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Visualization of Wave Phenomena by an Array of Coupled Oscillators

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Conference

2016 ASEE Annual Conference & Exposition

Location

New Orleans, Louisiana

Publication Date

June 26, 2016

Start Date

June 26, 2016

End Date

August 28, 2016

ISBN

978-0-692-68565-5

ISSN

2153-5965

Conference Session

Mathematics Division Technical Session 1

Tagged Division

Mathematics

Page Count

16

DOI

10.18260/p.27185

Permanent URL

https://peer.asee.org/27185

Download Count

747

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Paper Authors

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Günter Bischof Joanneum University of Applied Sciences

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Throughout his career, Dr. Günter Bischof has combined his interest in science and engineering application. He studied physics at the University of Vienna, Austria, and acquired industry experience as development engineer at Siemens Corporation. Currently he teaches Engineering Mathematics at Joanneum University of Applied Sciences. His research interests focus on automotive engineering, materials physics, and on engineering education.

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Thomas Singraber B.Sc. Joanneum University of Applied Sciences

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Thomas Singraber obtained his B.Sc. in Automotive Engineering at the FH Joanneum, University of Applied Sciences Graz, Austria. Currently he is working on finalizing his Master's Thesis at the same faculty with a company partner supplying components to top motorsport teams all over the world. During his time at the Formula Student team he focused his work on aerodynamics and chassis developement and achieved therefore practical knowledge on a wide spectrum of racing topics. On completion of his studies, he intends to pursue an interdisciplinary career in the automotive sector with a strong motorsport affiliation.

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Christian J. Steinmann HM&S IT-Consulting

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Christian Steinmann has an engineer degree in mathematics from the Technical University Graz, where he focused on software quality and software development process assessment and improvement. He is manager of HM&S IT-Consulting and provides services for SPiCE/ISO 15504 and CMMI for development as a SEI-certified instructor. He performed more than 100 process assessments in software development departments for different companies in the finance, insurance, research, automotive, and automation sector. Currently, his main occupation is a consulting project for process improvement for safety related embedded software development for an automobile manufacturer. On Fridays, he is teaching computer science introductory and programming courses at Joanneum University of Applied Sciences in Graz, Austria.

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Marton Szabo-Kass B.Sc. Joanneum University of Applied Sciences

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Marton Szabo-Kass obtained his B.Sc. degree in Automotive Engineering at Joanneum University of Applied Sciences in Graz, Austria, after which he is currently completing his M.Sc. studies at the same faculty. He gained practical experiences through leading the Formula Student racing team of his university and doing internships at high-tech automotive companies such as Porsche Motorsport. On completion of his studies, he intends to pursue an interdisciplinary career in the automotive and motorsport sector.

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Stefan Woerndl B.Sc. Joanneum University of Applied Sciences

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Stefan Woerndl obtained his B.Sc. degree in Automotive Engineering at the University of Applied Sciences Joanneum Graz. Currently he is completing his M.Sc. studies at the same faculty. Prior to this he gained some work experience as a technician, also in the automotive sector. On completion of his studies, he intends to pursue a career in research.

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Abstract

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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