The basic function.
The whole point of wind-vane self-steering gear is to steer the boat relative to the wind, so let's start there. Look at Figure one first, and imagine an airfoil section mounted vertically on the blue wind vane mount at the top of the figure. The wind would be blowing from right to left across this wind vane, and as long as the vane is perfectly aligned with this wind it will remain vertical. No error signal is generated in this case (we'll get to the "error signal" below) and the boat will stay happily on course because we previously rotated the whole structure shown in Figure one on it's vertical axis to align the wind vane with the wind while at the same time maintaining the desired course by hand.
But alas, things never remain static and we are bound to eventually drift off course relative to our wind unless we make corrections. Assume we drift a bit to the left of our desired course (counter-clockwise, looking down on the boat). Now the wind is no longer aligned head-on with the wind-vane, but is applying pressure on the side nearest us. The wind-vane mount is pivoted on the wind-vane shaft, which is mostly horizontal, so the wind-vane will be pushed away and cause the wind-vane shaft to also rotate. Note that I said "mostly" horizontal. Actually the wind-vane shaft is inclined 18 degrees from true horizontal, and this gives the wind-vane an important property: it's angle relative to the wind changes as it is rotated in such a way to again bring it into alignment with the wind. In other words, the magnitude of the rotation of the wind-vane is proportional to the magnitude of the wind direction error. This is a very desirable property to have in a wind-vane and is often implemented (look at an Aries, for example).
The wind-vane is also counterbalanced with the purple weight to a point where in a still room it will just barely remain upright.
The gears shown in Figure one serve one primary purpose only: they allow the entire wind-vane "head" to be rotated on it's vertical axis to set a course relative to the wind. But before delving into that particular mind-twister, let's just assume a fixed relationship on the course setting and skip all the gears for now. To do that, just picture the torque from the wind-vane shaft being transmitted through the green spring-looking heli-cal device straight through the mess of gears directly to the red "Signal shaft."
Now look at Figure two. The red "Signal shaft" passes torque through a second heli-cal device to a gray bevel gear which is driving a green bevel gear which is rigidly attached to the shaft of a servo rudder through suitable bearings and seals. The gray bevel gear is free to rotate within a bearing mounted in the blue structure, and the blue structure is free to rotate on an axis coincident with the gray bevel gear. In Figure two the right side of the figure points to the bow of the boat, so water will be flowing from right to left across the servo rudder. As long as the servo rudder is oriented directly into the water flow, it will remain vertical. If the red signal shaft rotates, however, thus rotating the gray bevel gear and subsequently the green bevel gear and servo rudder, the servo rudder will pivot either toward or away from the observer looking at the Figure. In the example above we had a course error to the left relative to the wind which caused the wind-vane to pivot away from the observer, which will cause the green bevel gear to rotate clockwise-looking-down, which will cause the red signal shaft to also rotate clockwise-looking-down, which will cause the gray bevel gear of Figure two to rotate clockwise-looking-forward, which will cause the green bevel gear of Figure two to rotate counter-clockwise-looking down, which will also cause the servo rudder to rotate counter-clockwise-looking down, which will cause the servo rudder to rotate clockwise-looking-forward on it's horizontal pivot . . . thus steering the boat to the right and applying power to the rudder actuator to steer the boat to the right thus correcting the original error. (Phew!) Note that as the servo rudder pivots on it's horizontal axis it also rotates the green bevel gear relative to the gray driving bevel gear (Figure two) in such a way to cancel the original error rotation. This cancellation effect as the servo rudder rotates on it's horizontal axis produces a proportional response at the servo rudder that is also highly desirable.
A simple way to set the desired course could be to put some sort of clutch in the signal shaft and simply decouple this clutch to rotate the wind-vane on a vertical axis. But I tend to get rather lazy when at sea, what with the sea-sickness and all, and I just don't want to be climbing aft and trying to set the wind-vane, then back to the center-cockpit to adjust the course with the wheel, then back aft to try again . . . and so forth. So, I was looking for a way to keep the arrangement described above but be able to just arbitrarily adjust the course using a worm gear on the entire wind-vane head assembly. I envision a small crank in the cockpit for this, with a long torque tube to a worm drive on the self-steerer. I did something similar on another boat I had and it was most convenient!
It should be obvious, however, that simply rotating the wind-vane head will really screw up the relationship between the wind-vane and the signal shaft if something isn't done about it. What we need here is a way to be able to rotate that head but at the same time cancel out the effect of that head rotation on the signal shaft. Which is the whole point of the mess of gears shown in Figure one. Here's how it works.
First let me explain what you are looking at in Figure one. There are two "differential" gear sets, the top one comprised of a green bevel and a teal bevel, with a red three-way yoke in between that has three gray bevel gears attached by bearings. The red yoke is directly attached to the signal shaft, so when the green bevel rotates, say, 180 degrees, and the teal bevel is stationary, then the signal shaft will rotate exactly half that distance, or 90 degrees, in the same direction. All the functioning of the self-steering apparatus is the same as described above, except that the magnitude of the rotation on the signal shaft is half what it was.
We want to preserve the relationship between the wind-vane and the servo rudder and also be able to rotate the wind-vane head on a vertical axis. This means that we have to cancel out the effect of the rotation of the wind-vane in some manner. We do this with the second "differential" gear set comprised of the teal bevel gear and the gray bevel gear with the purple three-way yoke in between that has three gray bevel gears attached by bearings. The teal "double bevel gear" is not attached to anything except the six small gray yoke bevel gears. The purple yoke is fixed relative to the hull of the boat, and the bottom gray bevel gear is fixed relative to the wind-vane head.
Now when we rotate the wind-vane head on it's vertical axis, the gray bevel gear drives the three gray bevel gears on the purple yoke (fixed to the boat) thus driving the teal double-bevel gear by the same number of degrees in the opposite direction! The green bevel gear is also effectively attached to the wind-vane head and it also drives the gears of it's red yoke, thus driving the "teal double bevel gear" in the opposite direction by the same number of degrees exactly as the lower differential gear set has done. Thus the two yokes stay stationary in their positions if the wind-vane and servo rudder remain stationary relative to their respective attachments, yet the wind-vane head is rotated as desired and the function described above for the wind-vane to effect steering has not changed at all.
A primary goal with this design was a device which used bearings that could be submerged in oil, with oil seals between the salt water and internal workings. It isn't at all clear in the drawings, but all parts do meet this goal. Another goal was a design where all "soft" parts subject to ultraviolet degradation would be shaded. The only such parts are the oil seals, and they can easily be recessed into dark areas. The wind vane itself will be exposed and probably will require painting, depending on the material used. Oil submersion means that materials such as hard steel for bearings and gears are protected from corrosion.