Damped Vibration

Measure displacement from equilibrium

For a linear spring and viscous damper, use displacement x from the static equilibrium position. A constant weight is already balanced there. With no external time-varying force, mass m, stiffness k, and damping coefficient c obey the free-vibration equation.

Two parameters organize the response

Natural angular frequency ωₙ has units rad/s; the damping ratio ζ has no units. Mass and stiffness must be positive, and c is nonnegative in this passive model.

Three regimes, plus the undamped limit

For 0 < ζ < 1 the response oscillates with a decaying amplitude. At ζ = 1 the characteristic roots coincide. For ζ > 1 there are two real decay rates and no periodic oscillation. At ζ = 0 the ideal system does not dissipate energy. Critical damping is the boundary of nonoscillatory response; an initial velocity can still cause a single crossing, so it is not a universal promise of zero overshoot.

Initial conditions still matter

Damping ratio determines the regime, but initial displacement and velocity set the actual curve. The damped angular frequency is lower than the undamped natural frequency.

Worked example

A 2 kg mass is attached to a 200 N/m spring and a 12 N·s/m damper. Classify its free response.

1. 1.Compute natural frequency and critical damping.

   $$\begin{array}{c}{\omega }_n=\sqrt{200/2}=10rad/s\\ c_{\mathrm{cr}}=2\sqrt{200(2)}=40\mathrm{N}s/m\end{array}$$

2. 2.Compare the damping coefficient with the critical value.

   $$\begin{array}{c}\zeta =12/40=0.30\\ {\omega }_d=10\sqrt{1-0.30^2}=9.54rad/s\end{array}$$

The system is underdamped: it oscillates at about 9.54 rad/s while its amplitude decays.