---
tags : Mechanics,Osciliation
title : 2020.4.18(15)
---
# Mechanics(15)
## forced Osciliation with Damping
Osciliation with viscous drag and force is written as

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

Asuume this solution as

$\large Acos\omega t+Bsin\omega t$

The situation which each terms complete equal independently is

$\large A = \frac{-2\gamma \omega f_0}{(\omega_0^2-\omega^2)^2+(2\gamma\omega)^2}$

$\large B = \frac{(\omega_0^2-\omega^2)f_0}{(\omega_0^2-\omega^2)^2+(2\gamma\omega)^2}$

Set $x(t)=asin(\omega t-\delta)$ ,

$\large a=\frac{f_0}{\sqrt{(\omega_0^2-\omega^2)^2+(2\gamma\omega)^2}}$

$\large tan\delta =\frac{2\gamma\omega}{\omega_0^2-\omega^2}$

Because of damping , ampitude isn't diverge, but increase near $\omega=\omega_0$.

We call the frequency which has the ampitude maximum as resonance frequency $\omega_{res}$ . When this become minimum , the frequency when $(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2$ become minimum is

$\large \omega_{res}=\sqrt{\omega_0^2-2\gamma^2}$

When $\gamma=0$,$\omega=\omega_0$. Become $\gamma$ bigger, the resonance frequency become smaller, the peak of ampitude become smaller ,and width become wider.When $\gamma>\frac{\omega_0}{\sqrt{2}}$, the ampitude become maximum in $\omega=0$, so this is NOT resonance.