Modeling a Diving Board

Abstract

The beam equation is a classic fourth-order partial differential equation used to de scribe the transverse displacement of a vibrating beam. One may encounter it in an introductory course on boundary value problems or mathematical physics. One par ticular type of beam modeled with this equation is the cantilever, which is a long, thin, rectangular bar with one end clamped and the other end free to move up and down [3]. Cantilever beams appear in a variety of applications, including bridge con struction [13], aircraft wing design [1], architecture [9,11], microwave imaging [15], temperature measurement [19], and virus detection [7]. This paper came about as the result of an undergraduate Honors Thesis project at Ball State University [16]. The goal of the thesis was to address the question: Since we can think of a diving board as a cantilever, how well does the beam equation model a diving board? To answer this question, we created a simple experimental set-up con sisting of a two-meter stick with one end clamped to a table and the free end set in motion. Vertical displacements at tape-marked positions along the beam were cap tured with a digital video camera, analyzed with the freely available World-in-Motion physics demonstration software [4], and compared to a model based on the standard beam equation. As is often the case with models of real-world phenomena, our initial model failed to match the measured data. Using the wave equation as a model [2], we first introduced a damping term to improve results and then a forcing term to arrive at a reasonable model.