Math Solutions

Discuss the cause of the Tacoma bridge disaster, in terms of waves, vibrations, and resonance. Elaborate the effects with relevant equations and formulae. The Tacoma bridge collapse can be attributed to the waves caused by the buildup of energetic vibrations. These energetic vibrations were built up from the bridge “taking energy from the steadily blowing wind” (Crowell). Eventually enough of these energetic vibrations built up to cause resonance within the system, causing the wave-like motion of the bridge. Energetic vibrations can be explained by using the example of a mass on a spring where potential and kinetic energy are involved. This system can be put into motion by “hitting the mass to put in kinetic energy or by pulling it to one side to put in potential energy” (Crowell) and will continue in this motion, trading the energy from potential to kinetic and back again, assuming there is no friction in the system. Knowing that the total energy is constant and that it is proportional to the square of the amplitude we are left to determine when the energy is in a certain state. With the Tacoma Bridge there were two different mechanisms involved in its collapse. “The first mechanism was the one responsible for the initial, relatively weak vibrations, and it involved resonance. As the wind moved over the bridge, it began acting like a kite or an airplane wing…The bridge [however] was probably destroyed by a different mechanism, in which its vibrations at its own natural frequency of 0.2 Hz set up an alternating pattern of wind gusts in the air immediately around it, which then increased the amplitude of the bridge's vibrations. This vicious cycle fed upon itself, increasing the amplitude of the vibrations until the bridge finally collapsed.” (Crowell) The sudden change in pressure the bridge underwent caused an up and down force on the bridge similar to the force on a waving flag. “This back-and-forth sequence of forces is exactly the kind of periodic driving force that would excite a resonance. The faster the wind, the more quickly the swirls would get across the bridge, and the higher the frequency of the driving force would be. At just the right velocity, the frequency would be the right one to excite the resonance” (Crowell)

SOURCES:
Crowell, B. (n.d.). Resonance. Light and Matter: open-source physics textbooks. Retrieved July 20, 2013, from http://www.lightandmatter.com/html_books