Tacoma Narrows Disaster: A Hypothesis (of Lazer and McKenna)

"Rock and Roll Bridge” by Ivars Peterson has?some fascinating physics and mathematics, and some interesting (but unfortunate) examples of bridge and ship disasters caused by oscillations exceeding the strength of materials. I’ve heard about the Tacoma Narrows disaster, but in isolation. The context of other, similar situations, with disaster and without disaster, is important cognitively and logically (and scientifically). We need examples, contrasts, and context to form valid conclusions and to reason well. (The article is from 1990; I do not know what, pro and con, has developed since then; I will have to do some research.)

Here are some excerpts from the article: "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 bridge, 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.'

…

"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.

...

"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.

...

"In 1854, winds completely destroyed the roadway of a 1,010-foot suspension bridge across the Ohio River at Wheeling,?W. Va. An eyewitness wrote that the structure lunged like a ship in a storm, finally crashing into the waters below.

"There are a number of other examples.?In some cases, the bridges didn't actually shake themselves to pieces, but the oscillations grew large enough that a traveler crossing the bridge would get seasick.

…

"There is, however, one possible energy source that could send such a bridge into large-scale oscillations. 'An earthquake is precisely the sort of energy source that will put you into the nonlinear mode,' McKenna says.

"Last fall, the Golden Gate bridge went into large-scale oscillations during the magnitude 7.1 Loma Prieta earthquake.?The bridge oscillated for about a minute, roughly four times longer than the earthquake itself lasted. One witness noted that the stays connecting the roadbed to the main cables were alternately slackening and tightening 'like strands of spaghetti' -- a sign that the bridge was in a nonlinear state. Fortunately, the bridge didn't start twisting, perhaps because the earthquake waves hit it head-on rather than obliquely, McKenna says.

…

"According to their simple model, gusts of wind initially act as a random buffeting force on a suspension bridge, causing the towers and main cable to go into a high-frequency periodic motion like that of a randomly struck guitar string. That motion initiates low-frequency, vertical oscillations that ripple the roadbed.

"The sudden transition from vertical oscillations to a twisting mode is more difficult to explain. Using computer sim-ulations, McKenna has shown that a rod suspended from cables that behave non-linearly can unexpectedly start twisting.?Analyses of more realistic models should bear this out, he says.

“ 'This, we feel, is the likely explanation of the destruction of the Tacoma Narrows bridge,' Lazer and McKenna conclude. 'An impact, due either to an unusually strong gust of wind, or to a minor structural failure, provided sufficient energy to send the bridge from one-dimensional to torsional (modes of oscillation].' The resulting twisting destroyed the bridge.

"The nonlinear theory also suggests an intriguing new design for light-weight, inexpensive suspension bridges.?In conventional suspension bridges, nothing keeps a stay from slackening during an oscillation. One possible answer is the installation of two sets of stays and cables: one set from which the roadway hangs and a second set that pulls down on the roadway from below.?That modification would make the forces acting on the bridge more symmetric and less nonlinear.

“ 'The mathematics predicts that this should work,' McKenna says. Such a bridge would be less likely to oscillate wildly.

"A suspension bridge built in 1850 across the Niagara River gorge provides a historical precedent for the efficacy of tie-down cables. Originally, the bridge was stabilized by a set of cables running from the roadway to the sides of the gorge. The structure survived without incident until the spring of 1864, when engineers temporarily removed the cables to keep them from snaring chunks of ice from the breakup of an unusually heavy ice jam. A heavy wind destroyed the bridge before the cables could be restored.

"Nonlinear effects similar to those influencing suspension bridges also arise in flexible ships, especially when they're lightly loaded and ride high in the water. In a storm, waves can lift much of the ship out of the water.?At that point, its behavior becomes non-linear, and subsequent wave action can cause the entire ship to oscillate.

“ 'Most ships probably won't go into this mode because they're very large and extremely rigid,' McKenna says.

"Nevertheless, nonlinear theory may explain a number of marine disasters, including the famous case of the Edmund Fitzgerald, which mysteriously sank in a storm on Lake Superior in November?1975. As a Great Lakes carrier, the vessel had considerably more flexibility than an oceangoing freighter. One explanation for the tragedy holds that the Edmund Fitzgerald dove into a 'wall of water' and?never recovered. Lazer and McKenna speculate instead that the ship sank after it went into a large-scale flexing motion.

“ 'This would also account for one of the most puzzling aspects of the case, namely why the ship was broken not at its midpoint but at two points approximately 80 feet from the midpoint,' Lazer and McKenna say. That's precisely what would happen if the vessel oscillated in one of the modes that appears as a solution of the simple nonlinear equation they use to model ship behavior."

We need to drive our thinking from real, concrete, individual things and events, and abstractions from them. Reason and science to do not function by "falsification" or by making up "pure theory" from which we could "deduce" real things. Reason and science are fruitful when done a la Aristotle, Galileo, and Newton, not a la Plato, Ptolemy, and Descartes.

Excerpts from "Rock and Roll Bridge” by Ivars Peterson (Science News, vol. 137, no. 22, 1990, pp. 344–46. JSTOR, https://doi.org/10.2307/3974221)

For more info read, e.g., “ Large Torsional Oscillations in Suspension Bridges Revisited: Fixing an Old Approximation” by P. J. McKenna (The American Mathematical Monthly, Jan., 1999, Vol. 106, No. 1 (Jan., 1999), pp. 1-18, https://www.jstor.org/stable/2589581)

Tacoma Narrows Bridge image from https://commons.wikimedia.org/wiki/File:Tacoma-narrows-bridge-collapse.jpg

#Science #Engineering #Physics #Logic #CriticalThinking #Epistemology #RootCause Analysis #RCA #FMEA #Education?