To begin to solve equation (12) for a building in free response, let us again consider a one-story structure.
In a structure with a single story, the mass, damping, and stiffness are scalars instead of matrices and the differential equations of motion are
, which means that
and
.
Recombining terms in the differential equation of motion leads to
or
.
The solution is the trivial solution; the nontrivial solution is
. We see that we are dealing with the natural frequency of oscillation of the system,
, just as we should in analyzing free response.
Another special case is when 
has a single value, when
equals zero.
This special level of damping is called critical damping,
.
It is customary to talk about damping in terms of a fraction of critical damping,
, where 
is called the damping ratio.
We can now rewrite the single-story differential equation (23) and
(equation 25) in terms of 
and instead of mass, damping, and stiffness:
.
If the damping ratio is between zero and one, then will be a complex conjugate pair.
This is called under-damped behavior and is characteristic of oscillatory response.
The imaginary part of is called the damped natural frequency
.
,
