7.4 Damped and Driven Oscillations (Simple Harmonic Motion)

Damped oscillations lose energy

In real systems, friction or resistive forces like damping (drag) forces remove mechanical energy from an oscillator over time. The amplitude of oscillation decreases exponentially, though the angular frequency may shift slightly from the undamped natural frequency. Damping force is often modeled as proportional to velocity: F = -bv, where b is the damping coefficient.

Three damping regimes

Depending on the strength of damping relative to the restoring force, a system can be underdamped (oscillates with decreasing amplitude), critically damped (returns to equilibrium fastest without oscillating), or overdamped (returns to equilibrium slowly without oscillating). Critical damping is often desired in engineering applications like shock absorbers.

Driven oscillations and resonance

A driven oscillator experiences a periodic external force in addition to the restoring force. When the driving frequency matches the system's natural frequency, resonance occurs, producing a dramatic increase in amplitude. At resonance, energy transfer from the driving force to the oscillator is maximized, limited only by damping present in the system.

Key terms

Damping force

A resistive force, often modeled as F = -bv, that removes energy from an oscillating system.

Underdamped

System oscillates with exponentially decreasing amplitude before settling at equilibrium.

Critically damped

System returns to equilibrium in the shortest possible time without oscillating.

Overdamped

System returns to equilibrium slowly without oscillating, due to strong damping.

Resonance

Large increase in amplitude that occurs when a driving force's frequency matches the system's natural frequency.

Try an AP-style question

What happens to the amplitude of a lightly damped oscillator over time?

1. It drops to zero instantly
2. It increases linearly
3. It stays constant
4. It decreases exponentially