Underdamped oscillations
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The position in an underdamped oscillation can be given by \[x(t) = A \cdot \mathrm{e}^{-\frac{\gamma}{2 m} \cdot t} \sin\left(\omega t + \phi_0\right),\] where the angular frequency \(\omega\) of the damped oscillation is obtained from \[\omega = \sqrt{\frac{k}{m} - \left(\frac{\gamma}{2 m}\right)^2} = \sqrt{\omega_0^2 - \left(\frac{\gamma}{2 m}\right)^2}.\] The angular frequency decreases as compared to the natural angular frequency due to damping. The oscillation is sinusoidal, but its amplitude decays exponentially in time. The time dependence of the amplitude is |
