Driven Oscillations
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1
What is the a(t) of a driven oscillation?
$a(t) = -\frac{k}{m} x(t) - \frac{b}{m} v(t) + \frac{F_d}{m}cos(\omega_d t)$
2
What is the x(t) of a driven oscillation, in terms of $
X_m$?
$x(t) = X_m(t) \cos(\omega_x t + \phi)$
3
What is the x(t) of a driven oscillation, in terms of A and B?
$x(t) = A \cos(\omega_d t) + B \sin(\omega_d t)$
4
What is the $\gamma$ for a driven oscillation?
$\gamma = \frac{b}{2m}$
5
What is the formula for tan($\phi$) for a driven oscillation?
$\tan(\phi) = -\frac{2\gamma \omega_d}{\omega_0^2 - \omega_d^2}$
6
What is the formula for A for a driven oscillation?
$A = \frac{F_d}{m} \cdot \frac{(\omega_0^2 - \omega_d^2)}{(\omega_0^2 - \omega_d^2)^2 + 4\gamma^2\omega_d^2}$
7
What is the formula for B for a driven oscillation?
$B = \frac{F_d}{m} \cdot \frac{2\gamma\omega_d}{(\omega_0^2 - \omega_d^2)^2 + 4\gamma^2\omega_d^2}$
8
What is the formula for $\omega_d$ for a driven oscillation?
$\omega_d = \sqrt{\omega_0^2 - \gamma ^2}$
9
What is the formula for $\omega_{res}$ for a driven oscillation?h
$\omega_{res} = \sqrt{\omega_0^2 - 2\gamma ^2}$
10
What is the condition for resonance for a driven oscillation?
$\omega_d \approx \omega_{\text{res}}$
11
What is the time constant ($\tau$) for a driven oscillation?
$\tau = \frac {m}{b}$
12
What is the quality factor (Q) for a driven oscillation?
$Q = \tau \omega_{\text{res}}$
13
What is the formula for the average power transferred from the driving force to the oscillater per cycle ($\langle P \rangle$) for a driven oscillation?
$\langle P \rangle = \frac{1}{2} F_d \omega_d B$
14
What is the quality factor (Q) for a driven oscillation in terms of energy?
$Q = 2\pi \frac{\text{initial energy stored in oscillating system}}{\text{energy dissipated in cycle}}$
15
What is the $\omega_0$ for a driven oscillation?
$\sqrt{\frac{k}{m}}$
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