• A bridge, like most constructions, is designed so as to prevent a resonance from being excited under operating conditions. As for all other civil buildings, in fact, the presence of one or more resonances in a frequency band that can be excited by natural events or by traffic exposes the structure to the risk of damage and, in the worst case, makes it unstable.
• Bridges, moreover, are structures that tend to extend much more in one dimension than in the others, which makes them particularly sensitive to transverse and torsional oscillations.
• Precisely for these reasons, engineers take particular care in studying the frequency response of these structures. Knowing the natural frequencies of the bridge, that is, the frequencies of the stimuli that can set it into resonance, and the ways in which the structure oscillates at these frequencies (that is, the normal modes), it is possible to adopt construction measures that damp the dangerous resonances.

The following two sections illustrate very briefly two cases in which the presence of resonances compromised the stability of two bridges, in the first case causing its collapse, in the second case not, thanks to careful work by the engineers.

These examples are particularly interesting from a physical point of view too, because they are striking but also completely different manifestations of the phenomenon of resonance, which can occur in much subtler and more complicated ways than is usually illustrated with simple examples such as that of the resonance of the harmonic oscillator.

The case of the Tacoma Narrows Bridge

Analysis

• This is not a classic case of resonance as defined in textbooks. Indeed, if we compare it with the simplest possible case, the resonance of the harmonic oscillator, it is immediately evident that this bridge was not brought down by the presence of an external periodic force at the resonance frequency of the torsional mode seen in the film. The wind, in fact, can be seen blowing at constant speed, and not in regular gusts.
• Nevertheless, it is clearly a case of resonance in the broad sense, because a small external force, through a complex mechanism, was able to accumulate energy progressively in the structure, inducing oscillations with a frequency equal to that of one of its normal modes and with increasing amplitude, until its structural integrity was compromised.
• The detailed mechanism by which a steady wind can trigger oscillations with those characteristics is rather complex, and explaining it in detail requires laborious aerodynamic calculations. Briefly, we can say that these are self-excited aeroelastic vibrations: the force of the wind affects the motion of the bridge, and the motion of the bridge in turn modifies the flow of the wind, creating oscillating forces that excite one or more normal modes of the structure. When particular conditions occur, the oscillation, instead of being damped, becomes unstable, with possible consequences for the integrity of the structure.
• The disaster, however, was not caused by ignorance of the resonance of the torsional mode seen in the film, but by ignorance of the mechanism through which a steady wind can trigger it. This mechanism is, ultimately, a particular type of positive feedback.
• The bridge was rebuilt in 1950 and, for anyone who has to cross it, let us say that it was also subjected to the appropriate aerodynamic tests in the wind tunnel.

The case of the Millennium Bridge

The Millennium Bridge is a footbridge over the Thames in the heart of London. It is a bridge 325 m long, divided into three sections, whose main feature is that the suspension cables, instead of being placed above the deck as in most suspension bridges, are placed at the sides, giving those who cross it a greater sense of openness. The bridge was designed to carry, under ordinary working conditions, 5000 people at the same time. The bridge was opened in June 2000, and the problem that led to its closure two days later appeared on the very day of the inauguration. After the bridge had filled with pedestrians walking in both directions, a lateral oscillation set in that induced the passers-by to change their gait until they all walked in unison. Although it did not prove destructive, the oscillation moved the deck from right to left by 7 cm, and made it necessary for a significant number of people to hold on to the railing to keep their balance.

Analysis

• This too is not a canonical case of resonance. In this case it is true that the exciting force is periodic (with a frequency below 1.3 Hz), but it is not properly an external force, since it is produced by the reaction of the pedestrians to the oscillation of the system, and is therefore self-induced.
• After a relatively small number of people had begun to walk across the bridge, a slight lateral oscillation induced the people arriving later to synchronize their step instinctively with the frequency of the oscillation, because this allowed a more comfortable walk. As the number of people increased, however, the force exerted by the passers-by, in resonance with the lateral mode, increased progressively, reinforcing the oscillation. Again an example of positive feedback. This feedback mechanism had not been considered in the numerical simulations of the structure, and that is why the lateral mode had been underestimated.
• Incredibly, self-induced excitation by pedestrians was not considered a potential source of danger in construction protocols until 2000. Tests on the Millennium Bridge showed that just 166 people walking simultaneously were able to suddenly excite the lateral mode.
• An excellent analysis of the resonance mechanism, together with films of the inauguration day and of the subsequent tests, can be found on the very site of the bridge's designers, who, on the basis of this experience, issued an international notice explicitly aimed at updating the testing protocols for footbridges.
• Watching the films, one can clearly see the oscillations of the first and second normal modes of two of the sections of the bridge, with frequencies between 0.5 and 1.0 Hz (see Normal modes of a string for a schematic representation of these modes).
• In 2002, after the appropriate studies, the structure was stabilized by adding 37 viscous dampers and 45 inertial dampers (see seismic dampers), which attenuate the resonance peak (see the figure and the discussion in resonance).