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

Theorem

Consider a cart $C$ of mass $m$ attached to a wall by means of a spring $S$.

Let $C$ be free to move along a straight line in a medium which applies a damping force $F_d$ whose magnitude is proportional to the speed of $C$.

Let the force constant of $S$ be $k$.

Let the displacement of $C$ at time $t$ from the equilibrium position be $\mathbf x$.

Let:

$a^2 = \dfrac k m$

$2 b = \dfrac c m$

Then the horizontal position of $C$ at time $t$ is determined by the roots of the auxiliary equation:

$m^2 + 2 b + a^2 = 0$

$\dfrac {\d^2 \mathbf x} {\d t^2} + \dfrac c m \dfrac {\d \mathbf x} {\d t} + \dfrac k m \mathbf x = 0$

With the given substitutions $a$ and $b$, this resolves to:

$\dfrac {\d^2 \mathbf x} {\d t^2} + 2 b \dfrac {\d \mathbf x} {\d t} + a^2 \mathbf x = 0$

Recall that $m_1$ and $m_2$ are the roots of the auxiliary equation:

$m^2 + 2 b + a^2 = 0$

$\ds m_1, m_2$

$=$

$\ds \dfrac {- 2 b \pm \sqrt {\paren {2 b}^2 - 4 a^2} } 2$

$\ds $

$=$

$\ds -b \pm \sqrt {b^2 - a^2}$

From the initial problem definition, we have that $k, m, c \in \R_{>0}$.

Hence $a, b \in \R_{>0}$.

Hence the nature of $m_1$ and $m_2$ is dependent upon whether $b > a$, $b = a$ or $b < a$.

$\blacksquare$

1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems: $(13)$