Forced Vibration of Cart attached to Wall by Spring/Problem Definition

Definition of Problem

 * CartOnSpringForcedVibration.png

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 $\mathbf F_d$ whose magnitude is proportional to the speed of $C$.

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

Let the constant of proportion of the damping force $\mathbf F_d$ be $c$.

Let there be applied to $C$ an external force which varies as a function of time as:
 * $\mathbf F_e = \mathbf F_0 \cos \omega t$

where $\mathbf F_0$ is constant.

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