Harmonic Drive Gear Ratios
I mentioned in a previous chapter that I was using a 51:1 gear ratio when calculating out the movements of my HD, and this caused some discussion. So this chapter will be a brief sideways digression into how the HD gear ratios work. For a better explanation of how the Harmonic Drive itself works, see any of their literature. However, their literature does skip a few vital (imho) details.
Attachment 272232
I had better start by explaining my and their terminology. Previously I have been referring to red/pink bits and blue bits and green bits. I need to relate those terms to the 'official' HD ones.
The key to the whole system is the Flexspline, which I was calling the red bit with a pink fill. It is flexible - slightly. It is connected to what I was calling the blue bit, but which is not shown here. The connection is via the outer back rim, with all those (16) holes. The blue bit is rigid, and includes the outer race parts for the Crossed Roller Bearing. The Harmonic Drive itself does not have to include the CRB, but packaged units do have a CRB.
The Circular Spline is what I had drawn as green. It too is rigid, and in my drawing there are a couple more parts attached to it, including the inner race for the CRB.
The interesting and tricky bit seems to be the Wave Generator, which gets more detail in the next drawing.
Attachment 272234
The first diagram in this chapter shows a ball race between the Wave generator and the Flexspline. It looks round. This drawing shows the ball race all right, but it is elliptical. A moment's thought rings alarm bells here: just how do you have a
elliptical ball race? Nonetheless, it
is elliptical, and that is crucial to the whole design.
The inner race can be elliptical fairly easily: it sits on the Wave Generator and the balls go around it. That's easy enough to do with some CNC grinding. But the outer race
has to flex as the Wave Generator goes around. A ball race flexes??? My mind boggled at first.
Attachment 272236
Yes indeed, the outer race has to flex. Here we have a 'printed' version of a Harmonic Drive (picture off the web, (c) Peter Heim), and you can clearly see the outer race distorted into an ellipse. I think the designer has put a bit too much ellipticity into the shape, but with that size teeth he had no choice. You could get a lower ellipticity by having a bigger number of teeth, as the difference between the inner and outer teeth rings is always just two. I count 32 teeth on the inner ring, so the reduction ratio should be (I think) 16:1. My HD is 50:1. I am not sure whether the plastic would support the finer teeth you would get with a 50 tooth version.
So how does the outer race in a real HD flex? You just make it very thin and out of spring steel. Very simple, and quite reasonable. Stay within the elastic limits of the steel and you are fine. They make springs that way too. Now try to find that explanation in the HD literature! (If you can, do please tell me where.) Very clever people, the Germans. (HD is a German company.)
Attachment 272238
Does it have to be made this way? No. You could make it with a couple of small ball race wheels as shown here, but the construction would be far weaker, and a bit more complex. But it would work.
You could also make it with just one ball race, so there is only one point of contact. That would allow the use of one ball race, which might be a whole lot simpler. I imagine that was tried and found lacking. I suspect the Flexspline did not like it, and failed after a while.
You could even make it like this, but beware! This design will have significant backlash at the input gears. We don't like backlash. Not good.
Well, how do they keep all the balls in line at the right separations? They use a Vespel TP ball cage. Vespel is a high-end engineering plastic by DuPont, able to handle the sideways forces the balls generate as they go around the ellipse. Unlike brass, which could fatigue and crack after a while, Vespel handles the flexing quite OK, and the grease, and the high temperatures sometimes encountered.
OK, moving right along to the original question of whether it is R:1 or R+1:1. Here we have the official explanation from the Harmonic Drive manual for the SHF & SHG drives of how the ratios work (PDF file on my disk drive, from the HD web site). Nice diagrams, but unfortunately the text under each diagram is wrong. Yes, I know that this is from the official company manual, but it is still wrong. Compare the text for each of the boxes with that in the first box. They are all the same, despite the arrangements being all different. They all say 'CS Fixed', 'Input and output in opposite directions', '1. Reduction Gearing', and so on. The text is right for the first box, and wrong for all the others.
Let me hasten to add here, that while the first two arrangements make engineering sense, none of the other arrangements would be viable in the real world. No engineer with an ounce of brains would even contemplate trying to use a HD unit to gear
up by R:1, or even R+1:R. You would simply smash the Flexspline.
How? I guess the graphics designer who created the catalog was not an engineer, hadn't a clue, and simply did a cut-and-paste from the first box to all the other, changed the big red arrows and the equation numbers, and forgot (or did not know) to change the text. And no-one did any serious proof reading. I suspect this mistake has mislead many people. (End of slight rant.)
I used the arrangement shown in box 2. That's equation 2. R+1:1.
Well, yes, very fine, but does it work like that in practice, or am I up the creek? Leaping several chapters ahead, I will simply say this. If I give a command 'g0 a3600' (I am working in degrees), the RT spins exactly 10 revolutions if I use the 51:1 ratio. On the other hand, a 50:1 ratio is actually 2% short, which is 7.2 degrees per revolution. Over 10 revs, that's 72 degrees of error. Kinda visible, yes? And yes, that's what happened to me at first.