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

Abstract

The spectacular collapse of the Tacoma Narrows Bridge has attracted the attention of engi- neers, physicists, and mathematicians in the last 74 years. There have been many attempts to explain this amazing event, but none is universally accepted. It is however well estab- lished that the main culprit was the unexpected appearance of torsional oscillations. We suggest a mathematical model for the study of the dynamical behavior of suspension bridges which provides a new explanation for the appearance of torsional oscillations dur- ing the Tacoma collapse. We show that internal resonances, which depend on the bridge structure only, are the source of torsional oscillations

The role of aerodynamic forces in a mathematical model for suspension bridges

“…Since our purpose is merely to describe the qualitative phenomenon, we may take EI π L 4 = 3µ π L 2 = 1 although these parameters may be fairly different in actual bridges. For the same reason, the choice of the nonlinearity is not of fundamental importance; it is shown in [2] that several different nonlinearities yield the same qualitative behavior for the solutions. Whence, as suggested by Plaut-Davis [12, Section 3.5], we take f (s) = s + s 3 .…”

“…By using suitable Poincaré maps, it has been proved that when enough energy is present within the structure a resonance may occur, leading to an energy transfer between oscillators, from vertical to torsional. The results in [2] are purely numerical. We found a similar answer in [3] by analyzing a different mathematical model, named fish-bone.…”

“…In two recent papers the onset of torsional oscillations was attributed to a structural instability. In [2] a model of suspension bridge composed by several coupled (second order) nonlinear oscillators has been proposed. By using suitable Poincaré maps, it has been proved that when enough energy is present within the structure a resonance may occur, leading to an energy transfer between oscillators, from vertical to torsional.…”

Stability of dynamic response of suspension bridges

“…A recent work by Arioli and Gazzola [24], as previous ones by different authors [21,22] investigates internal parametric resonance potentially suffered by suspension bridges. In order to catch this phenomenon, it is necessary to consider the fully non-linear equation of motion, hereby for the first time supplemented by the aeroelastic operator.…”

“…The authors of [21,22] used the continuous model proposed by Abdel-Ghaffar [11], and solved the system of equations by means of the multiple scale perturbative technique [23]. Recently, Arioli and Gazzola [24], trying to explain why torsional oscillations suddenly appeared before the Tacoma Narrows collapse, found out that, also in isolated systems, vertical oscillations may switch to torsional ones, as long as they become large enough. The problem was already tackled by other authors [25][26][27][28][29] but it seems that there is still an open issue regarding the complete explanation of the sudden appearance of large oscillations which led to collapse.…”