Chapter 7.3, Problem 1.4SP

Resonance

Consider an undamped system exhibiting simple harmonic motion. In the real world, we never truly have an undamped system; -some damping always occurs. For theoretical purposes, however, we could imagine a spring-mass system contained in a vacuum chamber. With no air resistance, the mass would continue to move up and down indefinitely.

The frequency of the resulting motion, given by $f=\frac{1}{T}=\frac{\omega }{2\pi }$ , is called the natural frequency of the system. If an external force acting on the system has a frequency close to the natural frequency of the system, a phenomenon called resonance results. The external forte reinforces and amplifies the natural motion of the system.

4. In the real world, there is always some damping. However, if the damping force is weak, and the external force is strong enough, real-world systems can still exhibit resonance. One of the most famous examples of resonance is the collapse of the Tacoma Narrows Bridge on November 7, 1940. The bridge had exhibited strange behavior ever since it was built. The roadway had a strange “bounce” to it. On the day it collapsed, a strong Windstorm caused the roadway to twist and ripple violently. The bridge was unable to withstand these forces and it ultimately collapsed. Experts believe the windstorm exerted forces on the bridge that were very close to its natural frequency, and the resulting resonance ultimately shook the bridge apart.

This website (http://www.openstaxcollege.org/l/20_TacomaNarrow) contains more information about the collapse of the Tacoma Narrows Bridge.

During the short time the Tacoma Narrows Bridge stood, it became quite a tourist attraction. Several people were on site the day the bridge collapsed, and one of them caught the collapse on film. Watch the video (http//www.openstaxcollege.org/l/20_TacomaNarr02) to see the collapse.