Difference between revisions of "Forest UCM Osc Damped"

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let
 
let
  
:<math>\frac{k}{m} = \omega_0 =</math> undamped oscillation frequency
+
:<math>\frac{k}{m} = \omega^2_0 =</math> undamped oscillation frequency
  
 
:<math>\frac{b}{m} \equiv 2 \beta =</math> damping constant
 
:<math>\frac{b}{m} \equiv 2 \beta =</math> damping constant
 +
 +
then
 +
 +
:<math>  \ddot x - 2 \beta \dot x- \omega^2_0x  = 0</math>: in 1-D
 +
  
 
As see in section [[Forest_UCM_Osc_SHM#Equation_of_motion]]
 
As see in section [[Forest_UCM_Osc_SHM#Equation_of_motion]]
  
 
[[Forest_UCM_Osc#Damped_Oscillations]]
 
[[Forest_UCM_Osc#Damped_Oscillations]]

Revision as of 12:40, 5 October 2014

1-D Damped Oscillaions

Equation of Motion

As in the case of air resistance, assume there is frictional force proportional to the velocity of the oscillation body.


[math] \sum \vec{F}_{ext} = -k\vec r - b \vec \dot v = m \vec \ddot r[/math]
[math] \sum F_{ext} = -kx - b \dot x = m \ddot x[/math]: in 1-D

or

[math] m \ddot x -kx - b \dot x = 0[/math]: in 1-D

or

[math] \ddot x - \frac{k}{m}x - \frac{b}{m} \dot x = 0[/math]: in 1-D


let

[math]\frac{k}{m} = \omega^2_0 =[/math] undamped oscillation frequency
[math]\frac{b}{m} \equiv 2 \beta =[/math] damping constant

then

[math] \ddot x - 2 \beta \dot x- \omega^2_0x = 0[/math]: in 1-D


As see in section Forest_UCM_Osc_SHM#Equation_of_motion

Forest_UCM_Osc#Damped_Oscillations