ẍ + 2ζωẋ + ω²x = 0

with initial conditions

x = 1

ẋ = 0 (where ẋ is shorthand for dx/dt)

for various values of ζ (ζ ≥ 0) showing how the damped natural frequency increases as damping increases (ζ becomes larger). If ζ < 1, then the system is said to be under-damped; if ζ = 1, the system is said to be critically damped and if ζ > 1, the system is said to be over-damped.

Derivation

The standard equation for damped harmonic motion is:^[1]

mẍ + cẋ + kx = 0.

where m, c and k have standard meanings - for example, in a damped spring system m is mass, c is the velocity damping constant and k is the spring constant.

This equation is transformed to the normalised form by setting:

ω² = k/m and ζ² = c²/4mk

The analytic solutions to the standard damped harmonic equation are:

• if ζ > 1 then x = c₁ e^α₁t + c₂ e^α₂t where α₁ and α₂ are the solutions to the equation α² + 2ωζα + ω² = 0
• if ζ = 1; then x = c₁ e^-ωt + c₂t e^-ωt
• if ζ < 1; then x = e^-ωζt (c₁ cos(γt) + c₂sin(γt)) where γ² = ω²(1 - ζ²)

where c₁ and c₂ are constants of integration.

Reference