Damped Harmonic Oscillator Formula
The equation of motion and solution for a damped harmonic oscillator.
Covers underdamped, critically damped, and overdamped regimes with real-world examples.
The Formula
ω' = √(ω₀² − γ²)
γ = b / (2m)
ω₀ = √(k/m)
A damped harmonic oscillator is a spring-mass system (or any oscillating system) where friction or resistance removes energy over time. The oscillations gradually decay. Three distinct behaviors occur depending on the ratio of damping to natural frequency.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| x(t) | Displacement at time t | meters (m) |
| A | Initial amplitude | meters (m) |
| γ | Damping coefficient = b/(2m) | s⁻¹ |
| b | Damping constant (viscous drag coefficient) | N·s/m |
| ω' | Damped angular frequency | rad/s |
| ω₀ | Natural (undamped) angular frequency = √(k/m) | rad/s |
| k | Spring constant | N/m |
| m | Mass | kg |
Three Damping Regimes
Underdamped (γ < ω₀): The system oscillates with gradually decreasing amplitude. This is the most common case — car suspensions, pendulum clocks, guitar strings after plucking.
Critically damped (γ = ω₀): The system returns to equilibrium as fast as possible without oscillating. This is the ideal case for shock absorbers and instrument gauges that need to respond quickly and settle without bouncing.
Overdamped (γ > ω₀): The system returns to equilibrium slowly without oscillating. Think of a door with a very strong hydraulic closer — it swings slowly back and barely moves.
Example 1 — Underdamped Car Suspension
A car suspension: k = 20,000 N/m, m = 500 kg, b = 2,000 N·s/m. Is it underdamped?
ω₀ = √(k/m) = √(20000/500) = √40 = 6.32 rad/s
γ = b/(2m) = 2000/(2×500) = 2 s⁻¹
Since γ (2) < ω₀ (6.32): underdamped
ω' = √(6.32² − 2²) = √(39.94 − 4) = √35.94 = 5.995 rad/s
The suspension oscillates at ω' ≈ 6 rad/s with exponential decay constant γ = 2 s⁻¹
Example 2 — RLC Circuit (Electrical Analog)
An RLC circuit with R = 100 Ω, L = 0.1 H, C = 10 μF. What regime?
ω₀ = 1/√(LC) = 1/√(0.1 × 10⁻⁵) = 1/√(10⁻⁶) = 1000 rad/s
γ = R/(2L) = 100/(0.2) = 500 s⁻¹
γ (500) < ω₀ (1000): underdamped
The circuit oscillates with damped frequency ω' = √(1000² − 500²) = √750000 ≈ 866 rad/s
When to Use It
Use the damped oscillator formula when:
- Designing vehicle suspension systems (want just-underdamped for comfort)
- Analyzing RLC circuits — R is the damping, L is inductance, C is capacitance
- Engineering earthquake-resistant buildings with tuned mass dampers
- Studying atomic force microscope cantilever response
- Modeling door closers, shock absorbers, and vibration dampers