2018-10-24

2018-12-10 本文已影响0人 carpediemmlf

Canonical form of Damped SHM

$\ddot{x}+\gamma\dot{x}+{\omega_0} ^2x=0$

the spontaneous frequency
$\omega_0 = \sqrt {\frac{k}{m}}$
the damping coefficient
$\gamma = \frac{b}{m}$
Quality factor: the number of radians of phase elapsed as the amplitude falls to $e^{-\frac{1}{2}}$
$Q=\frac{\omega_0}{\gamma}$
the system is critically damped when
$\gamma = 2 \omega_0$

damped oscillation frequency
$\omega_f = \sqrt {{\omega_0} ^2 - \frac{\gamma^2}{4}} = \omega_0 \sqrt {1- \frac{Q^2}{4}}$

$\ddot{x}+\gamma\dot{x}+{\omega_0} ^2x=\frac{F}{m}$

The response function (1. obtain the harmonic x solution; 2. divide solution by F). can naturally devise the velocity and acceleration response function
$R(\omega) = \frac{1}{m[({\omega_0}^2 - {\omega}^2) + i\gamma \omega]} = \frac{{(\omega_0}^2 - {\omega}^2) - i\gamma \omega}{m[({\omega_0}^2 - {\omega}^2)^2 + \gamma ^2 \omega ^2]}$
resonance frequency (differentiate to get)
$\omega_a = \omega_0 \sqrt {1 - \frac{\gamma^2}{2 {\omega_0} ^2}} = \omega_0 \sqrt {1 - \frac{1}{2Q^2}}$