# Intuitive understanding of the resonance of a bridge

Dynamics of strides, walking and corresponding force cycles: In the civil engineering literature, it is known that the resonant frequency of a bridge should not be the same as that of the strides of pedestrians. In this video, Stephen Strogatz clearly states that the frequency of the strides (a foot coming into contact with the ground and leaving) cycle is roughly $$2 \; Hz$$. Therefore, the frequency of walking (one left and one right stride cycle) is $$1\; Hz$$ while the frequency of strides is $$2 \;Hz$$. Consequently it is easily understood that the frequency of the lateral force cycle (side-ways forces exerted on the ground during walking) is $$1 \; Hz$$, since $$freq(\text{lateral force})=freq(\text{walking})=1\;Hz$$ and that the frequency of the vertical force cycle (downward forces exerted on the ground during walking) is $$2\;Hz$$, since $$freq(\text{vertical force})=freq(\text{stride})=2\;Hz$$. The discrepancy can be understood simply as follows. While in the lateral case, a left and right foot action are distinct and form a cycle together, in the vertical case, the left and right foot actions are indistinguishable and contribute two cycles when considered together.

The initial design and construction of the Millennium bridge turned out to have a side-ways sway resonant frequency of $$1\;Hz$$ (as discussed in the video). This led to near-catastrophic sway because the lateral force cycle of pedestrians has a frequency of $$1\;Hz$$.

However, in the video, Strogatz states two reasons which enabled the collective of walkers to cause resonance.

1. The first is the correspondence of the underlying individual dynamics of pedestrian walking with the side-ways sway resonant frequency.
2. The second is the emergent behavior of the synchronization of walking.

This reasoning presumes that the amplitude of lateral forces of a few walkers did not sufficiently excite the system, but the amplitude of the synchronized lateral forces of a number of them did.

Question: Can a single input force cycle of low magnitude (say, one person walking) at the resonant frequency ($$1\;Hz$$) or high force amplitude (say, a jackhammer operating at the resonant frequency ($$1\;Hz$$)) also cause the identical resonance and consequent near-catastrophic swaying?

• In response to the close vote I have modified the post to pose a singular focused question. – kbakshi314 Apr 13 at 3:07

Shouldn't an input force cycle of a low magnitude (say, one person walking) at the resonant frequency (1Hz) be sufficient to cause resonance and near-catastrophic swaying?

Only if the damping in the system is less than some threshold.

One person walking will indeed send some energy into the system. But the motion of the bridge will be damped by the materials as they move. Energy will leave the system as heat. When the power input equals the power absorbed by damping, the amplitude remains constant.

It is very likely that the power from a single person is insufficient to cause significant motion. It could probably be detected with instruments, but doesn't give enough amplitude to cause swaying that people on the structure would notice.

Can a single input force cycle of low magnitude (say, one person walking) at the resonant frequency (1𝐻𝑧) or high force amplitude (say, a jackhammer operating at the resonant frequency (1𝐻𝑧)) also cause the identical resonance

There must be some power input which would cause the same behavior. The question is whether a jackhammer is sufficient (maybe?), or if the required input can come from a single source without local damage (possibly). I don't know the necessary power. Certainly you could imagine some small number of power sources that would reproduce the behavior.

and consequent near-catastrophic swaying?

The swaying was out of the design limits, but I don't know if it was truly near-catastrophic.