Pohl's wheel
The Pohl wheel (named after its inventor Robert Wichard Pohl ) is a rotating pendulum with a variable eddy current brake . It is stored horizontally and is held in a rest position by a spiral spring , around which it can swing . An exciter with variable amplitude and frequency coupled to the spring makes it possible to generate forced vibrations .
The device is used for didactic purposes, for example in physics internships or lectures . Forced and damped vibrations as well as resonance phenomena are examined with it.
Special experiments with the Pohl wheel
The Pohl wheel and its way into chaos
The deflection of the wheel can be read off a scale attached above the wheel. So that the Pohl wheel behaves chaotically , an imbalance in the form of an additional weight has to be added to the upper half. The damping of the eddy current brake is set to a small value. The pendulum has two stable positions of rest in the unexcited state .
Test execution
In order to illustrate the transition of the rotary pendulum to chaotic behavior, one observes its oscillation at different damping currents and constant excitation frequency and amplitude.
Strong cushioning
It can be observed that the rotary pendulum oscillates periodically at high damping currents after a certain settling time, namely until there is resonance between the oscillator and the rotary pendulum. Resonance means that the natural frequency of the rotary pendulum coincides with the excitation frequency, that is, for example with the speed of an electric motor driving the crank. If the damping is reduced, a steady increase in the amplitude can be expected.
Medium attenuation
This is actually the case up to a certain degree of damping. If you reduce it further, the basic amplitude is split, the so-called bifurcation.
Since the natural frequency of the rotary pendulum depends on the amplitude, the oscillation frequency of the rotary pendulum deviates from the constant excitation frequency with increasing amplitude, so that there is no longer any resonance, which in turn reduces the amplitude. As a result, the pendulum comes into resonance with the exciter and the oscillation amplitude increases again.
Weak damping
If the attenuation is further reduced, a second bifurcation occurs. This means that the basic oscillation is now divided into four sub-oscillations and the period length is four times as long. If the attenuation is further reduced, a third bifurcation occurs. After that it is very difficult to hit the individual bifurcations as the distances between them become smaller and smaller. Finally, if the damping is very weak, even after a long settling time, there is no longer any periodic oscillation: the rotary pendulum oscillates irregularly or chaotically .
Window in chaos
If the damping current strength is further reduced, it can be observed that at certain damping strengths a periodic, stable oscillation suddenly occurs again. This phenomenon is known as the window in chaos . These states dissolve again with a slight change in the damping current strength. This alternation between chaos and order is called intermittency. In more general terms, a system behaves periodically for a long time until it suddenly shows chaotic behavior and then becomes periodic again.
Description of the rotary motion in phase space
In describing the movement of the rotary pendulum, one encounters a problem relating to the representation. Three quantities are necessary for the description: the deflection φ, the speed of the pendulum ω and the current phase of the oscillator , with a value between 0 and the period of the excitation . With these three quantities, the movement of the rotary pendulum can be fully described as a point in a three-dimensional phase space .
Since a three-dimensional phase space is difficult to represent, a trick is used by placing a plane in the phase space on which only the intersection points of the three-dimensional diagram are entered. If this level is chosen favorably, the result is an easy-to-display diagram that describes the development of the behavior of the system over time. The distribution of the intersection points shows whether the system is behaving chaotically or periodically. This experiment clearly shows that even small changes in the initial conditions can have a major impact on the end result.
literature
- Klaus Lüders, Robert Otto Pohl: Pohl's introduction to physics: Volume 1: Mechanics, acoustics and thermodynamics . Springer DE, October 17, 2008, ISBN 978-3-540-76337-6 , pp. 195ff (accessed on November 11, 2012).
- Hans Joachim Eichler , Heinz-Detlef Kronfeldt , Jürgen Sahm: The new basic physics internship . Springer DE, September 21, 2005, ISBN 978-3-540-21453-3 , pp. 75ff (accessed on November 11, 2012).
- Friedhelm Kuypers: Physics for Engineers and Natural Scientists . John Wiley & Sons, October 4, 2012, ISBN 978-3-527-66957-8 , pp. 144ff (accessed November 11, 2012).
- Ludwig Bergmann, Clemens Schaefer: Textbook of Experimental Physics: Mechanics, Relativity, Warmth . Walter de Gruyter, 1998, ISBN 978-3-11-012870-3 , pp. 654ff (accessed on November 11, 2012).
- Klaus G. Schröder: School experiments on chaos on the Pohl pendulum (accessed on July 3, 2013).