering 7

minus004

minus002

0

002

004

minus01

minus005

0

005

01

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

x1

(m)

x2

(m)

(a)

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

minus004

minus002

0

002

004

minus01

minus005

0

005

01

x1

(m)

x2

(m)

(b)

Figure 2 Direct numerical simulations of (3) for (a) 119899 = 112 and (b) 119899 = 118

Inspection of the first equation in (31) shows that the vibra-tion energy of cables decreases in one period Thereforecables cannot vibrate periodically when the deck is subjectedto the crowd excitation From the second equation in (31)if 1198963120590 minus 1205722lt 0 then 119863

1lt 0 At this point the deck has an

attenuate vibration If 1198963120590minus1205722gt 0119863

1= 0may hold Letting

1198631= 0 in the second equation in (31) and ignoring the sixth

or more power of 1198862(since 119886

2≪ 1 according to its physical

meaning) one obtains

712057421198864

212057321205901198963

3minus 3 (5120575120574

2+ 21205732

21205901198963

3) 1198862

2+ 24120575120573

2= 0 (32)

where 120575 = 1198963120590 minus 1205722 Solving (32) yields

119886lowast

2

=radic61205901198963

31205732

2+ 151205742120575 minus radic12120590119896

3

31205732

2(31205901198963

31205732

2minus 411205742120575) + 225120574

2

21205752

1412059012057421198963

31205732

= 2radic2radic1205732120575

21205901198963

31205732

2+ 51205742120575

(33)

For clarification the amplitude can be rewritten as

119886lowast

2= 4radic

1198982119896 (1198963120574 minus 1198882)

41205741198963

31198962 + 5 (11986411986021198713)119898

2(1198963120574 minus 1198882) (34)

where 120574 = 11989611198962119866(119891)119872

119901119892 and 119896

3120574 minus 1198882gt 0

(6) The first-order approximation solution of (3) isexpressed by

1199092(119905) = 119886

lowast

2cos (120596119905)

= 119886lowast

2cos(radic 1

1198982

[119896 minus3

8

119864119860

1198713(119886lowast

2)2

]119905)

(35)

where 119886lowast2is given in (34)

5 Numerical Simulations and Discussion

In this section we will verify the validity of our analyticalsolutions obtained in the last section by comparing theresults based on (35) with that derived by direct numericalsimulations for (3) The following parameters of deck andpedestrians are taken to calculate the periodic solutions in (3)

1198982= 21401 times 10

5(kg)

119896 = 7307361 times 106(kgs2)

1198882= 28262 times 10

4(kgs)

119866 (119891) = 10

1198961= 00987

1198962= 02

1198963= 18

(36)

In fact above values of parameters are used in [17 19] forthe first lateral model of the T-bridge in Japan Furthermorethe weight of a single person on the bridge is assumed to be70 (kg) the gravity acceleration 119892 = 98 (ms2) The modalmass of pedestrians119872

119901119892 can be given by119872

119901119892 = 119899times70times98

where 119899 is the number of the pedestrians on the bridgeAdditionally according to [19] the average length of thecables is taken as 119871 = 60 (m) However other parameters ofthe cables such as stiffness damping coefficient and massare not provided in [19] In order to compare the theoreticalresults of T-bridge between our cable-stayed bridge modeland Nakamurarsquos model following parameters of cables areassumed in the numerical simulations

1198981= 21401 times 10

3(kg)

119864119860 = 168 times 1010(N)

8 Mathematical Problems in Engineering

9955 996 9965 997 9975 998 9985 999 9995 1000

Numerical resultsAnalytical results

t (s)

minus3

minus2

minus1

0

1

2

3

4times10minus3

x2

(m)

(a)

9955 996 9965 997 9975 998 9985 999 9995 1000minus6

minus4

minus2

0

2

4

6

8

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(b)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(c)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3x2

(m)

(d)

Figure 3 Comparison between the first-order approximation solution equation (35) and direct numerical results of (3) for the cases of (a)119899 = 118 (b) 119899 = 125 (c) 119899 = 130 and (d) 119899 = 135

1198881= 28262 times 10

2(kgs)

1198790= 724 times 10

5(N)

(37)

Then we have

1205721= 01321 (1s)

1205722= 01321 (1s)

1205731= 676604 (1s2)

1205732= 341450 (1s2)

1205741= 39251 times 10

3(1s2)

1205742= 392505 (1s2)

(38)

Regarding the number 119899 as the bifurcation parameter one has

120590 =

11989611198962119866 (119891)119872

119901119892

1198982

= 0000063275734785119899

120575 = 1198963120590 minus 1205722= 000113896119899 minus 013201

(39)

From the analysis in Section 3 120575 lt 0 means the attenuationof the lateral vibration of the bridges while 120575 gt 0 means the

Mathematical Problems in Engineering 9

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

2 Mathematical Problems in Engineering

force exerted by pedestrians on footbridges However theforce exerted by pedestrians is generally tackled by using anempirical approach strict and exactmathematicalmodels arenot available yet

On the other side in almost all dynamic models pro-posed to describe the interaction between pedestrians anda footbridge the footbridge always is simplified as a single-degree-freedom system On an intuitive level such simpli-fication may not be rational to all types of footbridges forexample suspension bridges and cable-stayed bridges In factMcRobie et al [13] experimentally investigated the ldquolock-inrdquophenomenonby constructing a sectionmodelwith cables andbridge deck Zhou and Ji [14] theoretically and experimentallyanalyzed a generalized suspension system developed fromthe section model Their research showed that cables havegreat influence on free vibrations of the suspension systemwhereas the dynamic behaviour of the suspension systemunder crowd excitation has not been considered Cable-stayed bridges are important engineering structures widelyapplied all over the world A cable-stayed bridge consists ofa deck suspended from stay cables and supported on piersand towers Geurts et al [15] reported that Erasmus bridgein Netherlands opened to the public for just two monthsin 1996 had to be closed because of the large amplitudevibrations of the cables Lilien and Pinto Da Costa [16]found that a dynamic instability of the cables may occurdue to small periodic movements of the girder and themasts Fujino et al [5] argued that cables may have intensivevibrations induced by deck when the natural frequency ofthe cables approximates to half natural frequency of thedeck Nevertheless seldom theoretical analysis for cable-stayed bridges with nonlinear couples between cables anddeck under crowd excitation has been carried out

The paper focuses on the lateral vibrations of a cable-stayed bridge under crowd excitationThe cable-stayed bridgemodel consists of cables and deck which are nonlinearlycoupled Nakamurarsquos assumption [4] is adopted to model thelateral force induced by crowd excitation We will pull all ouremphasis on whether and how the cables affect the lateralvibrations of the deck under crowd excitation The rest ofthe paper is organized as follows the governing equationof the pedestrian-bridge system is established in Section 2the qualitative and quantitative analysis for the governingequation are presented in Sections 3 and 4 respectivelynumerical verifications and discussion are demonstrated inSection 5 conclusions are drawn in Section 6

2 The Governing Equation

The cable-stayed bridge model investigated in this paper isillustrated in Figure 1 in which line 119860119861 represents cablesand the rigid block of the deck Line 119860119861 has a length 119871mass119898

1 stiffness 119864119860 tension 119879

0 and damping coefficient 119888

1

For simplicity all the matters in line 119860119861 are assumed to betogether in the center of mass (point 119862) The rigid block hasa mass 119898

2 stiffness 119896 and damping coefficient 119888

2 Assume

that the lateral displacements of point 119862 and the rigid blockare 1199091and 119909

2 respectively

12 12

represent corresponding

A

C

Cables

x1

F

x2

B

m1

m2

k

Deck

c1

c2

Figure 1 The cable-stayed bridge diagram

velocities and accelerations respectively The lateral forceinduced by pedestrians on the bridge is denoted by 119865

According to Nakamurarsquos assumption [4] 119865 is given by

119865 = 11989611198962119872119901119892119866 (119891)119867 (

2) (1)

where coefficient 1198961is the ratio of lateral force to pedestriansrsquo

weight coefficient 1198962is the percentage of pedestrians who

synchronize to bridge vibration119872119901119892 is modal self-weight of

pedestrians 119866(119891) is a function describing how pedestrianssynchronizewith the natural frequency of a bridge and119867(

2)

is a function that describes the pedestriansrsquo synchronizationnature Nakamura assumed that the pedestrians synchronizeproportionally with the deck velocity

2at low velocities

When the deck velocity becomes large pedestrians feeluncomfortable and then they decrease their walking pacewhich limits the deck response at a certain level rather thaninfinite one Based on the analysis of Zhen et al [17] 119867(

2)

can be expressed as

119867(2) = tanh (119896

32) (2)

in which the coefficient 1198963indicates the saturation rate of the

pedestrian-induced forceBy applying Newtonrsquos second law of motion and consid-

ering the geometric nonlinearity of the cables the governingequation of the cable-stayed bridge under crowd excitation isderived as follows

1+ 12057211+ 12057311199091+ 1205741[1199093

1+ (1199091minus 1199092)3

] = 0

2+ 12057222+ 12057321199092+ 1205742(1199091minus 1199092)3

= 120590 tanh (11989632)

(3)

where 1205721= 11988811198981 1205722= 11988821198982 1205731= 211987901198981119871 1205732= 119896119898

2

1205741= 1198641198602119898

11198713 1205742= 1198641198602119898

21198713 and 120590 = (119896

11198962119866(119891)119872

119901119892)

1198982According to the physics meanings 120572

12 12057312 12057412gt 0

Furthermore we assume that 1205732gt 1205742in the following

analysis

Mathematical Problems in Engineering 3

3 Qualitative Analysis for the Cable-StayedBridge Model

By letting

1199091= 1199101

1= 1199102

1199092= 1199103

2= 1199104

(4)

(3) can be rewritten as

1199101= 1199102

1199102= minus12057311199101minus 12057211199102minus 1205741[1199103

1+ (1199101minus 1199103)3

]

1199103= 1199104

1199104= minus12057321199103minus 12057221199104+ 120590 tanh (119896

31199104) minus 1205742(1199101minus 1199103)3

(5)

Obviously (0 0 0 0) is an equilibrium of (5)The charac-teristic equation of (5) near the origin is

1205824+ (1205721minus 120575) 120582

3+ (1205731+ 1205732minus 1205721120575) 1205822

+ (12057211205732minus 1205751205731) 120582 + 120573

11205732= 0

(6)

where 120575 = 1205901198963minus 1205722 Since 120572

119894 120573119894 120574119894gt 0 119894 = 1 2 it is easy to

verify that (5) has a stable origin if 120575 lt 0 and an unstable oneif 120575 gt 0 When 120575 = 0 (6) has a pair of purely imaginary roots12058212= plusmn119894radic120573

2 Regarding 120582 as a function of 120575 in (6) and taking

the derivative of 120582 with respect to 120575 one has

119889120582

119889120575

=

120582 (1205822+ 1205721120582 + 1205731)

41205823 minus 3 (120575 minus 1205721) 1205822 minus 2 (120575120572

1minus 1205731minus 1205732) 120582 minus 120573

1120575 + 12057211205732

(7)

Substituting 120582 = plusmn119894radic1205732and 120575 = 0 into (7) yields

(119889120582

119889120575)

120575=0

=1

2gt 0 (8)

According to Hopf bifurcation theory system (5) under-goes a Hopf bifurcation near 120575 = 0 and a limit cycle appearswhen 120575 gt 0 This means that periodic vibrations occur in thecable-stayed bridgemodel under crowd excitation To furtherdetermine the stability of the limit cycle we employ centermanifold theory to analyze system (5)

If 120575 = 0 (6) has a pair of purely imaginary roots and twonegative real roots which satisfies the condition for centermanifold theory Assume that a local center manifold can beexpressed by

1199101= ℎ1(1199103 1199104)

1199102= ℎ2(1199103 1199104)

(9)

where ℎ1and ℎ2are vector functions of nonlinear terms of 119910

3

and 1199104 Assume that 119910

1and 1199102can be transformed into power

series

1199101= ℎ101199103

3+ ℎ111199102

31199104+ ℎ1211991031199102

4+ ℎ131199103

4

+ 119874 (1199103 1199104)

1199102= ℎ201199103

3+ ℎ211199102

31199104+ ℎ2211991031199102

4+ ℎ231199103

4

+ 119874 (1199103 1199104)

(10)

where ℎ1119894and ℎ

2119894(119894 = 0 1 2 3 ) are coefficients to be

determined 119874(sdot) represents the higher orders of 11991034 Substi-

tuting (10) into (5) and setting the coefficients of each powerof11991034

equal to zerowould yield the following set of equations

ℎ10=1

119860[(712057311205732minus 91205732

2) 12057411205722

1

+ 1205741(1205733

1minus 17120573

2

11205732+ 79120573

11205732

2minus 63120573

3

2)]

ℎ11=1

119860(312057321205722

1+ 1205732

1minus 612057311205732+ 21120573

2

2)

ℎ12=61205741

119860(12057311205722

1minus 1205732

1+ 10120573

11205732minus 91205732

2)

ℎ13= minus

612057411205721

119860(1205722

1minus 21205731+ 10120573

2)

(11)

where

119860 = 91205732

21205724

1+ 2 (minus18120573

11205732

2+ 10120573

2

11205732+ 90120573

3

2) 1205722

11205734

minus 201205733

11205732+ 118120573

2

11205732

2minus 180120573

11205733

2+ 81120573

4

2

(12)

Then the control equations on the center manifold are givenby

1199103= 1199104

1199104= minus12057321199103minus 12057221199104+ 120590 tanh 119896

31199104

minus 1205742[ℎ1(1199103 1199104) minus 1199103]3

(13)

By letting

119906 = 1199103

V = minus1

radic1205732

1199104

(14)

(13) can be rewritten as

= minusradic1205732V

V = radic1205732119906 + 119876

(15)

where

119876 = minus1205742

radic1205732

1199063minus1

312057321205903V3 + 119874 (1199063 V3) (16)

4 Mathematical Problems in Engineering

When 120575 = 1205901198963minus 1205722= 0 the first Lyapunov coefficient of

system (13) can be calculated

1198971=1

16(1205973119865

1205971199062120597V+1205973119865

120597V3)

minus1

16radic1205732

[1205972119865

120597119906120597V(1205972119865

1205971199062+1205972119865

120597V2)]

= minus1

81205732(1205722

1198963

)

3

lt 0

(17)

The first Lyapunov coefficient is less than zero which meansthat original system (5) undergoes a Hopf bifurcation near120575 = 0 and a stable limit cycle occurs To discuss the dynamicresponse of cables and its influence on the deck in greaterdetail we will analytically calculate the periodic solution insystem (3) in next section

4 Quantitative Analysis for the Cables-StayedBridge Model

In this section we use the energy method [18] to calculate theperiodic solution of (3)

(1) Denote

1198921(1199091) = 12057311199091+ 212057411199093

1

1198922(1199092) = 12057321199092minus 12057421199093

2

1198911(119909 ) = 120572

11+ 1205741(311990911199092

2minus 31199092

11199092minus 1199093

2)

1198912(119909 ) = 120572

22minus 120590 tanh (119896

32)

+ 1205742(1199093

1minus 31199092

11199092+ 311990911199092

2)

(18)

Considering the physics meanings of parameters in (3) andassuming that the vibration amplitudes are not too large(|11990912| lt 1119898) one has

11990911198921(1199091)1199091=0= 12057311199092

1+ 212057411199094

1gt 0

11990921198922(1199092)1199092=0= 12057321199092

2minus 12057421199094

2gt 0

(19)

Therefore the energy method can be applied to (3) to cal-culate its approximation periodic solution

(2) The potential energy functions of system (3) can beexpressed by

1198811(1199091) = int

1199091

0

(12057311199091+ 212057411199093

1) 1198891199091

=1

2(12057311199092

1+ 12057411199094

1)

1198812(1199092) = int

1199092

0

(12057321199092minus 12057421199093

2) 1198891199092

=1

4(212057321199092

2minus 12057421199094

2)

(20)

119881119894(119909) 119894 = 1 2 are even functions thus 119887

12= 0 hold [18] The

coordinate change formulas can be written as

119909119894= 119886119894cos (120579

119894) (119894 = 1 2)

1= plusmnradic2 (119881

1(1198861) minus 1198811(1198861cos (120579

1)))

= plusmn1198861

1003816100381610038161003816sin (1205791)1003816100381610038161003816radic1205731+3

212057411198862

1+1

212057411198862

1cos (2120579

1)

2= plusmnradic2 (119881

2(1198862) minus 1198812(1198862cos (120579

2)))

= plusmn1198862

1003816100381610038161003816sin (1205792)1003816100381610038161003816radic1205732minus3

412057421198862

2minus1

412057421198862

2cos (2120579

2)

(21)

Ignoring the terms of the harmonic expansions higher thanthe second the coordinate change formulas can be simplifiedas

1= minus1198861sin (1205791) 1198601(1 + 119860

2cos (2120579

1))

2= minus1198862sin (1205792) 1198611(1 + 119861

2cos (2120579

2))

(22)

where

1198601= radic1205731+3

212057411198862

1gt 0

0 lt 1198602=

12057411198862

1

41205731+ 612057411198862

1

lt 1

1198611= radic1205732minus3

412057421198862

2gt 0

minus1 lt 1198612= minus

12057421198862

2

81205732minus 612057421198862

2

lt 0

(23)

(3) Denote the energy of 1198981and 119898

2by 1198641and 119864

2

respectively 1198891198641119889119905 and 119889119864

2119889119905 can be calculated in which

the terms of the harmonic expansions higher than the secondwill be ignored

1198891198641

119889119905= minus1198911(119909 )

1

= 1198621+ 1198622cos (2120579

1)

+ 1198623[sin (120579

1minus 1205792) + sin (120579

1+ 1205792)]

+ 1198624sin (2120579

1minus 21205792) + 1198625sin (3120579

1minus 31205792)

+ 1198626sin (2120579

1) + 1198627sin (3120579

1minus 1205792)

+ 1198628sin (1205791minus 31205792) + 1198629sin (4120579

1minus 21205792)

≜ 1198651(119864 120579)

Mathematical Problems in Engineering 5

1198891198642

119889119905= minus1198912(119909 )

2

= 1198631+ 1198632cos (2120579

2)

+ 1198633[sin (120579

1minus 1205792) minus sin (120579

1+ 1205792)]

+ 1198634sin (2120579

1minus 21205792) + 1198635sin (3120579

1minus 31205792)

+ 1198636sin (2120579

2) + 1198637sin (3120579

1minus 1205792)

+ 1198638sin (1205791minus 31205792) + 1198639sin (2120579

1minus 41205792)

≜ 1198652(119864 120579)

(24)

where

1198621=1

4(21198602minus 2 minus 119860

2

2)1198602

11198862

11205721

1198622=1

8(4 + 3119860

2

2minus 81198602)1198602

112057211198862

1

1198623= minus

3

812057411198863

111986011198862+3

161205741(1198602minus 2)119860

11198863

21198861

1198624=3

812057411198862

111986011198862

2

1198625= minus

1

161205741119886111986011198863

21198602

1198626=3

412057411198862

111986011198862

2

1198627= minus

3

161205741(1198602+ 2)119860

111988621198863

1minus3

161205741119886111986011198863

21198602

1198628=1

161205741(1198602minus 2)119860

11198863

21198861

1198629=3

1612057411198862

111986011198862

21198602

1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

1198632=1

12(2 minus 7119861

2) 1198963

31198614

11198864

2120590 +

1

8(81198612minus 4) 119886

2

21198612

1(1198963120590

minus 1205722)

1198633=1

16((minus6 + 3119861

2) 11988621198863

1minus 61198863

21198861) 12057421198611

1198634=3

812057421198862

211986111198862

1

1198635= minus

1

161205742119886211986111198863

11198612

1198636= minus

3

412057421198862

211986111198862

1

1198637=1

16(1198612minus 2) 119886

21198863

112057421198611

1198638= minus

1

16(311988621198863

11198612+ (6 + 3119861

2) 1198863

21198861) 12057421198611

1198639=3

1612057421198862

211986111198862

11198612

(25)

(4) 1198891205791119889119905 and 119889120579

2119889119905 are given by

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)[1198671(1 + cos (2120579

2))

+ 1198672sin (2120579

1) + 1198673cos (2120579

1)

+ 1198674(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198675cos (2120579

1minus 21205792) + 1198676cos (3120579

1minus 31205792)

+ 1198677cos (120579

1minus 31205792) + 1198678cos (3120579

1minus 1205792)

+ 1198679cos (4120579

1minus 21205792)] + 119860

1(1 + 119860

2cos (2120579

1))

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)[1198691(1 + cos (2120579

1))

+ 1198692sin (2120579

2) + 1198693cos (2120579

2)

+ 1198694(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198695cos (2120579

1minus 21205792) + 1198696cos (3120579

1minus 31205792)

+ 1198697cos (120579

1minus 31205792) + 1198698cos (3120579

1minus 1205792)

+ 1198699cos (2120579

1minus 41205792)] + 119861

1(1 + 119861

2cos (2120579

2))

(26)

where

1198671=3

8(2 + 119860

2) 119860112057411198862

21198862

1

1198672= minus

1

8(4 + 119860

2

2)1198602

11198862

11205721

1198673= minus

3

4(1 + 119860

2) 11986011198862

11198862

21205741

1198674= minus

3

16((2 + 119860

2) 11988611198863

2+ (6 + 4119860

2) 1198863

11198862)11986011205741

1198675=3

8(1 + 119860

2) 11986011198862

11198862

21205741

1198676= minus

1

161205741119886111986011198863

21198602

1198677= minus

1

16(2 + 119860

2) 11988611198863

211986011205741

1198678= minus

3

16(11988611198863

21198602+ (2 + 3119860

2) 1198863

11198862)11986011205741

1198679=3

1612057411198862

111986011198862

21198602

6 Mathematical Problems in Engineering

1198691= minus

3

8(2 + 119861

2) 1198862

11198862

211986111205742

1198692=1

12(1198612minus 1) 119886

4

21198611

41198963

3+ (1

21198963120590 minus 48120572

2) 1198862

21198612

1

1198693= minus

3

4(1198612+ 1) 119861

11198862

21198862

11205742

1198694=3

16((2 + 119861

2) 11988621198863

1+ (4119861

2+ 6) 119886

3

21198861) 12057421198611

1198695= minus

3

8(1198612+ 1) 119861

11198862

21198862

11205742

1198696=1

161205742119886211986111198863

11198612

1198697=3

16(11988621198863

11198612+ (3119861

2+ 2) 119886

3

21198861) 12057421198611

1198698=1

16(2 + 119861

2) 11988621198863

112057421198611

1198699= minus

3

1612057421198862

211986111198862

11198612

(27)

(5) Letting

1205791= 120596119905 + 120601

1

1205792= 120596119905 + 120601

2

(28)

and substituting them into (24) (26) yields

1198891198641

119889119905= 1198621+ 1198623sin (120601

1minus 1206012) + 1198624sin (2120601

1minus 21206012)

+ 1198625sin (3120601

1minus 31206012) + [119862

2cos (2120601

1) + 1198623

sdot sin (1206011+ 1206012) + 1198626sin (2120601

1) + 1198627sin (3120601

1minus 1206012)

+ 1198628sin (120601

1minus 31206012) + 1198629sin (4120601

1minus 21206012)] cos (2120596119905)

+ [minus1198622sin (2120601

1) + 1198623cos (120601

1+ 1206012) + 1198626cos (2120601

1)

+ 1198627cos (3120601

1minus 1206012) minus 1198628cos (120601

1minus 31206012) + 1198629

sdot cos (41206011minus 21206012)] sin (2120596119905)

1198891198642

119889119905= 1198631+ 1198633sin (120601

1minus 1206012) + 1198634sin (2120601

1minus 21206012)

+ 1198635sin (3120601

1minus 31206012) + [119863

2cos (2120601

1) minus 1198633

sdot sin (1206011+ 1206012) + 1198636sin (2120601

2) + 1198637sin (3120601

1minus 1206012)

+ 1198638sin (120601

1minus 31206012) + 1198639sin (2120601

1minus 41206012)]

sdot cos (2120596119905) + [minus1198632sin (2120601

1) minus 1198633cos (120601

1+ 1206012)

+ 1198636cos (2120601

2) + 1198637cos (3120601

1minus 1206012) minus 1198638

sdot cos (1206011minus 31206012) minus 1198639cos (2120601

1minus 41206012)] sin (2120596119905)

(29)

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)1198862

1(1205731+ 212057411198862

1)1198601+ 1198671

+ 1198674cos (120601

1minus 1206012) + 1198675cos (2120601

1minus 21206012) + 1198676

sdot cos (31206011minus 31206012) + [119867

1cos (2120601

2) + 1198672sin (2120601

1)

+ (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) cos (2120601

1)

+ 1198674cos (120601

1+ 1206012) + 1198677cos (120601

1minus 31206012)

+ 1198678cos (3120601

1minus 1206012) + 1198679cos (4120601

1minus 21206012)]

sdot cos (2120596119905) + [minus1198671sin (2120601

2) + 1198672cos (2120601

1)

minus (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) sin (2120601

1)

minus 1198674sin (120601

1+ 1206012) + 1198677sin (120601

1minus 31206012)

minus 1198678sin (3120601

1minus 1206012) minus 1198679sin (4120601

1minus 21206012)]

sdot sin (2120596119905)

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)1198862

2(1205732minus 12057421198862

2) 1198611+ 1198691+ 1198694

sdot cos (1206011minus 1206012) + 1198695cos (2120601

1minus 21206012) + 1198696

sdot cos (31206011minus 31206012) + [1198691cos (2120601

1) + 1198692sin (2120601

2)

+ (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) cos (2120601

2)

+ 1198694cos (120601

1+ 1206012) + 1198697cos (120601

1minus 31206012)

+ 1198698cos (3120601

1minus 1206012) + 1198699cos (2120601

1minus 41206012)] cos (2120596119905)

+ [minus1198691sin (2120601

1) + 1198692cos (2120601

2)

minus (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) sin (2120601

2)

minus 1198694sin (120601

1+ 1206012) + 1198697sin (120601

1minus 31206012)

minus 1198698sin (3120601

1minus 1206012) minus 1198699sin (2120601

1minus 41206012)]

sdot sin (2120596119905)

(30)

If there exists a periodic solution with period 2120587120596 in (3)then the energy of the system should remain the same at ini-tial and finial positions in one period Integrating both sidesof (29) in one period (from 119905 = Δ to 119905 = Δ + 2120587120596) leads to

(1198641)2120587120596+Δ

minus (1198641)Δ= 1198621= minus

1

4[(1198602minus 1)2

minus 1]

sdot 1198602

11198862

11205721lt 0

(1198642)2120587120596+Δ

minus (1198642)Δ= 1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

(31)

Mathematical Problems in Engineering 7

minus004

minus002

0

002

004

minus01

minus005

0

005

01

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

x1

(m)

x2

(m)

(a)

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

minus004

minus002

0

002

004

minus01

minus005

0

005

01

x1

(m)

x2

(m)

(b)

Figure 2 Direct numerical simulations of (3) for (a) 119899 = 112 and (b) 119899 = 118

Inspection of the first equation in (31) shows that the vibra-tion energy of cables decreases in one period Thereforecables cannot vibrate periodically when the deck is subjectedto the crowd excitation From the second equation in (31)if 1198963120590 minus 1205722lt 0 then 119863

1lt 0 At this point the deck has an

attenuate vibration If 1198963120590minus1205722gt 0119863

1= 0may hold Letting

1198631= 0 in the second equation in (31) and ignoring the sixth

or more power of 1198862(since 119886

2≪ 1 according to its physical

meaning) one obtains

712057421198864

212057321205901198963

3minus 3 (5120575120574

2+ 21205732

21205901198963

3) 1198862

2+ 24120575120573

2= 0 (32)

where 120575 = 1198963120590 minus 1205722 Solving (32) yields

119886lowast

2

=radic61205901198963

31205732

2+ 151205742120575 minus radic12120590119896

3

31205732

2(31205901198963

31205732

2minus 411205742120575) + 225120574

2

21205752

1412059012057421198963

31205732

= 2radic2radic1205732120575

21205901198963

31205732

2+ 51205742120575

(33)

For clarification the amplitude can be rewritten as

119886lowast

2= 4radic

1198982119896 (1198963120574 minus 1198882)

41205741198963

31198962 + 5 (11986411986021198713)119898

2(1198963120574 minus 1198882) (34)

where 120574 = 11989611198962119866(119891)119872

119901119892 and 119896

3120574 minus 1198882gt 0

(6) The first-order approximation solution of (3) isexpressed by

1199092(119905) = 119886

lowast

2cos (120596119905)

= 119886lowast

2cos(radic 1

1198982

[119896 minus3

8

119864119860

1198713(119886lowast

2)2

]119905)

(35)

where 119886lowast2is given in (34)

5 Numerical Simulations and Discussion

In this section we will verify the validity of our analyticalsolutions obtained in the last section by comparing theresults based on (35) with that derived by direct numericalsimulations for (3) The following parameters of deck andpedestrians are taken to calculate the periodic solutions in (3)

1198982= 21401 times 10

5(kg)

119896 = 7307361 times 106(kgs2)

1198882= 28262 times 10

4(kgs)

119866 (119891) = 10

1198961= 00987

1198962= 02

1198963= 18

(36)

In fact above values of parameters are used in [17 19] forthe first lateral model of the T-bridge in Japan Furthermorethe weight of a single person on the bridge is assumed to be70 (kg) the gravity acceleration 119892 = 98 (ms2) The modalmass of pedestrians119872

119901119892 can be given by119872

119901119892 = 119899times70times98

where 119899 is the number of the pedestrians on the bridgeAdditionally according to [19] the average length of thecables is taken as 119871 = 60 (m) However other parameters ofthe cables such as stiffness damping coefficient and massare not provided in [19] In order to compare the theoreticalresults of T-bridge between our cable-stayed bridge modeland Nakamurarsquos model following parameters of cables areassumed in the numerical simulations

1198981= 21401 times 10

3(kg)

119864119860 = 168 times 1010(N)

8 Mathematical Problems in Engineering

9955 996 9965 997 9975 998 9985 999 9995 1000

Numerical resultsAnalytical results

t (s)

minus3

minus2

minus1

0

1

2

3

4times10minus3

x2

(m)

(a)

9955 996 9965 997 9975 998 9985 999 9995 1000minus6

minus4

minus2

0

2

4

6

8

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(b)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(c)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3x2

(m)

(d)

Figure 3 Comparison between the first-order approximation solution equation (35) and direct numerical results of (3) for the cases of (a)119899 = 118 (b) 119899 = 125 (c) 119899 = 130 and (d) 119899 = 135

1198881= 28262 times 10

2(kgs)

1198790= 724 times 10

5(N)

(37)

Then we have

1205721= 01321 (1s)

1205722= 01321 (1s)

1205731= 676604 (1s2)

1205732= 341450 (1s2)

1205741= 39251 times 10

3(1s2)

1205742= 392505 (1s2)

(38)

Regarding the number 119899 as the bifurcation parameter one has

120590 =

11989611198962119866 (119891)119872

119901119892

1198982

= 0000063275734785119899

120575 = 1198963120590 minus 1205722= 000113896119899 minus 013201

(39)

From the analysis in Section 3 120575 lt 0 means the attenuationof the lateral vibration of the bridges while 120575 gt 0 means the

Mathematical Problems in Engineering 9

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

3 Qualitative Analysis for the Cable-StayedBridge Model

By letting

1199091= 1199101

1= 1199102

1199092= 1199103

2= 1199104

(4)

(3) can be rewritten as

1199101= 1199102

1199102= minus12057311199101minus 12057211199102minus 1205741[1199103

1+ (1199101minus 1199103)3

]

1199103= 1199104

1199104= minus12057321199103minus 12057221199104+ 120590 tanh (119896

31199104) minus 1205742(1199101minus 1199103)3

(5)

Obviously (0 0 0 0) is an equilibrium of (5)The charac-teristic equation of (5) near the origin is

1205824+ (1205721minus 120575) 120582

3+ (1205731+ 1205732minus 1205721120575) 1205822

+ (12057211205732minus 1205751205731) 120582 + 120573

11205732= 0

(6)

where 120575 = 1205901198963minus 1205722 Since 120572

119894 120573119894 120574119894gt 0 119894 = 1 2 it is easy to

verify that (5) has a stable origin if 120575 lt 0 and an unstable oneif 120575 gt 0 When 120575 = 0 (6) has a pair of purely imaginary roots12058212= plusmn119894radic120573

2 Regarding 120582 as a function of 120575 in (6) and taking

the derivative of 120582 with respect to 120575 one has

119889120582

119889120575

=

120582 (1205822+ 1205721120582 + 1205731)

41205823 minus 3 (120575 minus 1205721) 1205822 minus 2 (120575120572

1minus 1205731minus 1205732) 120582 minus 120573

1120575 + 12057211205732

(7)

Substituting 120582 = plusmn119894radic1205732and 120575 = 0 into (7) yields

(119889120582

119889120575)

120575=0

=1

2gt 0 (8)

According to Hopf bifurcation theory system (5) under-goes a Hopf bifurcation near 120575 = 0 and a limit cycle appearswhen 120575 gt 0 This means that periodic vibrations occur in thecable-stayed bridgemodel under crowd excitation To furtherdetermine the stability of the limit cycle we employ centermanifold theory to analyze system (5)

If 120575 = 0 (6) has a pair of purely imaginary roots and twonegative real roots which satisfies the condition for centermanifold theory Assume that a local center manifold can beexpressed by

1199101= ℎ1(1199103 1199104)

1199102= ℎ2(1199103 1199104)

(9)

where ℎ1and ℎ2are vector functions of nonlinear terms of 119910

3

and 1199104 Assume that 119910

1and 1199102can be transformed into power

series

1199101= ℎ101199103

3+ ℎ111199102

31199104+ ℎ1211991031199102

4+ ℎ131199103

4

+ 119874 (1199103 1199104)

1199102= ℎ201199103

3+ ℎ211199102

31199104+ ℎ2211991031199102

4+ ℎ231199103

4

+ 119874 (1199103 1199104)

(10)

where ℎ1119894and ℎ

2119894(119894 = 0 1 2 3 ) are coefficients to be

determined 119874(sdot) represents the higher orders of 11991034 Substi-

tuting (10) into (5) and setting the coefficients of each powerof11991034

equal to zerowould yield the following set of equations

ℎ10=1

119860[(712057311205732minus 91205732

2) 12057411205722

1

+ 1205741(1205733

1minus 17120573

2

11205732+ 79120573

11205732

2minus 63120573

3

2)]

ℎ11=1

119860(312057321205722

1+ 1205732

1minus 612057311205732+ 21120573

2

2)

ℎ12=61205741

119860(12057311205722

1minus 1205732

1+ 10120573

11205732minus 91205732

2)

ℎ13= minus

612057411205721

119860(1205722

1minus 21205731+ 10120573

2)

(11)

where

119860 = 91205732

21205724

1+ 2 (minus18120573

11205732

2+ 10120573

2

11205732+ 90120573

3

2) 1205722

11205734

minus 201205733

11205732+ 118120573

2

11205732

2minus 180120573

11205733

2+ 81120573

4

2

(12)

Then the control equations on the center manifold are givenby

1199103= 1199104

1199104= minus12057321199103minus 12057221199104+ 120590 tanh 119896

31199104

minus 1205742[ℎ1(1199103 1199104) minus 1199103]3

(13)

By letting

119906 = 1199103

V = minus1

radic1205732

1199104

(14)

(13) can be rewritten as

= minusradic1205732V

V = radic1205732119906 + 119876

(15)

where

119876 = minus1205742

radic1205732

1199063minus1

312057321205903V3 + 119874 (1199063 V3) (16)

4 Mathematical Problems in Engineering

When 120575 = 1205901198963minus 1205722= 0 the first Lyapunov coefficient of

system (13) can be calculated

1198971=1

16(1205973119865

1205971199062120597V+1205973119865

120597V3)

minus1

16radic1205732

[1205972119865

120597119906120597V(1205972119865

1205971199062+1205972119865

120597V2)]

= minus1

81205732(1205722

1198963

)

3

lt 0

(17)

The first Lyapunov coefficient is less than zero which meansthat original system (5) undergoes a Hopf bifurcation near120575 = 0 and a stable limit cycle occurs To discuss the dynamicresponse of cables and its influence on the deck in greaterdetail we will analytically calculate the periodic solution insystem (3) in next section

4 Quantitative Analysis for the Cables-StayedBridge Model

In this section we use the energy method [18] to calculate theperiodic solution of (3)

(1) Denote

1198921(1199091) = 12057311199091+ 212057411199093

1

1198922(1199092) = 12057321199092minus 12057421199093

2

1198911(119909 ) = 120572

11+ 1205741(311990911199092

2minus 31199092

11199092minus 1199093

2)

1198912(119909 ) = 120572

22minus 120590 tanh (119896

32)

+ 1205742(1199093

1minus 31199092

11199092+ 311990911199092

2)

(18)

Considering the physics meanings of parameters in (3) andassuming that the vibration amplitudes are not too large(|11990912| lt 1119898) one has

11990911198921(1199091)1199091=0= 12057311199092

1+ 212057411199094

1gt 0

11990921198922(1199092)1199092=0= 12057321199092

2minus 12057421199094

2gt 0

(19)

Therefore the energy method can be applied to (3) to cal-culate its approximation periodic solution

(2) The potential energy functions of system (3) can beexpressed by

1198811(1199091) = int

1199091

0

(12057311199091+ 212057411199093

1) 1198891199091

=1

2(12057311199092

1+ 12057411199094

1)

1198812(1199092) = int

1199092

0

(12057321199092minus 12057421199093

2) 1198891199092

=1

4(212057321199092

2minus 12057421199094

2)

(20)

119881119894(119909) 119894 = 1 2 are even functions thus 119887

12= 0 hold [18] The

coordinate change formulas can be written as

119909119894= 119886119894cos (120579

119894) (119894 = 1 2)

1= plusmnradic2 (119881

1(1198861) minus 1198811(1198861cos (120579

1)))

= plusmn1198861

1003816100381610038161003816sin (1205791)1003816100381610038161003816radic1205731+3

212057411198862

1+1

212057411198862

1cos (2120579

1)

2= plusmnradic2 (119881

2(1198862) minus 1198812(1198862cos (120579

2)))

= plusmn1198862

1003816100381610038161003816sin (1205792)1003816100381610038161003816radic1205732minus3

412057421198862

2minus1

412057421198862

2cos (2120579

2)

(21)

Ignoring the terms of the harmonic expansions higher thanthe second the coordinate change formulas can be simplifiedas

1= minus1198861sin (1205791) 1198601(1 + 119860

2cos (2120579

1))

2= minus1198862sin (1205792) 1198611(1 + 119861

2cos (2120579

2))

(22)

where

1198601= radic1205731+3

212057411198862

1gt 0

0 lt 1198602=

12057411198862

1

41205731+ 612057411198862

1

lt 1

1198611= radic1205732minus3

412057421198862

2gt 0

minus1 lt 1198612= minus

12057421198862

2

81205732minus 612057421198862

2

lt 0

(23)

(3) Denote the energy of 1198981and 119898

2by 1198641and 119864

2

respectively 1198891198641119889119905 and 119889119864

2119889119905 can be calculated in which

the terms of the harmonic expansions higher than the secondwill be ignored

1198891198641

119889119905= minus1198911(119909 )

1

= 1198621+ 1198622cos (2120579

1)

+ 1198623[sin (120579

1minus 1205792) + sin (120579

1+ 1205792)]

+ 1198624sin (2120579

1minus 21205792) + 1198625sin (3120579

1minus 31205792)

+ 1198626sin (2120579

1) + 1198627sin (3120579

1minus 1205792)

+ 1198628sin (1205791minus 31205792) + 1198629sin (4120579

1minus 21205792)

≜ 1198651(119864 120579)

Mathematical Problems in Engineering 5

1198891198642

119889119905= minus1198912(119909 )

2

= 1198631+ 1198632cos (2120579

2)

+ 1198633[sin (120579

1minus 1205792) minus sin (120579

1+ 1205792)]

+ 1198634sin (2120579

1minus 21205792) + 1198635sin (3120579

1minus 31205792)

+ 1198636sin (2120579

2) + 1198637sin (3120579

1minus 1205792)

+ 1198638sin (1205791minus 31205792) + 1198639sin (2120579

1minus 41205792)

≜ 1198652(119864 120579)

(24)

where

1198621=1

4(21198602minus 2 minus 119860

2

2)1198602

11198862

11205721

1198622=1

8(4 + 3119860

2

2minus 81198602)1198602

112057211198862

1

1198623= minus

3

812057411198863

111986011198862+3

161205741(1198602minus 2)119860

11198863

21198861

1198624=3

812057411198862

111986011198862

2

1198625= minus

1

161205741119886111986011198863

21198602

1198626=3

412057411198862

111986011198862

2

1198627= minus

3

161205741(1198602+ 2)119860

111988621198863

1minus3

161205741119886111986011198863

21198602

1198628=1

161205741(1198602minus 2)119860

11198863

21198861

1198629=3

1612057411198862

111986011198862

21198602

1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

1198632=1

12(2 minus 7119861

2) 1198963

31198614

11198864

2120590 +

1

8(81198612minus 4) 119886

2

21198612

1(1198963120590

minus 1205722)

1198633=1

16((minus6 + 3119861

2) 11988621198863

1minus 61198863

21198861) 12057421198611

1198634=3

812057421198862

211986111198862

1

1198635= minus

1

161205742119886211986111198863

11198612

1198636= minus

3

412057421198862

211986111198862

1

1198637=1

16(1198612minus 2) 119886

21198863

112057421198611

1198638= minus

1

16(311988621198863

11198612+ (6 + 3119861

2) 1198863

21198861) 12057421198611

1198639=3

1612057421198862

211986111198862

11198612

(25)

(4) 1198891205791119889119905 and 119889120579

2119889119905 are given by

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)[1198671(1 + cos (2120579

2))

+ 1198672sin (2120579

1) + 1198673cos (2120579

1)

+ 1198674(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198675cos (2120579

1minus 21205792) + 1198676cos (3120579

1minus 31205792)

+ 1198677cos (120579

1minus 31205792) + 1198678cos (3120579

1minus 1205792)

+ 1198679cos (4120579

1minus 21205792)] + 119860

1(1 + 119860

2cos (2120579

1))

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)[1198691(1 + cos (2120579

1))

+ 1198692sin (2120579

2) + 1198693cos (2120579

2)

+ 1198694(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198695cos (2120579

1minus 21205792) + 1198696cos (3120579

1minus 31205792)

+ 1198697cos (120579

1minus 31205792) + 1198698cos (3120579

1minus 1205792)

+ 1198699cos (2120579

1minus 41205792)] + 119861

1(1 + 119861

2cos (2120579

2))

(26)

where

1198671=3

8(2 + 119860

2) 119860112057411198862

21198862

1

1198672= minus

1

8(4 + 119860

2

2)1198602

11198862

11205721

1198673= minus

3

4(1 + 119860

2) 11986011198862

11198862

21205741

1198674= minus

3

16((2 + 119860

2) 11988611198863

2+ (6 + 4119860

2) 1198863

11198862)11986011205741

1198675=3

8(1 + 119860

2) 11986011198862

11198862

21205741

1198676= minus

1

161205741119886111986011198863

21198602

1198677= minus

1

16(2 + 119860

2) 11988611198863

211986011205741

1198678= minus

3

16(11988611198863

21198602+ (2 + 3119860

2) 1198863

11198862)11986011205741

1198679=3

1612057411198862

111986011198862

21198602

6 Mathematical Problems in Engineering

1198691= minus

3

8(2 + 119861

2) 1198862

11198862

211986111205742

1198692=1

12(1198612minus 1) 119886

4

21198611

41198963

3+ (1

21198963120590 minus 48120572

2) 1198862

21198612

1

1198693= minus

3

4(1198612+ 1) 119861

11198862

21198862

11205742

1198694=3

16((2 + 119861

2) 11988621198863

1+ (4119861

2+ 6) 119886

3

21198861) 12057421198611

1198695= minus

3

8(1198612+ 1) 119861

11198862

21198862

11205742

1198696=1

161205742119886211986111198863

11198612

1198697=3

16(11988621198863

11198612+ (3119861

2+ 2) 119886

3

21198861) 12057421198611

1198698=1

16(2 + 119861

2) 11988621198863

112057421198611

1198699= minus

3

1612057421198862

211986111198862

11198612

(27)

(5) Letting

1205791= 120596119905 + 120601

1

1205792= 120596119905 + 120601

2

(28)

and substituting them into (24) (26) yields

1198891198641

119889119905= 1198621+ 1198623sin (120601

1minus 1206012) + 1198624sin (2120601

1minus 21206012)

+ 1198625sin (3120601

1minus 31206012) + [119862

2cos (2120601

1) + 1198623

sdot sin (1206011+ 1206012) + 1198626sin (2120601

1) + 1198627sin (3120601

1minus 1206012)

+ 1198628sin (120601

1minus 31206012) + 1198629sin (4120601

1minus 21206012)] cos (2120596119905)

+ [minus1198622sin (2120601

1) + 1198623cos (120601

1+ 1206012) + 1198626cos (2120601

1)

+ 1198627cos (3120601

1minus 1206012) minus 1198628cos (120601

1minus 31206012) + 1198629

sdot cos (41206011minus 21206012)] sin (2120596119905)

1198891198642

119889119905= 1198631+ 1198633sin (120601

1minus 1206012) + 1198634sin (2120601

1minus 21206012)

+ 1198635sin (3120601

1minus 31206012) + [119863

2cos (2120601

1) minus 1198633

sdot sin (1206011+ 1206012) + 1198636sin (2120601

2) + 1198637sin (3120601

1minus 1206012)

+ 1198638sin (120601

1minus 31206012) + 1198639sin (2120601

1minus 41206012)]

sdot cos (2120596119905) + [minus1198632sin (2120601

1) minus 1198633cos (120601

1+ 1206012)

+ 1198636cos (2120601

2) + 1198637cos (3120601

1minus 1206012) minus 1198638

sdot cos (1206011minus 31206012) minus 1198639cos (2120601

1minus 41206012)] sin (2120596119905)

(29)

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)1198862

1(1205731+ 212057411198862

1)1198601+ 1198671

+ 1198674cos (120601

1minus 1206012) + 1198675cos (2120601

1minus 21206012) + 1198676

sdot cos (31206011minus 31206012) + [119867

1cos (2120601

2) + 1198672sin (2120601

1)

+ (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) cos (2120601

1)

+ 1198674cos (120601

1+ 1206012) + 1198677cos (120601

1minus 31206012)

+ 1198678cos (3120601

1minus 1206012) + 1198679cos (4120601

1minus 21206012)]

sdot cos (2120596119905) + [minus1198671sin (2120601

2) + 1198672cos (2120601

1)

minus (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) sin (2120601

1)

minus 1198674sin (120601

1+ 1206012) + 1198677sin (120601

1minus 31206012)

minus 1198678sin (3120601

1minus 1206012) minus 1198679sin (4120601

1minus 21206012)]

sdot sin (2120596119905)

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)1198862

2(1205732minus 12057421198862

2) 1198611+ 1198691+ 1198694

sdot cos (1206011minus 1206012) + 1198695cos (2120601

1minus 21206012) + 1198696

sdot cos (31206011minus 31206012) + [1198691cos (2120601

1) + 1198692sin (2120601

2)

+ (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) cos (2120601

2)

+ 1198694cos (120601

1+ 1206012) + 1198697cos (120601

1minus 31206012)

+ 1198698cos (3120601

1minus 1206012) + 1198699cos (2120601

1minus 41206012)] cos (2120596119905)

+ [minus1198691sin (2120601

1) + 1198692cos (2120601

2)

minus (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) sin (2120601

2)

minus 1198694sin (120601

1+ 1206012) + 1198697sin (120601

1minus 31206012)

minus 1198698sin (3120601

1minus 1206012) minus 1198699sin (2120601

1minus 41206012)]

sdot sin (2120596119905)

(30)

If there exists a periodic solution with period 2120587120596 in (3)then the energy of the system should remain the same at ini-tial and finial positions in one period Integrating both sidesof (29) in one period (from 119905 = Δ to 119905 = Δ + 2120587120596) leads to

(1198641)2120587120596+Δ

minus (1198641)Δ= 1198621= minus

1

4[(1198602minus 1)2

minus 1]

sdot 1198602

11198862

11205721lt 0

(1198642)2120587120596+Δ

minus (1198642)Δ= 1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

(31)

Mathematical Problems in Engineering 7

minus004

minus002

0

002

004

minus01

minus005

0

005

01

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

x1

(m)

x2

(m)

(a)

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

minus004

minus002

0

002

004

minus01

minus005

0

005

01

x1

(m)

x2

(m)

(b)

Figure 2 Direct numerical simulations of (3) for (a) 119899 = 112 and (b) 119899 = 118

Inspection of the first equation in (31) shows that the vibra-tion energy of cables decreases in one period Thereforecables cannot vibrate periodically when the deck is subjectedto the crowd excitation From the second equation in (31)if 1198963120590 minus 1205722lt 0 then 119863

1lt 0 At this point the deck has an

attenuate vibration If 1198963120590minus1205722gt 0119863

1= 0may hold Letting

1198631= 0 in the second equation in (31) and ignoring the sixth

or more power of 1198862(since 119886

2≪ 1 according to its physical

meaning) one obtains

712057421198864

212057321205901198963

3minus 3 (5120575120574

2+ 21205732

21205901198963

3) 1198862

2+ 24120575120573

2= 0 (32)

where 120575 = 1198963120590 minus 1205722 Solving (32) yields

119886lowast

2

=radic61205901198963

31205732

2+ 151205742120575 minus radic12120590119896

3

31205732

2(31205901198963

31205732

2minus 411205742120575) + 225120574

2

21205752

1412059012057421198963

31205732

= 2radic2radic1205732120575

21205901198963

31205732

2+ 51205742120575

(33)

For clarification the amplitude can be rewritten as

119886lowast

2= 4radic

1198982119896 (1198963120574 minus 1198882)

41205741198963

31198962 + 5 (11986411986021198713)119898

2(1198963120574 minus 1198882) (34)

where 120574 = 11989611198962119866(119891)119872

119901119892 and 119896

3120574 minus 1198882gt 0

(6) The first-order approximation solution of (3) isexpressed by

1199092(119905) = 119886

lowast

2cos (120596119905)

= 119886lowast

2cos(radic 1

1198982

[119896 minus3

8

119864119860

1198713(119886lowast

2)2

]119905)

(35)

where 119886lowast2is given in (34)

5 Numerical Simulations and Discussion

In this section we will verify the validity of our analyticalsolutions obtained in the last section by comparing theresults based on (35) with that derived by direct numericalsimulations for (3) The following parameters of deck andpedestrians are taken to calculate the periodic solutions in (3)

1198982= 21401 times 10

5(kg)

119896 = 7307361 times 106(kgs2)

1198882= 28262 times 10

4(kgs)

119866 (119891) = 10

1198961= 00987

1198962= 02

1198963= 18

(36)

In fact above values of parameters are used in [17 19] forthe first lateral model of the T-bridge in Japan Furthermorethe weight of a single person on the bridge is assumed to be70 (kg) the gravity acceleration 119892 = 98 (ms2) The modalmass of pedestrians119872

119901119892 can be given by119872

119901119892 = 119899times70times98

where 119899 is the number of the pedestrians on the bridgeAdditionally according to [19] the average length of thecables is taken as 119871 = 60 (m) However other parameters ofthe cables such as stiffness damping coefficient and massare not provided in [19] In order to compare the theoreticalresults of T-bridge between our cable-stayed bridge modeland Nakamurarsquos model following parameters of cables areassumed in the numerical simulations

1198981= 21401 times 10

3(kg)

119864119860 = 168 times 1010(N)

8 Mathematical Problems in Engineering

9955 996 9965 997 9975 998 9985 999 9995 1000

Numerical resultsAnalytical results

t (s)

minus3

minus2

minus1

0

1

2

3

4times10minus3

x2

(m)

(a)

9955 996 9965 997 9975 998 9985 999 9995 1000minus6

minus4

minus2

0

2

4

6

8

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(b)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(c)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3x2

(m)

(d)

Figure 3 Comparison between the first-order approximation solution equation (35) and direct numerical results of (3) for the cases of (a)119899 = 118 (b) 119899 = 125 (c) 119899 = 130 and (d) 119899 = 135

1198881= 28262 times 10

2(kgs)

1198790= 724 times 10

5(N)

(37)

Then we have

1205721= 01321 (1s)

1205722= 01321 (1s)

1205731= 676604 (1s2)

1205732= 341450 (1s2)

1205741= 39251 times 10

3(1s2)

1205742= 392505 (1s2)

(38)

Regarding the number 119899 as the bifurcation parameter one has

120590 =

11989611198962119866 (119891)119872

119901119892

1198982

= 0000063275734785119899

120575 = 1198963120590 minus 1205722= 000113896119899 minus 013201

(39)

From the analysis in Section 3 120575 lt 0 means the attenuationof the lateral vibration of the bridges while 120575 gt 0 means the

Mathematical Problems in Engineering 9

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

4 Mathematical Problems in Engineering

When 120575 = 1205901198963minus 1205722= 0 the first Lyapunov coefficient of

system (13) can be calculated

1198971=1

16(1205973119865

1205971199062120597V+1205973119865

120597V3)

minus1

16radic1205732

[1205972119865

120597119906120597V(1205972119865

1205971199062+1205972119865

120597V2)]

= minus1

81205732(1205722

1198963

)

3

lt 0

(17)

The first Lyapunov coefficient is less than zero which meansthat original system (5) undergoes a Hopf bifurcation near120575 = 0 and a stable limit cycle occurs To discuss the dynamicresponse of cables and its influence on the deck in greaterdetail we will analytically calculate the periodic solution insystem (3) in next section

4 Quantitative Analysis for the Cables-StayedBridge Model

In this section we use the energy method [18] to calculate theperiodic solution of (3)

(1) Denote

1198921(1199091) = 12057311199091+ 212057411199093

1

1198922(1199092) = 12057321199092minus 12057421199093

2

1198911(119909 ) = 120572

11+ 1205741(311990911199092

2minus 31199092

11199092minus 1199093

2)

1198912(119909 ) = 120572

22minus 120590 tanh (119896

32)

+ 1205742(1199093

1minus 31199092

11199092+ 311990911199092

2)

(18)

Considering the physics meanings of parameters in (3) andassuming that the vibration amplitudes are not too large(|11990912| lt 1119898) one has

11990911198921(1199091)1199091=0= 12057311199092

1+ 212057411199094

1gt 0

11990921198922(1199092)1199092=0= 12057321199092

2minus 12057421199094

2gt 0

(19)

Therefore the energy method can be applied to (3) to cal-culate its approximation periodic solution

(2) The potential energy functions of system (3) can beexpressed by

1198811(1199091) = int

1199091

0

(12057311199091+ 212057411199093

1) 1198891199091

=1

2(12057311199092

1+ 12057411199094

1)

1198812(1199092) = int

1199092

0

(12057321199092minus 12057421199093

2) 1198891199092

=1

4(212057321199092

2minus 12057421199094

2)

(20)

119881119894(119909) 119894 = 1 2 are even functions thus 119887

12= 0 hold [18] The

coordinate change formulas can be written as

119909119894= 119886119894cos (120579

119894) (119894 = 1 2)

1= plusmnradic2 (119881

1(1198861) minus 1198811(1198861cos (120579

1)))

= plusmn1198861

1003816100381610038161003816sin (1205791)1003816100381610038161003816radic1205731+3

212057411198862

1+1

212057411198862

1cos (2120579

1)

2= plusmnradic2 (119881

2(1198862) minus 1198812(1198862cos (120579

2)))

= plusmn1198862

1003816100381610038161003816sin (1205792)1003816100381610038161003816radic1205732minus3

412057421198862

2minus1

412057421198862

2cos (2120579

2)

(21)

Ignoring the terms of the harmonic expansions higher thanthe second the coordinate change formulas can be simplifiedas

1= minus1198861sin (1205791) 1198601(1 + 119860

2cos (2120579

1))

2= minus1198862sin (1205792) 1198611(1 + 119861

2cos (2120579

2))

(22)

where

1198601= radic1205731+3

212057411198862

1gt 0

0 lt 1198602=

12057411198862

1

41205731+ 612057411198862

1

lt 1

1198611= radic1205732minus3

412057421198862

2gt 0

minus1 lt 1198612= minus

12057421198862

2

81205732minus 612057421198862

2

lt 0

(23)

(3) Denote the energy of 1198981and 119898

2by 1198641and 119864

2

respectively 1198891198641119889119905 and 119889119864

2119889119905 can be calculated in which

the terms of the harmonic expansions higher than the secondwill be ignored

1198891198641

119889119905= minus1198911(119909 )

1

= 1198621+ 1198622cos (2120579

1)

+ 1198623[sin (120579

1minus 1205792) + sin (120579

1+ 1205792)]

+ 1198624sin (2120579

1minus 21205792) + 1198625sin (3120579

1minus 31205792)

+ 1198626sin (2120579

1) + 1198627sin (3120579

1minus 1205792)

+ 1198628sin (1205791minus 31205792) + 1198629sin (4120579

1minus 21205792)

≜ 1198651(119864 120579)

Mathematical Problems in Engineering 5

1198891198642

119889119905= minus1198912(119909 )

2

= 1198631+ 1198632cos (2120579

2)

+ 1198633[sin (120579

1minus 1205792) minus sin (120579

1+ 1205792)]

+ 1198634sin (2120579

1minus 21205792) + 1198635sin (3120579

1minus 31205792)

+ 1198636sin (2120579

2) + 1198637sin (3120579

1minus 1205792)

+ 1198638sin (1205791minus 31205792) + 1198639sin (2120579

1minus 41205792)

≜ 1198652(119864 120579)

(24)

where

1198621=1

4(21198602minus 2 minus 119860

2

2)1198602

11198862

11205721

1198622=1

8(4 + 3119860

2

2minus 81198602)1198602

112057211198862

1

1198623= minus

3

812057411198863

111986011198862+3

161205741(1198602minus 2)119860

11198863

21198861

1198624=3

812057411198862

111986011198862

2

1198625= minus

1

161205741119886111986011198863

21198602

1198626=3

412057411198862

111986011198862

2

1198627= minus

3

161205741(1198602+ 2)119860

111988621198863

1minus3

161205741119886111986011198863

21198602

1198628=1

161205741(1198602minus 2)119860

11198863

21198861

1198629=3

1612057411198862

111986011198862

21198602

1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

1198632=1

12(2 minus 7119861

2) 1198963

31198614

11198864

2120590 +

1

8(81198612minus 4) 119886

2

21198612

1(1198963120590

minus 1205722)

1198633=1

16((minus6 + 3119861

2) 11988621198863

1minus 61198863

21198861) 12057421198611

1198634=3

812057421198862

211986111198862

1

1198635= minus

1

161205742119886211986111198863

11198612

1198636= minus

3

412057421198862

211986111198862

1

1198637=1

16(1198612minus 2) 119886

21198863

112057421198611

1198638= minus

1

16(311988621198863

11198612+ (6 + 3119861

2) 1198863

21198861) 12057421198611

1198639=3

1612057421198862

211986111198862

11198612

(25)

(4) 1198891205791119889119905 and 119889120579

2119889119905 are given by

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)[1198671(1 + cos (2120579

2))

+ 1198672sin (2120579

1) + 1198673cos (2120579

1)

+ 1198674(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198675cos (2120579

1minus 21205792) + 1198676cos (3120579

1minus 31205792)

+ 1198677cos (120579

1minus 31205792) + 1198678cos (3120579

1minus 1205792)

+ 1198679cos (4120579

1minus 21205792)] + 119860

1(1 + 119860

2cos (2120579

1))

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)[1198691(1 + cos (2120579

1))

+ 1198692sin (2120579

2) + 1198693cos (2120579

2)

+ 1198694(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198695cos (2120579

1minus 21205792) + 1198696cos (3120579

1minus 31205792)

+ 1198697cos (120579

1minus 31205792) + 1198698cos (3120579

1minus 1205792)

+ 1198699cos (2120579

1minus 41205792)] + 119861

1(1 + 119861

2cos (2120579

2))

(26)

where

1198671=3

8(2 + 119860

2) 119860112057411198862

21198862

1

1198672= minus

1

8(4 + 119860

2

2)1198602

11198862

11205721

1198673= minus

3

4(1 + 119860

2) 11986011198862

11198862

21205741

1198674= minus

3

16((2 + 119860

2) 11988611198863

2+ (6 + 4119860

2) 1198863

11198862)11986011205741

1198675=3

8(1 + 119860

2) 11986011198862

11198862

21205741

1198676= minus

1

161205741119886111986011198863

21198602

1198677= minus

1

16(2 + 119860

2) 11988611198863

211986011205741

1198678= minus

3

16(11988611198863

21198602+ (2 + 3119860

2) 1198863

11198862)11986011205741

1198679=3

1612057411198862

111986011198862

21198602

6 Mathematical Problems in Engineering

1198691= minus

3

8(2 + 119861

2) 1198862

11198862

211986111205742

1198692=1

12(1198612minus 1) 119886

4

21198611

41198963

3+ (1

21198963120590 minus 48120572

2) 1198862

21198612

1

1198693= minus

3

4(1198612+ 1) 119861

11198862

21198862

11205742

1198694=3

16((2 + 119861

2) 11988621198863

1+ (4119861

2+ 6) 119886

3

21198861) 12057421198611

1198695= minus

3

8(1198612+ 1) 119861

11198862

21198862

11205742

1198696=1

161205742119886211986111198863

11198612

1198697=3

16(11988621198863

11198612+ (3119861

2+ 2) 119886

3

21198861) 12057421198611

1198698=1

16(2 + 119861

2) 11988621198863

112057421198611

1198699= minus

3

1612057421198862

211986111198862

11198612

(27)

(5) Letting

1205791= 120596119905 + 120601

1

1205792= 120596119905 + 120601

2

(28)

and substituting them into (24) (26) yields

1198891198641

119889119905= 1198621+ 1198623sin (120601

1minus 1206012) + 1198624sin (2120601

1minus 21206012)

+ 1198625sin (3120601

1minus 31206012) + [119862

2cos (2120601

1) + 1198623

sdot sin (1206011+ 1206012) + 1198626sin (2120601

1) + 1198627sin (3120601

1minus 1206012)

+ 1198628sin (120601

1minus 31206012) + 1198629sin (4120601

1minus 21206012)] cos (2120596119905)

+ [minus1198622sin (2120601

1) + 1198623cos (120601

1+ 1206012) + 1198626cos (2120601

1)

+ 1198627cos (3120601

1minus 1206012) minus 1198628cos (120601

1minus 31206012) + 1198629

sdot cos (41206011minus 21206012)] sin (2120596119905)

1198891198642

119889119905= 1198631+ 1198633sin (120601

1minus 1206012) + 1198634sin (2120601

1minus 21206012)

+ 1198635sin (3120601

1minus 31206012) + [119863

2cos (2120601

1) minus 1198633

sdot sin (1206011+ 1206012) + 1198636sin (2120601

2) + 1198637sin (3120601

1minus 1206012)

+ 1198638sin (120601

1minus 31206012) + 1198639sin (2120601

1minus 41206012)]

sdot cos (2120596119905) + [minus1198632sin (2120601

1) minus 1198633cos (120601

1+ 1206012)

+ 1198636cos (2120601

2) + 1198637cos (3120601

1minus 1206012) minus 1198638

sdot cos (1206011minus 31206012) minus 1198639cos (2120601

1minus 41206012)] sin (2120596119905)

(29)

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)1198862

1(1205731+ 212057411198862

1)1198601+ 1198671

+ 1198674cos (120601

1minus 1206012) + 1198675cos (2120601

1minus 21206012) + 1198676

sdot cos (31206011minus 31206012) + [119867

1cos (2120601

2) + 1198672sin (2120601

1)

+ (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) cos (2120601

1)

+ 1198674cos (120601

1+ 1206012) + 1198677cos (120601

1minus 31206012)

+ 1198678cos (3120601

1minus 1206012) + 1198679cos (4120601

1minus 21206012)]

sdot cos (2120596119905) + [minus1198671sin (2120601

2) + 1198672cos (2120601

1)

minus (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) sin (2120601

1)

minus 1198674sin (120601

1+ 1206012) + 1198677sin (120601

1minus 31206012)

minus 1198678sin (3120601

1minus 1206012) minus 1198679sin (4120601

1minus 21206012)]

sdot sin (2120596119905)

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)1198862

2(1205732minus 12057421198862

2) 1198611+ 1198691+ 1198694

sdot cos (1206011minus 1206012) + 1198695cos (2120601

1minus 21206012) + 1198696

sdot cos (31206011minus 31206012) + [1198691cos (2120601

1) + 1198692sin (2120601

2)

+ (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) cos (2120601

2)

+ 1198694cos (120601

1+ 1206012) + 1198697cos (120601

1minus 31206012)

+ 1198698cos (3120601

1minus 1206012) + 1198699cos (2120601

1minus 41206012)] cos (2120596119905)

+ [minus1198691sin (2120601

1) + 1198692cos (2120601

2)

minus (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) sin (2120601

2)

minus 1198694sin (120601

1+ 1206012) + 1198697sin (120601

1minus 31206012)

minus 1198698sin (3120601

1minus 1206012) minus 1198699sin (2120601

1minus 41206012)]

sdot sin (2120596119905)

(30)

If there exists a periodic solution with period 2120587120596 in (3)then the energy of the system should remain the same at ini-tial and finial positions in one period Integrating both sidesof (29) in one period (from 119905 = Δ to 119905 = Δ + 2120587120596) leads to

(1198641)2120587120596+Δ

minus (1198641)Δ= 1198621= minus

1

4[(1198602minus 1)2

minus 1]

sdot 1198602

11198862

11205721lt 0

(1198642)2120587120596+Δ

minus (1198642)Δ= 1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

(31)

Mathematical Problems in Engineering 7

minus004

minus002

0

002

004

minus01

minus005

0

005

01

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

x1

(m)

x2

(m)

(a)

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

minus004

minus002

0

002

004

minus01

minus005

0

005

01

x1

(m)

x2

(m)

(b)

Figure 2 Direct numerical simulations of (3) for (a) 119899 = 112 and (b) 119899 = 118

Inspection of the first equation in (31) shows that the vibra-tion energy of cables decreases in one period Thereforecables cannot vibrate periodically when the deck is subjectedto the crowd excitation From the second equation in (31)if 1198963120590 minus 1205722lt 0 then 119863

1lt 0 At this point the deck has an

attenuate vibration If 1198963120590minus1205722gt 0119863

1= 0may hold Letting

1198631= 0 in the second equation in (31) and ignoring the sixth

or more power of 1198862(since 119886

2≪ 1 according to its physical

meaning) one obtains

712057421198864

212057321205901198963

3minus 3 (5120575120574

2+ 21205732

21205901198963

3) 1198862

2+ 24120575120573

2= 0 (32)

where 120575 = 1198963120590 minus 1205722 Solving (32) yields

119886lowast

2

=radic61205901198963

31205732

2+ 151205742120575 minus radic12120590119896

3

31205732

2(31205901198963

31205732

2minus 411205742120575) + 225120574

2

21205752

1412059012057421198963

31205732

= 2radic2radic1205732120575

21205901198963

31205732

2+ 51205742120575

(33)

For clarification the amplitude can be rewritten as

119886lowast

2= 4radic

1198982119896 (1198963120574 minus 1198882)

41205741198963

31198962 + 5 (11986411986021198713)119898

2(1198963120574 minus 1198882) (34)

where 120574 = 11989611198962119866(119891)119872

119901119892 and 119896

3120574 minus 1198882gt 0

(6) The first-order approximation solution of (3) isexpressed by

1199092(119905) = 119886

lowast

2cos (120596119905)

= 119886lowast

2cos(radic 1

1198982

[119896 minus3

8

119864119860

1198713(119886lowast

2)2

]119905)

(35)

where 119886lowast2is given in (34)

5 Numerical Simulations and Discussion

In this section we will verify the validity of our analyticalsolutions obtained in the last section by comparing theresults based on (35) with that derived by direct numericalsimulations for (3) The following parameters of deck andpedestrians are taken to calculate the periodic solutions in (3)

1198982= 21401 times 10

5(kg)

119896 = 7307361 times 106(kgs2)

1198882= 28262 times 10

4(kgs)

119866 (119891) = 10

1198961= 00987

1198962= 02

1198963= 18

(36)

In fact above values of parameters are used in [17 19] forthe first lateral model of the T-bridge in Japan Furthermorethe weight of a single person on the bridge is assumed to be70 (kg) the gravity acceleration 119892 = 98 (ms2) The modalmass of pedestrians119872

119901119892 can be given by119872

119901119892 = 119899times70times98

where 119899 is the number of the pedestrians on the bridgeAdditionally according to [19] the average length of thecables is taken as 119871 = 60 (m) However other parameters ofthe cables such as stiffness damping coefficient and massare not provided in [19] In order to compare the theoreticalresults of T-bridge between our cable-stayed bridge modeland Nakamurarsquos model following parameters of cables areassumed in the numerical simulations

1198981= 21401 times 10

3(kg)

119864119860 = 168 times 1010(N)

8 Mathematical Problems in Engineering

9955 996 9965 997 9975 998 9985 999 9995 1000

Numerical resultsAnalytical results

t (s)

minus3

minus2

minus1

0

1

2

3

4times10minus3

x2

(m)

(a)

9955 996 9965 997 9975 998 9985 999 9995 1000minus6

minus4

minus2

0

2

4

6

8

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(b)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(c)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3x2

(m)

(d)

Figure 3 Comparison between the first-order approximation solution equation (35) and direct numerical results of (3) for the cases of (a)119899 = 118 (b) 119899 = 125 (c) 119899 = 130 and (d) 119899 = 135

1198881= 28262 times 10

2(kgs)

1198790= 724 times 10

5(N)

(37)

Then we have

1205721= 01321 (1s)

1205722= 01321 (1s)

1205731= 676604 (1s2)

1205732= 341450 (1s2)

1205741= 39251 times 10

3(1s2)

1205742= 392505 (1s2)

(38)

Regarding the number 119899 as the bifurcation parameter one has

120590 =

11989611198962119866 (119891)119872

119901119892

1198982

= 0000063275734785119899

120575 = 1198963120590 minus 1205722= 000113896119899 minus 013201

(39)

From the analysis in Section 3 120575 lt 0 means the attenuationof the lateral vibration of the bridges while 120575 gt 0 means the

Mathematical Problems in Engineering 9

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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 5

1198891198642

119889119905= minus1198912(119909 )

2

= 1198631+ 1198632cos (2120579

2)

+ 1198633[sin (120579

1minus 1205792) minus sin (120579

1+ 1205792)]

+ 1198634sin (2120579

1minus 21205792) + 1198635sin (3120579

1minus 31205792)

+ 1198636sin (2120579

2) + 1198637sin (3120579

1minus 1205792)

+ 1198638sin (1205791minus 31205792) + 1198639sin (2120579

1minus 41205792)

≜ 1198652(119864 120579)

(24)

where

1198621=1

4(21198602minus 2 minus 119860

2

2)1198602

11198862

11205721

1198622=1

8(4 + 3119860

2

2minus 81198602)1198602

112057211198862

1

1198623= minus

3

812057411198863

111986011198862+3

161205741(1198602minus 2)119860

11198863

21198861

1198624=3

812057411198862

111986011198862

2

1198625= minus

1

161205741119886111986011198863

21198602

1198626=3

412057411198862

111986011198862

2

1198627= minus

3

161205741(1198602+ 2)119860

111988621198863

1minus3

161205741119886111986011198863

21198602

1198628=1

161205741(1198602minus 2)119860

11198863

21198861

1198629=3

1612057411198862

111986011198862

21198602

1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

1198632=1

12(2 minus 7119861

2) 1198963

31198614

11198864

2120590 +

1

8(81198612minus 4) 119886

2

21198612

1(1198963120590

minus 1205722)

1198633=1

16((minus6 + 3119861

2) 11988621198863

1minus 61198863

21198861) 12057421198611

1198634=3

812057421198862

211986111198862

1

1198635= minus

1

161205742119886211986111198863

11198612

1198636= minus

3

412057421198862

211986111198862

1

1198637=1

16(1198612minus 2) 119886

21198863

112057421198611

1198638= minus

1

16(311988621198863

11198612+ (6 + 3119861

2) 1198863

21198861) 12057421198611

1198639=3

1612057421198862

211986111198862

11198612

(25)

(4) 1198891205791119889119905 and 119889120579

2119889119905 are given by

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)[1198671(1 + cos (2120579

2))

+ 1198672sin (2120579

1) + 1198673cos (2120579

1)

+ 1198674(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198675cos (2120579

1minus 21205792) + 1198676cos (3120579

1minus 31205792)

+ 1198677cos (120579

1minus 31205792) + 1198678cos (3120579

1minus 1205792)

+ 1198679cos (4120579

1minus 21205792)] + 119860

1(1 + 119860

2cos (2120579

1))

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)[1198691(1 + cos (2120579

1))

+ 1198692sin (2120579

2) + 1198693cos (2120579

2)

+ 1198694(cos (120579

1+ 1205792) + cos (120579

1minus 1205792))

+ 1198695cos (2120579

1minus 21205792) + 1198696cos (3120579

1minus 31205792)

+ 1198697cos (120579

1minus 31205792) + 1198698cos (3120579

1minus 1205792)

+ 1198699cos (2120579

1minus 41205792)] + 119861

1(1 + 119861

2cos (2120579

2))

(26)

where

1198671=3

8(2 + 119860

2) 119860112057411198862

21198862

1

1198672= minus

1

8(4 + 119860

2

2)1198602

11198862

11205721

1198673= minus

3

4(1 + 119860

2) 11986011198862

11198862

21205741

1198674= minus

3

16((2 + 119860

2) 11988611198863

2+ (6 + 4119860

2) 1198863

11198862)11986011205741

1198675=3

8(1 + 119860

2) 11986011198862

11198862

21205741

1198676= minus

1

161205741119886111986011198863

21198602

1198677= minus

1

16(2 + 119860

2) 11988611198863

211986011205741

1198678= minus

3

16(11988611198863

21198602+ (2 + 3119860

2) 1198863

11198862)11986011205741

1198679=3

1612057411198862

111986011198862

21198602

6 Mathematical Problems in Engineering

1198691= minus

3

8(2 + 119861

2) 1198862

11198862

211986111205742

1198692=1

12(1198612minus 1) 119886

4

21198611

41198963

3+ (1

21198963120590 minus 48120572

2) 1198862

21198612

1

1198693= minus

3

4(1198612+ 1) 119861

11198862

21198862

11205742

1198694=3

16((2 + 119861

2) 11988621198863

1+ (4119861

2+ 6) 119886

3

21198861) 12057421198611

1198695= minus

3

8(1198612+ 1) 119861

11198862

21198862

11205742

1198696=1

161205742119886211986111198863

11198612

1198697=3

16(11988621198863

11198612+ (3119861

2+ 2) 119886

3

21198861) 12057421198611

1198698=1

16(2 + 119861

2) 11988621198863

112057421198611

1198699= minus

3

1612057421198862

211986111198862

11198612

(27)

(5) Letting

1205791= 120596119905 + 120601

1

1205792= 120596119905 + 120601

2

(28)

and substituting them into (24) (26) yields

1198891198641

119889119905= 1198621+ 1198623sin (120601

1minus 1206012) + 1198624sin (2120601

1minus 21206012)

+ 1198625sin (3120601

1minus 31206012) + [119862

2cos (2120601

1) + 1198623

sdot sin (1206011+ 1206012) + 1198626sin (2120601

1) + 1198627sin (3120601

1minus 1206012)

+ 1198628sin (120601

1minus 31206012) + 1198629sin (4120601

1minus 21206012)] cos (2120596119905)

+ [minus1198622sin (2120601

1) + 1198623cos (120601

1+ 1206012) + 1198626cos (2120601

1)

+ 1198627cos (3120601

1minus 1206012) minus 1198628cos (120601

1minus 31206012) + 1198629

sdot cos (41206011minus 21206012)] sin (2120596119905)

1198891198642

119889119905= 1198631+ 1198633sin (120601

1minus 1206012) + 1198634sin (2120601

1minus 21206012)

+ 1198635sin (3120601

1minus 31206012) + [119863

2cos (2120601

1) minus 1198633

sdot sin (1206011+ 1206012) + 1198636sin (2120601

2) + 1198637sin (3120601

1minus 1206012)

+ 1198638sin (120601

1minus 31206012) + 1198639sin (2120601

1minus 41206012)]

sdot cos (2120596119905) + [minus1198632sin (2120601

1) minus 1198633cos (120601

1+ 1206012)

+ 1198636cos (2120601

2) + 1198637cos (3120601

1minus 1206012) minus 1198638

sdot cos (1206011minus 31206012) minus 1198639cos (2120601

1minus 41206012)] sin (2120596119905)

(29)

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)1198862

1(1205731+ 212057411198862

1)1198601+ 1198671

+ 1198674cos (120601

1minus 1206012) + 1198675cos (2120601

1minus 21206012) + 1198676

sdot cos (31206011minus 31206012) + [119867

1cos (2120601

2) + 1198672sin (2120601

1)

+ (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) cos (2120601

1)

+ 1198674cos (120601

1+ 1206012) + 1198677cos (120601

1minus 31206012)

+ 1198678cos (3120601

1minus 1206012) + 1198679cos (4120601

1minus 21206012)]

sdot cos (2120596119905) + [minus1198671sin (2120601

2) + 1198672cos (2120601

1)

minus (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) sin (2120601

1)

minus 1198674sin (120601

1+ 1206012) + 1198677sin (120601

1minus 31206012)

minus 1198678sin (3120601

1minus 1206012) minus 1198679sin (4120601

1minus 21206012)]

sdot sin (2120596119905)

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)1198862

2(1205732minus 12057421198862

2) 1198611+ 1198691+ 1198694

sdot cos (1206011minus 1206012) + 1198695cos (2120601

1minus 21206012) + 1198696

sdot cos (31206011minus 31206012) + [1198691cos (2120601

1) + 1198692sin (2120601

2)

+ (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) cos (2120601

2)

+ 1198694cos (120601

1+ 1206012) + 1198697cos (120601

1minus 31206012)

+ 1198698cos (3120601

1minus 1206012) + 1198699cos (2120601

1minus 41206012)] cos (2120596119905)

+ [minus1198691sin (2120601

1) + 1198692cos (2120601

2)

minus (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) sin (2120601

2)

minus 1198694sin (120601

1+ 1206012) + 1198697sin (120601

1minus 31206012)

minus 1198698sin (3120601

1minus 1206012) minus 1198699sin (2120601

1minus 41206012)]

sdot sin (2120596119905)

(30)

If there exists a periodic solution with period 2120587120596 in (3)then the energy of the system should remain the same at ini-tial and finial positions in one period Integrating both sidesof (29) in one period (from 119905 = Δ to 119905 = Δ + 2120587120596) leads to

(1198641)2120587120596+Δ

minus (1198641)Δ= 1198621= minus

1

4[(1198602minus 1)2

minus 1]

sdot 1198602

11198862

11205721lt 0

(1198642)2120587120596+Δ

minus (1198642)Δ= 1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

(31)

Mathematical Problems in Engineering 7

minus004

minus002

0

002

004

minus01

minus005

0

005

01

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

x1

(m)

x2

(m)

(a)

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

minus004

minus002

0

002

004

minus01

minus005

0

005

01

x1

(m)

x2

(m)

(b)

Figure 2 Direct numerical simulations of (3) for (a) 119899 = 112 and (b) 119899 = 118

Inspection of the first equation in (31) shows that the vibra-tion energy of cables decreases in one period Thereforecables cannot vibrate periodically when the deck is subjectedto the crowd excitation From the second equation in (31)if 1198963120590 minus 1205722lt 0 then 119863

1lt 0 At this point the deck has an

attenuate vibration If 1198963120590minus1205722gt 0119863

1= 0may hold Letting

1198631= 0 in the second equation in (31) and ignoring the sixth

or more power of 1198862(since 119886

2≪ 1 according to its physical

meaning) one obtains

712057421198864

212057321205901198963

3minus 3 (5120575120574

2+ 21205732

21205901198963

3) 1198862

2+ 24120575120573

2= 0 (32)

where 120575 = 1198963120590 minus 1205722 Solving (32) yields

119886lowast

2

=radic61205901198963

31205732

2+ 151205742120575 minus radic12120590119896

3

31205732

2(31205901198963

31205732

2minus 411205742120575) + 225120574

2

21205752

1412059012057421198963

31205732

= 2radic2radic1205732120575

21205901198963

31205732

2+ 51205742120575

(33)

For clarification the amplitude can be rewritten as

119886lowast

2= 4radic

1198982119896 (1198963120574 minus 1198882)

41205741198963

31198962 + 5 (11986411986021198713)119898

2(1198963120574 minus 1198882) (34)

where 120574 = 11989611198962119866(119891)119872

119901119892 and 119896

3120574 minus 1198882gt 0

(6) The first-order approximation solution of (3) isexpressed by

1199092(119905) = 119886

lowast

2cos (120596119905)

= 119886lowast

2cos(radic 1

1198982

[119896 minus3

8

119864119860

1198713(119886lowast

2)2

]119905)

(35)

where 119886lowast2is given in (34)

5 Numerical Simulations and Discussion

In this section we will verify the validity of our analyticalsolutions obtained in the last section by comparing theresults based on (35) with that derived by direct numericalsimulations for (3) The following parameters of deck andpedestrians are taken to calculate the periodic solutions in (3)

1198982= 21401 times 10

5(kg)

119896 = 7307361 times 106(kgs2)

1198882= 28262 times 10

4(kgs)

119866 (119891) = 10

1198961= 00987

1198962= 02

1198963= 18

(36)

In fact above values of parameters are used in [17 19] forthe first lateral model of the T-bridge in Japan Furthermorethe weight of a single person on the bridge is assumed to be70 (kg) the gravity acceleration 119892 = 98 (ms2) The modalmass of pedestrians119872

119901119892 can be given by119872

119901119892 = 119899times70times98

where 119899 is the number of the pedestrians on the bridgeAdditionally according to [19] the average length of thecables is taken as 119871 = 60 (m) However other parameters ofthe cables such as stiffness damping coefficient and massare not provided in [19] In order to compare the theoreticalresults of T-bridge between our cable-stayed bridge modeland Nakamurarsquos model following parameters of cables areassumed in the numerical simulations

1198981= 21401 times 10

3(kg)

119864119860 = 168 times 1010(N)

8 Mathematical Problems in Engineering

9955 996 9965 997 9975 998 9985 999 9995 1000

Numerical resultsAnalytical results

t (s)

minus3

minus2

minus1

0

1

2

3

4times10minus3

x2

(m)

(a)

9955 996 9965 997 9975 998 9985 999 9995 1000minus6

minus4

minus2

0

2

4

6

8

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(b)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(c)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3x2

(m)

(d)

Figure 3 Comparison between the first-order approximation solution equation (35) and direct numerical results of (3) for the cases of (a)119899 = 118 (b) 119899 = 125 (c) 119899 = 130 and (d) 119899 = 135

1198881= 28262 times 10

2(kgs)

1198790= 724 times 10

5(N)

(37)

Then we have

1205721= 01321 (1s)

1205722= 01321 (1s)

1205731= 676604 (1s2)

1205732= 341450 (1s2)

1205741= 39251 times 10

3(1s2)

1205742= 392505 (1s2)

(38)

Regarding the number 119899 as the bifurcation parameter one has

120590 =

11989611198962119866 (119891)119872

119901119892

1198982

= 0000063275734785119899

120575 = 1198963120590 minus 1205722= 000113896119899 minus 013201

(39)

From the analysis in Section 3 120575 lt 0 means the attenuationof the lateral vibration of the bridges while 120575 gt 0 means the

Mathematical Problems in Engineering 9

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

6 Mathematical Problems in Engineering

1198691= minus

3

8(2 + 119861

2) 1198862

11198862

211986111205742

1198692=1

12(1198612minus 1) 119886

4

21198611

41198963

3+ (1

21198963120590 minus 48120572

2) 1198862

21198612

1

1198693= minus

3

4(1198612+ 1) 119861

11198862

21198862

11205742

1198694=3

16((2 + 119861

2) 11988621198863

1+ (4119861

2+ 6) 119886

3

21198861) 12057421198611

1198695= minus

3

8(1198612+ 1) 119861

11198862

21198862

11205742

1198696=1

161205742119886211986111198863

11198612

1198697=3

16(11988621198863

11198612+ (3119861

2+ 2) 119886

3

21198861) 12057421198611

1198698=1

16(2 + 119861

2) 11988621198863

112057421198611

1198699= minus

3

1612057421198862

211986111198862

11198612

(27)

(5) Letting

1205791= 120596119905 + 120601

1

1205792= 120596119905 + 120601

2

(28)

and substituting them into (24) (26) yields

1198891198641

119889119905= 1198621+ 1198623sin (120601

1minus 1206012) + 1198624sin (2120601

1minus 21206012)

+ 1198625sin (3120601

1minus 31206012) + [119862

2cos (2120601

1) + 1198623

sdot sin (1206011+ 1206012) + 1198626sin (2120601

1) + 1198627sin (3120601

1minus 1206012)

+ 1198628sin (120601

1minus 31206012) + 1198629sin (4120601

1minus 21206012)] cos (2120596119905)

+ [minus1198622sin (2120601

1) + 1198623cos (120601

1+ 1206012) + 1198626cos (2120601

1)

+ 1198627cos (3120601

1minus 1206012) minus 1198628cos (120601

1minus 31206012) + 1198629

sdot cos (41206011minus 21206012)] sin (2120596119905)

1198891198642

119889119905= 1198631+ 1198633sin (120601

1minus 1206012) + 1198634sin (2120601

1minus 21206012)

+ 1198635sin (3120601

1minus 31206012) + [119863

2cos (2120601

1) minus 1198633

sdot sin (1206011+ 1206012) + 1198636sin (2120601

2) + 1198637sin (3120601

1minus 1206012)

+ 1198638sin (120601

1minus 31206012) + 1198639sin (2120601

1minus 41206012)]

sdot cos (2120596119905) + [minus1198632sin (2120601

1) minus 1198633cos (120601

1+ 1206012)

+ 1198636cos (2120601

2) + 1198637cos (3120601

1minus 1206012) minus 1198638

sdot cos (1206011minus 31206012) minus 1198639cos (2120601

1minus 41206012)] sin (2120596119905)

(29)

1198891205791

119889119905=

1

1198862

1(1205731+ 212057411198862

1)1198862

1(1205731+ 212057411198862

1)1198601+ 1198671

+ 1198674cos (120601

1minus 1206012) + 1198675cos (2120601

1minus 21206012) + 1198676

sdot cos (31206011minus 31206012) + [119867

1cos (2120601

2) + 1198672sin (2120601

1)

+ (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) cos (2120601

1)

+ 1198674cos (120601

1+ 1206012) + 1198677cos (120601

1minus 31206012)

+ 1198678cos (3120601

1minus 1206012) + 1198679cos (4120601

1minus 21206012)]

sdot cos (2120596119905) + [minus1198671sin (2120601

2) + 1198672cos (2120601

1)

minus (1198862

1(1205731+ 212057411198862

1)11986011198602+ 1198673) sin (2120601

1)

minus 1198674sin (120601

1+ 1206012) + 1198677sin (120601

1minus 31206012)

minus 1198678sin (3120601

1minus 1206012) minus 1198679sin (4120601

1minus 21206012)]

sdot sin (2120596119905)

1198891205792

119889119905=

1

1198862

2(1205732minus 12057421198862

2)1198862

2(1205732minus 12057421198862

2) 1198611+ 1198691+ 1198694

sdot cos (1206011minus 1206012) + 1198695cos (2120601

1minus 21206012) + 1198696

sdot cos (31206011minus 31206012) + [1198691cos (2120601

1) + 1198692sin (2120601

2)

+ (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) cos (2120601

2)

+ 1198694cos (120601

1+ 1206012) + 1198697cos (120601

1minus 31206012)

+ 1198698cos (3120601

1minus 1206012) + 1198699cos (2120601

1minus 41206012)] cos (2120596119905)

+ [minus1198691sin (2120601

1) + 1198692cos (2120601

2)

minus (1198862

2(1205732minus 12057421198862

2) 11986111198612+ 1198693) sin (2120601

2)

minus 1198694sin (120601

1+ 1206012) + 1198697sin (120601

1minus 31206012)

minus 1198698sin (3120601

1minus 1206012) minus 1198699sin (2120601

1minus 41206012)]

sdot sin (2120596119905)

(30)

If there exists a periodic solution with period 2120587120596 in (3)then the energy of the system should remain the same at ini-tial and finial positions in one period Integrating both sidesof (29) in one period (from 119905 = Δ to 119905 = Δ + 2120587120596) leads to

(1198641)2120587120596+Δ

minus (1198641)Δ= 1198621= minus

1

4[(1198602minus 1)2

minus 1]

sdot 1198602

11198862

11205721lt 0

(1198642)2120587120596+Δ

minus (1198642)Δ= 1198631=1198612

11198862

2

24[(81198612minus 3) 119861

2

11198862

21198963

3120590

minus 12 (1198612minus 1) (119896

3120590 minus 1205722)]

(31)

Mathematical Problems in Engineering 7

minus004

minus002

0

002

004

minus01

minus005

0

005

01

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

x1

(m)

x2

(m)

(a)

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

minus004

minus002

0

002

004

minus01

minus005

0

005

01

x1

(m)

x2

(m)

(b)

Figure 2 Direct numerical simulations of (3) for (a) 119899 = 112 and (b) 119899 = 118

Inspection of the first equation in (31) shows that the vibra-tion energy of cables decreases in one period Thereforecables cannot vibrate periodically when the deck is subjectedto the crowd excitation From the second equation in (31)if 1198963120590 minus 1205722lt 0 then 119863

1lt 0 At this point the deck has an

attenuate vibration If 1198963120590minus1205722gt 0119863

1= 0may hold Letting

1198631= 0 in the second equation in (31) and ignoring the sixth

or more power of 1198862(since 119886

2≪ 1 according to its physical

meaning) one obtains

712057421198864

212057321205901198963

3minus 3 (5120575120574

2+ 21205732

21205901198963

3) 1198862

2+ 24120575120573

2= 0 (32)

where 120575 = 1198963120590 minus 1205722 Solving (32) yields

119886lowast

2

=radic61205901198963

31205732

2+ 151205742120575 minus radic12120590119896

3

31205732

2(31205901198963

31205732

2minus 411205742120575) + 225120574

2

21205752

1412059012057421198963

31205732

= 2radic2radic1205732120575

21205901198963

31205732

2+ 51205742120575

(33)

For clarification the amplitude can be rewritten as

119886lowast

2= 4radic

1198982119896 (1198963120574 minus 1198882)

41205741198963

31198962 + 5 (11986411986021198713)119898

2(1198963120574 minus 1198882) (34)

where 120574 = 11989611198962119866(119891)119872

119901119892 and 119896

3120574 minus 1198882gt 0

(6) The first-order approximation solution of (3) isexpressed by

1199092(119905) = 119886

lowast

2cos (120596119905)

= 119886lowast

2cos(radic 1

1198982

[119896 minus3

8

119864119860

1198713(119886lowast

2)2

]119905)

(35)

where 119886lowast2is given in (34)

5 Numerical Simulations and Discussion

In this section we will verify the validity of our analyticalsolutions obtained in the last section by comparing theresults based on (35) with that derived by direct numericalsimulations for (3) The following parameters of deck andpedestrians are taken to calculate the periodic solutions in (3)

1198982= 21401 times 10

5(kg)

119896 = 7307361 times 106(kgs2)

1198882= 28262 times 10

4(kgs)

119866 (119891) = 10

1198961= 00987

1198962= 02

1198963= 18

(36)

In fact above values of parameters are used in [17 19] forthe first lateral model of the T-bridge in Japan Furthermorethe weight of a single person on the bridge is assumed to be70 (kg) the gravity acceleration 119892 = 98 (ms2) The modalmass of pedestrians119872

119901119892 can be given by119872

119901119892 = 119899times70times98

where 119899 is the number of the pedestrians on the bridgeAdditionally according to [19] the average length of thecables is taken as 119871 = 60 (m) However other parameters ofthe cables such as stiffness damping coefficient and massare not provided in [19] In order to compare the theoreticalresults of T-bridge between our cable-stayed bridge modeland Nakamurarsquos model following parameters of cables areassumed in the numerical simulations

1198981= 21401 times 10

3(kg)

119864119860 = 168 times 1010(N)

8 Mathematical Problems in Engineering

9955 996 9965 997 9975 998 9985 999 9995 1000

Numerical resultsAnalytical results

t (s)

minus3

minus2

minus1

0

1

2

3

4times10minus3

x2

(m)

(a)

9955 996 9965 997 9975 998 9985 999 9995 1000minus6

minus4

minus2

0

2

4

6

8

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(b)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(c)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3x2

(m)

(d)

Figure 3 Comparison between the first-order approximation solution equation (35) and direct numerical results of (3) for the cases of (a)119899 = 118 (b) 119899 = 125 (c) 119899 = 130 and (d) 119899 = 135

1198881= 28262 times 10

2(kgs)

1198790= 724 times 10

5(N)

(37)

Then we have

1205721= 01321 (1s)

1205722= 01321 (1s)

1205731= 676604 (1s2)

1205732= 341450 (1s2)

1205741= 39251 times 10

3(1s2)

1205742= 392505 (1s2)

(38)

Regarding the number 119899 as the bifurcation parameter one has

120590 =

11989611198962119866 (119891)119872

119901119892

1198982

= 0000063275734785119899

120575 = 1198963120590 minus 1205722= 000113896119899 minus 013201

(39)

From the analysis in Section 3 120575 lt 0 means the attenuationof the lateral vibration of the bridges while 120575 gt 0 means the

Mathematical Problems in Engineering 9

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

minus004

minus002

0

002

004

minus01

minus005

0

005

01

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

x1

(m)

x2

(m)

(a)

200 400 600 800 10000t (s)

200 400 600 800 10000t (s)

minus004

minus002

0

002

004

minus01

minus005

0

005

01

x1

(m)

x2

(m)

(b)

Figure 2 Direct numerical simulations of (3) for (a) 119899 = 112 and (b) 119899 = 118

Inspection of the first equation in (31) shows that the vibra-tion energy of cables decreases in one period Thereforecables cannot vibrate periodically when the deck is subjectedto the crowd excitation From the second equation in (31)if 1198963120590 minus 1205722lt 0 then 119863

1lt 0 At this point the deck has an

attenuate vibration If 1198963120590minus1205722gt 0119863

1= 0may hold Letting

1198631= 0 in the second equation in (31) and ignoring the sixth

or more power of 1198862(since 119886

2≪ 1 according to its physical

meaning) one obtains

712057421198864

212057321205901198963

3minus 3 (5120575120574

2+ 21205732

21205901198963

3) 1198862

2+ 24120575120573

2= 0 (32)

where 120575 = 1198963120590 minus 1205722 Solving (32) yields

119886lowast

2

=radic61205901198963

31205732

2+ 151205742120575 minus radic12120590119896

3

31205732

2(31205901198963

31205732

2minus 411205742120575) + 225120574

2

21205752

1412059012057421198963

31205732

= 2radic2radic1205732120575

21205901198963

31205732

2+ 51205742120575

(33)

For clarification the amplitude can be rewritten as

119886lowast

2= 4radic

1198982119896 (1198963120574 minus 1198882)

41205741198963

31198962 + 5 (11986411986021198713)119898

2(1198963120574 minus 1198882) (34)

where 120574 = 11989611198962119866(119891)119872

119901119892 and 119896

3120574 minus 1198882gt 0

(6) The first-order approximation solution of (3) isexpressed by

1199092(119905) = 119886

lowast

2cos (120596119905)

= 119886lowast

2cos(radic 1

1198982

[119896 minus3

8

119864119860

1198713(119886lowast

2)2

]119905)

(35)

where 119886lowast2is given in (34)

5 Numerical Simulations and Discussion

In this section we will verify the validity of our analyticalsolutions obtained in the last section by comparing theresults based on (35) with that derived by direct numericalsimulations for (3) The following parameters of deck andpedestrians are taken to calculate the periodic solutions in (3)

1198982= 21401 times 10

5(kg)

119896 = 7307361 times 106(kgs2)

1198882= 28262 times 10

4(kgs)

119866 (119891) = 10

1198961= 00987

1198962= 02

1198963= 18

(36)

In fact above values of parameters are used in [17 19] forthe first lateral model of the T-bridge in Japan Furthermorethe weight of a single person on the bridge is assumed to be70 (kg) the gravity acceleration 119892 = 98 (ms2) The modalmass of pedestrians119872

119901119892 can be given by119872

119901119892 = 119899times70times98

where 119899 is the number of the pedestrians on the bridgeAdditionally according to [19] the average length of thecables is taken as 119871 = 60 (m) However other parameters ofthe cables such as stiffness damping coefficient and massare not provided in [19] In order to compare the theoreticalresults of T-bridge between our cable-stayed bridge modeland Nakamurarsquos model following parameters of cables areassumed in the numerical simulations

1198981= 21401 times 10

3(kg)

119864119860 = 168 times 1010(N)

8 Mathematical Problems in Engineering

9955 996 9965 997 9975 998 9985 999 9995 1000

Numerical resultsAnalytical results

t (s)

minus3

minus2

minus1

0

1

2

3

4times10minus3

x2

(m)

(a)

9955 996 9965 997 9975 998 9985 999 9995 1000minus6

minus4

minus2

0

2

4

6

8

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(b)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(c)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3x2

(m)

(d)

Figure 3 Comparison between the first-order approximation solution equation (35) and direct numerical results of (3) for the cases of (a)119899 = 118 (b) 119899 = 125 (c) 119899 = 130 and (d) 119899 = 135

1198881= 28262 times 10

2(kgs)

1198790= 724 times 10

5(N)

(37)

Then we have

1205721= 01321 (1s)

1205722= 01321 (1s)

1205731= 676604 (1s2)

1205732= 341450 (1s2)

1205741= 39251 times 10

3(1s2)

1205742= 392505 (1s2)

(38)

Regarding the number 119899 as the bifurcation parameter one has

120590 =

11989611198962119866 (119891)119872

119901119892

1198982

= 0000063275734785119899

120575 = 1198963120590 minus 1205722= 000113896119899 minus 013201

(39)

From the analysis in Section 3 120575 lt 0 means the attenuationof the lateral vibration of the bridges while 120575 gt 0 means the

Mathematical Problems in Engineering 9

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

9955 996 9965 997 9975 998 9985 999 9995 1000

Numerical resultsAnalytical results

t (s)

minus3

minus2

minus1

0

1

2

3

4times10minus3

x2

(m)

(a)

9955 996 9965 997 9975 998 9985 999 9995 1000minus6

minus4

minus2

0

2

4

6

8

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(b)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3

x2

(m)

(c)

9955 996 9965 997 9975 998 9985 999 9995 1000minus8

minus6

minus4

minus2

0

2

4

6

8

10

Numerical resultsAnalytical results

t (s)

times10minus3x2

(m)

(d)

Figure 3 Comparison between the first-order approximation solution equation (35) and direct numerical results of (3) for the cases of (a)119899 = 118 (b) 119899 = 125 (c) 119899 = 130 and (d) 119899 = 135

1198881= 28262 times 10

2(kgs)

1198790= 724 times 10

5(N)

(37)

Then we have

1205721= 01321 (1s)

1205722= 01321 (1s)

1205731= 676604 (1s2)

1205732= 341450 (1s2)

1205741= 39251 times 10

3(1s2)

1205742= 392505 (1s2)

(38)

Regarding the number 119899 as the bifurcation parameter one has

120590 =

11989611198962119866 (119891)119872

119901119892

1198982

= 0000063275734785119899

120575 = 1198963120590 minus 1205722= 000113896119899 minus 013201

(39)

From the analysis in Section 3 120575 lt 0 means the attenuationof the lateral vibration of the bridges while 120575 gt 0 means the

Mathematical Problems in Engineering 9

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

n = 125

n = 118

0

0002

0004

0006

0008

001

0012

times1092 4 6 8 100

EAL3

n = 135

n = 130

x2

(m)

(a) 1199092 versus 1198641198601198713

0918

092

0922

0924

0926

0928

093

0932

0934

times109

f(H

z)

2 4 6 8 100EAL3

n = 125

n = 118n = 135

n = 130

(b) 119891 versus 1198641198601198713

Figure 4 The (a) amplitude (1199092) and (b) frequency (119891) of deck always decrease with 1198641198601198713 which is increased based on (35) for different

numbers of pedestrians on the bridge

occurrence of the lateral vibration of the bridge No matterwhat the motion of the deck is the cables vibration always isattenuated Thus 119899 = 115 is the critical point correspondingto the lateral vibration of the deck The time history curvesof displacements of the cables and deck for cases of 119899 = 112and 119899 = 118 are presented in Figure 2 by direct numericalsimulations for (3) which demonstrates the correctness ofour analysis in Section 3 The initial condition in numericalsimulations is given by 119909

1(0) = 0

1(0) = 0 119909

2(0) = 008

and 2(0) = 0

In the following the analytical results based on (35) arecompared with that obtained by direct numerical simulationsof (3) for cases of 119899 = 118 125 130 and 135 respectivelywhich are shown in Figure 3 It shows good agreementbetween the two approaches for 119899 near bifurcation point 119899 =115 However the precision of the first-order approximationsolution is lost when 119899 is considerably larger than the valueof the bifurcation point In this case second-order or moreorder approximation is needed to be calculated for higherprecision of the analytical solution

Since the validity of (35) has been verified in the rest ofthe section we will consider the behavior of system equa-tion (3) on varying the parameters based on (35) From(35) the amplitude of the lateral vibration of a cable-stayedbridge under crowd excitation has nothing to do with themass tension and damping coefficient of the cables But theexistence of the cables can decrease the amplitude of the decksince the term of 1198641198601198713 appears in the denominator Thismeans that the theoretical results of the vibration amplitudeof the deck derived by using (3) are less than that obtainedby using Nakamurarsquos model in [19] For the first lateral modelof the T-bridge the theoretical result is higher by 49 thanthe measured data Nakamura considered that the reasonis the uncertainty of measurements of parameters such asmodal mass stiffness and damping coefficient of the deckHowever only using a dating error is difficult to explain such

a big difference between theoretical results and measureddata Equation (35) shows that neglecting the structuraltype of the cable-stayed bridge may be an important causeof the difference between theoretical and measured resultsFurthermore from (35) the frequency of the lateral vibrationof the deck also decreases due to the existence of the cablesThe curves about the change of the amplitude and frequencyof the deck with the term of 1198641198601198713 for different numbersof pedestrians are given in Figure 4 based on (35) in whichthe parameters of deck and pedestrians presented in (36)are adopted From Figure 4 the more the pedestrians on thebridge are the faster the vibrational frequency decreases with1198641198601198713 which is increased

Figures 5 and 6 present the curves about the change ofthe amplitude and frequency of the deck with the terms of 119896

2

1198963 respectively for different numbers of pedestrians based on

(35) in which all other parameters of deck and cables are thesame as that presented in (36) and (37) It can be seen fromFigures 5 and 6 that changes of 119896

2 1198963have significant effect on

the amplitude of the deck and however almost have nothingto do with its vibrational frequency

6 Conclusions

In this paper the lateral vibrations of a cable-stayed bridgeunder crowd excitation are investigated theoretically andnumerically In our study the cable-stayed bridge is simplifiedas a string and a rigid block and the pedestrian-inducedforce model is satisfied with Nakamurarsquos assumption Thegoverning equation of our model is established by takingthe geometric nonlinear property of the cables into accountCenter manifold theory is employed to determine the criticalcondition that periodic vibrations of the bridge occur Thenfirst-order approximation solution of periodic vibrations ofthe deck is calculated by using the energymethod the validityof which is verified by using direct numerical simulations

10 Mathematical Problems in Engineering

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

times10minus3

n = 125

n = 118n = 135

n = 130

2

4

6

8

10

12

14

16

0403 03502502k2

x2

(m)

(a) 1199092 versus 1198962

n = 125

n = 118n = 135

n = 130

093

093

093

093

093

093

093

f(H

z)

0403 03502502k2

(b) 119891 versus 1198962

Figure 5 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

2based on (35) for different numbers of pedestrians

on the bridge

n = 125

n = 118n = 135

n = 130

times10minus3

2

3

4

5

6

7

8

9

10

20 22 24 26 28 3018k3

x2

(m)

(a) 1199092 versus 1198963

n = 125

n = 118n = 135

n = 130

20 22 24 26 28 3018k3

093

093

093

093

093

093

093

093

093

f(H

z)

(b) 119891 versus 1198963

Figure 6 The curves of (a) amplitude (1199092) and (b) frequency (119891) of deck change with 119896

3based on (35) for different numbers of pedestrians

on the bridge

Our analysis shows that cables can have no periodic vibra-tion even if the deck loses its stability and begins to sway butcables can affect the amplitude and frequency of the lateralvibration of the deck From the first-order approximationsolution equations (34) and (35) the mass damping coeffi-cient and tension of the cables have no effect on the lateralvibrations of the deckHowever existence of the cables alwaysreduces the amplitude and frequency of the lateral vibrationsof the deckWith the bifurcation parameter far away from the

bifurcation point the cables have a growing influence on bothamplitude and frequency of the deck Our analysis resultsmay be used to explain why themeasured results for T-bridge(a cable-stayed bridge) derived by Nakamura and Kawasaki[19] are much less than their theoretical results calculated byregarding the T-bridge as a single-degree-of-freedom systemThis indicates that the structure types of the footbridgescannot be easily ignored in the study of pedestrian-footbridgeinteraction

Mathematical Problems in Engineering 11

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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

Conflict of Interests

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

Authorsrsquo Contribution

All authors carried out the proofreading of the paper Allauthors conceived of the study and participated in its designand coordination All authors read and approved the finalpaper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant nos 11472160) the State KeyProgram of National Natural Science of China (Grant no11032009) and PhD start-up fund of Shanghai Publishingand Printing College 95-A (1030114203) Furthermore theauthors appreciate very much the helpful advice of thereviewers

References

[1] P Dallard A J Fitzpatrick A Flint et al ldquoThe London mil-lennium footbridgerdquo Structural Engineer vol 79 no 22 pp 17ndash33 2001

[2] P Dallard T Fitzpatrick A Flint et al ldquoLondon millenniumbridge pedestrian-induced lateral vibrationrdquo Journal of BridgeEngineering vol 6 no 6 pp 412ndash417 2001

[3] Y Fujino B M Pacheco S-I Nakamura and P WarnitchaildquoSynchronization of human walking observed during lateralvibration of a congested pedestrian bridgerdquo Earthquake Engi-neering and Structural Dynamics vol 22 no 9 pp 741ndash7581993

[4] S-I Nakamura ldquoModel for lateral excitation of footbridges bysynchronous walkingrdquo Journal of Structural Engineering vol130 no 1 pp 32ndash37 2004

[5] Y Fujino P Warnitchai and B M Pacheco ldquoAn experimentaland analytical study of autoparametric resonance in a 3DOFmodel of cable-stayed-beamrdquo Nonlinear Dynamics vol 4 no2 pp 111ndash138 1993

[6] H Bachmann A J Pretlove and H Rainer ldquoDynamic forcesfrom rhythmical human body motionsrdquo in Vibration Problemsin Structure Practical Guidelines Birkhauser Basel Switzer-land 1995

[7] A N Blekherman ldquoAutoparametric resonance in a pedestriansteel arch bridge solferino bridge Parisrdquo Journal of Bridge Engi-neering vol 12 no 6 pp 669ndash676 2007

[8] T M Roberts ldquoLateral pedestrian excitation of footbridgesrdquoJournal of Bridge Engineering vol 10 no 1 pp 107ndash112 2005

[9] S H Strogatz D M Abrams A McRobie B Eckhardt and EOtt ldquoCrowd synchrony on theMillennium BridgerdquoNature vol438 no 7064 pp 43ndash44 2005

[10] F Venuti L Bruno and N Bellomo ldquoCrowd dynamics on amoving platform mathematical modelling and application tolively footbridgesrdquoMathematical and Computer Modelling vol45 no 3-4 pp 252ndash269 2007

[11] S Lenci and L Marcheggiani ldquoCritical threshold and underly-ing dynamical phenomena in pedestrian-induced lateral vibra-tions of footbridgesrdquo Journal of Mechanics of Materials andStructures vol 6 no 7-8 pp 1031ndash1051 2012

[12] G Piccardo and F Tubino ldquoParametric resonance of flexiblefootbridges under crowd-induced lateral excitationrdquo Journal ofSound and Vibration vol 311 no 1-2 pp 353ndash371 2008

[13] AMcRobie GMorgenthal J Lasenby andM Ringer ldquoSectionmodel tests on humanmdashstructure lock-inrdquo Proceedings of theICEmdashBridge Engineering vol 156 no 2 pp 71ndash79 2003

[14] D Zhou and T J Ji ldquoDynamic characteristics of a generalisedsuspension systemrdquo International Journal of Mechanical Sci-ences vol 50 no 1 pp 30ndash42 2008

[15] C Geurts T Vrouwenvelder P V Staalduinen and J ReusinkldquoNumerical modeling of rain-wind-indued vibration erasmusBridge Rotterdamrdquo Structural Engineering International vol 8no 2 pp 129ndash135 1998

[16] J L Lilien andA Pinto Da Costa ldquoVibration amplitudes causedby parametric excitation of cable stayed structuresrdquo Journal ofSound and Vibration vol 174 no 1 pp 69ndash90 1994

[17] B Zhen W P Xie and J Xu ldquoNonlinear analysis for the lateralvibration of footbridges induced by pedestriansrdquo Journal ofBridge Engineering vol 18 no 2 pp 122ndash130 2013

[18] L Li and Y Hongling ldquoEnergy method for computing periodicsolutions of strongly nonlinear autonomous systemswithmulti-degree-of-freedomrdquo Nonlinear Dynamics vol 31 no 1 pp 23ndash47 2003

[19] S Nakamura and T Kawasaki ldquoA method for predicting thelateral girder response of footbridges induced by pedestriansrdquoJournal of Constructional Steel Research vol 65 no 8-9 pp1705ndash1711 2009

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