Exposing undergraduate engineering students to nonlinear differential equations using a practical approach in project based learning environments

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 Poincar 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.