Let's say we have:

- coil shaped spring couplings
- springs *not* used to be expanded or compressed, but rather to couple 2 axles (correct word for longish cylinders within bearings allowing them to rotate?)

- so it's about rotational coupling. I.e. the driven axle will somewhat lag behind the driving one, most of the time.

- for now, say we have 2 axles, one on the motor, and one which is to be driven

- the special motor outputs periodically *varying* torque - connected by such a spring to the other axle
- The motor also has a flywheel attached
- The 2nd axle has a disc weight on it, rather small compared to the motor's flywheel

How to calculate the reactions of this system, e.g. the rotational movements of the 2nd axle depending on the unsteady motor output?

And if we then make the system more complicated by introducing

- a 3rd and 4th axle

- the 3rd is coupled by 2 gears, probably with a small amount of backlash, to the 2nd,

- and the 4th is coupled by another spring to the 3rd one.
- Also, the disc weight is now on the 4th axle, not the 2nd.

How does the 4th axle behave, pertaining to rotation angle over time? As may be obvious, the flywheel and springs are there to dampen any unevenness in rotation at the target axle. But it's probably not doing so perfectly, and I'm interested in the remaining unsteadiness.
At least in my funny mind, the springs and flywheel are sort of an equivalent to low pass filters, which dampen the pulses coming out of the motor, but then perhaps there's also some sort of ringing going on or other effects, at least not 100% of the higher frequency content removed by the "filters".

connected by such a spring to the other axle

You mean like a big rubber band? That would work like a torsional spring. The motor would twist it up (apply torque), the springiness would untwist it, applying torque to the connected axle. Assuming the backward turning resistance of the engine always exceeded the turning resistance of the axle, the spring would turn the axle, not the motor.

Coil springs use F=k*s. I don't recall the analogous formula for tortional springs,. You can look it up. So your engine will input a torque into the spring system, then your spring will output a torque to the axle.

Eventually it should hit a steady state where the spring has "loaded up" and is essentially a solid link for purposes of forward thrust over smooth surfaces. IE torque in = torque out.

Ah, silly me, I could have included illustrations.

I only now discover the two springs on the left, holding the disc on the motor together with what I guess is a flywheel - yet another coupling I overlooked.
I was initially talking about a coupling like on the center right of the image. You can see (red arrow) the 90° bent end of the coil spring stuck through a hole in the small disk, so rotational force will turn that spring. The whole apparatus has several of couplings like that, but since that's the main axle, this one is especially big.

As for that funky motor, here's a gif of how it works:

Pay attention to the flipping of the poles of the electro-magnets. A 1-phase sine voltage is applied, with opposite polarities, to those two coils, i.e. the fields repeatedly flip. The fixed magnets on the rotor are relatively low in number, you can imagine how that thing will output not a constant, but rather "pulsating" torque. (it can't even start by itself, needs a starter motor)
EDIT: Also, this is driven directly by mains voltage frequency, which also has some fluctuation.

What is all this?
It is a electro-mechanical, "tonewheel" organ (e.g. Hammond).
I am currently going along all routes which I suspect provide reasons for why my emulation sounds so much more sterile than the real thing.
There are groups of {2 tonewheels sharing one axle}, and each tonewheel is coupled by an own spring to that axle. Then, each of those 2-tonewheel-axles is coupled by 2 gears with a certain ratio to another axle, which is in turn coupled to the motor by that big spring. (or maybe it's even more complex, I haven't seen a complete system drawing, only pieces here and there)
You see, the whole system is quite a bit more ;) So in my initial post I started with subsets of it to start getting the hang of that first.

Well, there has to come some variation from somewhere in there, to produce that lively sound (and I don't mean the rotational speaker cabinet).
I suspect that all that multiple spring loaded monstrosity has some minute variations in the movement of each single tonewheel (each one has different mass and spreading of that mass into space, so fluctuations in torque may not make them react exactly the same on their springs).
Maybe I'm wrong and the effect of this is so minute that it does not actually contribute to the sound, but I'd like to explore this :-)