In the previous section, we saw the model that accounts for the geometry of the bridge has solutions with large oscillations and these oscillations can persist for long periods of time. I think you would agree that a bridge undergoing such violent torsional oscillations would be susceptible to failure. So we have answered, at least in part, the question of how large amplitude oscillations can arise in these bridges. Its simply due to the geometry of their construction.
We have not addressed how large amplitude torsional oscillations arise. In a recent paper by P. J. McKenna1, he proposed that the geometry coupled to a more physically realistic description of the cables allowed the vertical oscillations to generate horizontal oscillations.
McKenna's cable idea is very simple. In the derivation of the previous equations, it was assumed that the suspending cables of the bridge always act like Hooke's law springs. As everyone that has tried to push a string knows, if you grab one end of a cable, you can move the other end by pulling, but if you try to move the other end by pushing, the cable simply folds into itself. McKenna argues that the suspending cables of these bridges should act this way too. Therefore, they cannot be Hooke's law springs because they don't create a force when they are shortened.
Hooke's law springs create forces proportional to the displacement of the spring past equilibrium, or
where k is the spring constant and x is the displacement from equilibrium of the spring. McKenna's cable springs are Hookes law springs when x is positive, but when x is negative they do not generate a force, therefore,
If your interested in how the cable springs theoretically induced large torsional oscillations copy this Maple Worksheet, into your workspace and load it into Maple.