The demise of the Tacoma Narrows bridge was casused by aeroelastic flutter. But isn't that just a special case of resonance?

Flutter is only possible if you have similar structural and aerodynamic frequencies. One without the other would produce much lower amplitudes.

Look at a mass-spring system suspended on an eccentric tappet which sits on the edge of a small rotating wheel. When the wheel turns, it raises and lowers the top of the spring, and the mass on the bottom will barely move. Now change the rotation frequency to something close to the resonance frequency of the mass-spring system, and the mass will produce wild oscillations.

Now use a beam (which also has its eigenfrequencies in bending and torsion), and attach to it an airflow which will periodically separate, as in a Karman vortex street. Normally, nothing happens. Now change wind speed such that the separation frequency is close to one of the structural eigenfrequencies of the beam, and you will get wild excitations again. It gets worse if the deformation will induce flow changes, because now the resonance will be self-propelling over a wider frequency range. This is flutter.

The deformation stores elastic energy, and if the aerodynamic forces change such that they support the elastic motion, they will add a little energy with each cycle, such that the eventual amplitude will become immense, right to the point of failure.

Flutter and structural resonance are inseparable. One is part of the other.

The paper by Yusuf Billah and Robert Scanlan (cited in the wikipedia article on the Tacoma Narrows Bridge 1940) distinguishes between resonance as a response to a driving force and what the authors call "self-excitation" or "negative damping." They demonstrate that the Karman Vortex Street (which occurs at the trailing edge of the deck) was not the cause of the collapse : in the conditions which pertained, it was the wrong frequency, and it is self-limiting : ie above a certain limit as amplitude increases the vortices decrease. As windspeed increases the frequency of vortex-shedding also changes. Such oscillations had been reported previously (leading to the deck being dubbed "Galloping Gertie"), and although alarming they never caused damage.

Instead the authors attribute the collapse to aero-dynamic flutter which occurs at the leading edge of the deck. The difference is that the torsional oscillation of the bridge causes the flutter wake, rather being caused by it. Furthermore, unlike the vortex street this effect is not self-limiting, and increases without limit as windspeed increases.

So there was no external fixed driving frequency, hence (strictly speaking) no resonance between an external driving force and a natural oscillation of the bridge. The high winds provided the energy to increase the amplitude of oscillations, but the oscillations of the bridge did not "resonate" with any frequency in the wind (eg periodic gusts), nor in any aerodynamic effect (vortex street).

http://www.ketchum.org/billah/Billah-Scanlan.pdf

The flutter that occurred at the Tacoma Narrows Bridge can’t be accurately described as just a special type of resonance. Just because the situation occurred at a resonant frequency does not mean that the resonance was the cause. I hate to disagree with Peter Kämpf. Flutter often involves the convergence of two resonant frequencies, but the situation he’s describing is not what occurred at the bridge.

When saying that a failure was caused by resonance what is meant is forced resonance, where an outside force which has a regular frequency interacts with an object’s internal elastic resonance to cause a failure. The classic example of this is breaking a wine glass by singing at its resonant frequency. The two eigenfrequencies interact to cause excessive elastic deformation. That was not what occurred here, since the wind is random and does not have any type of sinusoidal pattern.

Flutter can also involve two separate eigenmodes whose frequencies interact to cause an internal mutual forced resonance. Peter Kämpf describes that situation in an airplane wing here. This convergence of eigenfrequencies did not occur at the Tacoma Narrows Bridge.

What downed the bridge was a positive feedback loop. The fact that it occurred at a resonant frequency is not relevant, because it was not being forced by convergence with another frequency.

There were two eigenmodes involved. The first was the vertical mode which was induced by the wind and went on for several months without any damage to the bridge. The wind caused a lift on the bridge which was irregular in nature. This caused a vibration in the bridge. This vibration did occur at its natural resonant frequency somewhere around 1Hz. The strength of the wind did not alter the frequency of this mode, only its amplitude. The stronger the wind the higher the undulation, but always around the same frequency. Vortex shedding and the Karman vortex street may have helped create this initial vibration, but it would be created by any movement of the structure induced by the wind. It would occur at the same frequency regardless of the cause. The nature of wind would mean the frequency of the vortex street would be all over the place and would not converge with the elastic mode with any regularity.

The second eigenmode that occurred was the twisting deformation mode. The strong winds on the day of the collapse caused the vertical undulation to be at a very high amplitude, enough that the bridge was closed to traffic. But the whole time this vibration was going on it only occurred in the vertical mode. The undulation was straight up and down and there was no twisting. As long as the vibration stayed in the vertical mode the bridge would have probably survived the day.

All that was needed, though was for something to trip off the twisting mode. The vibration on one side of the bridge getting out of phase, or anything that led to the two sides getting off of alignment would be enough. According to this article the impetus for the twisting was the snapping of one of the support cables. In this manner the extreme amplitude of the first mode caused the fatal second mode by overstressing the cables. Once this twisting motion was set off in such high winds the collapse was imminent. It did not need to interact with the first mode. In fact their frequencies did not coincide. What it set up was a self-enforcing positive feedback system in the twisting mode. As soon as the bridge twisted, even a small amount, it then had an angle of attack with respect to the wind. This significantly increased the lift created and the portion at the center of the twist would lift higher than the rest. This would increase until the elastic properties would snap it back down. The momentum of the structure caused it to overshoot its equilibrium point then creating an angle of attack in the opposite direction. The lifting force with this newly induced angle of attack set up a positive feedback, where each oscillation would create a slightly higher angle, and thus more lift, than the last. This occurred with a sinusoidal eigenfrequency that was different from and independent of the first vertical vibration.

The positive feedback was what brought down the structure. The bridge was able to withstand the vertical loads even with a snapped cable. Had the snapped cable caused a progressive failure of other cables one would expect this to occur quite rapidly, as occurred in the collapse of the Silver Bridge in West Virginia. What the snapped cable resulted in was the twisting deformation of the bridge. It was not designed with enough stiffening in the structure which would have damped the twisting motion and kept it from getting out of control. It simply could not prevent the positive feedback from perpetuating and eventually the bridge failed under the ever increasing loads.

None of this involved a forced resonance from an outside source. Although the two modes each had their own resonant frequencies, they did not coincide with each other and cause the failure. This case of aeroelastic flutter was not caused by resonance.

The article I cited earlier has an excellent description of the difference and explains why the incorrect resonance hypothesis became the dominant explanation.