Hey! Pretty good forms for the gears already, but there are a few tricks you can use to make it work better.

First, for animations, "most" animations should be at least 3 frames, as the human mind can easily recognize something bouncing between two images. By using 3 frames however, the mind has trouble recognizing the pattern, and accepts it more easily as an animation. I think that's what the main cause of the "flickering". (Flickering can be a useful technique, but it doesn't convey motion very well.)

Here's an example of what I mean.

2 Frames@20FPS       3 Frames@30FPS

                    

On the making of the gears, here's a quick shortcut for most round objects to get the shape right.

Working on 1/4 or 1/8 of a round object makes getting the shape right much easier, since it quarters the work, and makes sure all sides are the same. Also, it makes the refining step much easier, as I did all of this in ~5-10 minutes. This technique should also be used to make the base gear.

Your large gear is pretty good, but could be better refined with this method, but you can use the small gear provided if you want. (Also, I held shift in the example, because many programs will lock rotation to angles that work well, but if you need more frames, don't hold shift.) (Also also, sometimes you'll need to put the quarters together, refine it while it's the full circle, take a quarter again, and repeat until you like the result.)

As for the little cog needing to move 3x faster, you can do two things:

   1)Since it's moving so fast, you could have the larger gear only move once every 3 frames. This is the simplest method, and works for other speeds as well (movement every 2Fr=2xSpeed, 4Fr=4xSpeed, alternating 2Fr/3Fr=2.5xSpeed etc...). This is the method I believe you already used, but you moved the large gear every 2 frames from what I saw, so it was 2x Speed.

   2)For a more complex method, you would need to make 1 rotation(or quarter rotation) take X Times as many frames to move that distance. So, for every 3 frames the small gear takes to repeat, the large gear should take 9 frames(at 3x speed). This is the same as option 1, but just requires more work, but it does make for a smoother animation.

I hope this helps, and good luck with the work!