Forced Oscillations and Resonance

When an external periodic force continuously drives an oscillating system, the resulting motion is called a forced oscillation. The frequency at which the driving force oscillates is the driving frequency (f_d), and the system's inherent frequency of free oscillation is the natural frequency (f₀). The relationship between these two frequencies determines the amplitude of the response, leading to one of the most important phenomena in physics: resonance.

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Forced Oscillations

Consider a mass on a spring that is being driven by an external periodic force — perhaps the top of the spring is attached to a mechanical oscillator that moves up and down at a controllable frequency f_d.

When the driving force is first applied, there is a brief transient phase where the system adjusts. After this, the system settles into a steady state where:

• The system oscillates at the driving frequency f_d, not at its natural frequency
• The amplitude of oscillation depends on how close f_d is to f₀
• The phase relationship between the driving force and the displacement depends on f_d

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Resonance

Resonance occurs when the driving frequency equals the natural frequency:

f_d = f₀