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In my physics class it was suggested that ancient armies had a rough understanding of the idea of a resonant frequency and so they "broke step" when crossing bridges so as to avoid a very high $Q$. I am not particularly clear on how this high $Q$ corresponds to increased force downward on the bridge (which is what I assume would make the bridge break), but my main question is whether this is really within reason - using reasonable estimates is it really possible that soldiers marching in step, at the resonant frequency on a reasonably sturdy bridge (made of, say, a strong wood) could actually make the bridge break?

Edit 1: Actually, I think I remember this having more to do with the resonant frequency of the material of the bridge and not its strength, but I don't really know what that even means physically.

Edit 2: If it helps to narrow down the problem suppose we're talking about a very primitive bridge that is made of $z$ thick plywood and a super-strong supporting beam right in the middle of the bridge. If it matters, let the length of the bridge be $l$ and the width be $w$.

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For an account of modern instances of resonance damaging structures, see the Sketpics SE post listed by Dr. RedGrittyBrick listed in the comments. I don't know of any historians' recording of events such as you describe, so hopefully another answer can do this.

As for understanding the $Q$-factor and its effect on resonance: classical resonances comprise three four things

  1. Two "good" (i.e. not leaky) energy reservoirs - for example (i) the elastic potential energy stored in a sprung spring and (ii) the kinetic energy of a mass bouncing on the spring;
  2. A means of shuttling energy back and forth between these two reservoirs - for example: Newton's second law shows how the spring's force on the mass drains the latter's kinetic energy and "banks" it in the spring's kinetic energy, and contrawise;
  3. The coupling of an outside force to the energy stores.

The damping factor $\zeta = \frac{1}{2\,Q}$ is a measure of the proportion of this shuttling energy is lost each cycle (actually it is per "radian", so its the proportion lost each $\frac{1}{2\pi}$ of a cycle) through nonideal leaking. It is thus proportional to the rate that the outside force must work (i.e. the power input) needed to keep a resonant system going. The bigger the $Q$, the smaller this minimum power is. Any power input above this minimum will be stored in the system, whose stored energy can thus slowly build up to huge, dangerous values (think of "trickle charging" the system - indeed resonant parallel inductor-capacitor circuits are called "tank" circuits for this reason). In the mass on a lossless spring (infinite $Q$), the spring force precisely matches the $F = m\,a$ requirements of the mass for the latter to undergo its oscillation at all times. Thus, even the tiniest force, as long as it stays in-phase with the mass, can raise the latter's kinetic energy as though it were the only force on the mass, so ultimately the system's energy is unbounded.

Of course, with big oscillations, something must be imparting the force on the spring/mass system. This is simply Newton's second law applied to the centre of mass. In a mass on a spring, this force must be imparted by stresses in the base of the spring, which will eventually snap as dangerous resonance sets in. A cantilevered beam is more complicated, but can be idealized to a toy model of a point mass on a massless springy beam. At resonance with increasing amplitude, the bending moment supplied by stresses in the cantilever's bearing overwhelms the cantilever's matter, and the cantilever snaps. $Q$ then simply measures how easy it is for this dangerous situation to arise.

Although I can't give historical citations, one could make the reasonable inference that if resonance failures happen in this age of sophisticated building regulations and engineering standards, it almost certainly happened in the middle ages. And what was called a "bridge" in those days was sometimes very flimsy by today's standards. It wouldn't take a great deal of thought to come up with the idea of breaking step - if you try walking across a rope suspension bridge and if you are like me and fear heights, you will certainly feel and swiftly change your gait to avoid resonance even if you can't spell the word!

Edit: The best citations of bridge failure from resonance I could find are the collapse of the Broughton Suspension Bridge and the Angers Bridge, whilst supposedly the Albert Bridge in central London bears a sign ordering that soldiers should break step on crossing the bridge. I daresay a London poster on this site could verify that claim.

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