Investigation of the Transition from Order to Chaos by a Numerical Simulation of Pohl’s Wheel

One reason for the success of classical physics is the ability to predict the evolution of systems whose equations of motion and initial condi tions are known. Unfortunately, this ability becomes lost when systems behave in a chaot ic way. Initially it was believed that chaotic behavior is due to the complexity of a syst em, but eventually it has transpired that this behavior appears even in very simple systems. There for , it has become important to gain insight into the manner of how the transition from rder to chaos takes place. In the meantime, it is known that this transition follows some general patterns with which the system indicates the breakdown of the deterministic behavi or. A simple and convenient experimental setup for the inv stigation 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 d isk with a homogeneous mass distribution, attached to a rotation axis through its center of g ravity. 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, it is possible to adjust damping of the wheel using an eddy current brake. B y 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 affe ct chaotic behavior of the system for a certain choice of parameters. As the search for the parameters that cause the tra nsition into chaos has turned out to be quite intricate, the experiment should be first numerical ly simulated and later verified in the real experimental setup. For this purpose, a computer pr ogram has been developed within the framework of an undergraduate student project, whic h s mulates and visualizes the forced oscillations of Pohl’s wheel. 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 the adjust able parameters of the torsional pendulum can be modified interactively, which facilitates gr eatly the identification of parameter sets leading to the transition between regular and chaot ic motion. In this paper the theoretical background, the appro ach to the problem and the outcome of the student project is presented. The dynamic visual ou tput of the program can increase and enhance the understanding of the transition of nonl i ear dynamic systems into chaotic states and is therefore well suited as a teaching aid.