Damped Harmonic Oscillator Reference Cheat Sheet

Key Facts

• The damped oscillator equation for linear damping is $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$.
• The natural angular frequency without damping is $\omega_0 = \sqrt{\frac{k}{m}}$.
• The damping constant used in many solutions is $\gamma = \frac{b}{2m}$.
• For underdamped motion, the displacement is $x(t) = A_0 e^{-\gamma t}\cos(\omega_d t + \phi)$.
• The damped angular frequency is $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$ for an underdamped oscillator.
• The motion is underdamped when $\gamma < \omega_0$, critically damped when $\gamma = \omega_0$, and overdamped when $\gamma > \omega_0$.
• The amplitude envelope decreases according to $A(t) = A_0 e^{-\gamma t}$.
• The mechanical energy decreases approximately as $E(t) = E_0 e^{-2\gamma t}$ because energy is proportional to amplitude squared.

Vocabulary

Damping

Damping is the loss of mechanical energy from an oscillator due to resistive forces such as friction, drag, or internal resistance.

Damping force

A damping force is often modeled as $F_d = -bv$, where $b$ is the damping coefficient and the negative sign means the force opposes velocity.

Natural angular frequency

The natural angular frequency $\omega_0 = \sqrt{\frac{k}{m}}$ is the angular frequency the system would have with no damping.

Damped angular frequency

The damped angular frequency $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$ is the oscillation rate of an underdamped system.

Critical damping

Critical damping occurs when $\gamma = \omega_0$, giving the fastest return to equilibrium without oscillation.

Quality factor

The quality factor $Q = \frac{\omega_0}{2\gamma}$ measures how weakly damped an oscillator is, with larger $Q$ meaning slower energy loss.