June 15, 2019
June 15, 2019
October 19, 2019
Engineering Physics and Physics
One reason for the success of classical physics is the ability to predict the evolution of systems whose equations of motion and initial conditions are known. Unfortunately, this ability gets lost when systems behave in a chaotic way. Initially it was believed that chaotic behavior is due to the complexity of a system, but eventually it turned out that it appears even in very simple systems. Therefore it became important to gain insight into the manner of how the transition from order to chaos takes place. Now it is known that this transition follows some general patterns with which the system indicates the breakdown of the deterministic behavior.
A simple and convenient experimental setup for the investigation of such a transition between deterministic and chaotic behavior is Pohl’s wheel, a torsional pendulum named after Robert Wichard Pohl. It consists of a ring-shaped copper disk with a homogeneous mass distribution, attached to a rotation axis through its center of gravity. The wheel is elastically bound to an equilibrium position by a spiral spring. The other end of that spring is attached to a motor via an eccentric-and-rod mechanism. This motor provides an additional, external, periodic torque to the pendulum with a selectable angular frequency. Furthermore, an adjustable damping of the wheel is provided by an eddy current brake. By means of an additional mass eccentrically fixed to the wheel, which leads to an imbalance of the copper disk, the restoring force becomes nonlinear. This nonlinearity can effect chaotic behavior of the system for a certain choice of parameters.
As the search for the parameters that cause the transition into chaos turns out to be quite intricate, the experiment should be first numerically simulated and later verified in the real experimental setup. For this purpose a computer program, which simulates and visualizes the forced oscillations of Pohl’s wheel, has been developed within the framework of an undergraduate student project. The program, written in C#, offers a graphical user interface that provides a display of the moving torsional pendulum, a graph of the wheel’s deflection angle over time and a phase-space diagram. All adjustable parameters of the torsional pendulum can be modified interactively, which facilitates the identification of parameter sets leading to the transition between regular and chaotic motion.
In this paper the theoretical background, the approach to the problem and the outcome of the student project will be presented. The dynamic visual output of the program can increase and enhance the understanding of the transition of nonlinear dynamic systems into chaotic states and is therefore well suited as a teaching aid. The program is freely available and can be downloaded from our institution’s home page.
Bischof, G., & Klatzer, M., & Müller, C., & Reifer, D., & Steinmann, C. J. (2019, June), Investigation of the Transition from Order to Chaos by a Numerical Simulation of Pohl’s Wheel Paper presented at 2019 ASEE Annual Conference & Exposition , Tampa, Florida. 10.18260/1-2--33025
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