Calculate natural frequency and damping ratio of a single degree-of-freedom vibrating system.
ζ < 1: underdamped (oscillates with decreasing amplitude). ζ = 1: critically damped (returns to equilibrium fastest without oscillation). ζ > 1: overdamped (slow return, no oscillation). Most structures aim for ζ = 0.02-0.05.
If an external force matches the natural frequency (resonance), vibration amplitude grows dramatically and can cause structural failure. Engineers design systems so natural frequencies don't coincide with expected excitation frequencies.