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]\displaystyle{ \sum \vec{F}_{ext} = -k\vec r - b \vec \dot v = m \vec \ddot r }[/math]

[math]\displaystyle{ \sum F_{ext} = -kx - b \dot x = m \ddot x }[/math]: in 1-D

or

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

or

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

let

[math]\displaystyle{ \frac{k}{m} = \omega^2_0 = }[/math] undamped oscillation frequency

[math]\displaystyle{ \frac{b}{m} \equiv 2 \beta = }[/math] damping constant

then

[math]\displaystyle{ \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, you can determine solutions to the above by writing the analogous auxilary eqation.