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In the presence of damping but without external forces, the solution to the linear vibration problem takes the form (compare with free vibration in Chapter 2) where Ω, the natural frequency of the damped point-mass spring system, is given by:

When the damping coefficient c > 2m ω₀, energy dissipation is so large that vibration motion cannot be sustained. The motion becomes that of the exponentially decay type instead of vibration. The value of c at which this happens ( c_c = 2mω₀) is called the critical damping. The more commonly used damping ratio is the damping coefficient expressed as a fraction of its value at critical damping

In terms of the frequency (in Hz) and the damping ratio, the damped natural frequency is given by: