Forced Oscillations and Resonance

Periodic steady-state response to harmonic external drives, resonance curves, and Q factor.

General idea of the topic

When an external periodic force acts on an oscillator, after a transient the system oscillates at the frequency of the force, not its own. The amplitude of that response depends on how close the force frequency is to the natural frequency and on the damping.

The topic studies first the forced oscillation, with its steady-state solution and its phase lag with respect to the force, and then resonance: when the frequencies nearly coincide, the amplitude reaches a maximum that damping limits.

What you will truly understand

By the end you will be able to compute the amplitude and phase of a forced oscillator, locate the resonance frequency and estimate its width from the quality factor.

You will also know why small damping gives a tall, narrow resonance and large damping a low, wide one.

Recommended learning route

1. Forced Oscillation: equation with a driving term, steady-state response, amplitude and phase.
2. Resonance: amplitude maximum near the natural frequency, resonance curve and quality factor.

Physical criterion you must acquire

The essential criterion is to distinguish three frequencies: that of the force, the natural one of the undamped system and the resonance one, slightly lower when there is damping. The energy entering each cycle is balanced by what friction dissipates, and at resonance the phase between force and displacement is 90°.

Connection with real problems

• bridges and buildings under wind and earthquakes.
• musical instruments and sound boxes.
• radios and tuners.
• magnetic resonance.

Final goal of the topic

The final goal is to predict the response of an oscillator to a periodic force, locate resonance and assess its risks and uses.