Simulation of Multiple-Degree-of-Freedom Oscillatory Systems within an Undergraduate Project-based Learning Environment

Abstract

Coupled oscillations can be found throughout the physical world on both micro and macro levels, from oscillating molecules to lattice vibrations in solids, up to the oscillations of macroscopic mechanical or electrical systems. Despite the fact that the dynamics of such systems is governed by forces originating from a variety of potentials, the harmonic-oscillator potential approximation can be used for almost every system close to equilibrium, which makes it fundamental in many fields of physics. The equations of motion of single harmonic-oscillators as well as of one-dimensional linear elastic multiple-degree-of-freedom systems can be solved analytically, which enables a quantitative study of idealized model systems and, furthermore, some qualitative insight into the behavior of more complex real-life systems. Multi-dimensional multiple-degree-of-freedom systems are, in general, no longer accessible to analytical solutions. A perpendicular spring configuration, for instance, introduces a nonlinearity of the Duffing type and can lead to chaotic behavior. In order to engage our students with the analysis of multiple-degree-of-freedom oscillatory systems, an interdisciplinary undergraduate student research project was initiated, which encompassed the development of computer programs for the simulation and visualization of elastically coupled particles aligned in a straight line, as well as for the simulation of two-dimensional arrays of coupled oscillators. The equation of motion of one-dimensional oscillatory systems was solved numerically and - for small systems - analytically in order to test the quality of the numerical integration. In the case of two-dimensional arrays, the conservation of total energy was used for validation. Three teams of three students each took up the challenge and worked simultaneously and competitively on that project, with the additional complication that the team members had to work in different locations due to the Covid-19 pandemic. The integration of the coupled systems of differential equations was programmed in C#, with a graphical user interface that provides a display of the vibrating systems, graphs of the mass displacements over time, and phase-space diagrams. The dynamic visual output of the program was designed to provide a playful insight into the behavior of multiple-degree-of-freedom lumped-mass systems. In this paper, the theoretical background, the approach to the problem and the outcome of the undergraduate student projects are presented and discussed.