# Forcing Frequency for Resonance of Cart attached to Wall by Spring

## Contents

## Theorem

### Problem Definition

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 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:

- $(1): \quad m \dfrac {\d^2 \mathbf x} {\d t^2} + c \dfrac {\d \mathbf x} {\d t} + k \mathbf x = \mathbf F_0 \cos \omega t$

Let:

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

This is in the form:

- $(2): \quad \dfrac {\d^2 y} {\d x^2} + 2 b \dfrac {\d y} {\d x} + a^2 x = K \cos \omega x$

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 $b$:

\(\displaystyle \omega\) | \(=\) | \(\displaystyle \sqrt {\dfrac k m - 2 \paren {\dfrac c {2 m} }^2}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sqrt {\dfrac k m - \dfrac {c^2} {2 m^2} }\) |

By definition of frequency of simple harmonic motion:

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

$\blacksquare$

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.20$: Problem $1$