---
tags : Mechanics,Osciliation
title : 2020.4.16(11)
---
# Mechanics(11)
## Damped osciliation
### 2~nd~ Over damping
When viscous drag is large , and $\omega_0 < \gamma$, set as $\sigma = \sqrt{\gamma^2 - \omega_0^2} ,$

$\large \ddot{f} - \sigma^2f = 0$

$f=e^{\sigma t}$ and$f=e^{-\sigma t}$ complete this equasion , so general solution is

$\large a_+e^{-(\gamma-\sigma)t}+a_-e^{-(\gamma - \sigma)t}$

This solution is a sum of 2 exponential function which close to 0 in different relaxation times $\frac{1}{\gamma -\sigma}$ ,and $\frac{1}{\gamma+\sigma}$.

Because of second term damp quicker than first one , so the influence of first one become large.

This graph shows such as a dinamics of a pendulum in water.