Enlarge / London's Millennium Bridge had issues with excessive shaking and swaying when it first opened in June 2000.Alberto Pezzali/NurPhoto/Getty Images

When London's Millennium Bridge first opened in June 2000, the city was alarmed to discover that the motion of crowds of pedestrians crossing it gave rise to significant shaking and swaying. Londoners nicknamed it "Wobbly Bridge." Officials shut it down after just two days, and the bridge remained closed for the next two years until appropriate modifications could be made to stop the swaying.

It's not an entirely unknown phenomenon: there's a sign dating back to 1873 on London's Albert Bridge warning military troops to break their usual lock-step motion when crossing. The culprit was not Millennium Bridge's design. Rather, it was due to a weird synchronicity between the bridge's lateral (sideways) sway and pedestrians' gaits.

A new paper in Biology Letters sheds further light on this by simulating the biomechanics of large crowds of people walking on a bridge. While there have been many different approaches to studying these fascinating dynamics over the years—including a lab-based treadmill recreation of people walking across Millennium Bridge by Cambridge University engineer Allan McRobie—this is a significantly improved model of how people adjust their gait when walking on a wobbly surface, according to co-author Varun Joshi of Ohio State University. It suggests that one might not even need synchronization to cause the shaking.

Synchronized swaying

It turns out that people walking on a bridge that starts to shift will instinctively adjust their stride to match the bridge's swaying motion as it lurches sideways. This will be familiar to anyone who has tried to walk on a fast-moving train and needed to find steady footing as the train wobbled from side to side. But on a bridge, this exacerbates the problem, giving rise to additional small sideways oscillations that amplify the swaying.

The result is a positive feedback loop (the technical term is "synchronous lateral excitation"). Get a large enough crowd matching their stride to the bridge's motion, and the swaying can become dangerously severe, as happened with the Millennium Bridge. Approximately 90,000 people crossed the bridge on opening day, with around 2,000 people on it at any given time.

"Wobbling and synchrony are inseparable. They emerge together, once the crowd reaches a critical size."

It's an example of an emergent collective phenomenon. In fact, Cornell University mathematician Steven Strogatz co-authored a 2005 Nature paper with McRobie and two others that modeled the dynamics of the Millennium Bridge as a weakly damped and driven harmonic oscillator. According to Strogatz, the bridge was driven to sway sideways by the pedestrians as they walked across it. "Their periodic footfalls pumped energy into the bridge and caused it to move from side to side, which in turn caused the people to adjust their gaits to conform to the movement of the bridge," he says.

Over time, the pedestrians inadvertently fell into sync with each other and thereby caused the bridge to wobble even more severely. The spontaneous synchrony of the crowd was similar to what happens with the highly synchronized flashing of fireflies or firing of neurons in the brain. "Wobbling and synchrony are inseparable," Strogatz et al wrote. "They emerge together, as dual aspects of a single instability mechanism, once the crowd reaches a critical size."

As for this latest paper, "The authors use a much more realistic and biomechanically inspired model of human walking than my colleagues and I were able to muster back in 2005," says Strogatz. "At that time, very little was known about how people adjust their gait when walking on a surface that moves beneath their feet, as the Millennium Bridge did when it wobbled from side to side on its opening day."

In a 2015 simulation, Joshi and his co-author, Manoj Srinivasan—both of whom have backgrounds in studying human locomotion—found that people walking in synchrony with a sideways-swaying bridge lowers the metabolic energy cost of the motion. So it was a natural reaction, they reasoned, for people to start to synchronize their gaits with the bridge's motion. However, this model didn't factor in the energy cost of stabilizing one's gait.

Their latest study looks at a different biomechanical principle. Human beings are so-called "stable" walkers. We want to walk without falling, and will make adjustments based on the feedback we receive from our environment. Since "humans are top-heavy objects, often modeled as an inverted pendulum," the authors write, we need that feedback to stabilize us; otherwise we'd lose our balance pretty quickly. So they incorporated that feedback into their simulations.

"The bridge can begin to wobble even at lower numbers of pedestrians. The crowd does not synchronize, yet the bridge spontaneously begins to move."

The improved model correctly predicts some phenomena that the 2005 model couldn't account for, like the wobbling of footbridges even in the absence of this crowd synchrony. Also, the onset of crowd synchrony and the onset of bridge wobbling are not simultaneous. They occur at different numbers of pedestrians.

"In our model, crowd synchrony and bridge wobbling always went hand-in-hand; in their model, they can occur together but don't have to," says Strogatz. "It's only when there are a sufficient number of people on the bridge that the crowd become synchronized. The bridge can begin to wobble even at lower numbers of pedestrians, in which case the crowd does not synchronize, yet the bridge spontaneously begins to move."

As for the metabolic energy cost, once the virtual bridge began to shake, the simulated walkers widened their steps—a much less energy-efficient gait. But the authors argue that perhaps over time, people would figure out how to minimize the energy expenditure.

They next hope to incorporate crowd dynamics into their simulations. "We have all these individual bipeds walking on a bridge, but they can only interact with each other through the bridge," Joshi says of the current model. In reality, people try to avoid colliding with others, or suddenly change directions, for example. Ultimately, he hopes to learn more about the biomechanics of how we can adapt so quickly to unusual situations, like a moving surface.

Engineers fixed the Millennium Bridge's swaying issues by retrofitting the structure with 37 energy dissipating dampers to control the horizontal movement, and another 52 inertial dampers to control the vertical movement. The bridge hasn't had a significant wobble problem since it reopened in February 2002. But a fictional version was destroyed by marauding Death Eaters in the 2009 film Harry Potter and the Half-Blood Prince.

Destruction by Death Eaters

This is a fairly decent fictional portrayal of a different bridge dynamic, calling to mind the infamous collapse of the Tacoma Narrows Bridge (aka "Galloping Gertie") in 1940. For years, the popular explanation of what happened was forced resonance: the strong wind matched the bridge's natural resonance frequency and the resulting positive feedback loop got so strong, the bridge broke apart.

It was actually a bit more complicated than that. A phenomenon called "vortex shedding" set the bridge to undulate (or gallop). Those undulations snapped one of the suspension cables, so the bridge was lopsided and began twisting along its center axis along with the galloping motion. Technically, it's known as aerodynamically induced self-excitation, or "flutter"— a self-sustaining vibrational feedback loop. Every twist of the bridge amplified the wind's effect instead of dampening it, until the pent-up energy got so strong the entire structure collapsed.

The specific circumstances might be different, but in principle that's what we see in the Harry Potter scene. Death Eaters take out key structural supports, causing the bridge to gallop and twist, and the resulting feedback loop gets so strong that a center portion of the bridge collapses.

Death Eaters may be fictional, but the complicated real-world dynamics between bridges and the pedestrians walking across them are not. That's why so many scientists continue to be fascinated by the problem. The better we understand these dynamics in general, the better (and safer) future bridge designs will be.

DOI: Biology Letters, 2018. 10.1098/rsbl.2018.0564 (About DOIs).

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