Natural Frequency of Underdamped Cart attached to Wall by Spring

Problem Definition
Let $C$ be underdamped.

Let $C$ be pulled aside to $x = x_0$ and released from stationary at time $t = 0$.

Then the natural frequency of $C$ can be expressed as:
 * $\nu = \dfrac 1 {2 \pi} \sqrt {\dfrac k m - \dfrac {c^2} {4 m^2} }$

Proof
From Period of Oscillation of Underdamped Cart attached to Wall by Spring:
 * $T = \dfrac {2 \pi} {\sqrt {\dfrac k m - \dfrac {c^2} {4 m^2} } }$

where $T$ is the period of oscillation of $C$.

Then by definition of the natural frequency of $C$: