Modeling a Building during an Earthquake to Demonstrate Resonance

Spring-mass systems are presented as an application of second-order, constant coefficient differential equations in many differential equations textbooks. The phenomenon of resonance can then be analysed after nonhomogeneous differential equations are introduced. With some simplifying assumptions, the movement of the roof of a one-story building can be modeled as a spring-mass system and ground vibrations, such as those from an earthquake, can then be viewed as an external force acting on this system. In this paper, I will discuss a project I wrote and use in which students analyse the movement of a building resulting from an earthquake. This requires students to think in terms of amplitude/displacement (rather than position) and observe how this is affected by the frequency of an external force.