Oscillator driving frequencies

[Saado]

I've attached a graph to this post. Why is it that the periodic external frequency applied never starts at 0 on graphs like these?

 

[AlephZero]

It does start at 0 on the graph you posted.
$\omega_A$ is the input frequency (or periodic external frequency). When $\omega_A = 0$, the frequency ratio $\omega_A\omega_0 = 0$.

 

[Saado]

Sorry my bad I meant the amplitude of the oscillations.

 

[AlephZero]

If the force has constant amplitude, at low frequencies the $m\ddot x$ term is small compared with the $kx$ term in the equation of motion, so the amplitude is approximately $F/k$. That is why the amplitude doesn't go to 0 as the frequency goes to 0.

Note, these type of plots can be drawn in different ways, corresponding to different physical situations:

1. The force has constant amplitude, like your attachment
2. The force amplitude is proportional to the frequency
3. The force amplitude is proportional to the frequency squared (for example the unbalanced force on a rotating object)

You can also plot how the velocity, and acceleration of the system changes with frequency.

Most of those plots do go to 0 when the frequency is 0.