Single degree of freedom damped free vibrations

Governing Equation
Governing equation for the motion of single of freedom oscilator can be written as:

This is a homogeneous second-order linear ordinary differential equation with constant coefficients that can be rearranged as follows:

After substitution of \omega = \sqrt{\frac{k}{m}} Eqn. (2) becomes

which yields the following characteristic equation

with a solution

Critically-Damped Systems
If the radical term in Eqn. (5)is set equal to zero then c/2m = \omega and the critical value of damping coefficient, c_c can be expressed as

Undercritically-Damped Systems
If damping is less than critical, i.e. if c < c_c (in other words if c < 2 m \omega ), it is convenient to define the damping in the terms of damping ratio, which is defined as

Introducing \xi into Eq. (5) yields

where

is the free-vibration frequency of damped system.