Difference between revisions of "Forest UCM Osc Damped"

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or
 
or
  
:<math>  m \ddot x + kx + b \dot x = 0</math>: in 1-D
+
:<math>  m \ddot x + kx + b \dot x = 0</math>
  
 
or
 
or
  
:<math>  \ddot x + \frac{k}{m}x + \frac{b}{m} \dot x = 0</math>: in 1-D
+
:<math>  \ddot x + \frac{k}{m}x + \frac{b}{m} \dot x = 0</math>
  
  
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then  
 
then  
  
:<math>  \ddot x + 2 \beta \dot x + \omega^2_0x  = 0</math>: in 1-D
+
:<math>  \ddot x + 2 \beta \dot x + \omega^2_0x  = 0</math>
  
  
 
As see in section [[Forest_UCM_Osc_SHM#Equation_of_motion]], you can determine solutions to the above  
 
As see in section [[Forest_UCM_Osc_SHM#Equation_of_motion]], you can determine solutions to the above  
by writing the analogous auxilary eqation.
+
by writing the analogous auxilary equation:
 +
 
 +
:<math>  O^2 + 2 \beta O + \omega^2_0  = 0 \;\;\;\;\;\; O \equiv \frac{d}{dt}</math>
 +
 
 +
Using the quadratic formula
 +
 
 +
: <math>O = \frac{- 2\beta \pm \sqrt{(2\beta)^2 -4\omega}}{2} =  - \beta \pm \sqrt{\beta^2 -\omega}</math>
  
 
[[Forest_UCM_Osc#Damped_Oscillations]]
 
[[Forest_UCM_Osc#Damped_Oscillations]]

Revision as of 12:48, 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.


\sum \vec{F}_{ext} = -k\vec r - b \vec \dot v = m \vec \ddot r
Fext=kxb˙x=m¨x: in 1-D

or

m¨x+kx+b˙x=0

or

¨x+kmx+bm˙x=0


let

km=ω20= undamped oscillation frequency
bm2β= damping constant

then

¨x+2β˙x+ω20x=0


As see in section Forest_UCM_Osc_SHM#Equation_of_motion, you can determine solutions to the above by writing the analogous auxilary equation:

O2+2βO+ω20=0Oddt

Using the quadratic formula

O=2β±(2β)24ω2=β±β2ω

Forest_UCM_Osc#Damped_Oscillations