This isn't really about the electronics at all, but rather about balancing a spinning object. I'm trying to understand the physics, and maybe someone here will have an answer.

Anyway, suppose you'll be spinning a perfect, massless disc which has various weights mounted on it at various distances from the axis of rotation. The disc plane is vertical, and the axis of rotation is horizontal. And you manage to arrange the weights so that the disc assembly is in balance at rest. By that I mean that no matter where you position the disc, when you release it, it stays put. It has no tendency to rotate to a "heavy" spot. There's no pendulum effect.

The question is - when you spin up the disc, will it still be in balance? It seems to me it would not be. But apparently they balance drone propellers statically that way, and they remain balanced when spinning. So maybe I'm wrong. Does anyone know for sure? Does anyone have a link to an explanation of all this that a civilian can understand?

A disk that is statically balanced while horizontal will not be in balance when vertical and spun because the weights are now not all the same distance from the center of the earth, therefore they may have identical mass, but will have different WEIGHTS.

Most drone propellers must be balanced dynamically so they have identical lift for each blade. Your spinning disk is not the same.
Paul

But what about a disk that's statically balanced when it's vertical? I've done some additional research since I posted, and apparently if the disk is stable at rest at every vertical position, then its center of mass coincides with the axis of rotation, and when it spins up in the same orientation it should remain in balance and not vibrate. Does that sound right?

I understand that everything changes when it's no longer a single plane, and dynamic balancing is required.

You can google this .. a beam can statically balance , by having say two weights on one side and one appropriately positioned on the other side .
When spun up the forces created are now different on each side - it is unbalanced dynamically .

Dynamic balance takes into account the distribution along the axis, not just the torque on the axis - however if all the masses are coplanar with the axis as a normal, dynamic balance and static balance are the same, as such a plane's principal moments of inertia
are at 0 degrees or 90 degrees to the rotation axis, the condition for dynamic balance (well, also that the CoM is on the axis of course)

Just to clarify, if the mass is in multiple planes, then dynamic balancing would require something equivalent to balancing each plane independently, but the entire assembly would still have to be balanced at rest. Right? The center of mass would still have to coincide with the axis of rotation.

I have a situation where a long rod is mounted between two flanges, with one flange in a bearing, and the other driven by a motor. The flanges appear to be "perfect". I had assumed that the rod was of uniform density, so if I mounted it in the geometric center of the flanges, the assembly would be pretty well balanced. But when I put the rod/flanges assembly on a flat surface, it insists on rolling forcefully to one position, which presumably is the heavy side. To get it to remain where I place it, with no tendency to become a pendulum, I have to move the rod 1mm off-center on one axis. Then it should not vibrate when spinning if the rod is at least consistently non-uniform throughout it's length. But of course if it isn't, the assembly would still be dynamically out of balance, and would vibrate.

The problem is that I have no way to dynamically balance the assembly.

Yes - although you only need to bring the principal moments of inertia into line with the rotation axis, the individual planes can be unbalanced if they cancel each other out - that
only adds bending moments to the shaft. However if each plane is balanced then each has its principle moments aligned so the net effect is also aligned.

For an object freely spinning in space you have to spin about the maximum MoI for stable rotation, but if the shaft constrains the rotation any of them will do. Some shapes have enough symmetry that the frame of the principle moments of inertia is not unique allowing an infinite set of balanced axes, for instance a circular disc can be spun balanced
about any diagonal, or about its proper axis, but not any other direction

moment of inertia is defined relative to an axis of rotation - in most circumstances the
centre of mass should be on that axis (certainly that's a requirement for balance).