Position of Cart attached to Wall by Spring under Damping/Underdamped

Problem Definition
Let:
 * $a^2 = \dfrac k m$
 * $2 b = \dfrac c m$

Let $b < a$.

Then the horizontal position of $C$ at time $t$ can be expressed as:
 * $x = e^{-b t} \left({C_1 \cos \alpha t + C_2 \sin \alpha t}\right)$

where:
 * $C_1$ and $C_2$ depend upon the conditions of $C$ at time $t = 0$
 * $\alpha = \sqrt {a^2 - b^2}$

Such a system is defined as being underdamped.


 * Underdamped.png

Proof
When $b < a$, we have $b^2 - a^2 < 0$ and so:

So from Solution of Constant Coefficient Homogeneous LSOODE: Complex Roots of Auxiliary Equation:


 * $\mathbf x = e^{-b t} \left({C_1 \cos \alpha t + C_2 \sin \alpha t}\right)$

where:
 * $\alpha = \sqrt {a^2 - b^2}$