Mathematicians explain bridge oscillations caused by nonsynchronous movements of pedestrians

An international team of scientists led by Professor Igor Belykh of Georgia State University (USA), who is also Head of the Dynamic Chaos Laboratory at the UNN Institute of Information Technologies, Mathematics and Mechanics, has proved that pedestrians can rock the bridge even if they move out of step and at different speeds.

The oscillation model proposed by the researchers is based on the example of London's Millennium Bridge. In 2000, about two thousand pedestrians were on the bridge at the same time during its grand opening. Their movements caused the bridge to wobble. Based on the common notion of oscillatory systems, the bridge designers decided that the frequency at which people made their footsteps became 'locked-in' to the natural frequency of the bridge, which caused resonance. The bridge was then reinforced with additional structures to damp vibrations and reopened later.

However, according to the study by Igor Belykh's team, it is not only people walking in step that can rock the bridge, but also a large number of pedestrians moving out of step at different speeds. The scientists developed a mathematical model showing that the oscillating bridge-people system lost its balance because the pedestrians were moving about trying to keep their balance. In this case, people instinctively lean, some on their right foot and some on their left foot. Contrary to expectations, these movements do not cancel each other out, but instead add up to energy that can be transmitted to the bridge and increase its vibrations.

Igor Belykh explains it this way: "We all know from childhood that soldiers crossing a bridge should not walk in step; to prevent the bridge from swaying, their movements should not be synchronised.  Our work shows that this is no guarantee for the stability of the bridge, and soldiers can unexpectedly initiate its instability. These small initial oscillations will force the soldiers to change the frequency and width of their stride, thus increasing the oscillation of the bridge even more. As a result, the soldiers will walk in step as if they were passengers walking on the deck of a cruise liner caught in a storm and swaying sideways. That is, the synchronisation of pedestrians' steps on the bridge is not the cause but a consequence of its significant oscillations. Which is exactly what happened in the case of the Millennium Bridge in London".

The work was published in December 2021 in Nature Communications, a journal of the Nature Publishing Group. The research was carried out at the Dynamic Chaos Laboratory in the Department of Control Theory and Dynamical Systems of Lobachevsky University. It was supported by a grant from the Russian Ministry of Science and Higher Education. Igor Belykh’s co-authors were British scientists from the University of Cambridge, University of Bristol and University of Leicester.

An important role in this research was played by the observations of Professor John Macdonald from Bristol University's Department of Civil Engineering. He discovered that during a balloon fiesta on Clifton Bridge in Bristol, groups of people without synchronised steps provoked its oscillations.

All these observations and the mathematical models supporting them will form the basis of the author's methodology for calculating the parameters of bridges that can be used in bridge engineering and design. The scientists plan to incorporate these approaches into the standard software package that bridge builders around the world use today.

"Quite often, as was the case with the London Bridge, designers want to make their bridge more elegant and visually appealing at the expense of having fewer dampers.  It would seem that if you design a bridge with a natural frequency far removed from that of pedestrians’ motion, the number of dampers can be reduced.

Our work shows that this is a dangerous approach: instability can occur at other frequencies. There is no reliable alternative to significant bridge damping. Our calculations precisely estimate the level of damping required, depending on the size and properties of the bridge," says Igor Belykh.

The next goal of the research team is to come up with a formula for calculating the critical number of pedestrians and the range of dangerous frequencies for a given bridge. To solve this task, the researchers of the Dynamic Chaos Laboratory at the UNN Institute of Information Technologies, Mathematics and Mechanics intend to use the Lobachevsky supercomputer.