Resonance and modes I use a metal body (let's say cooking pot) in which I induce eddy current via induction at certain frequency which I can change as I wish.
Every object has it's own accustic resonant frequency. If exciting frequency is the same as the accustic resonant frequency of the body, then the sound produced by eddy currents will also be strong at that frequency.
If I drive the coil at half the frequency (or 1/4th the frequency) of an accustic resonance of the cooking pot), then the sound will still be loud at the resonant frequency.
But does the same effect happen if I drive it with double frequency of the cooking pot accustic resonant frequency? Do I still hear loud frequency at the accustic resonant frequency, or does this only apply if the exciting frequency is lower(half, one fourth...) of the accustic resonant frequency?
 A: This is a bit complicated and it depends on the details of the driving.  If you are driving with a pure sinewave and there are no nonlinear aspects that introduce overtones, then the pot will only resonate at its resonant frequencies (which are many, not just one).  Driving at 1/2 or at double will not excite a resonance.  
But you may be driving with something other than a pure sinewave or there may be  nonlinearity which effectively makes it not a sinewave.  If the cooking pot is resting on a surface, then it will respond differently to a push than to a pull, and this non-linearity introduces overtones of the driving frequency. This is similar to what you hear when part of a speaker has come loose and it buzzes.  Harmonics (multiples of the frequency) not present in the original are being generated. So in your case, if you drive the pot at 1/4 of a resonant frequency, there may easily be some nonlinearity that will create a harmonic of 4 times the driving frequency.  That harmonic is what makes the pot resonate.
But nonlinearities don't create subharmonics, so I would not expect you to be able to excite a particular resonance by driving it at say 3x that resonant frequency.
However you speak of a pot's acoustic resonance in the singular.  Any real pot will have many many resonant frequencies (also called vibrational modes or normal modes) so you might well find it resonating when you try higher frequencies but the pot is now vibrating in a different resonance mode. 
