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The Equations of Motion

Oscillations and Resonances

Nonlinear Dynamics

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The Harmonic Oscillator

Nonlinear Oscillations

Nonlinear Resonance

Parametric Resonance

Oscillations and Resonances

A pendulum is a prototypical system of an oscillator. For small angles its equation of motion can be linearized leading to the so-called harmonic oscillator. The harmonic oscillator is a central topic in physics. It has a unique property that is missing in any nonlinear oscillator: The period of oscillation does not depend on the amplitude of oscillation. This is a direct consequence of the superposition principle of linear dynamics. Without this property, it would be impossible to build precise clocks, because controlling the oscillation amplitude with the necessary precision is extremely difficult. The so-called eigenfrequency is a parameter of the system independent from its state. Note, that the prefix eigen is a german word meaning "own". Since the eigenfrequncy is independent from the internal state, it is an ideal fingerprint for identifying a harmonic oscillator. Any method obtaining these fingerprints is called spectroscopy. Most of these methods are based on resonance which is a fundamental behavior of a harmonic oscillator when it is driven by a periodic force. Presumably you have some intuitive knowledge about the resonance phenomenon from swinging a swing: In order to swing very high you have to move forward and backward with roughly the same frequency as the swing would move forward and backward alone.

The pendulum is a nonlinear oscillator. Although nonlinear oscillators share many properties with a harmonic oscillator there is an important difference: The frequency of a nonlinear oscillator depends on the amplitude of oscillation. This leads to bistability and hysteresis because of the foldover effect. Furthermore, resonances not only occur at the frequency of the oscillator but also at integer fractions (superharmonic resonance) and at twice the frequency (parametric resonance).







© 1998 Franz-Josef Elmer,  elmer@ubaclu.unibas.ch last modified Sunday, July 19, 1998.