# Rock and Roll Bridge: A New Analysis Challenges the Common Explanation for a Famous Collapse

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Rock and Roll Bridge

Startling scenes of rippling pavement, featured in a classic film that captured the 1940 destruction of the Tacoma Narrows suspension bridge in Washington state, rank among the most dramatic and widely known images in science and engineering. This old film, a staple of most elementary physics courses, has left an indelible impression on countless students over the years.

Many of those students also remember the standard explanation for the disaster. Both textbooks and instructors usually attribute the bridge's collapse to the phenomenon of resonance. Like a mass hanging from a spring, a suspension bridge oscillates at a natural frequency. In the case of the Tacoma Narrows brdige, so the explanation goes, the wind blowing past the bridge generated a train of vortices that produced a fluctuating force in tune with the bridge's natural frequency, steadily increasing the amplitude of its oscillations until the bridge finally failed.

"This explanation has enormous appeal in the mathematical and scientific community," observes applied mathematician P. Joseph McKenna of the University of Connecticut in Storrs. "It is plausible, remarkably easy to understand, and makes a nice example in a differential-equations class."

But the explanation is flawed, he says.

Resonance is actually a very precise phenomenon. Anyone who has seen sound waves shatter glass knows how closely the forcing frequency must match an object's natural frequency. It's hard to imagine that such precise, steady conditions existed during the powerful storm that hit the bridge, McKenna says.

Furthermore, the structure displayed a number of different types of oscillations. Initially, its roadway merely undulated vertically. Then the bridge abruptly switched its oscillation mode, and the roadway started to twist. It was this extreme twisting that actually led to the bridge's demise.

Indeed, even the 1941 report of the commission that investigated the disaster concludes: "It is very improbable that resonance with alternating vortices plays an important role in the oscillations of suspension bridges."

If simple resonance doesn't explain the Tacoma Narrows destruction, what does? Fascinated by that question, McKenna and Alan C. Lazer of the University of Miami in Coral Gables, Fla., have spent the last six years developing an alternative mathematical model that may help elucidate the catastrophic collapse.

"What distinguishes suspension bridges, we claim, is their fundamental nonlinearity," Lazer and McKenna state in a paper to appear in a forthcoming SIAM REVIEW.

Linear differential equations, such as those typically used by engineers to model the behavior of structures such as bridges, embody the idea that a small force leads to a small effect and a large force leads to a large effect. Nonlinear differential equations, such as those studied by Lazer and McKenna, have more complicated solutions. Often, a small force can lead to either a small effect or a large effect. And exactly what happens in a given situation may be quite unpredictable.

Lazer and McKenna say their new theory provides key insights into why suspension bridges oscillate the way they do. It applies not only to the Tacoma Narrows bridge and San Francisco's Golden Gate bridge--which may be prone to large-scale, potentially destructive oscillations during earthquakes -- but also to large, glexible structures, such as space stations, giant space-based robot arms and certain types of ships. The theory even suggests ways of constructing extremely light, flexible bridges that won't oscillate wildly.

Suspension bridges have a long history of large-scale oscillations and catastrophic failure under high and even moderate winds. The earliest recorded problem involved a 260-foot-long footbridge constructed in 1817 across the River Tweed in Scotland. A gale destroyed that bridge six months after its completion. …