|Ivars Peterson's MathTrek|
February 1, 1999
Many students also remember the standard explanation for the disaster. Physics 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 vortexes 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.
That explanation, however, is incomplete and flawed. Engineers Robert H. Scanlan of Johns Hopkins University in Baltimore and K. Yusuf Billah of Princeton University sought to set the record straight by providing a detailed account of what actually went wrong and why in an article in the February, 1991 American Journal of Physics.
One issue is the source of the periodic impulses that would produce resonance. There is no definite periodicity generally associated with wind gusts or gales. Moreover, in the prevailing 42-mile-per-hour winds, the bridge wasn't likely to shed vortexes at a rate that would excite the twisting oscillations that finally drove the bridge to destruction.
Scanlan and Billah focused on a second, complex type of vortex, one associated with the motion of the structure itself. In this case, the mechanism responsible for large oscillations is self-excitation -- an interaction between the bridge's motion and the vortexes produced by that motion --rather than forced resonance. Unlike the wind-induced vortexes that put the bridge into an up-and-down, roller-coaster motion, the additional motion-induced vortexes would get damped out at low amplitudes but predominate at large amplitudes of oscillation.
The destructive twisting mode didn't kick in until a minor structural failure created asymmetrical conditions. Once the twisting started, it got completely out of control in about 45 minutes.
The standard engineering explanations, however, haven't satisfied applied mathematician P. Joseph McKenna of the University of Connecticut. He has spent nearly two decades developing alternative mathematical models to account for the undulations and gyrations shown by the ill-fated bridge. What distinguishes suspension bridges is their fundamental nonlinearity, McKenna argues.
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, however, can 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.
In a 1990 report, McKenna and colleague Alan C. Lazer of the University of Miami contended that a complete mathematical explanation for the Tacoma Narrows disaster must isolate the factors that make suspension bridges prone to large-scale oscillations; show how a bridge could go into large oscillations as the result of a single gust and at other times remain motionless even in high winds; and demonstrate how large vertical oscillations could rapidly change to a twisting motion.
One significant detail, McKenna asserted, lies in the behavior of the vertical strands of wire, or stays, connecting the roadbed to a bridge's main cable. Civil engineers usually assume that the stays always remain in tension under a bridge's weight, in effect acting as stiff springs. That allows them to use relatively simple, linear differential equations to model the bridge's behavior.
When a bridge starts to oscillate, however, the stays begin alternately loosening and tightening. That produces a nonlinear effect, changing the nature of the force acting on the bridge. When the stays are loose, they exert no force, and only gravity acts on the roadbed. When the stays are tight, they pull on the bridge, countering the effect of gravity.
Solutions of the nonlinear differential equations that correspond to such an asymmetric situation suggest that, for a wide range of initial conditions, a given push can produce either small or large oscillations. Lazer and McKenna went on to argue that the alternate slackening and tightening of cables might also explain the large twisting oscillations experienced by a suspension bridge.
McKenna revisits the Tacoma Narrows Bridge collapse in the January American Mathematical Monthly--with a new twist. He offers an alternative scenario as a possible cause of the bridge's extreme torsional motion.
The basic model is extremely simple: A rigid horizontal rod supported by two vertical springs at each side to represent a cross section of a suspension bridge. This geometry provides the framework that underlies equations expressing the rod's vertical and twisting motions.
Normally, those equations are simplified (or linearized) by assuming that the oscillations are small and near equilibrium. Indeed, engineers nowadays design and construct bridges so that oscillations remain strictly limited. Despite the inherent approximations, the linear theory works under those conditions.
However, "redesigning the bridge to remove the offending behavior is not the same as mathematically understanding its cause," McKenna insists. Linearizing the equations has the effect of removing a large class of interesting, large-amplitude motions from consideration.
Taking advantage of the computational power now available for solving nonlinear equations accurately, McKenna investigated the original nonlinear model, which he calls the trigonometric oscillator. He discovered that, for various initial conditions and pushes, the equations can lead to startlingly different types of motion.
In fact, "the trigonometric oscillator can have several different periodic responses to the same periodic forcing term," he emphasizes. "Which response eventually results can be determined by a single transient event such as a large. . .push."
In effect, one ends up with multiple solutions to the equations and chaotic behavior.
Combining those results with the effect of the slackening of cables for brief, transitory periods because of vertical movement, McKenna comes to a new and unexpected conclusion. "A purely vertical nontorsional motion in which the cables lose tension can become disastrously unstable even in the presence of tiny torsional forces, setting up a rapid transition to large-amplitude torsional motion," he says.
"We believe that we have discovered a convincing explanation for the mystery of the sudden transition to torsional motion," McKenna contends. "A large vertical motion had built up, there was a small push in the torsional direction to break symmetry, the instability occurred, and small aerodynamic torsional periodic forces were sufficient to maintain the large periodic torsional motions."
No mathematical model is ever perfect, however. As Alan M. Turing once noted, "This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of the greatest importance in the present state of knowledge."
What is also certain is that the Tacoma Narrows Bridge disaster will continue to fascinate.
Copyright 1999 by Ivars Peterson
Berreby, D. 1992. The great bridge controversy. Discover (February):26.
Billah, K.Y., and R.H. Scanlan. 1991. Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks. American Journal of Physics 59(February):118.
Lazer, A.C., and P.J. McKenna. 1990. Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis. SIAM Review 32(December):537.
McKenna, P.J. 1999. Large torsional oscillations in suspension bridges revisited: Fixing an old approximation. American Mathematical Monthly 106(January):1.
McKenna, P.J., and W. Walter. 1990. Travelling waves in a suspension bridge. SIAM Journal of Applied Mathematics 50(June):703.
Peterson, I. 1990. Rock and roll bridge. Science News 137(June 2):344.
Petroski, H. 1991. Still twisting. American Scientist 79(September-October):398.
Scanlan, R.H. 1982. Developments in low-speed aeroelasticity in the civil engineering field. AIAA Journal 20(June):839.
Schwarz, F.D. 1993. Why theories fall down. Invention & Technology (Winter):6.
Additional commentaries on the cause of the Tacoma Narrows Bridge failure are available at http://www.icaen.uiowa.edu/~hawkeng/spring_97/articles/gertie.html, and http://www.tfhrc.gov/pubrds/winter96/p96w46.htm.
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