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Exposing Undergraduate Engineering Students to Nonlinear Differential Equations Using a Practical Approach in Project-based Learning Environments

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Conference

2020 ASEE Virtual Annual Conference Content Access

Location

Virtual On line

Publication Date

June 22, 2020

Start Date

June 22, 2020

End Date

June 26, 2021

Conference Session

Mathematics Division Technical Session 1: Best Practices in Engineering Math Education

Tagged Division

Mathematics

Page Count

18

DOI

10.18260/1-2--34652

Permanent URL

https://strategy.asee.org/34652

Download Count

74

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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 is an associate professor at Joanneum University
of Applied Sciences and teaches engineering and applied mathematics.

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Maximilian Brauchart Joanneum University of Applied Sciences

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Maximilian Brauchart is currently studying Automotive Engineering at the Joanneum University of Applied Sciences. Prior to his studies, he attended a grammar school in Graz.

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Patrick Jenni Joanneum University of Applied Sciences

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Patrick Jenni is currently studying Automotive Engineering at the University of Applied Sciences JOANNEUM Graz.
After completing his studies, he intends to work in development.

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Jeremias Pirker Joanneum University of Applied Sciences

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Jeremias Pirker is currently studying Automotive Engineering at the University of Aplied Sciences Joanneum Graz. After completion of his studies, he would like to participate in the development of technical innovations.

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Julian Sachslehner Joanneum University of Applied Sciences

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Julian Sachslehner is currently studying Automotive Engineering at the University of Applied Sciences Joanneum Graz. Before he started his studies he worked as a car mechanic and gained some practical experience in the automotive sector.

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Christian J. Steinmann Joanneum University of Applied Sciences

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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 HMS IT-Consulting and provides services for SPiCE/ISO 15504 and CMMI for development as certified Automotive SPiCE assessor. 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 software development in the automobile sector. On Fridays, he is teaching computer science introductory and programming courses at Joanneum University of Applied Sciences in Graz, Austria.

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Tobias Markus Zörweg Joanneum University of Applied Sciences

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Tobias Markus Zörweg is currently studying Automotive Engineering at the University of Applied Sciences Joanneum Graz.
After completion of his studies, he aspires to work in development.

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Abstract

The increasing complexity inherent in the technologies currently being developed has profound implications for engineering education. Our students have to face increasingly complex adaptive systems organized in recursive, nonlinear feedback loops. A more pronounced attention to nonlinear systems could thus enable our students to understand not only the usefulness but also the limits of linearity, and facilitate the comprehension of more complex systems. This is especially true for mathematics, as differential equations play a fundamental role in the modeling and analysis of complex systems. Courses in ordinary differential equations are part of the majority of undergraduate engineering curricula, and typically cover analytical solution methods for first-order and linear higher-order differential equations, as well as an introduction to numerical methods like the explicit Euler integration. Increasingly, nonlinear differential equations are included in the curricula, with a focus on phase portraits, stability, and local linearization, based on standard systems like the predator-prey equations. The present work introduces an application of nonlinear differential equations in a classical engineering problem that is well-suited for undergraduate education, namely the optimal design of a dynamic vibration absorber. Such absorbers are mounted in a wide variety of structures to reduce the amplitude of mechanical vibrations, and are frequently designed as coupled spring-mass-damper systems. Linear absorbers can be tuned to suppress the resonance frequency of the structure, but unfortunately only within a relatively small frequency bandwidth. The introduction of nonlinearity into the damper system can overcome that drawback and increase the attenuation bandwidth. A perpendicular spring configuration, for instance, introduces a nonlinearity of the Duffing type and can increase the absorber’s effective bandwidth significantly. The mathematical modelling and numerical simulation of such a nonlinear absorber was carried out within the framework of a multidisciplinary undergraduate student project. The integration of the coupled system of differential equations was programmed in C#, with a graphical user interface that provides a display of the vibrating system, a graph of the mass displacements over time, the system’s response function, phase-space diagrams, and Poincaré sections. The adjustable parameters of the vibration absorber can be modified interactively during the simulation, which facilitates the identification of parameter sets leading to instabilities that can intensify the system’s vibration instead of suppressing it. The dynamic visual output of the program is expected to increase the students’ understanding of tuned vibration absorbers and of the transition of nonlinear dynamic systems into chaotic states. In this paper, the theoretical background, the approach to the problem and the outcome of the undergraduate student project are presented and discussed.

Bischof, G., & Brauchart, M., & Jenni, P., & Pirker, J., & Sachslehner, J., & Steinmann, C. J., & Zörweg, T. M. (2020, June), Exposing Undergraduate Engineering Students to Nonlinear Differential Equations Using a Practical Approach in Project-based Learning Environments Paper presented at 2020 ASEE Virtual Annual Conference Content Access, Virtual On line . 10.18260/1-2--34652

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