Mechanical Movement – Doubts…

I'll vote for alternately rotates at two different angular velocities within and each rotation.

I think yesyen is correct about the illustration in his post, and you are correct about the illustration in your post. The radii in your picture are much bigger in proportion to the overall size and this allows the gears to stay engaged without moving the axes. This is not true of yesyen's gears

I think they will stay engaged in both cases.

As the smaller gear face contact area recedes, the larger gear face contact area will be moving closer, so they will remain meshed.

I think.

These are even weirder, (skip to 0.38).

No change in center distance is needed.

When I look again at the drawings I think the OP is correct. Even though I posted a video of someone who had made working gears of that general shape, the gears in the OP illustrations are different in important ways.

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The squares in the OP illustration are noticeably different in size from one another. More importantly, the number of teeth on the larger square is greater than the number on the smaller square. This means the alignment will shift away from the corner-to-face arrangement it had in the illustration, to a corner-to-corner alignment which will definitely cause a difference in center distance....if it doesn't first slip as it approached face-to-face alignment.

See this

Also, the 507 mechanisms has been partially animated. If you search the intenet, you will see that many mechanisms have been animated. Start by Googling 'animated mechanical mechanisms'

For this to work, the sum¹ of the diagonal of the left square, plus the perpendicular of the right square, must equal the sum¹ of the perpendicular of the left square plus the diagonal of the right or they become disengaged at some point.

1. Allowing for the curvature and overlap of the teeth.

I take your points after verifying on the board, if only both the gears of same size.

For different sized gears (diameters), on board they have interference (one eating into other) on rotation, which means in real they would get locked-up, unless the centers are allowed to slide. Am I right in saying this?