The Beam Lab

Perhaps the most difficult lab is one that explores the geometry, material properties, and deformation of loaded cantilevered and simply-supported beams. The lab is certainly the longest, taking up to three weeks to properly complete. This lab was designed in collaboration with a colleague in aerospace engineering who teaches an upper-division engineering course in deformation mechanics. (This course is required of many engineering majors.)

The mathematical basis for the lab is fairly simple: the so-called ``beam equation'' is a fourth order linear differential equation that may be solved exactly by performing four integrals and applying four (independent) boundary conditions to determine the constants of integration. The complication arises in connecting the expressions for the differential equation and boundary conditions to the physical situation. In engineering terms, what does it mean that the second and third derivatives at the free end of a cantilevered beam are zero? How do the dimensions, cross-section, and material properties of beam affect its deformation?

In this lab, the students completely analyze the beam problem with the help of a ``beam simulator'' that allows students to simulate the bending of any beam with a rectangular cross-section. The students compare the prediction of the (linear) model with actual experimental data, and discover some of the parameter regimes in which the beam equation is a good model of real beam deformation. Data from this lab is also used in multi-variable calculus for a series of discussions about rates of change in multi-variable functions.

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Frederick J. Wicklin <fjw@geom.umn.edu>