I suggested that the electromechanical spark erosion machine was worth a look because of its simplicity and where a machine was only needed to remove broken taps et al it would be quite adequate .
Spark erosion technology starts with a bare wire being tapped up and down by hand but can can be developed through numerous levels of sophistication into exotic machinery .
Responses to some comments raised :
(1) Speed of operation .
The pulsing rate may not be as slow as suggested at 3Hz . For a tap removing device the moving parts would be very lightweight and properly set up the actual movement of the cutting tip would be very small . Given that very powerful solenoid force would also be available I think that 20 – 50 Hz would be achievable and that is quite adequate for the purpose . In smaller sizes the mechanism is after all not much different to a door buzzer .
(2) Improving the simple system .
Ther are many ways of improving the performance of the simple system but I think that it would be better to either leave it simple and easy to make or start in a different place and build an advanced machine .
(3) Electronics .
As soon as any design starts to need more sophisticated control and monotoring functions then a microcontroller is the easiest option but the certainly the same things can be done in other ways .
(4) Loudspeaker cone .
There is a further type of spark erosion device which is viable in small sizes only and these use a solenoid driven by a relatively low power transformed down and isolated mains 50Hz supply . The cutting current is supplied separately . A combination of the mechanical oscillation caused by the 50Hz supply , the effect of inertia and a bit of good luck cause the cutting tool to bounce up and down on the work and cut quite effectively . This comes quite close to the idea of using a loudspeaker cone .
More:
Spark erosion machines are in active use in industry doing work over a surprising range of sizes . Larger ones are used to make things like plastic moulding dies and smaller ones are used to cut holes in hard materials down to only a few thou diameter .