Forced oscillations

We call an oscillation forced (or driven) oscillation when there is a periodic driving force applied to a damped oscillation to replace lost mechanical energy. For example, when a child on a swing kept in motion by pushes at regular time intervals.

The expression for periodic force can be

\[F(t) = F_0 \sin\left(\omega t\right).\]

With this, the net force acting on the object is

\[F(t) - k x - \gamma v.\]

Applying Newton's second law, we get

\[m\frac{\mathrm{d}^2x}{\mathrm{d}t^2} = F_0 \sin\left(\omega t\right) - \gamma \frac{\mathrm{d}x}{\mathrm{d}t} - k x.\]

Dividing by \(m\) and rearranging,

\[\frac{\mathrm{d}^2x}{\mathrm{d}t^2} + \frac{\gamma}{m} \frac{\mathrm{d}x}{\mathrm{d}t} + \omega_0^2 x = F_0 \sin\left(\omega t\right).\]

Here, as before, \(\omega_0\) is the natural angular frequency of the undamped oscillator:

\[\omega_0 := \sqrt{\frac{k}{m}}.\]

In the beginning, there will be a transient motion, which will be a mixture of the damped oscillation and the effect of the driving force, but in the long run, the steady-state motion will be a motion dictated by the periodic driving force.

The steady-state solution of the equation for forced oscillations, which describes the oscillations in the long run, is

\[x(t) = A \cdot \cos\left(\omega t + \phi\right),\]

where the amplitude \(A\) depends on the natural angular frequency \(\omega_0\), the angular frequency \(\omega\) of the driving force and the damping coefficient \(\gamma\) according to

\[A(\omega) = \frac{F_0 / m}{\sqrt{\left(\omega^2 - \omega_0^2\right)^2 + \left(\frac{\gamma \omega}{m}\right)^2}}.\]