How to design a contactless mechanical shaker
A high-powered electromagnet excited with some  sinusoidal signal?

Sounds like a PA speaker / subwoofer to me!

So, instead of building a high-powered oscillator, I'd simply use a soundcard, or a low-powered one, and feed it to a cheap power amplifier for speakers.

You'll mostly find amplifiers for 4Ω impedance loads, so to achieve 50A, you'd need P=I²/R=2500 A²/4Ω = 625 W of speaker power. (Often, stereo amps come with the ability to drive twice as much output power with the outputs bridged to just one speaker, or in your case, electromagnet.)

You can buy stage-grade PAs for that at reputable sources (Thomann etc), or you could build your own class-D amp (considering you really don't need to switch overly fast to drive something at 80 Hz).

You could of course also just build/get a class-AB amplifier, and add negative, phase-delaying feedback just so that you get your 80 Hz oscillation. That's cool, because it's then an oscillator on its own, but seriously, feeding in an external 80 Hz reference is both easier and more stable, precise and reliable.