Forced Vibration of Cart attached to Wall by Spring

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Theorem

Problem Definition

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 $\mathbf x$.


Then the motion of $C$ is described by the second order ODE:

$m \dfrac {\d^2 \mathbf x} {\ d t^2} + c \dfrac {\d \mathbf x} {\d t} + k \mathbf x = \mathbf F_0 \cos \omega t$


Proof

Let $\mathbf F_s$ be the force on $C$ exerted by the spring $S$.

Let $\mathbf F_d$ be the damping force.

Let $\mathbf F_e$ be the external force on $C$.

By Newton's Second Law of Motion, the total force on $C$ equals its mass times its acceleration:

$\mathbf F_s + \mathbf F_d + \mathbf F_e = m \mathbf a$

By Acceleration is Second Derivative of Displacement with respect to Time:

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

By definition, the velocity $\mathbf v$ is defined as:

$\mathbf v = \dfrac {\d \mathbf x} {\d t}$

By Hooke's Law:

$\mathbf F = -k \mathbf x$

By definition of the damping force:

$\mathbf F_d = - c \mathbf v$

By definition of the external force:

$\mathbf F_e = \mathbf F_0 \cos \omega t$

So:

\(\ds m \mathbf a\) \(=\) \(\ds -k \mathbf x - c \mathbf v + \mathbf F_0 \cos \omega t\)
\(\ds \leadsto \ \ \) \(\ds m \dfrac {\d^2 \mathbf x} {\d t^2}\) \(=\) \(\ds -k \mathbf x - c \dfrac {\d \mathbf x} {\d t} + \mathbf F_0 \cos \omega t\)
\(\ds \leadsto \ \ \) \(\ds m \dfrac {\d^2 \mathbf x} {\d t^2} + c \dfrac {\d \mathbf x} {\d t} + k \mathbf x\) \(=\) \(\ds \mathbf F_0 \cos \omega t\)

$\blacksquare$


Sources

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