---
tags : Mechanics,Osciliation
title : 2020.4.17(13)
---
# Mechanics(13)
## Forced Osciliation
This is kinematics of a pendulum which is forced from outside to osciliation direction such as osciliatig the side of the pendulum holizontally.
Now, let's think about the patern which add sine wave vibration External force with frequency $\omega$.

$\large m\frac{d^2x}{dt^2} =-kx+F_0sin\omega t$

:::warning
### Remark
We call a liner differential equation with forced vibration external force $F_0$ as nonhomogeneous equation.Now, I write about apecial solution.
:::
divide $m$

$\large \ddot{x}+\omega_0^2x=f_0sin\omega t$

$(\omega_0^2=\frac{k}{m} , f_0=\frac{F_0}{m})$

Assume the solution as $x(t)=asin\omega t$,

$\large a=\frac{f_0}{\omega_0^2-\omega^2}$

When the frequency of external force is larger than original frequency $(\omega > \omega_0)$, $a$ becomes minus.In oter words, this means osciliatig reverce against external force.

Ampitude $|a|$ increace when $\omega$ close to $\omega_0$, and diverge when $\omega$ = $\omega_0$.

Forced flequency clocse to original frequency , the ampitude become bigger.This is "**Resonance**".

:::info
General solution of unhomogeneous equation is its special solution with general solution of homogeneous equation.(Equation without forced vibration external force)
:::

$\huge x(t)=Acos\omega_0t+Bsin\omega_0t+\frac{f_0}{\omega_0^2-\omega^2}sin\omega t$