Mechanical oscillations

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  • Harmonic motion equation and its solution:$$\ddot{x} +{\omega}^2_0 x=0, x= a\cos(\omega_0 t+ \alpha),\tag{1} $$where coois the natural oscillation frequency.
  • Damped oscillation equation and its solution:$$ \ddot{x}+2 \beta \dot{x} +\omega^2_0 x=0, x=a_0 e^{-\beta t} \cos(\omega t + \alpha)$$ where $\beta$ is the damping coefficient, o) is the frequency of damped oscillations:$$ \omega = \sqrt{\omega^2_0 - \beta^2}.$$
  • Logarithmic damping decrement ? and quality factor Q:$$ \lambda= \beta T, Q=\frac{\pi}{\lambda},$$ where $$T =\frac{2\pi}{\omega}.$$
  • Forced oscillation equation and its steady-state solution:$$\ddot{x}+2\beta \dot{x} + \omega^2_0 x= f_0\cos \omega t, x=a \cos (\omega t- \phi), $$ where $$a=\frac{f_0}{\sqrt{( \omega^2_0 - \omega^2)^2 + 4\beta^2 \omega^2 }} , \tan \phi=\frac{2 \beta \omega }{\omega^2_0 - 2 \beta^2 }.$$
  • Maximum shift amplitude occurs at $$ \omega_{res} = \sqrt{\omega^2_0 - 2 \beta^2}. $$

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