OSCILLATIONS AND WAVES - MECHANICAL OSCILLATIONS

MECHANICAL OSCILLATIONS

1. Harmonic motion equation and its solution:

$x\, +{\omega }^{2}_{0}x\, =0\, ,\, x\, =\, acos({\omega }_{0}t\, +\alpha )$

where ${\omega }_{0}$ is the natural oscillation frequency

2. Damped oscillation equation and its solution:

$x\, +2\beta x+{\omega }^{2}_{0}x\, =0\, ,\, x\, =\, {a}_{0}{e}^{\beta t}cos({\omega }_{0}t\, +\alpha )$

where $\beta$ is the damping coefficient, $\omega$ is the frequency of damped oscillation;

$\omega \, =\, \sqrt {{\omega }^{2}_{0}-{\beta }^{2}}$

3. Logarithmic damping decrement ? and quality factor Q:

$\lambda =\beta T\, ,\, Q\, =\, {\pi /\lambda }$

where $T\, =2\, {\pi /\omega }$

4.Forced oscillation equation and its steady-state solution:

$x\, +2\beta x+{\omega }^{2}_{0}x\, ={f}_{0}cos\omega t,\, x\, =\, acos({\omega }_{\, }t\, -\phi ),$

Where

$a\, =\, \frac {{f}_{o}} {\sqrt {{({\omega }^{2}_{0}-{\omega }^{2})}^{2}+4{\beta }^{2}{\omega }^{2}}}\, \, \, \, \, tan\phi \, =\, \frac {2\beta \omega } {{\omega }^{2}_{0}-{\omega }^{2}}$

5.Maximum shift amplitude occurs at

${\omega }_{res}\, =\, \sqrt {{\omega }^{2}_{0}-2{\beta }^{2}}$

Post By : Faiz Ahmad Faiz 24 Apr, 2020 130 views Physics

OSCILLATIONS AND WAVES - MECHANICAL OSCILLATIONS

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