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]
or
- [math] \ddot x + \frac{k}{m}x + \frac{b}{m} \dot x = 0[/math]
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]
As see in section Forest_UCM_Osc_SHM#Equation_of_motion, you can determine solutions to the above
by writing the analogous auxilary equation:
- [math] \left ( O^2 + 2 \beta O + \omega^2_0 \right ) x = 0 \;\;\;\;\;\; O \equiv \frac{d}{dt}[/math]
Setting the term in parentheses to zero and 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