The field from a permanent magnet is the magnetic equivalent of DC; zero frequency. It shouldn't bother a bit, in terms of receiving. Now if you were transmitting through it, you might degauss it, but the power levels in receiving are so low as to not be a concern.
The inductance and frequency are almost certainly totally unsuited to the drive level provided. Unless you are providing 1000's of amps of drive (at which stage you'd need welding goggles to view the inductor) then you need a different arrangement.
The current through an ideal inductor will increase linearly with time when a constant voltage is applied.
Here
L = 220 uH = 2.2E-4
t = 1/2 of 10 Hz cycle = 0.05 = 5E-2second
V = 12 Volt
- I = V.t/L = 12 x 5E-2 / 2.2E-4 = 2727 Amps
Most available power supplies are not capable of this sort of current :-).
For operation at around 10 Hz you need a MUCH larger inductance value. The inductor may not be much larger physically but it will have a very large number of turns comparatively. For a physically very small inductor the wire will become very fine, resistance will be high and resistance will start to play a very significant part.
To use the existing inductor at sensible power and current levels you would need to increase the frequency substantially and the resultant pulses would not be able to b sensed as vibration. For example, to reduce peak current to ~=2.7A would require operation at 10 kHz.
To start to experiment in this area the easiest way is probably to dismantle a small relay. This will usually have a laminated core, DC operates currents in the 10's of mA's to 100s of mAs range and would produce "pull in" forces that are noticeable.
Note that you can use a spring loaded "pole piece" to get mechanical vibration. Your magnet will successively provide attraction and repulsion on subsequent cycles. As the magnet will also be attracted to the core material you may still need to utilise a spring if you want approximately symmetrical attraction and repulsion forces.
* ADDED COMMENTS *
There are a number of comments on other answers. I opine that the basic core shape is OK but that the material and amp-turns are wrong. So -
Start with pulling almost any small relay apart as per my suggestion. They have done the work of optimising core and using VERY fine wire and many turns and ... .
Once you have a feel for it try your own. The core you are using MAY work with many many turns of the finest wire you can find BUT ideally you want a core with MUCH higher "permeability". This will greatly increase your inductance for a given ampere-turn product. A core of soft iron wires would work very well. Metal (iron) powder also.
Best Answer
(1) A small airgap when at closest approach will (should) make a substantial difference. This can easily be achieved with a small layer of non magnetic material if desired. Try tape, paper etc initially. A more permaent solution can follow. This applies even with a non-magnetic material for the door as the plate the magnet attracts may suffice.
(2) A low force spring should be able to be made to work. Magnet-on attracts plate and compresses spring. Magnet off - spring is designed to exceed remaining attraction.
Again, start informally - you can eg use bubble wrap air bubble with suitable mechanical placement and sellotape to hold it.