Apply the damping ratio formula ζ = b/(2√(mk)) to any mass–spring system. Enter the mass, spring constant and damping coefficient to classify the regime and plot the displacement x(t) inside its ±A₀e^(−t/τ) decay envelope.
The Damped Harmonic Oscillator Envelope Diagram solves m·x'' + b·x' + k·x = 0 in closed form. Enter the mass, spring constant and damping coefficient and it applies the damping ratio formula zeta = b/(2*sqrt(mk)) to classify the system as undamped, underdamped, critically damped or overdamped.
The chart shows the displacement x(t) together with the pair of decay envelopes ±A0·e^(-t/tau), so it is easy to see how the natural frequency, the damped frequency, the time constant and the quality factor combine to shape the motion.
The damping ratio formula is zeta = b/(2*sqrt(mk)), where b is the damping coefficient, m the mass and k the spring constant. Because 2*sqrt(mk) is the critical damping value, zeta measures the actual damping as a fraction of critical damping.
Underdamped (zeta < 1) systems oscillate while their amplitude decays. Critically damped (zeta = 1) systems return to equilibrium in the shortest possible time without oscillating. Overdamped (zeta > 1) systems also do not oscillate, but return slowly.
The envelope is the pair of curves +A0*e^(-t/tau) and -A0*e^(-t/tau) with tau = 2m/b. The oscillating displacement touches the envelope once every half period, so it shows how fast the amplitude dies away independently of the oscillation.
The natural angular frequency omega_0 = sqrt(k/m) is the rate with no damping. Damping lowers it to omega_d = omega_0*sqrt(1 - zeta^2). Once zeta reaches 1 there is no oscillation at all and omega_d is reported as zero.
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