Abstract Modern pedestrian and suspension bridges are designed using industry standard packages, yet disastrous resonant vibrations are observed, necessitating multimillion dollar repairs. Recent examples include pedestrian-induced vibrations during the opening of the Solférino Bridge in Paris in 1999 and the increased bouncing of the Squibb Park Bridge in Brooklyn in 2014. The most prominent example of an unstable lively bridge is the London Millennium Bridge, which started wobbling as a result of pedestrian-bridge interactions. Pedestrian phase locking due to footstep phase adjustment is suspected to be the main cause of its large lateral vibrations; however, its role in the initiation of wobbling was debated. We develop foot force models of pedestrians’ response to bridge motion and detailed, yet analytically tractable, models of crowd phase locking. We use biomechanically inspired models of crowd lateral movement to investigate to what degree pedestrian synchrony must be present for a bridge to wobble significantly and what is a critical crowd size. Our results can be used as a safety guideline for designing pedestrian bridges or limiting the maximum occupancy of an existing bridge. The pedestrian models can be used as “crash test dummies” when numerically probing a specific bridge design. This is particularly important because the U.S. code for designing pedestrian bridges does not contain explicit guidelines that account for the collective pedestrian behavior.

INTRODUCTION Collective behavior in mechanical systems was first discovered by C. Huygens some 350 years ago (1). He observed two pendulum clocks, suspended on a wooden beam, spontaneously synchronize. The pendula oscillated, remaining locked in antiphase, whereas the beam remained still. Pendula with the same support coupling mechanism can also oscillate in-phase, in turn making the beam vibrate in antiphase (2–4). Notably, increasing the number of in-phase synchronized pendula attached to the supporting beam leads to larger amplitudes of the swinging beam (5, 6). Originating from Huygens’s experiment in simple mechanical systems, the interplay between network structure and cooperative dynamics has been extensively studied (7–12), as cooperative dynamics and phase locking have been shown to play an important role in the function or dysfunction of a wide spectrum of biological and technological networks (13–16), including complex mechanical structures and pedestrian bridges (17). Many bridges have experienced dramatic vibrations or even have collapsed because of the effects of mechanical resonance [see the related Wikipedia entry (18) for a long list of bridge failures]. There were two major causes for the dangerous vibrations: (i) pedestrian excitation of laterally unstable bridges (19–21) and (ii) wind-induced vibrations of suspension and girder bridges, including the collapse of the Tacoma Narrows Bridge (22–26). Some of the most well-known cases of unstable pedestrian bridges are the Toda Park Bridge in Japan (27), the Solférino Bridge in Paris (28), the London Millennium Bridge (29), the Maple Valley Great Suspension Bridge in Japan (30), the Singapore Airport’s Changi Mezzanine Bridge (31), the Clifton Suspension Bridge in Bristol, U.K. (32), and the Pedro e Inês Footbridge in Portugal (33). A recent example is the Squibb Park Bridge in Brooklyn (34), described in a quote from a 2014 New York Times report (35): “This barely two-year-old wooden structure, which cost $5 million, was purposefully designed to bounce lightly with the footsteps of visitors-reminiscent of trail bridges—but over time the movement has become more conspicuous.” The bridge started to move too much, and not just up and down, but also side to side. The increased bouncing and swaying were a safety concern for pedestrians walking 50 feet over the park. The Brooklyn Bridge Park Corporation sought $3 million in a lawsuit against HNTB Corporation, the bridge’s original designers. A new contractor, Arup, the British engineering firm that designed the London Millennium Bridge, was then hired to develop and oversee a plan to improve the stability of the Squibb Park Bridge. Arup fixed a truss-like structure beneath the bridge and added large mass dampers that reduced bouncing more than half of what it was before [see the 2017 New York Times report for more details (36)]. Three years after it initially closed for repairs, the bouncy Squibb Park Bridge reopened to the public in April 2017 (37). Interest in pedestrian collective dynamics was significantly amplified by the London Millennium Bridge, which started wobbling after its opening in 2000 (19–21). Despite significant interest, the interaction between walking pedestrians and the London Millennium Bridge has still not been fully understood. The wobbling of the London Millennium Bridge is suspected to have been initiated by a cooperative lateral excitation caused by pedestrians falling into step with the bridge’s oscillations (29, 38–42). It was noted though that phase locking on the London Millennium Bridge was not perfect, and pedestrians repeatedly tuned and detuned their footstep phase with the lateral bridge motion (43). This effect of lateral excitation of footbridges by synchronous walking was studied in recent papers (29, 38–42), including modeling the crowd synchrony by phase oscillator models (40–42). These models explain how a relatively small synchronized crowd can initiate wobbling. As nice as these phase models are, they do not fully capture a bifurcation mechanism of the abrupt onset of wobbling oscillations that occurred when the number of pedestrians exceeded a critical value [about 165 pedestrians on the London Millennium Bridge (29)], especially in the limiting case of identical walkers for which the phase oscillator network has no threshold for instability, and therefore the bridge would start wobbling for even a very small crowd size (42). Rigorous analysis of the phase model system (40–42) is based on the heterogeneity of pedestrians’ frequencies, which is a natural assumption; however, the frequencies’ density distribution is continuous, thereby implicitly assuming an infinite number of pedestrians. Here, we develop a biomechanically inspired model of pedestrians’ response to bridge motion and detailed, yet analytically tractable, models of crowd synchrony and phase locking that take into account the timing and impact of pedestrian foot forces and assume a finite number of pedestrians. Our analysis predicts the critical number of pedestrians and its dependence on the frequencies of human walking and the natural frequency of the London Millennium Bridge remarkably well. Our results support the general view that pedestrian lock-in was necessary for the London Millennium Bridge to develop large-amplitude wobbling. However, our observations also suggest that the popular explanation that the wobbling of the London Millennium Bridge was initiated by crowd synchrony among pedestrians may be an oversimplification. Surprisingly, the U.S. code and industry standard packages for designing pedestrian bridges do not explicitly address an impact of crowd collective behavior but rather relies on a higher nominal static pedestrian load and testing via the application of a periodic external force (44). The Guide Specifications for the Design of Pedestrian Bridges [(44), chapter 6] set lower limits on the fundamental frequency of a bridge in the vertical and horizontal directions to 3.0 and 1.3 Hz, respectively. However, these limits might not be respected, provided that there is “phasing of loading from multiple pedestrians on the bridge at the same time” (44). The code gives no specifications on how to model and describe these phase loads. In light of this, our biomechanical models of pedestrian locomotion and their bidirectional interaction with a lively bridge can be incorporated into industry standard bridge simulation packages to provide a more accurate account for the emergence of undesired crowd-induced resonant vibrations. The layout of this paper is as follows: In the “Pendulum Models of Walker-Bridge Interactions” section, we first introduce an inverted pendulum model of pedestrian balance, where the control of the position of foot placement plays a significant role in lateral stabilization. We also propose a simplified, more analytically tractable model with a van der Pol–type oscillatory mechanism of each pedestrian’s gait. In the “Predicting the Onset of Bridge Wobbling” section, we study bidirectional interactions between the van der Pol–type pedestrian models and a bridge and derive explicit analytical conditions under which phase locking and bridge wobbling appear in the system when the crowd size exceeds a threshold value. In the “Nonidentical Inverted Pendulum Walkers” section, we study the pedestrian-bridge interaction model, where pedestrians’ lateral gaits are described by an inverted pendulum model. This analysis also reveals the threshold effect that induces bridge wobbling at a sufficiently large crowd size.

PENDULUM MODELS OF WALKER-BRIDGE INTERACTIONS We model lateral oscillations of the bridge and side-to-side movement of walkers by a mass-spring-damper system (Fig. 1). The bridge is represented by a platform of mass M that moves in the horizontal direction with damping coefficient d. One side of the platform is attached to the support via a spring with elasticity k. The platform is subjected to horizontal forces, caused by the walker response to lateral bridge motion. The walkers are modeled by n self-sustained oscillators, representing walker lateral balance and capable of adjusting their footfall forces and amplitudes in response to the lateral wobbling of the bridge. The modeling equations read (1)where the x i equation describes the oscillatory side-to-side motion of the ith walker. The oscillatory mechanism and response of each walker’s gait to the bridge’s vibrations are modeled by the function f(x i , x i ), and the effect of the bridge on the ith walker is described by the inertial feedback term . The y equation describes bridge oscillations. Without loss of generality, each walker is assumed to have mass m; heterogeneity of the walkers will be introduced by nonidentical gait frequencies. The ith walker applies sideways force to the bridge. The parameter h = d/(2(M + nm)) is the normalized, dimensionless damping coefficient. is the natural frequency of the bridge loaded with n walkers. The ratio r = m/(M + nm) represents the strength of the coupling between the walkers and the bridge. Model 1 takes into account the role of walkers’ footfall forces and their adaptation. It extends the models (40–42) that describe the crowd synchrony, where walkers are modeled by phase oscillators and therefore only account for the timing of each walker’s gait. As in earlier studies (40–42), no person-person interactions, including visual communication between the walkers and moving in dense crowds at a slower pace, are included in the model (Eq. 1). Fig. 1 A Huygens-type setup as a mechanical model for lateral vibrations of a bridge. (Left). The platform with mass M, a spring, and a damper represents lateral vibrations of the bridge y. Pedestrians are modeled by self-sustained oscillators, representing walker lateral balance that are capable of adjusting the position of their centers of mass and are subjected to lateral bridge motion. (Middle) Inverted pendulum model of pedestrian lateral movement. Variable x is the lateral position of pedestrian’s center of mass. The constant p defines the lateral position of the center of pressure of the foot. L is the inverted pendulum length. (Right) The corresponding limit cycle in the inverted pendulum model (Eq. 3) along with its acceleration time series. Parameters are as in Fig. 8. Function , determines a self-sustained oscillatory mechanism of the ith walker gait dynamics and takes into account the role of walkers’ footfall force and gait adaptation to bridge oscillations. While we anticipate that various forms of , ensuring self-excitation of walkers’ movement, can induce significant bridge oscillations via Huygens’ excitation mechanism (1–6), we propose two particular types of function to formulate two foot force walker models. The first biomechanically inspired oscillator model is an extension of the inverted pendulum pedestrian models (32, 43, 45, 46). The second, more analytically tractable model uses a simple van der Pol–type excitation mechanism. Represented by two different , both models reveal similar effects and predict the critical number of walkers [n = 165, (29)], beyond which the sharp onset of wobbling occurred on the London Millennium Bridge. Our study of the inverted pendulum model also indicates that crowd phase locking stably appears in the system and supports the observation of crowd synchrony on the London Millennium Bridge, especially during large-amplitude vibrations. However, our simulations indicate that the initiation of wobbling is accompanied by multiple phase slips and nonsynchronous adjustments of pedestrian footsteps such that a phase lock-in mechanism might not necessarily be the main cause of initial small-amplitude wobbling. Inverted pendulum walker model Many studies in biomechanics have confirmed that an inverted pendulum model (43, 45) can be successfully applied to the analysis of whole-body balance in the lateral direction of human walking (47–49). The most popular biomechanical model (43) of the individual walker on a rigid floor is an inverted pendulum that is composed of a lumped mass m at the center of mass supported by rigid massless legs of length L (see Fig. 1, middle). Its equations read (43) (2)where x and p are the horizontal displacements of the center of mass and center of pressure of the foot, respectively, and , with g being the acceleration due to gravity and L being the distance from the center of mass to the center of pressure. Although this model captures the main time characteristics of human walking such as lateral displacement, velocity, and acceleration rather precisely [see the study of Bocian et al. (43) and that of Macdonald (46) for details], it has several limitations, especially when used in the coupled walker-bridge model (Eq. 1). From a dynamical systems theory perspective, this is a conservative system with no damping; its phase portrait is formed by three fixed points: a center at the origin, which is confined in a diamond-shape domain, bounded by the separatrices of two saddles at (p, 0) and (−p, 0). The periodic motion is then governed by closed conservative curves of the center fixed point. As a result, the amplitude of a walker’s lateral gait depends on the initial conditions (via the corresponding level of the closed curves of the center) and is determined by the size of the first step, although this step might be too big or too small. Note −x, not +x in Eq. 2. This implies that the solutions of each of the two systems for the right and left feet, comprising the piecewise linear system (Eq. 2), are not cosine and sine functions as with the harmonic oscillator but instead hyperbolic functions. Therefore, the closed curves, “glued” from the solutions of the two systems, have a distinct diamond-like shape, yielding realistic time series for lateral velocity and acceleration of human walking (see Fig. 1, right). We propose the following modification of the pedestrian model (Eq. 2) to make it a self-sustained oscillator (3)where λ is a damping parameter, and parameter ν controls the period of the limit cycle. The auxiliary parameter a controls the amplitude of the limit cycle (lateral foot displacement) via the difference (p − a) [see the study of Belykh et al. (50) for details]. Note that System 3 has a stable limit cycle (Fig. 1, right), which coincides with a level of the nonlinear center of the conservative System 2 for ν = ω 0 . The period of the limit cycle can be calculated explicitly as the limit cycle is composed of two glued solutions for the right and left foot systems where x = a cosh νvt and ẋ = aνv sinh νvt (for ν = ω 0 ) correspond to one side of the combined limit cycle, glued at x = 0. The use of System 3 instead of the conservative System 2 in the coupled system allows one to analyze the collective behavior of the walker-bridge interaction much more effectively, yet preserves the main realistic features of human walking. System 3 can also be written in a more convenient form (4)defined on a cylinder and obtained by shifting x to x + p. As a result, the two saddles shift to x = 0 and x = 2p, and the limit cycle becomes centered around x = p. In the following, we will use Eq. 3 as the main model of the individual walker in our numerical studies of the coupled walker-bridge model (Eq. 1) with , defined via the right-hand side (RHS) of Model 3. The analytical study of the coupled model (Eqs. 1 to 4) is somewhat cumbersome and will be reported in a more technical publication; we will further simplify the walker-bridge model (Eqs. 1 to 4), incorporating a van der Pol–type excitation mechanism and making the analysis more elegant and manageable. Van der Pol–type walker model An oscillatory mechanism of each walker’s gait, modeled by the function in Eq. 4, can be similarly described by the van der Pol–type term . Therefore, the walker-bridge model (Eqs. 1 to 4) can be transformed into the following system (5)where the dimensionless time τ = νt, , and Ω = Ω 0 /ν. Walkers are assumed to have different parameters ω i , randomly distributed within the range [ω − , ω + ]. In relation to the frequency of pedestrian walking, parameter ω i does not directly set the actual footfall frequency of the ith pedestrian because this frequency is also controlled by parameters a and λ. In the absence of bridge movement, the ith van der Pol–type oscillator has a limit cycle with a frequency that lies within or near the interval (ω i , 1) (see Materials and Methods for the justification). Note that the equations for the limit cycle of the individual van der Pol–type oscillator cannot be found explicitly, yet we will be able to find phase-locked solutions in the coupled pedestrian-bridge model (Eq. 5). It is important to emphasize that this model is an extension of the London Millennium Bridge reduced (phase) models (40–42), where the dynamics of each walker is modeled by a simple phase oscillator with phase Θ i . In terms of Model 5, the Eckhardt et al. model (41, 42) reads , where ω i is the natural stepping frequency of walker i, c i is a coupling constant reflecting the strength of the response of walker i to a lateral force, and c is the maximum sideways force of walker i applied to the bridge. The frequencies ω i are randomly chosen from a continuous density distribution P(ω), thereby assuming an infinite number of pedestrians. The Eckhardt et al. model allows for a reduction to a Kuramoto-type network and derivation of analytical estimates on the degree of coherence among the phase oscillators that causes the bridge oscillations to amplify. That is, the wobbling begins as a Kuramoto-type order parameter gradually increases in time when the size of the synchronized group increases in time (40–42). However, these phase models, based on the widespread notion of footstep phase adjustment as the main mechanism of synchronous lateral excitation, have the abovementioned limitations, including the absence of a threshold on the critical number of identical pedestrians, initiating abrupt wobbling. The presence of the feedback term in the x i - equation of Model 5 has been the main obstacle, preventing the rigorous transformation of Eq. 5 to an analytically tractable model in polar coordinates. Because these limitations call for more detailed models, also taking into account foot forces and position of foot placement, we use the biomechanically inspired model (Eqs. 1 to 4) and its van der Pol–type simplification (Eqs. 1 to 5) to make progress toward a mathematical understanding of the mechanism of the London Millennium Bridge’s vibrations. We begin with rigorous analysis of the model (Eqs. 1 to 5) and its implications to the London Millennium Bridge. We will derive an explicit bound on the critical crowd size needed for the onset of wobbling in the case of both identical and nonidentical pedestrians. We will also numerically validate this bound for both the inverted pendulum and the van der Pol–type models and obtain an excellent fit.

NONIDENTICAL INVERTED PENDULUM WALKERS We return to the most realistic case of bioinspired inverted pendulum pedestrian-bridge model (Eqs. 1 to 3) with a 10% parameter mismatch (see Fig. 8). We randomly choose the parameters ω i from the interval [0.6935, 0.7665], which is centered around ω = 0.73 used in the above numerical simulations of the identical van der Pol–type pedestrian models (Eq. 5) (cf. Fig. 4). We choose other free individual pedestrian parameters as follows: λ i = λ = 2. 8, p = 2, and a = 1. This choice, together with the fixed parameters for the London Millennium Bridge (Figs. 4 and 8), gives the threshold value around n c = 160. Therefore, we consistently obtain nearly the same threshold for both van der Pol–type and inverted pendulum models. It is worth noticing that numerical simulations of the inverted pendulum pedestrian-bridge model (Eqs. 1 to 3) with identical oscillators (ω i = ω = 0.73) indicate the same threshold effect and nearly the same dependence of the bridge amplitude on the crowd size as in the nonidentical oscillator case of Fig. 8 (left) and therefore are not shown. Fig. 8 i ∈ [0.6935, 0.7665] (10% mismatch). Inverted pendulum model ( Eqs. 1 to 3 ) of nonidentical pedestrians with randomly chosen parameters ω∈ [0.6935, 0.7665] (10% mismatch). Diagrams similar to Fig. 4. The onset of bridge oscillations is accompanied by a drop in the average phase difference between the pedestrian’s foot adjustment. The initial drop corresponding to the initiation of the bridge wobbling and partial phase locking is less significant, compared to the well-established phase locking at larger crowd sizes over 200 pedestrians. The individual pedestrian parameters are λ = 2.8, p = 1, and a = 1. Other parameters are as in Fig. 4. Models 1 to 3 with identical walkers produce similar curves with nearly the same critical crowd size; however, the phase difference drops to 0 (because complete phase locking is possible for identical oscillators). Figures 8 and 9 also indicate that well-developed phase locking is inevitably present when the number of pedestrians exceeds n c ; however, the abrupt initiation of bridge wobbling right at the edge of instability around the critical number does not exactly coincide with total phase locking among the pedestrians and is accompanied by multiple phase slips and detuning. This suggests that pedestrian phase locking is crucial for a highly inertial system such as the London Millennium Bridge weighing over 113 tons to develop significant wobbling. However, its initiation mechanism at the edge of instability can be more complicated than simple phase locking. In particular, the balance control based on the lateral position of foot placement can initiate low-amplitude bridge wobbling before the onset of crowd synchrony at larger crowd sizes, as suggested by Macdonald (46). Fig. 9 Diagrams for the mismatched inverted pendulum pedestrian-bridge model ( Eqs. 1 to 3 ), similar to Fig. 5 The full videos for both identical and mismatched inverted pendulum models are given in the Supplementary Materials.

CONCLUSION The history of pedestrian and suspension bridges is full of dramatic events. The most recent examples are (i) the pedestrian-induced vibrations during the opening of the Solférino Bridge in Paris in 1999 (28) and the London Millennium Bridge in 2000 and (ii) the increased swaying of the Squibb Park Bridge in Brooklyn in 2014 (34) and the Volga Bridge (56) in the Russian city of Volgograd (formerly Stalingrad) (cost $396 million and 13 years to build), which experienced wind-induced resonance vibrations in 2011 soon after its opening and was shut down for expensive repairs. Parallels between wind and crowd loading of bridges have been widely discussed (57) because intensive research on the origin of resonant vibrations, caused by the pedestrian-bridge interaction and wind-induced oscillations, can have an enormous safety and economic impact. Here, we have contributed toward understanding the dynamics of pedestrian locomotion and its interaction with the bridge structure. We have proposed two models where the individual pedestrian is represented by a biomechanically inspired inverted pendulum model (Eq. 3) and its simplified van der Pol–type analog. We have used the parameters close to those of the London Millennium Bridge to verify the popular explanation that the wobbling of London Millennium Bridge was initiated by phase synchronization of pedestrians falling into step with the bridge’s oscillations. The analysis of both models has indicated the importance of the inclusion of foot force impacts for a more accurate prediction of the threshold effect. Surprisingly, the pedestrian-bridge system with a van der Pol–type oscillator as an individual pedestrian model allows for the rigorous analytical analysis of phase-locked solutions, although periodic solutions in the individual van der Pol–type oscillator cannot be found in closed form. Although we have used the van der Pol–type model of pedestrian gait to analytically elucidate the nontrivial relation between the pedestrians’ and bridge dynamics, the numerical analysis of both the inverted pendulum and the van der Pol–type models reveals the same threshold effect for the sudden onset of bridge wobbling when the crowd size exceeds a critical value. This threshold effect is also present in the case of identical pedestrians such that a critical number of pedestrians are necessary for the bridge to abruptly develop significant wobbling. This is in contrast with the widespread view that identical pedestrians should always excite a bridge to wobble, with the amplitude of wobbling gradually increasing with the crowd size. Our numerical study of the inverted pendulum pedestrian-bridge model confirms that crowd phase locking was necessary for the London Millennium Bridge to wobble significantly, especially at intermediate and large amplitudes. However, our simulations indicate that the initiation of wobbling is accompanied by tuning and detuning of pedestrian footstep such that a phase lock-in mechanism might not necessarily be the main cause of initial small-amplitude wobbling. A rigorous analysis of the inverted pendulum-bridge model to reveal secondary harmonics at which the wobbling can be initiated without total phase locking is a subject of future study. The initiation of wobbling without crowd phase locking was previously observed during periods of instability of the Singapore Airport’s Changi Mezzanine Bridge (28) and the Clifton Suspension Bridge (32). Both bridges experienced crowd-induced vibrations at a bridge frequency different from the averaged frequency of the pedestrians, while the pedestrians continued to walk without visible phase locking (46). Our recent results (50) on the ability of a single pedestrian to initiate bridge wobbling when switching from one gait to another may shed light on the initiation of small wobbling without crowd synchrony. More detailed models of pedestrian-bridge interaction that incorporate additional factors, such as, for example, an extra degree of freedom that accounts for the knee control in the individual pedestrian model and person-to-person visual communication and pace slowing in dense crowds, may also suggest an additional insight into the origin of this small-amplitude wobbling. From the dynamical systems perspective, this is an important piecewise smooth problem that requires careful mathematical study of the dynamics of nonsmooth oscillators (pedestrians), with the nonlinear coupling within the crowd and a bidirection interaction with a bridge structure. Our theory and models should help engineers to better understand the dynamical impact of crowd collective behavior and guarantee the comfort level of pedestrians on a bridge. This requirement represents a major challenge because the natural frequency of an aesthetically pleasing lively bridge often falls into a critical frequency range of pedestrian phase locking. These frequencies cannot be identified through the conventional linear calculations that might lead to faulty designs.

MATERIALS AND METHODS Bounds for a limit cycle in the van der Pol–type walker model In the absence of bridge movement ( ) in the pedestrian-bridge model (Eq. 5), the equation of motion for the ith pedestrian becomes (28) Applying the technique from Belykh et al. (50), we will prove the existence of a limit cycle in System 28 and give estimates on its frequency. We introduce as a directing function for System 28. Its derivative along the trajectories of System 28 reads . This derivative is 0 on the circle in the phase plane of System 28 and is positive outside the circle C 1 and negative inside it. Without loss of generality, we assume that ω i < 1. We choose two levels of the directing function V: ellipses and , which inscribe and circumscribe the circle C 1 , respectively. Therefore, the derivative is positive (negative) on the level V 1 (V 2 ), except for the intersection points with the circle C 1 where . As a result, the trajectories of System 28 enter the annulus E = {V 1 < V < V 2 }, which contains a stable limit cycle. This claim also holds true for ω i > 1, with V 1 and V 2 interchanged. Observe that x = a sin ω i t and are solutions of the differential equations, governing V 1 and V 2 , respectively. Hence, the frequency of rotation along the two curves is ω i . Notice that the rotation along other level functions of V within E also has frequency ω i . At the same time, the frequency of rotation along the circle C 1 equals 1, as x = a sin t is a solution of the differential equation , determining C 1 . Although the limit cycle is bounded by the two level curves V 1 and V 2 , it also intersects the circle C 1 . Thus, its frequency lies within or near the interval (ω i , 1). In the limiting case of ω i → 1, the levels V 1 and V 2 and the circle C approach each other and merge; subsequently, the frequency of the limit cycle is 1. Notice that when the damping parameter λ is small, System 28 is nearly the harmonic oscillator, and therefore, the frequency of the limit cycle is close to ω i . On the other hand, when λ is large and the van der Pol term becomes dominant, the frequency is close to 1. Numerical simulations All simulations were performed by integrating the systems of differential equations using an eighth-order Runge-Kutta method with a time step of 0.01. The number of equations for each simulation depends explicitly on the number of pedestrians on the bridge. To eliminate transient behaviors, numerical simulations were run for a final time on the order of 100 times the period of the bridge and pedestrian oscillations (t final = 5000).

SUPPLEMENTARY MATERIALS Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/11/e1701512/DC1 movie S1. Identical van der Pol.avi is related to Fig. 5. movies S2 and S3. Inverted pendula identical.avi and inverted pendula 10% mismatch.avi are related to Fig. 9.

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Acknowledgments: We acknowledge guidance with existing bioinspired inverted pendulum models of pedestrian gait and the role of balance control from A. Champneys and J. Macdonald, both from the University of Bristol, U.K. We thank D. Feliciangeli, a bridge engineer with Mott Macdonald Group, for pointing us to the U.S. Guide Specifications for the design of pedestrian bridges and useful discussions. Funding: This work was supported by the U.S. National Science Foundation under grant no. DMS-1616345 (to I.B. and R.J.), the Russian Science Foundation under grant no. 14-12-00811, and the Russian Foundation for Fundamental Research under grant no. 15-01-08776 (to V.B.). Author contributions: I.B. conceived the project and drafted the manuscript. I.B. and V.B. developed the models and contributed to their theoretical analysis. R.J. performed simulations. All authors contributed to the editing of the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.