See Resonance if you are not familiar with it
The Resonant frequency of a system is defined in two ways which though similar are not equivalent
One way is to define it as the frequency at which the system must be driven in order for the oscillations to have the maximum amplitude .
Another way to define it is as the frequency at which the system must be driven in order for there to maximum energy transfer between the driving oscillator and the system
In the case of an undamped system, the resonant frequency is just the natural frequency of the system. However the moment you introduce damping this is not so anymore(I still have to figure out a satisfactory physical reason behind this).
If you have an undamped system described by the
differential equation:
d^2(x)/dt^2 + g*d(x)/dt + w0^2*x = 0
then the
natural frequency when it is undergoing
free oscillations is:
sqrt(w0^2 - g^2/4)
However the maximum amplitude is obtained at
sqrt(w0^2 - g^2/2)
and the maximum energy transfer at w0.