Damping

Underdamped, critically damped, and overdamped regimes in real dissipative oscillators.

General idea of the topic

In a real oscillator there is friction or resistance from the medium that dissipates energy, so the amplitude decreases with time. With a friction force proportional to velocity, F = −bv, the equation of motion has three behaviors depending on the ratio between b and the critical value 2√(km).

The topic studies each regime: the underdamped one, where the system oscillates with decreasing amplitude; the critical one, which returns to equilibrium as fast as possible without oscillating; and the overdamped one, which returns without oscillating but more slowly.

What you will truly understand

By the end you will be able to classify a damped oscillator from its parameters, compute the damped frequency and the decay time, and choose the right damping for each application.

You will also be able to interpret the quality factor as a measure of how long the oscillation lasts.

Recommended learning route

1. Underdamped Regime: oscillation with a decreasing exponential envelope and a frequency slightly below the natural one.
2. Critical Damping: value of b giving the fastest return to equilibrium without oscillating.
3. Overdamped Regime: slow, monotonic return to equilibrium when friction exceeds the critical value.

Physical criterion you must acquire

The essential criterion is to compare b with 2√(km), or γ = b/(2m) with ω₀. If γ < ω₀, it oscillates; if γ = ω₀, it is critical; and if γ > ω₀, it does not oscillate. More damping is not always better: the critical one gives the fastest response, and above it the system becomes slower.

Connection with real problems

• vehicle shock absorbers.
• soft-close doors.
• measuring instruments.
• buildings with seismic dampers.

Final goal of the topic

The final goal is to identify the regime of a damped oscillator, compute its evolution and size the damping according to the use.