Transformer frequency

Frequency of induced emf in the secondary coil is the same as the frequency of the voltage applied to the primary coil is the same as frequency of magnetic flux coupling the windings. By Faraday law of electromagnetic induction : emf = N⋅dφ/dt. No relative motion between parts* of transformer, transformer is the "static" converter of electromagnetic energy.

* small vibrations of turns and other parts can be neglected

 

Then, I believe you don't understand Faraday's law of em induction.

 

If you know that law and know the fact that parts of a classic transformer don't move (they are static), why did you mention movement between coils:

?

 

If the coils rotate/move one to each other and at least one is passed by current i(t)≠0 you'll have so called:

1. emf of rotation (frequency of which, f_r, coincides with frequency of mechanical rotation/motion)
2. emf of transformation (freq. of which, f_t, coincides with frequency of the current)

Total voltage is the supereposition of two emfs.

 

As a pair of examples, imagine this:
1. Two solenoidal coils are placed parallel, on directly above the other. Now, if you apply an AC current through the top solenoid, i=sin(wt), then it will produce a time varying magnetic field, phi = some_constant sin(wt).
Some of the top solenoid's field will cut into the second solenoid. Assuming the second solenoid isn't tied to a load, it will have a voltage induced based upon the change in magnetic field, dphi / dt, or V_coil2 = some_other_constant cos(wt).

Both coils are running at w (frequency)

2. Assume the top solenoid (from 1 ) had a steady DC current going though it. If the two solenoids are still, the amount of field induced in the second coil will be steady and dphi / dt = 0, no voltage is across the second coil.
However, if the top coil begins to rotate at some rate, wr, then there is a voltage induced in the second solenoid at V_coil2=a_constant cos(wrt) as the flux from the top solenoid begins to rotate.