The Problem

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Make a light pendulum with a small magnet at the free end.

An adjacent electromagnet connected to an AC power source of a much higher frequency than the natural frequency of the pendulum can lead to undamped oscillations with various amplitudes.

Study and explain the phenomenon.

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Key Conditions

Maximum amplitude near natural frequency

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Difference to FOrced Harmonic Oscillator

Harmonic

This Oscillator

Various undamped amplitudes driven by higher frequencies.

Driving force only time dependent.

Smaller sized EM field means force diminishes rapidly with displacement.

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Setup

Setup

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Rigid Arm

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Keeps magnet orientation constant.

Bob confined to arc with 1 degrees of freedom.

Setup

Solenoid Orientation

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Possible configurations:

Setup

Solenoid Specs

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Setup

Magnet Orientation

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Possible configurations:

Setup

Bearing, Pendulum Suspension

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If friction is too large,

weaker electromagnetic field could not compensate the energy dissipated by friction.

Setup

Pendulum interpretation: suspension needs to limit the motion onto a 2D plane.

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This setup is similar to that of a pendulum by Doubochinski.

Dynamical System

Doubochinski pendulum is known for having discrete stable amplitudes.

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Proposed Mechanisms

1.  Velocity modulation

2.  Phase regulation

Proposed by Doubochinski et al

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Proposed Mechanisms

1.  Velocity modulation

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Proposed Mechanisms

2.  Phase regulation

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Damping

For practicality, we investigated the pendulum in an overdamped configuration for initial amplitudes at angles within 90deg from vertical.

Overdamped

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Approx 20 Hz

Over Damped

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Overdamped

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Conclusions from overdamp

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In the overdamped configuration, velocity modulation had a great affect, but the phase regulation was not effective.

From different Frequencies

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Other frequencies (15 to 50Hz) resembled an ordinary damped oscillator.

Note the smaller fluctuations

Suggests that the smaller amplitudes increased interaction time amplified the velocity modulation, but kept shifting in and out of phase.

From different Frequencies

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Other frequencies (15 to 50Hz) resembled an ordinary damped oscillator.

Fluctuations present, but less 

Suggests that the smaller amplitudes increased interaction time amplified the velocity modulation, but kept shifting in and out of phase.

Conclusion

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For overdamped configuration of the Doubochinski's pendulum, phase regulation seems less apparent.

Increasing frequency creates smaller fluctuations, whose effect diminishes with frequency.