In June of 2000, London's Millennium Bridge opened. Several days later, it closed for a period of two years. What happened in the initial days the bridge was opened? The bridge turned out to be susceptible to a phenomenon called "synchronous lateral excitation." As summarized here, "the natural sway motion of people walking caused small sideways oscillations in the bridge, which in turn caused people on the bridge to sway in step, increasing the amplitude of the bridge oscillations and continually reinforcing the effect." This is a phenomenon known as resonance, which can be modeled using a second order constant coefficient linear differential equation with an inhomogeneity. This clip shows the synchronous walking and the bridge shaking. The bridge debacle spawned various mathematical modeling efforts, including this one.
By the end of this lesson, you should be able to:
- Solve linear, inhomogeneous constant coefficient equations via undetermined coefficients - [movie] [notes]
- Read Polking, Boggess and Arnold, Section 4.5.
- Take a look at these resources if you want to seem some example problems with solutions.
- Watch pencasts (linked above).
- Do post-class problems.