On November 7, 1940, the Tacoma Narrows Bridge collapsed. An explanation based on resonance was offered at that time, but in 1990, a new model was considered [1]. Assuming that a cable cannot be described with the usual Hooke's law, it set up a nonlinear model with vertical and torsional motion. This Demonstration uses another equation presented in 1999 [2] that avoided linearizing the torsional equation to avoid removing a wide variety of behavior. A sinusoidal forcing term was introduced with amplitude and frequency , while corresponds to the initial angle. Cases with slight increases for small torsional forcing with a transitioning to large oscillations can be observed, which is what presumably happened at Tacoma Narrows.

For the torsional motion, the equation and the initial conditions are

.

References

[1] A. C. Lazer and P. J. McKenna, "Large Amplitude Periodic Oscillations in Suspension Bridges: Some New Connections with Nonlinear Analysis," SIAM Review, 32(4), 1990 pp. 537–578.

[2] P. J. McKenna, "Large Torsional Oscillation in Suspension Bridges Revisited: Fixing an Old Approximation," American Mathematical Monthly, 106(1), 1999 pp. 1–18.