It is defined as the ratio of actual damping co-efficient to the critical damping coefficient.

i.e. $\xi = \frac{c}{c_c}$

Hence $\xi = \frac{c}{2mw_n}$

$\therefore$ $\xi w_n = \frac{c}{2m}$

Then the roots of characteristics equation can be written as

$s_1 = (\xi w_n) + \sqrt{ (\xi w_n)^2 - w^2_n}$

$= [ …