---
tags : Mechanics,Osciliation
title : 2020.4.18(14)
---
# Mechanics(14)
## Start of Osciliation
This is a conversation that a point mass kinematics when force the point mass which stopping in $t=0$ forced vibration external force $f_0sin\omega _0t$ in resonance frequence.

In a general solution of forced osciliation in general frequence $\omega$ , set as first condition,

$\large x(0)=A=0 , \dot{x}(0)=B\omega_0+\frac{f_0\omega}{\omega_0^2-\omega^2}=0$

Therefore, $B=-\frac{\omega}{\omega_0}\frac{f_0}{\omega_0^2-\omega^2}$ , substitute this to the solution ,

$\large x(t)=\frac{f_0}{\omega_0^2-\omega^2}(sin\omega t-\frac{\omega}{\omega_0}sin\omega_0t)$

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

Set as the frequency of external force $\omega=\omega+\epsilon$, think as close to original frequency($\epsilon \to 0$) ,

$\large x(t)=\frac{-f_0}{(2\omega_0+\epsilon)}(sin\omega_0tcos\epsilon t+cos\omega_0tsin\epsilon t-\frac{\omega_0+\epsilon}{\omega_0}sin\omega_0t)$

About $\epsilon$, use as
:::info

$cos\epsilon t\approx1-\frac{(\epsilon t)^2}{2}\approx 1$ ,$sin\epsilon t\approx\epsilon t$
:::

$\large x(t)=\lim_{\epsilon \to 0} \frac{-f_0}{2\omega_0\epsilon}(\epsilon tcos\omega_0t-\frac{\epsilon}{\omega_0}sin\omega_0t)$

$\huge =-\frac{f_0}{2\omega_0}(tcos\omega_0t-\frac{1}{\omega_0}sin\omega_0t)$

The first term means the ampitude increase in time proportion.