# Dynamic Analysis for Lateral Vibrations of Footbridges under Crowd Based on a Two-Degrees-of-Freedom Model

Authors: Bin Zhen; Wenbang Qi; Liang Chang
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Nakamura’s model is widely used to describe lateral vibrations of a footbridge induced by crowd. The predicted responses of Nakamura’s model were compared with measured data of T-bridge and M-bridge in Japan to demonstrate the validity. However, the predicted responses based on Nakamura’s model almost always were stronger than measured data. Considering that both T-bridge and M-bridge are cable-stayed bridges, it seems to be not precise enough to simplify a cable-stayed bridge as a single degree of freedom system in Nakamura’s model. In this paper, we establish a two-degrees-of-freedom model to describe lateral vibrations of a cable-stayed bridge. The cables have one degree, and the bridge deck has the other. Additionally, in this model we introduce a time delay in interaction between the bridge and pedestrians. By employing the center manifold theory, we find that a subcritical Hopf bifurcation occurs in the two-degrees-of-freedom model. We theoretically and numerically illustrate that the cables and time delay have significant influence on the lateral vibration amplitude of a footbridge under crowd. The appropriate increases of tension in the cables and time delay both can decrease the lateral vibration amplitude. The analysis for the proposed two-degrees-of-freedom model shows that the predicted responses of Nakamura’s model can better agree with the measured date if we take the influence of cables and time delay into account.


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## Body

## 1. Introduction

Excessive lateral vibrations of footbridges induced by pedestrians have attracted attention from all over the world in the last decade. The London Millennium Bridge is the most famous footbridge closed on 13 June 2000 due to its huge lateral vibration amplitude [1]. It took 20 months to suppress the excessive vibrations by using Taylor dampers and tuned mass dampers [2]. Nowadays, we have known that excessive lateral vibrations almost occur in all kind of footbridges, such as arch bridges, truss bridges, cable-stayed bridges, and suspension bridges [3–5]. There are two key points in dealing with the problem of excessive vibrations in footbridges induced by pedestrians. One is how to describe the lateral dynamics of pedestrians walking on a bridge. The lateral component of the force is about 4-5% of the body weight for men and little for women when a pedestrian walks on a stationary surface [6]. Experiments and measurements [7, 8] showed that the peak value of lateral force induced by a pedestrian increases with the walking speed, that is, the step frequency. Matsumoto et al. [8] and Bachmann [9] suggested that the lateral force induced by a pedestrian walking on a stationary surface can be approximately expressed by using a cosine function with constant amplitude. However, the problem becomes complex if a pedestrian walks on a moving surface. It was found that the step frequency of a pedestrian is effected by the driving frequency of the moving surface in a large range of frequencies [10, 11]. Piccardo and Tubino [12] considered that the lateral force induced by pedestrians under such condition can be expressed by using a cosine function, the amplitude of which depends on the lateral displacement of the moving surface when the lateral amplitude of the moving surface is large. Other researchers argued that the lateral force should refer to the velocity of the moving surface [1, 5].The other point of interest in excessive lateral vibrations of footbridges is how a footbridge increases its lateral amplitude from small to large under crowd. Fujino et al. [3] considered one possible reason is that the lateral frequency of external force induced by pedestrians is very close to that of the bridge, which is called “direct resonance.” Another possible reason is “internal resonance,” which results from structural nonlinearities of the bridge [4]. If the frequency of lateral mode of the bridge is half that of its vertical mode and the external force applied on the vertical mode satisfies the direct resonance condition, excessive lateral vibrations of the bridge occur due to vertical loads. Except for the two kinds of mechanisms, dynamic interaction mechanism was investigated to understand how the footbridge and pedestrians interact with one another. Based on the experimental tests of the London Millennium Bridge, Dallard et al. [1] proposed that the bridge can be simplified as a single degree of freedom system and the lateral force induced by pedestrians is proportional to the velocity of the bridge. In Dallard’s model the influence of the pedestrians is viewed as a velocity feedback, which infinitely increases the bridge response. Roberts [13, 14] assumed that the interactive force between footbridge and pedestrians varies harmonically about 1 Hz. The worst case occurs when the pedestrians synchronize with the bridge in the lateral direction. The critical condition of synchronization is that the lateral displacement of the pedestrians is greater than that of the bridge. Nakamura et al. [5] also considered that the lateral force induced by pedestrians is relevant to the lateral velocity of a bridge. Different from Dallard’s model, Nakamura et al. allowed schematization of the self-limiting nature of the synchronization phenomenon by considering that the pedestrians will reduce walking speed or stop walking when the lateral amplitude of the bridge is sufficiently large. In fact, the problem of pedestrian-footbridge interaction is so complex that the mechanism still is not very clear.In practical engineering, Dallard’s model and Nakamura’s model are very useful. They intended to establish a practical method instead of a pure mathematical method. Strogatz et al. [15] gave a more mathematically accurate model than that of Dallard and Nakamura by containing harmonic terms in the loading function. However, it is not easy to use during application due to the hypothetical equation. According to Dallard’s model, a footbridge infinitely increases its lateral amplitude under pedestrians, which is not in accordance with observations. Nakamura’s model is more rational since the calculated response of a footbridge always is finite. Nevertheless, the lateral responses predicted numerically by using Nakamura’s model are almost stronger than the measured data [16]. Zhen et al. [17] considered that there is a time delay when the external force induced by pedestrians applies on the bridge. By introducing such assumption in Nakamura’s model, the predicted results can better agree with the measured data. In addition, T-bridge and M-bridge measured in [16] are cable-stayed bridges, which were regarded as a single degree of freedom system in Nakamura’s model. On an intuitive level, it needs at least two degrees of freedom to describe the dynamics of cable-stayed bridges. The bridge deck has one, and the cables the other. In this paper, we intend to analyze the possible reason why Nakamura’s model always predicts stronger amplitudes than the measured data. A footbridge considered here is simplified as a two-degrees-of-freedom system, and the expression of the interactive force induced by pedestrians adopts the tanh function containing a discrete time delay suggested by Zhen et al. [17]. Excessive lateral vibrations of a footbridge correspond to the bifurcating cycle due to a Hopf bifurcation in the two-degrees-of-freedom model. We will employ center manifold theory to analyze the critical conditions of Hopf bifurcation in the two-degrees-of-freedom model and then derive the analytical expression of bifurcating limit cycle. From the two-degrees-of-freedom model, we will discuss the influence of cables and time delay on the lateral amplitude of a footbridge induced by pedestrians. We find that the existence of cables and time delay decreases the lateral amplitude of a footbridge under some conditions.The rest of the paper is organized as follows: a two-degrees-of-freedom model describing dynamical behavior of a footbridge induced by pedestrians is established in Section2. The characteristic roots of the two-degrees-of-freedom system are discussed to derive the Hopf bifurcation conditions in Section 3. The center manifold theory is employed in Section 4 to investigate the stability of bifurcating cycle. The approximate amplitude of the bifurcating periodic solution is given in Section 5. Numerical simulations are provided to demonstrate the correctness of theoretical analysis in Section 6. Conclusions are drawn in Section 7.


## 2. The Two-Degrees-of-Freedom Model

Consider the two-degrees-of-freedom system presented in Figure1, in which m1 and m2 represent concentrated masses of cables and bridge deck, respectively. The cables have a length 2L, stiffness EA, tension T0, and damping coefficient c1. The stiffness and damping coefficient of the bridge deck separately are k and c2. The displacements of the cables and bridge deck are denoted by x1 and x2, respectively. Then the velocities and accelerations of the cables and bridge deck can be written as x˙1,2 and x¨1,2, respectively. Fp is the lateral force induced by pedestrians, which has the following expression [5, 17]:(1)Fp=k1k2MpgGftanh⁡k3x˙2t-τ,where k1 is the ratio of lateral force to pedestrians’ weight; k2 is the percentage of pedestrians who synchronize to bridge vibration; k3 is the saturation rate of the pedestrian-induced force; Mpg is modal self-weight of pedestrians; G(f) is a function describing how pedestrians synchronize with the natural frequency of a bridge; τ is the time delay of interactive force between the bridge and pedestrians. According to Lagrange method, the governing equation of the TDOF system can be established(2)x¨1+α1x˙1+β1x1+γ1x13+x1-x23=0,x¨2+α2x˙2+β2x2+γ2x1-x23=σtanh⁡k3x˙2t-τ,where(3)α1=c1m1,α2=c2m2,β1=2T0L,β2=km2,γ1=EA2m1L3,γ2=EA2m2L3,σ=k1k2GfBMpgm2.Obviously, α1,2>0, β1,2>0, and γ1,2>0. In addition, we assume that β2>γ2.Figure 1
The two-degrees-of-freedom model.


## 3. Characteristic Roots Analysis for System (2)

By lettingx1=y1, x˙1=y2, x2=y3,and x˙2=y4, system (2) can be rewritten as(4)y˙1=y2,y˙2=-β1y1-α1y2-γ1y13+y1-y33,y˙3=y4,y˙4=-β2y3-α2y4+σtanh⁡k3y4t-τ-γ2y1-y33.The origin (0,0,0,0) is the unique equilibrium of system (4). The characteristic equation of system (4) near the origin is given by(5)λ4+p1λ3+p2λ2+p3λ+p4+p5λ3+p6λ2+p7λe-λτ=0,where(6)p1=α1+α2,p2=β1+β2+α1α2,p3=α1β2+β1α2,p4=β1β2,p5=-σk3,p6=-α1σk3,p7=-β1σk3.


### 3.1. The Case ofτ=0

We first consider the case ofτ=0. Under such case, (5) becomes(7)λ4+α1-ξλ3+β1+β2-α1ξλ2+α1β2-ξβ1λ+β1β2=0,where ξ=σk3-α2. Since we focus on the excessive vibrations, we need to consider the instable case of the origin in system (4). Obviously, λ=0 is not a root of (7). It is easy to verify that (7) just has a pair of imaginary roots λ=±iβ2 when ξ=0. Regarding λ as a function of ξ in (7) and taking the derivative of λ with respect to ξ, we have(8)dλdξ=λλ2+α1λ+β14λ3-3ξ-α1λ2-2ξα1-β1-β2λ-β1ξ+α1β2.Substituting λ=±iβ2 and ξ=0 into (8) yields(9)dλdξ∣ξ=0=12>0.It means that a pair of imaginary roots cross the imaginary axis from left to right when ξ increases from ξ<0 to ξ>0. Excessive lateral vibrations correspond to the instability of the origin in (4); then in the following analysis we limit our discussion in the case of ξ>0.


### 3.2. The Case ofτ>0

We have proven that the origin of system (4) is unstable when ξ>0 and τ=0 in previous section. Next, we consider the characteristic roots of system (4) in the case of ξ>0 and τ>0. We substitute λ=iω(ω>0) into (5), and separate the real and imaginary parts(10)ω4-p2ω2+p4-p6ω2cosωτ+-p5ω3+p7ωsinωτ=0,-p1ω3+p3ω+-p5ω3+p7ωcosωτ+p6ω2sinωτ=0.Eliminating τ from (10) and denoting w=ω2, we have(11)fw=w4+q1w3+q2w2+q3w+q4=0,where(12)q1=p12-2p2-p52,q2=2p5p7+2p4+p22-p62-2p1p3,q3=p32-2p2p4-p72,q4=p42.According to the physical meanings, it is easy to check that q1<0, q2>0, q3<0, and q4>0. Taking derivative with respect to w in (11), we have(13)f′w=4w3+3q1w2+2q2w+q3.Obviously, f′(w)<0 for w<0. Considering f(0)=q4>0 and f(-∞)→+∞, (11) has no negative root. If (11) has real roots, they must be positive. According to Descartes rule, (11) has a minimum of zero and maximum of 4 positive real roots, and the number of positive roots to (11) always is even. Assume that w=ω02 is one positive real root. From (10) we have(14)cos⁡ω0τ=--p1ω02+p3-p5ω02+p7-p6ω04-p2ω2+p4-p5ω02+p72+p62ω02=α2σk3.If α2>σk3, then cos(ω0τ)>1, which is impossible. If α2≤σk3, it is possible that the stability of the origin of system (4) depends on the size of time delay τ. From (14), the critical value of time delay changing the stability of the origin is given by(15)τj+=1ω02jπ+arccos⁡α2σk3,j=0,1,2,⋯,τj-=1ω02j+1π-arccosα2σk3,j=0,1,2,⋯.In order to determine the crossing direction of the pure imaginary roots ±iω0, we regard λ as a function of τ in (5) and take derivative with respect to τ(16)dλdτ=λ′τ=4λ3+3p1λ2+2p2λ+p3p5λ2+p6λ+p7λ2e-λτ+3p5λ2+2p6λ+p7p5λ2+p6λ+p7λ2-τλ.The crossing direction depends on the sign of real part of (17)dλdτ∣τ=τj+,τj-,j=0,1,2,⋯.Next, we separately discuss the two cases of τ=τj+ and τ=τj-. (i)
Whenτ=τj+,(18)cos⁡ω0τj+=α2σk3,sin⁡ω0τj+=σk32-α22σk3≡δσk3,δ>0.Under such condition, the sign of real part ofλ′(τj+),j=0,1,2,⋯, depends on the sign of the following polynomial:(19)-4δω02ω05+r1ω04+r2ω03+r3ω02+r4ω0+r5≡-4δω03M1ω0,wherer1=(1/4δ)(3δ2+α1α2), r2=-(1/4)(2β2-α1α2-3α12+6β1), r3=-(1/4δ)α1α2(β2-β1)-(1/2)(2β1-α12)δ, r4=(1/4)(2β12+β1α1α2-α12β2+2β1β2)δ, and r5=-(1/4δ)α1α2β1β2+(1/4)β12δ.(ii)
Whenτ=τj-,(20)cos⁡ω0τj-=α2σk3,sin⁡ω0τj-=-σk32-α22σk3=-δσk3.Under such condition, the sign of real part ofλ′(τj-),j=0,1,2,⋯, depends on the sign of the following polynomial:(21)4δω02ω05-r1ω04+r2ω03-r3ω02+r4ω0-r5≡4δω03M2ω0,Obviously,M1(ω0)=-M2(-ω0) holds for any ω0>0. If M1(ω0)≠0, the two signs of M1,2(ω0) always are opposite. We denote the real part of (dλ/dτ)τ=τj+,τj- by Re⁡(dλ/dτ)τ=τj+,τj-. Since τj+<τj-,j=0,1,2,⋯, if(22)Re⁡λ′τj+>0,Re⁡λ′τj-<0,j=0,1,2,⋯the origin of system (4) always is unstable if ξ=σk3-α2>0. From (19), Re⁡λ′τj+>0 holds only when M1(ω0)<0. If M1(ω0)>0, we have(23)Re⁡λ′τj+<0,Re⁡λ′τj->0,j=0,1,2,⋯.Considering the practical engineering, we focus on the first stability change of the origin in system (4). Under condition (23), the stability of origin in system (4) changes when τ crosses τ=τ0+.In conclusion, ifξ=σk3-α2>0 and τ=0 system (4) has an unstable origin, which means that the lateral vibrations of the footbridge presented in Figure 1 occur. When τ varies near τ=τ0+ and M1>0 (M1>0 is defined in (19)), a Hopf bifurcation appears in system (4). In the next section, we will employ the center manifold theory to analyze the dynamical behavior of system (4) at τ=τ0+ under the condition M1>0.


## 3.1. The Case ofτ=0

We first consider the case ofτ=0. Under such case, (5) becomes(7)λ4+α1-ξλ3+β1+β2-α1ξλ2+α1β2-ξβ1λ+β1β2=0,where ξ=σk3-α2. Since we focus on the excessive vibrations, we need to consider the instable case of the origin in system (4). Obviously, λ=0 is not a root of (7). It is easy to verify that (7) just has a pair of imaginary roots λ=±iβ2 when ξ=0. Regarding λ as a function of ξ in (7) and taking the derivative of λ with respect to ξ, we have(8)dλdξ=λλ2+α1λ+β14λ3-3ξ-α1λ2-2ξα1-β1-β2λ-β1ξ+α1β2.Substituting λ=±iβ2 and ξ=0 into (8) yields(9)dλdξ∣ξ=0=12>0.It means that a pair of imaginary roots cross the imaginary axis from left to right when ξ increases from ξ<0 to ξ>0. Excessive lateral vibrations correspond to the instability of the origin in (4); then in the following analysis we limit our discussion in the case of ξ>0.


## 3.2. The Case ofτ>0

We have proven that the origin of system (4) is unstable when ξ>0 and τ=0 in previous section. Next, we consider the characteristic roots of system (4) in the case of ξ>0 and τ>0. We substitute λ=iω(ω>0) into (5), and separate the real and imaginary parts(10)ω4-p2ω2+p4-p6ω2cosωτ+-p5ω3+p7ωsinωτ=0,-p1ω3+p3ω+-p5ω3+p7ωcosωτ+p6ω2sinωτ=0.Eliminating τ from (10) and denoting w=ω2, we have(11)fw=w4+q1w3+q2w2+q3w+q4=0,where(12)q1=p12-2p2-p52,q2=2p5p7+2p4+p22-p62-2p1p3,q3=p32-2p2p4-p72,q4=p42.According to the physical meanings, it is easy to check that q1<0, q2>0, q3<0, and q4>0. Taking derivative with respect to w in (11), we have(13)f′w=4w3+3q1w2+2q2w+q3.Obviously, f′(w)<0 for w<0. Considering f(0)=q4>0 and f(-∞)→+∞, (11) has no negative root. If (11) has real roots, they must be positive. According to Descartes rule, (11) has a minimum of zero and maximum of 4 positive real roots, and the number of positive roots to (11) always is even. Assume that w=ω02 is one positive real root. From (10) we have(14)cos⁡ω0τ=--p1ω02+p3-p5ω02+p7-p6ω04-p2ω2+p4-p5ω02+p72+p62ω02=α2σk3.If α2>σk3, then cos(ω0τ)>1, which is impossible. If α2≤σk3, it is possible that the stability of the origin of system (4) depends on the size of time delay τ. From (14), the critical value of time delay changing the stability of the origin is given by(15)τj+=1ω02jπ+arccos⁡α2σk3,j=0,1,2,⋯,τj-=1ω02j+1π-arccosα2σk3,j=0,1,2,⋯.In order to determine the crossing direction of the pure imaginary roots ±iω0, we regard λ as a function of τ in (5) and take derivative with respect to τ(16)dλdτ=λ′τ=4λ3+3p1λ2+2p2λ+p3p5λ2+p6λ+p7λ2e-λτ+3p5λ2+2p6λ+p7p5λ2+p6λ+p7λ2-τλ.The crossing direction depends on the sign of real part of (17)dλdτ∣τ=τj+,τj-,j=0,1,2,⋯.Next, we separately discuss the two cases of τ=τj+ and τ=τj-. (i)
Whenτ=τj+,(18)cos⁡ω0τj+=α2σk3,sin⁡ω0τj+=σk32-α22σk3≡δσk3,δ>0.Under such condition, the sign of real part ofλ′(τj+),j=0,1,2,⋯, depends on the sign of the following polynomial:(19)-4δω02ω05+r1ω04+r2ω03+r3ω02+r4ω0+r5≡-4δω03M1ω0,wherer1=(1/4δ)(3δ2+α1α2), r2=-(1/4)(2β2-α1α2-3α12+6β1), r3=-(1/4δ)α1α2(β2-β1)-(1/2)(2β1-α12)δ, r4=(1/4)(2β12+β1α1α2-α12β2+2β1β2)δ, and r5=-(1/4δ)α1α2β1β2+(1/4)β12δ.(ii)
Whenτ=τj-,(20)cos⁡ω0τj-=α2σk3,sin⁡ω0τj-=-σk32-α22σk3=-δσk3.Under such condition, the sign of real part ofλ′(τj-),j=0,1,2,⋯, depends on the sign of the following polynomial:(21)4δω02ω05-r1ω04+r2ω03-r3ω02+r4ω0-r5≡4δω03M2ω0,Obviously,M1(ω0)=-M2(-ω0) holds for any ω0>0. If M1(ω0)≠0, the two signs of M1,2(ω0) always are opposite. We denote the real part of (dλ/dτ)τ=τj+,τj- by Re⁡(dλ/dτ)τ=τj+,τj-. Since τj+<τj-,j=0,1,2,⋯, if(22)Re⁡λ′τj+>0,Re⁡λ′τj-<0,j=0,1,2,⋯the origin of system (4) always is unstable if ξ=σk3-α2>0. From (19), Re⁡λ′τj+>0 holds only when M1(ω0)<0. If M1(ω0)>0, we have(23)Re⁡λ′τj+<0,Re⁡λ′τj->0,j=0,1,2,⋯.Considering the practical engineering, we focus on the first stability change of the origin in system (4). Under condition (23), the stability of origin in system (4) changes when τ crosses τ=τ0+.In conclusion, ifξ=σk3-α2>0 and τ=0 system (4) has an unstable origin, which means that the lateral vibrations of the footbridge presented in Figure 1 occur. When τ varies near τ=τ0+ and M1>0 (M1>0 is defined in (19)), a Hopf bifurcation appears in system (4). In the next section, we will employ the center manifold theory to analyze the dynamical behavior of system (4) at τ=τ0+ under the condition M1>0.


## 4. Center Manifold Reduction for System (4)

For convenience, by lettingyi=ui(tτ)/k3,i=1,2,3,4,τ=τ0++μ, μ is a real small fluctuation and system (4) can be rewritten as(24)u˙1=τ0++μu2,u˙2=τ0++μ-β1u1-α1u2-γ1u13+u1-u33,u˙3=τ0++μu4,u˙4=τ0++μ-β2u3-α2u4+σu4+σk3tanh⁡u4t-1-γ2u1-u33.Linearizing system (24) around its origin yields(25)U˙=L1U+L2Ut-1,where U=u1,u2,u3,u4T and(26)L1=τ0++μ0100-β1-α100000100-β2-α2,L2=τ0++μ000000000001000σk3.For anyϕ=(ϕ1,ϕ2,ϕ3,ϕ4)T∈(C-1,0,R4), we define a linear operator(27)Lμϕ=L1ϕ0+L2ϕ-1.By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions η(θ,μ) in [-1,0]→R4, such that(28)Lμϕ=∫-10dηθ,μϕθ.In fact, if η(θ,μ)=L1δ(θ)+L2δ(θ+1) is chosen, where δ(θ) is the Dirac function, then (27) is satisfied. For ϕ∈(C1[-1,0],R4), define(29)Aμϕ=dϕθdθ,ϕ∈-1,0,∫-10dηθ,μϕθ,θ=0.For ϕ∈(C1[-1,0],R4), let(30)hμ,ϕ=τ0++μ0-γ1ϕ103+ϕ10-ϕ3030-13σk3ϕ4-13-γ2ϕ10-ϕ303+⋯.Further defined is(31)Rμϕ=0,ϕ∈-1,0,hμ,ϕ,θ=0.Thus, (24) can be transformed into the following functional differential equation:(32)U˙t=AμUt+RμUt,where Ut=u1t,u2t,u3t,u4tTand Ut=U(t+θ), θ∈[-1,0].Forϕ∈(C1[-1,0],R4∗), define(33)A∗ψs=-dψsds,s∈0,1,∫-10dηTξ,0ψ-ξ,s=0,where ηT is the transpose of the matrix η. For ϕ∈(C1[-1,0],R4) and ψ∈[C1[0,1],R4∗], define a bilinear form(34)ψs,ϕθ=ψ¯T0ϕ0-∫θ=-10∫ξ=0θψ¯Tξ-θdηθϕξdξ,where η=η(θ,0) and then A(μ) and A∗ are adjoint operators. By the preceding discussions, ±iω0τ0+ are the eigenvalues of A(0) and therefore also the eigenvalues of A∗. Suppose that q(θ) and q∗(θ) are the eigenvectors of A(0) and A∗, respectively, corresponding to the eigenvalues iω0τ0+ and -iω0τ0+, namely,(35)A0qθ=iω0τ0+qθ,A∗q∗s=-iω0τ0+q∗s.By a direct calculation(36)qθ=1iω01iω0eiω0τ0+θ,θ∈-1,0,q∗s=B¯1iω0β11iω0β2eiω0τ0+θ,s∈0,1.Let <q∗,q>=1(37)1B=2+2ω02β2+τ0+σk3ω02β2e-iω0τ0+.It is easy to check that <q∗,q¯>=0. Following the algorithms given by Hassard et al. [18], let(38)zt=q∗,Ut,where Ut=(u1t,u2t,u3t,u4t) is a solution of (24) with μ=0. Then(39)z˙t=q∗,Ut=iω0τ0+zt+q¯∗0h^z,z¯,where(40)h^z,z¯=h0,Wz,z¯,θ+2Re⁡ztqθ,Wz,z¯,θ=Ut-2Re⁡ztqθ.W ( z , z ¯ , θ ) also can be written as(41)Wz,z¯,θ=W20θz22+W11θzz¯+W02θz¯22+⋯.In fact,z and z¯ are local coordinates of the center manifold in the direction of q and q∗, respectively. Rewrite (39) as(42)z˙t=iω0τ0+zt+gz,z¯,where(43)gz,z¯=q¯∗0h^z,z¯=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.Substituting (32) and (39) into W˙=U˙t-z˙q-z¯˙q¯ yields(44)W˙=AW-2Re⁡q¯∗0h^qθ,θ∈-1,0,AW-2Re⁡q¯∗0h^qθ+h^,θ=0.Letting W˙=AW+H(z,z¯,θ), where(45)Hz,z¯,θ=H20θz22+H11θzz¯+H02θz¯22+⋯,and taking the derivative with respect with t on both sides, one has(46)W˙=Wzz˙+Wz¯z¯˙=AW+Hz,z¯,θ.Substituting (41), (42), and (45) into (46) and comparing the coefficients of z and z¯ in (46), we have(47)A-2iω0τ0+W20θ=-H20θ,AW11θ=-H11θ.According to (43),(48)gz,z¯=Bτ0+1-iω0β11-iω0β20-γ1u103+u10-u3030-13σk3u4-13-γ2u10-u303,where Ut(θ)=u1tθ,u2tθ,u3tθ,u4tθT=W(z,z¯,θ)+zq(θ)+z¯q¯(θ) and then(49)u1t0=z+z¯+W2010z22+W1110zz¯+W0210z¯22+⋯,u3t0=z+z¯+W2030z22+W1130zz¯+W0230z¯22+⋯,u4t-1=iω0ze-iω0τ0+-iω0z¯eiω0τ0++W204-1z22+W1130zz¯+W0230z¯22+⋯.It follows that(50)gz,z¯=-2Bσk3ω04τ0+β2eiω0τ0++6iω0γ1β1z2z¯2+⋯,where B is defined in (37). Comparing the coefficients in (50), it follows that(51)g20=g11=g02=0,g21=-2Bσk3ω04τ0+β2eiω0τ0++6iω0γ1β1.Then(52)C10=i2ω0g20g11-2g112-13g022+g212=g212,μ2=-Re⁡C10Re⁡λ′τ0+,β2=2Re⁡C10.It is easy to check that Re⁡{C1(0)}<0. Recalling (23), therefore μ2<0 and β2<0 hold in system (4). According to [19], the Hopf bifurcation is subcritical and the bifurcated periodic solution is stable. From the analysis in the this and the last sections, we find that if τ=0 and ξ=σk3-α2>0, the footbridge has lateral vibrations; if M1>0 (M1>0 is defined in (19)), when the time delay τ increases to cross τ=τ0+, a subcritical Hopf bifurcation happens. The amplitude of periodic solution in system (4) decreases with the time delay increasing in the region of (0,τ0+). And if the time delay varies in the region of (τ0+,τ0-), the origin in system (4) is stable and the amplitude of the periodic solution is zero.


## 5. The Approximate Amplitude of Periodic Solution in System (4)

From the analysis above ( (42), (43), and (50)), when τ=τ0+ the dynamic equation of system (4) on the center manifold can be written as(53)z˙t=iω0τ0+zt+-2Bσk3ω04τ0+β2eiω0τ0++6iω0γ1β1z2z¯2.Consider that there is a small change μ near τ=τ0+. According to the analytical results, if μ>0, the amplitude of the periodic solution in system (4) is not zero. The small change μ near τ=τ0+ results in a change of the real part of the characteristic roots near the imaginary axis, which we defined by Δ(μ). The governing equation of system (4) on center manifold can be written as(54)z˙t=Δμ+iω0τ0+zt+-2Bσk3ω04τ0+β2eiω0τ0++6iω0γ1β1ω03z2z¯.Obviously, Δ(μ) is a continuous function of μ and Δ(0)=0. Since μ is small, we expand Δ(μ) in its Taylor series near μ=0(55)Δμ=Δ0+Δ′0-μ+⋯=Δ′0-μ+⋯.From (16), we can calculate the expression of Δ′(0)(56)Δ′0=h1h3+h2h4h32+h42,where(57)h1=-ω02-ω02α2p5+α2p7+p6ω0δ,h2=ω02-p6ω0α2-ω02δp5+δp7,h3=δτ0+p5ω03+-3α2p5+α2τ0+p6-3p1k3σω02+2p6-τ0+p7δω0+α2p7+p3k3σ,h4=α2τ0+p5-4k3σω03+3p5-τ0+p6δω02+-α2τ0+p7+2p2k3σ+2p6α2ω0-δp7,where pi,i=1,2,…,7, and δ are defined in (5) and (20), respectively.Substitutingz=ρeiϕ into Eq. (54), yields(58)W˙=ρ˙=ρ-Δ′0μτ0++B1ρ2,ϕ˙=ρω0τ0++B2ρ2,where(59)B1=Re⁡-Bσk3ω04τ0+β2eiω0τ0++iω0γ1β1ω03,B2=Im⁡-Bσk3ω04τ0+β2eiω0τ0++iω0γ1β1ω03,Re⁡ and Im⁡ represent real and imaginary parts, respectively. Periodic solution of system (4) occurs only when ρ˙=0 in (58). The amplitude of the periodic solution can be obtained by letting the right side of the first equation in (58) be equal to zero(60)ρ=Δ′0μτ0+B1.


## 6. Numerical Simulations and Discussions

To verify the correctness of the analyze results in Sections4 and 5, we carry out numerical simulations for system (2) by using the parameters of T-bridge [16]: m2=2.1401×105(kg), c2=2.8262×104(kg/s), k=7.307361×106(kg/s2), Mpg=1.4969×105(N), k1=0.0987, and k2=0.2, G(fB)=1. In addition, we let m1=2.1401×103(kg), c1=2.8262×102(kg/s), T0=7.24×105(N), EA/2L3=1.68×106(kg/s2), and k3=18 and L=10(m). Then we have α1=α2=0.1321(1/s), β1=1.4480×105(N/m), β2=34.1450(1/s2), γ1=7.8501×102(1/s2), γ2=7.8501(1/s2), σ=k1k2G(fB)Mpg/m2=0.0138, ξ=σk3-α2=0.1165>0, and δ=(k3σ)2-α22=0.2105.View the time delayτ as the bifurcating parameter in system (2). For the above given parameters, (11) has two different positive real roots: ω1=5.739 and ω2=5.9496. Then we take ω0=5.9496. From (15), we have τ0+=0.1699 and τ0-=0.8862. It is easy to check that M1(ω0)>0, which is defined in (19). Therefore,(61)λ′τ0+<0,g21τ0+=-0.0113+0.1145i,μ2τ0+=-Re⁡g21τ0+λ′τ0+<0,β2τ0+=2Re⁡g21τ0+<0,λ′τ0->0,g21τ0-=-0.0609+0.0089i,μ2τ0-=-Re⁡g21τ0-λ′τ0-<0,β2τ0-=2Re⁡g21τ0-<0.A subcritical Hopf bifurcation happens in system (2) near τ=τ0+=0.1699; and a supercritical Hopf bifurcation appears in system (2) near τ=τ0-=0.8862. With the time delay increasing from 0 to 0.1699, the amplitude of periodic solution in system (2) decreases. When the time delay varies in the region of (0.1699,0.8862), the amplitude reduces to zero. Once the time delay exceeds 0.8862, the amplitude begins to be not zero again. The bifurcating diagrams are presented in Figure 2.Figure 2
The bifurcating diagram of Hopf bifurcation in system (2), in which solid-line and dot-line represent stable and unstable solutions, respectively.We consider thatτ increases from 0.03 to 0.15, and the initial conditions are taken as x1(0)=0.2, x˙1(0)=0 and x2(0)=0.01, x˙2(0)=0. The influence of the time delay on the amplitude of periodic solution in system (2) is depicted in Figure 3. The amplitude of the concentrated mass m1 always approaches zero regardless of the value of time delay. The amplitude of m2 keeps decreasing with the time delay increasing. Figure 3 shows that the periodic solution indeed decreases its amplitude when time delay varies in the region (0,τ0+). Next, we choose τ=0.3 (τ0+=0.1699<0.3<τ0-=0.8862) in system (2) to carry out numerical simulations. The initial conditions are still chosen as x1(0)=0.2 and x˙1(0)=0 and x2(0)=0.01 and x˙2(0)=0. The numerical results are presented in Figure 4. Both m1 and m2 approach their equilibrium positions, which illustrates the correctness of analysis results in previous sections. Then, we select τ=0.95 (τ0-=0.8862<0.95) to numerically calculate the time history curves of m1 and m2 in system (2), in which the initial conditions keep unchanged. The numerical results are provided in Figure 5, which verifies that the amplitude of periodic solution in system (2) becomes not zero again once the time delay is greater than τ0-=0.8862. Since m1 in system (2) has no periodic vibration for any value of time delay, we numerically calculate the amplitude of periodic vibrations of m2 for different time delays and compare with the analytical results based on (60). The initial conditions for numerical calculations are still given by x1(0)=0.01, x˙1(0)=0 and x2(0)=0.01, and x˙2(0)=0. τ increases from 0.11 to 0.17. The result of the comparison is given in Figure 6, which indicates the theoretical analysis results have enough precision in the amplitude calculation of periodic solutions in system (2). From the numerical simulations, it is clear that the increase of time delay always reduces the amplitude of vibrations of the bridge in the region of (0,τ0+).Figure 3
The influence of the time delay on the amplitude of periodic vibration in system (2) when τ increases from 0.03 to 0.15. (a) The time history curves of m1, the amplitude always approaches to zero. (b) The amplitude change of m2.
(a)
The time history curves ofm1 (b)
The amplitude change ofm2 with τFigure 4
The time history curves ofm1,2 in system (2) when τ=0.3 (a) m1 and (b) m2.
(a)
The time history curves ofm1 (b)
The time history curves ofm2Figure 5
The time history curves ofm1,2 in system (2) when τ=0.95 (a) m1 and (b) m2.
(a)
The time history curves ofm1 (b)
The time history curves ofm2Figure 6
The comparison between numerical and analytical results for the amplitude of periodic vibration ofm2 in system (2).Finally, we discuss the influence of cables on the amplitude of the bridge according to (60). It is easy to verify that the expression EA/(2L3) does not appear in (60). All bridge parameters used in the numerical simulations in this section are fixed except for T0. Figure 7 shows how the parameter T0 affects the amplitude of the bridge for different values of μ. In Figure 7(a), we let μ=0.05. This means that τ=τ0+-μ=0.1699-0.05=0.1199. In Figure 7(b), we let μ=0.1 and then τ=τ0+-μ=0.0699. The parameter T0 always varies in the region of (2×102,2×105). It is apparent that the amplitude decreases with T0 increasing. The decay of the amplitude is quick when T0 is not very large and stops when T0 is large enough. This means that the existence of cables reduces the amplitude of a footbridge.Figure 7
The influence ofT0 on the amplitude of periodic solution in system (2) for different values of μ.
(a)
μ = 0.05 (b)
μ = 0.1


## 7. Conclusions

Excessive lateral vibrations of footbridges induced by pedestrians have been investigated from many perspectives. Nakamura’s model is a famous model describing the interactive dynamics between footbridge and pedestrians. According to Nakamura’s method, predicted responses are easily compared with measured results. Nevertheless, the predicted amplitudes for T-bridge and M-bridge in Japan derived by using Nakamura’s model usually were larger than the measurements. In fact, T- and M-bridges are cable-stayed bridges, which are not appropriate to be simplified as a single degree of freedom system. It is more reasonable to use at least two degrees of freedom to describe the dynamical behavior of a cable-stayed bridge. The cables have one, and the bridge deck has the other. In order to find the cause why the predicted results of Nakamura’s model are larger than the measure data, we in this paper establish a two-degrees-of-freedom model to analyze the dynamics of a cable-stayed footbridge induced by pedestrians. We still use Nakamura’s force model to describe the lateral force induced by pedestrians. In addition, we consider a time delay in the interaction between the footbridge and the pedestrians. Excessive lateral vibrations correspond to a Hopf bifurcation in the two-degrees-of-freedom model. Then we first discuss the occurrence conditions of the Hopf bifurcation based on the characteristic roots analysis. Then we employ the center manifold theory to determine the direction of the Hopf bifurcation and the amplitude of the bifurcating limit cycle. Numerical simulations are carried out to demonstrate the correctness of the theoretical analysis.According to analysis for the two-degrees-of-freedom model presented in this paper, we find that the cables and the time delay in interaction between footbridge and pedestrians have significant effects on the amplitude of bridge deck. The material parameters of cables do not affect the lateral vibration amplitude of the footbridge. But the increase of the tension in cables can distinctly reduce the amplitude of the footbridge when the tension is not very large. If the tension is large enough, the change of tension almost has no influence on the amplitude. The time delay is another key factor affecting the amplitude of the footbridge. The time delay can induce a subcritical Hopf bifurcation in the two-degrees-of-freedom model. With the time delay increasing between 0 and the critical time delay causing the subcritical Hopf bifurcation, the amplitude of the footbridge keeps decreasing. Therefore, we draw the conclusion that if we consider the influence of cables and time delay in interaction in Nakamura’s model, its predicted responses can better agree with the measured data.



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