The mass on the spring obeys the linear, constant-coefficient second-order ODE m y'' + c y' + k y = 0. Substituting the trial solution y = ert turns the ODE into the characteristic (auxiliary) equation:
m r² + c r + k = 0
r = ( -c ± √(c² − 4mk) ) / (2m)
The discriminant c² − 4mk decides everything about the motion, without ever numerically integrating the ODE — the exact closed-form solution follows directly from the roots:
- Overdamped (c² > 4mk): two real negative roots r₁, r₂. y(t) = A er₁t + B er₂t — the mass creeps back to zero with no oscillation.
- Critically damped (c² = 4mk): one repeated real root r = -c/2m. y(t) = (A + Bt) ert — fastest return to rest without overshoot.
- Underdamped (c² < 4mk): complex conjugate roots r = -ζω₀ ± iωd. y(t) = e-ζω₀t (A cos ωdt + B sin ωdt) — a decaying oscillation.
- Undamped (c = 0): purely imaginary roots ±iω₀. y(t) = A cos ω₀t + B sin ω₀t — oscillates forever.
Here ω₀ = √(k/m) is the natural frequency, ζ = c / (2√(mk)) is the damping ratio, and ωd = ω₀√(1−ζ²) is the damped frequency. The complex-plane panel plots r₁ and r₂ directly — real roots land on the horizontal axis, complex pairs sit symmetric about it, and moving from the left half-plane toward the imaginary axis is exactly what "less damping" means. This is the same math behind car suspensions, door closers, and any circuit with resistance, inductance and capacitance (RLC).