To get started on the math, don't think of gears but simple
levers. The vane at it's lowest position is a simple case so start with that.
That's interesting, Dan, and a bit of a surprise. I mean, I realise that you can think of gears as levers, but it's not immediately obvious to me how I would use it. Is that to work out forces, or velocities, or can you do a fair bit of it that way?
See, I was working out the horizontal component of the velocites, because I figured that this was vital in understanding whether a portion of the vane is providing thrust at a particular position. I've done enough of this kind of maths recently enough to be fairly sure I'd worked that out for any point on the main wheel (particularly so as to get it for the vane axle at any radius, and at any angle so I could see how it behaved elsewhere). I got its velocity relative to the ground as
V
g = V
c + (sin a)*(R
1+R
2)*V
c/R
w
where V
g is the velocity of the larger gear axle, V
c is the velocity of the cart, a is the angle subtended by the gear axle from the rearward direction as it rotates round the wheel axle, R
1 and R
2 are the radii of the small and large gears respectively (their sum being 3R
1), and R
w is the radius of the wheel.
This obviously simplifies a lot for the axle at the bottom of its rotation, to
- 3*(R
1/R
w)*V
c relative to the cart.
It might be a rather slow job without programming it all to plot in 2D, and I might be wasting a lot of time. Better methods probably require more background about aero and mech. I made one attempt to add the velocity of points along the vane due to their circular motion on the gear, but I got it wrong. My intention was to get some idea of how long parts were moving backwards w.r.t. the cart and for how long and at what speeds, but it's all very complicated!
Another case to solve is finding the maximum chord for the vanes. Draw the diagram with the edges of two vanes just touching leaving out everything that isn't relevant and fill in the lengths and angles that you know. You should then be able to see the solution.
I'm not even sure how I'd go about that at this time. But I also feel that it is also academic: if the chord was so long that the vanes even nearly touched, it would be yet another source of turbulence and inefficiency. The ideal is to minimise that effect, yet we need vanes, so that has to be balanced with getting the vane to move in the most advantageous part of the wheel's motion, I would suggest even sacrificing some of that potentially useful chord to reduce the churning and sheer.
The other consideration I mentioned before is how large a wheel you have beyond the gears. That can of course reduce to radius 5*R
1, and at that point the chord can't extend beyond the gear radius or it hits the ground.
But you've got me intruigued about how to work that maximum out now! I have to work out at what angle the gears are when the vanes pass. It will be different I suppose for my 4-vane. When I'd got an understanding of the general case, I was hoping to move on to comparing the different vane numbers, and also how it compares with the reversed half-rotation (with more gears) - the vane kind of chops down at the bottom, moves longer in a roughly vertical motion, but this would add more backward velocity to the upper part (in faster wind) and slow the bottom (potentially even reversing), so it looks unhelpful.
Anyway, thanks for those pointers. That's been very helpful. I'm happy if you want to share more about it, or if you don't. I have worked out more complicated stuff at one time from first principles, but I sure am rusty. I've done a fair bit of programming.
Hvae you got any favourite programs (anyone) for sketching gear trains and so on - preferably something fairly simple without all the rendering stuff. Not a full-blown CAD program, but a simple dynamic simulation program, where I could just put a gear here, another there, fix that point to there, and see some forces, velocities, etc., and preferably animate it. I keep thinking I'm going to have to get some mechano or spirograph! Those animated drawings on the thread - are they done the laborious way?
ETA: If you extend the vane to nearly the centre, where you said you'd leave room for a thin axle, I figure that that part of the vane is moving forwards w.r.t. the cart. God knows how much! The part at the edge of the gear is only moving slightly backwards as it rolls round the inner gear (or is it momentarily stationary - yes, I think so - the inner gear is fixed to the cart chassis, and the gear top does to it what a wheel does to the ground, slows to zero as it touches - though it will be different before and after). These are all reasons why it's a bad idea to make the vanes too long, I think.
