Visualizing PDE solutions and Fourier Transforms with Flexural Waves

Abstract

The concepts of Partial Differential Equations and Fourier Transforms are taught in the Mathematics courses of the first university years of both basic sciences and engineering. Both concepts are of capital importance for the training of future researchers, academics and engineers since a great number of physical phenomena present their theoretical modeling by means of these two relevant branches of Mathematics. In this work we use the one-dimensional flexural waves in a solid elastic beam to contextualize a simple physical system that helps us to understand, as well as to deepen and to visualize in a simple way the intricacies of these two parts of the Mathematics curriculum of the first University courses. In particular, flexural waves are governed by a wave equation, which, unlike electromagnetic (vector) or acoustic (scalar) waves, is of fourth order. This has profound implications for how these waves propagate, such as dispersion. Normally, in the first university courses, the equations of the harmonic oscillator, i.e. of second order, are solved. By means of three project based learning tutorials, here the students are confronted with the solution of various boundary problems, using the method of separation of variables with transcendental equations that are not in general analytical. The students must obtain the ratios using Newton or secant type numerical methods. Moreover, a spatial Fourier transform would be used to obtain the dispersion relation of the flexural waves. An experimental system consisting of a free-free aluminum beam excited by a shaker will be scanned by a laser vibrometer to obtain the modal displacements and compared with those obtained theoretically to visualize the mathematical solutions.

Keywords

• partial differential equations
• fourier transforms
• flexural waves
• eigenfrequencies
• eigenmodes
• wave equation
• wave scattering