Forcing Frequency for Resonance of Cart attached to Wall by Spring

Problem Definition
Then the resonance frequency of $C$ is:


 * $\nu_R = \dfrac 1 {2 \pi} \sqrt {\dfrac k m - \dfrac {c^2} {2 m^2} }$

Proof
From Forced Vibration of Cart attached to Wall by Spring, the equation of motion of $C$ is:
 * $(1): \quad m \dfrac {\mathrm d^2 \mathbf x} {\mathrm d t^2} + c \dfrac {\mathrm d \mathbf x} {\mathrm d t} + k \mathbf x = \mathbf F_0 \cos \omega t$

Let:
 * $a^2 = \dfrac k m$
 * $2 b = \dfrac c m$
 * $K = \dfrac {\mathbf F_0} m$

This is in the form:
 * $(2): \quad \dfrac {\mathrm d^2 y} {\mathrm d x^2} + 2 b \dfrac {\mathrm d y} {\mathrm d x} + a^2 x = K \cos \omega x$

where:
 * $K \in \R: k > 0$
 * $a, b \in \R_{>0}: b < a$

From Condition for Resonance in Forced Vibration of Underdamped System, $C$ is in resonance when:
 * $\omega = \sqrt {a^2 - 2 b^2}$

Thus, substituting back for $a$ and $b$:

By definition of frequency of simple harmonic motion:
 * $\nu_f = \dfrac 1 {2 \pi} \sqrt {\dfrac k m - \dfrac {c^2} {2 m^2} }$