The global structure of periodic solutions to a suspension bridge mechanical model

Abstract

We study two systems of nonlinearly coupled ordinary di erential equations that govern the vertical and torsional motions of a cross section of a suspension bridge. We observe n umerically that the structure of the set of periodic solutions changes considerably when we smooth the nonlinear terms. The smoothed nonlinearities describe the force that we wish to model more realistically and the resulting periodic solutions more accurately replicate the phenomena observed at the Tacoma Narrows Bridge on the day of its collapse. The main conclusion is that purely vertical periodic forcing can result in subharmonic primarily torsional motion.

A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge

“…As pointed out in [11], the torsional oscillations that preceded the collapse were never observed until the day of the collapse. Our model explains why torsional oscillations may be seen, or may be hidden, or may even not appear, independently of the force applied to the bridge.…”

“…We show that, when these thresholds are reached, there is a sudden transfer of energy within the different fundamental vibrations of the bridge, just as observed at the TNB, see Eq. (11) below. This enables us to conclude that the bridge behaves driven by its own internal features;…”

Mathematical Models for Suspension Bridges

“…1 but this behavior becomes visible only for large s which are outside the range considered in [195]. McKenna-Moore [194] studied the bifurcation and stability properties of periodic solutions of (4.2) both for (4.6) and (4.7).…”

Bifurcation and stability properties of periodic solutions to two nonlinear spring-mass systems

“…In this model, motivated by [11] and [12], we construct a smoothed version of the system. We take the following things into account in this construction.…”

“…In [11] and [12], McKenna, O'Tuama, and Moore found that smoothing the nonlinearity yields a significant qualitative change in the structure of the set of periodic solutions to the nonlinearly coupled vertical-torsional system.…”