Forest UCM Osc Damped

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.

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

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

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

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:

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

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

$O = \frac{- 2\beta \pm \sqrt{(2\beta)^2 -4\omega^2_0}}{2} = - \beta \pm \sqrt{\beta^2 -\omega^2_0}$

$\left ( O + \beta + \sqrt{\beta^2 -\omega^2_0} \right ) \left ( O + \beta - \sqrt{\beta^2 -\omega^2_0}\right ) x = 0$

You have change the second order differential equation into two first order differential equations

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

$\Rightarrow \frac{dx}{x} = \left ( -\beta - \sqrt{\beta^2 -\omega^2_0} \right ) dt$

$x= e^{\left (- \beta - \sqrt{\beta^2 -\omega^2_0} \right )t}$

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

$\Rightarrow \frac{dx}{x} = \left ( - \beta + \sqrt{\beta^2 -\omega^2_0} \right ) dt$

$x= e^{\left ( - \beta + \sqrt{\beta^2 -\omega^2_0} \right )t}$

constructing a complete solution from the two solutions (orthogonal functions) above.

$x= \left ( C_1 e^{ \sqrt{\beta^2 -\omega^2_0} t} + C_2 e^{ -\sqrt{\beta^2 -\omega^2_0} t} \right) e^{- \beta t}$

Undamped oscillator

If $\beta$ = 0

Then

$x= \left ( C_1 e^{ \sqrt{\beta^2 -\omega^2_0} t} + C_2 e^{ -\sqrt{\beta^2 -\omega^2_0} t} \right) e^{- \beta t}$

$= \left ( C_1 e^{ i\omega_0 t} + C_2 e^{ -i\omega_0 t} \right)$ the SHM solution derived before at Forest_UCM_Osc_SHM#Equation_of_motion

Under damped Oscillator

$\beta \lt \omega_0$

In this case the term

$\sqrt{\beta^2 -\omega^2_0} =\sqrt{(-1)(\omega^2_0- \beta^2 } = i \sqrt{\omega^2_0- \beta^2 } \eqiv i \omega_1$

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

$x= \left ( C_1 e^{ \sqrt{\beta^2 -\omega^2_0} t} + C_2 e^{ -\sqrt{\beta^2 -\omega^2_0} t} \right) e^{- \beta t}$

Over damped Oscillator

$\beta \gt \omega_0$

Critically damped Oscillator

$\beta = \omega_0$

$x= \left ( C_1 e^{ \sqrt{\beta^2 -\omega^2_0} t} + C_2 e^{ -\sqrt{\beta^2 -\omega^2_0} t} \right) e^{- \beta t}$

Forest_UCM_Osc#Damped_Oscillations