Damped Oscillation

All three are solutions of mẍ + bẋ + kx = 0, distinguished by the discriminant b² − 4mk. Underdamped (b² < 4mk): the system oscillates with exponentially decaying amplitude e^(−γt) at damped frequency ω_d = √(ω₀² − γ²), where γ = b/2m. Critically damped (b² = 4mk): a double real root, no oscillation, fastest possible return to equilibrium. Overdamped (b² > 4mk): two distinct real roots, the system creeps slowly back without crossing zero. Door closers and analog meter movements are usually slightly underdamped or critical; honey-thick dashpots are overdamped.

Critical damping sits at the boundary where the discriminant is zero. The system has a double real root at λ = −γ = −b/2m, so the response is x(t) = (A + Bt) e^(−γt). For overdamped systems the slower of the two real roots dominates and decays slower than e^(−γt); for underdamped systems the envelope decays at e^(−γt) but the system oscillates around zero before settling within tolerance. Critical damping minimizes the worst-case settling time. That is why analog ammeters, automatic door closers, and many control loops target critical damping — fast response, no overshoot.

Q (quality factor) is the dimensionless ratio Q = ω₀ m / b = √(mk)/b. Physically Q is roughly 2π times the energy stored divided by the energy lost per cycle; equivalently Q ≈ π / logarithmic-decrement, or the number of radians the oscillator rings through before its energy decays by e^(−1). Q > 1/2 means underdamped. Practical examples: a car shock absorber Q ≈ 0.5 (near-critical), a guitar string Q ≈ 1000, a quartz crystal oscillator Q ≈ 10⁵, a superconducting microwave cavity Q ≈ 10¹¹. Measured from the FWHM of the resonance peak (Q = ω₀ / Δω) or by counting ring-down cycles.

Car shock absorbers (dampers) are dashpots that target a Q near 0.5 — slightly underdamped or near-critical. The wheel-on-spring system has a natural frequency around 1 to 2 Hz; without damping a single bump would set the car bouncing for many seconds. The shock pumps oil through orifices, dissipating kinetic energy as heat. Too soft (high Q): floaty, motion-sickness ride. Too stiff (low Q): harsh, transmits every road imperfection. Performance dampers add velocity-dependent valving that softens at low speed and stiffens at high speed for both ride comfort and handling control.

Kirchhoff's voltage law for a series RLC circuit gives L Q̈ + R Q̇ + Q/C = 0, structurally identical to mẍ + bẋ + kx = 0 with the mapping m ↔ L, b ↔ R, k ↔ 1/C. Same three regimes apply: underdamped (R² < 4L/C), critical (R² = 4L/C), overdamped (R² > 4L/C). Underdamped RLC circuits ring at ω_d = √(1/LC − R²/4L²), the basis of analog tuned radios and quartz crystal time bases. Critical damping is desirable in instrumentation amplifiers and meter movements where overshoot is unacceptable.

Logarithmic decrement δ = ln(x_n / x_{n+1}) is the natural log of the ratio of two successive oscillation peaks. For an underdamped oscillator δ = γT_d = 2πγ/ω_d, where T_d is the damped period. Useful because you can read peak amplitudes off a strip-chart or oscilloscope trace and recover both Q (Q ≈ π/δ for high Q) and the damping coefficient without solving the differential equation. Forms the basis of many ring-down measurements in mechanical engineering, acoustics, and seismology where you want a fast, low-instrument estimate of damping.