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.

$\sum \vec{F}_{ext} = -k\vec r - b \vec \dot v = m \vec \ddot r$

$\sum F_{ext} = -kx - b \dot x = m \ddot x$ : in 1-D

or

$m \ddot x + kx + b \dot x = 0$

or

$\ddot x + \frac{k}{m}x + \frac{b}{m} \dot x = 0$

let

$\frac{k}{m} = \omega^2_0 =$ undamped oscillation frequency

$\frac{b}{m} \equiv 2 \beta =$ damping constant

then

$\ddot x + 2 \beta \dot x + \omega^2_0x = 0$

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

$\left ( O^2 + 2 \beta O + \omega^2_0 \right ) x = 0 \;\;\;\;\;\; O \equiv \frac{d}{dt}$

Setting the term in parentheses to zero and using the quadratic formula

$O = \frac{- 2\beta \pm \sqrt{(2\beta)^2 -4\omega}}{2} = - \beta \pm \sqrt{\beta^2 -\omega}$