Oscillations and Resonance

By Derek Kverno

Abstract:

    Resonance occurs in any oscillatory system when the frequency of the driving force is near the natural frequency of the system.  At the resonant frequency of a linear system the amplitude of oscillation is at its maximum value.  Therefore, in a linear system of oscillation there is a direct correspondence between the amplitude of oscillation and the driving frequency, and thus a periodic input always results in a periodic output.  However, a non-linear system may oscillate at various amplitudes for one particular driving frequency because the elements of the driving force may not vary linearly with the space parameters.  Therefore, there is no unique maximum resonant amplitude in a non-linear oscillatory system and a periodic input may not result in a periodic output.  In such a non-linear oscillatory system one may observe chaotic behavior in the output of the system.  This paper examines normal resonance in a linear oscillatory system within the study of vibrating plates, a Helium-Neon Laser and a Carbon Dioxide Laser.  It then explores resonance in a non-linear oscillatory system, which may result in chaotic behavior, within the study of a pendulum and an RLC circuit.

Table of Contents:

Discussion of Oscillation and Resonance in Linear and Non-Linear Systems

Oscillations and Resonance in Linear Systems

1. Vibrating Plates
2. Helium-Neon Laser
3. Carbon Dioxide Laser

Oscillations and Resonance in Non-Linear Systems

1. Pendulum
2. RLC Circuit

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