Your idea of adding motion to the inductor making it a variable inductance adds another dimension to switch mode energy storage / retrieval by making it parametric, e.g. CARA, but a serious question: where do you suppose any extra energy will come from?
I don't know where from and I don't know if it will come at all. That's what the CARA experiment is designed to investigate. If I was to guess, maybe from the same "place" that keeps the perpetual atomic motion. as the motion induced into the variable inductance structure is also energy dissipated by the L/C circuit.
Ideal L/C circuits do not dissipate any energy - only the resistance does. The L in this circuit is the ratio of magnetic flux per electric current. When the ferrite is far away mostly all of the magnetic flux is caused by the current flowing in the coil. When the ferrite is is close by most of the magnetic flux is caused by atomic currents in the polarized ferrite. That ferrite polarization energy can be recovered, when the coil is discharged into the capacitor. Inverting buck-boost power supplies recover over 95% of that energy every day using stationary cores and synchronous rectification. Whether the mechanical work performed by the polarized ferrite is "for free" remains to be seen. We do so little real experiments to increase our understanding on this forum. Mostly we just replicate other people's devices without understanding how they are supposed to work... The CARA investigation is different. It's goal is to increase knowledge methodically, with the hope that an anomaly is found ...but nobody besides Itsu wants to participate. Most likely because it is hard, unexciting work now and when they read that we can recover only ~70% of the energy at the moment, they are just not interested, without even noticing that the motion of the core is not even accounted for at this stage, which is the major point of the entire endeavor.
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