Summary: When people discuss large torsional oscillations of suspension bridges like the one at the Tacoma Narrows, they invariably linearize the trigonometry out of the problem, unwittingly making a small-angle assumption. In this paper, I re-derive the equations for a torsionally oscillating plate, choosing the physical constants in accordance with the historical record.
I show how nonlinearity from the trigonometry embedded in the problem leads naturally to large amplitude motions sustained by small forces. These motions are an excellent match for the historical record. I also show how large vertical (nontorsional) motions can lead, via a sudden instability, into the purely torsional motions of the type recorded on famous historical film footage.