Large Torsional Oscillations in Suspension Bridges Visited Again: Vertical Forcing Creates Torsional Response

1. The Tacoma Narrows Bridge. In class we discussed a mathematical model that might expalin the collapse of the Broughton Bridge as a column of soldiers marched across it. It does not explain the collapse of the Tacoma Narrows Bridge, however, as that was caused by torsional, or twisting, oscillations. Consider a roadway of width 2L hanging between two suspended cables, as pictured below. (This model considers a 1-dimensional slice of the bridge, ignoring its length, in order to model side-to-side motion.) At rest, the roadway hangs at a certain equilibrium height due to gravity; let y(t) denote the distance the center of the roadway hangs below this equilibrium at time t. Let θ(t) be the angle the roadway makes with the horizontal. There are two suspension cables, holding the sides of the bridge at heights of y(t) + L sin θ(t) and y(t) − L sin θ(t) away from the center equilibrium height. −8 0 8 −8 0 8 y θ According to Hooke's law (for springs), the restoring force of the cables will be proportional to the distance of the bridge sides from equilibrium, but McKenna and Tuama hypothesized that cables pull back with more force when stretched than they push forward with when compressed. This led them to the following set of ordinary differential 1