New Orleans, Louisiana
June 26, 2016
June 26, 2016
August 28, 2016
Mechanical engineering curricula typically include courses in classical mechanics and continuum mechanics. Classical mechanics is governed by the Newtonian axioms, which lead to ordinary differential equations as the equations of motion. The mathematical description of continuum mechanics, on the other hand, is based on partial differential equations, describing the conservation laws and the constitutive relations. The underlying theories of ordinary and partially differential equations are usually covered in different mathematics courses, and a typical approach to a first discussion of partial differential equations in engineering mathematics is the heuristic derivation of the transversal wave equation of a vibrating string. Another approach, the continuum limit of the loaded string, leads to the one-dimensional longitudinal wave equation. Both approaches start from the ordinary differential equations of Newtonian mechanics and lead to partial differential equations of continuum mechanics. The advantage of the continuum limit of a chain of masses connected by springs is that it is easily comprehensible for students and, in addition, many materials respond to small perturbations just as if they were a system of coupled oscillators. This harmonic oscillator response to perturbations leads in a continuum model to the appearance of wave phenomena. For the visualization of such wave phenomena, a computer program that simulates a two-dimensional spring-mass system has been developed within an undergraduate student project. The model consists of a rectangular lattice of regularly spaced point masses connected to each other and to the rigid boundary by a network of massless springs. The force on each mass is computed due to its spring connections with its neighbors, along with external forces such as gravity. Energy dissipation can be added to the model on demand by viscoelastic damping. The motion of each particle is governed by Newton’s second law, which requires the solution of a system of coupled ordinary differential equations. This is done in the C programming language via a variety of implemented numerical integration schemes. The computer program allows the visualization of the motion of the point masses, which can be initiated by the displacement of an arbitrary number of masses via mouse drag. The motion of a single mass connected by springs to the adjacent walls closely resembles the behavior of the well-known harmonic oscillator. The introduction of additional masses brings particle interaction into effect, which leads to energy transfer in the system. When the number of involved mass points is increased, the moving particles increasingly appear to the observer as a continuous system. This is due to a scale change that links the microstructure of coupled particles with the macroscopic behavior of a continuous material, which includes wave behavior like interference effects. The dynamic visual output of the program can increase and enhance understanding of various wave phenomena and is therefore well suited as both a teaching aid and an analysis tool.
Bischof, G., & Singraber, T., & Steinmann, C. J., & Szabo-Kass, M., & Woerndl, S. (2016, June), Visualization of Wave Phenomena by an Array of Coupled Oscillators Paper presented at 2016 ASEE Annual Conference & Exposition, New Orleans, Louisiana. 10.18260/p.27185
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