Understand F_0 & Omega in Driven Oscillators

[oneplusone]

I found this equation for driven oscillators, and am unsure what the F_0 means, and how it's used. Can someone please briefly explain this?
Also, I don't get what the difference is between the initial and final omega. (how can there be a final omega if it's constantly increasing?)

 

[jhae2.718]

$F_0$ is the amplitude of a sinusoidal forcing function, $\omega_0$ is the undamped natural frequency of the system, and $\omega$ is the frequency of the forcing function, which looks something like $f(t)=f_0\sin(\omega t)$ or $f(t)=f_0\cos(\omega t)$.

The definition of $\omega$ shown is also the damped natural frequency of the system.

 

[oneplusone]

So are the units for F_0 meters? (So it's not a force?)
Thanks.

 

[jhae2.718]

$F_0$ has dimensions of force. You can easily see this by looking at the dimension of each part of the expression. Consider $$f(t)=f_0 \cos(\omega t).$$ The dimension of $t$ is of course time, and the dimension of $\omega$ must be angle/time so that $\omega t$ has dimensions of an angle (e.g. radians). This must be so, as, for example, what is the cosine of a second? It's meaningless. Since $f(t)$ is a force, and $\cos(\omega t)$ is dimensionless, it must be that $f_0$ has the dimensions of force, e.g. Newtons for SI.