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

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

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

$= \left ( C_1 e^{i\omega_1 t} + C_2 e^{ -i\omega_1t} \right) e^{- \beta t}$

$= Ae^{- \beta t} \cos(\omega_1 t -\delta)$

There are two terms above, the first term is an exponential decay and the secons is the usual harmonic oscilator.

They combine to produce oscillations whoase amplitudes decay with time.

Over damped Oscillator

$\beta \gt \omega_0$

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

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

For the overdamped case you have two exponentials

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