Harmonic Oscillator driven by a Dirac delta-like force Consider that there is no damping for simplicity.
As we know, a driving force of the form $\sin(\omega t)$ will make the oscillator at steady state vibrates at the external frequency $\omega$. 
What about a force of the form $\delta(t-t')$ but distributed evenly in time? it's called a Dirac Comb or impulse train.
Will it preserve the natural frequency or will it vibrates at a frequency $1/T$ where $T$ is the period between pulses?
 A: Well, I finally pull it out.
I used Green's functions and it was pretty straightforward,
For a harmonic oscillator, you have to solve:
$(\frac{d^2}{dt^2} + 2b\frac{d}{dt} + \omega^2)G(t-t')= \delta(t-t')$
The solution is  for $t>t' $:
$$
G(t-t')= exp(-b(t-t'))\frac{\sin(\omega'(t-t'))}{\omega'}
$$
where $\omega' = \sqrt{\omega^2-b^2}$ 
The solution is:
$$
y(t)= \int{f(t')exp(-b(t-t'))\frac{\sin(\omega'(t-t'))}{\omega'}dt'}
$$
Using $f(t) = \sum{\delta(t-nT)}$ the integral becomes super easy and you can interchange the sum and the integral since the sum does not depend on t':
Finally:
$$
y(t)= \sum{exp(-b(t-nT))\frac{\sin(\omega'(t-nT))}{\omega'}}
$$
So what we got is as many sine functions as delta diracs the comb has, and vibrating at the natural frequency (just like a guitar) regardless if you are plucking it with a determinated frequency.
A: The magnitude of the transfer function (amount of vibration vs amount of excitation) of a harmonic oscillator with damping is shown below (from this Wikipedia article). The Dirac Delta function is white in the frequency domain; meaning that it has equal excitation at all frequencies.  So, the motion of the harmonic oscillator is simply the white noise times the transfer function.  The response will be dominated, as you surmised, by the resonant frequency of the oscillator.
Damping will make the motion die away over time, so hitting it with a pulse train will give overtones at the frequency of the pulses and its harmonics.  The strength of the overtones will depend on the strength of the damping and the strength of the impulses.  

