Damped SHM
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1
What is the x(t) of an underdamped oscillation?
$x(t) = Ce^{-\gamma t} \cos(\omega_1 t + \phi)$
2
What is the x(t) of a critically damped oscillation?
$x(t) = (C_1 + C_2 t) e^{-\gamma t}$
3
What is the x(t) of an over damped oscillation?
$x(t) = C_1 e^{-\left( \gamma - \sqrt{\gamma^2 - \omega_0^2} \right)t} + C_2 e^{-\left( \gamma + \sqrt{\gamma^2 - \omega_0^2} \right)t}$
4
What is the a(t) for any damped oscillation?
$a(t) = -\frac{k}{m} x(t) - \frac{b}{m} v(t)$
5
What is the equation of the damping coefficient ($\gamma$)?
$\gamma = \frac{b}{2m}$
6
What is the equation of the damped angular velocity ($\omega_1$)?
$\omega_1 = \sqrt{\omega_0^2 - \gamma^2}$
7
What is the condition for underdamping in terms of $\gamma\ and\ \omega_0$?
$\gamma^2 - \omega_0^2 < 0$
8
What is the condition for critical damping in terms of $\gamma\ and\ \omega_0$?
$\gamma^2 - \omega_0^2 = 0$
9
What is the condition for over damping in terms of $\gamma\ and\ \omega_0$?
$\gamma^2 - \omega_0^2 > 0$
10
What is the $E_{\text{mech}}(t)$ of an underdamped oscillation?
$E_{\text{mech}}(t) = \frac{1}{2} k x(t)^2 = E_{\text{mech}}(0) e^{-\frac{t}{\tau}}$
11
What is the quality factor (Q) of any damped oscillation?
Q = $\tau\omega_1$
12
What is the time constant ($\tau$) for any damped oscillation?
$\tau = \frac{m}{b}$
13
What is the quality factor Q for any damped oscillation?
$Q = 2\pi \frac{\text{initial energy stored in oscillating system}}{\text{energy dissipated in cycle}}$
14
What does the time constant (τ) represent?
Time taken for the energy of the damped oscillation to reduce to $e^{-1}$ or 36.8% of its initial value
15
What do the three types of damped oscillations look like?