1-D Damped Oscillations
Newton's 2nd Law
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]
Solve for the Equation of Motion
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}}{2} = - \beta \pm \sqrt{\beta^2 -\omega^2}[/math]
- [math] \left ( O + \beta + \sqrt{\beta^2 -\omega^2} \right ) \left ( O + \beta - \sqrt{\beta^2 -\omega^2}\right ) x = 0 [/math]
You have change the second order differential equation into two first order differential equations
- [math] \left ( \frac{d}{dt} + \beta + \sqrt{\beta^2 -\omega^2} \right ) x = 0 [/math]
- [math] \Rightarrow \frac{dx}{x} = \left ( -\beta - \sqrt{\beta^2 -\omega^2} \right ) dt [/math]
- [math] x= e^{\left (- \beta - \sqrt{\beta^2 -\omega^2} \right )t} [/math]
- [math] \left (\frac{d}{dt} + \beta - \sqrt{\beta^2 -\omega^2}\right ) x = 0 [/math]
- [math] \Rightarrow \frac{dx}{x} = \left ( - \beta + \sqrt{\beta^2 -\omega^2} \right ) dt [/math]
- [math] x= e^{\left ( - \beta + \sqrt{\beta^2 -\omega^2} \right )t} [/math]
constructing a complete solution from the two solutions (orthogonal functions) above.
- [math] x= \left ( C_1 e^{ \sqrt{\beta^2 -\omega^2} t} + C_2 e^{ -\sqrt{\beta^2 -\omega^2} t} \right) e^{- \beta t} [/math]
Forest_UCM_Osc#Damped_Oscillations