Salient White Elephant

March 29, 2009

Eye of the Cat

From an aerodynamic perspective, the ideal vertical axis wind turbine rotor is the “H Rotor”:

H Rotor

Unfortunately, from a mechanical point of view, the H Rotor is terrible. This is so because as its blades bow out, seeking the troposkein shape, a tremendous load is applied to the horizontal beams that support the blades. This load is in exactly the direction that the beams are least able to support (tending to pull the ends of the beams together).

Mother Nature Doesn't Always Care What We Want

Now you know why the only H Rotors you see on YouTube are on small turbines rather than on utility scale machines. The small H Rotor is essentially a brute force solution to the problem. That is, if you’re only interested in small machines, then you can just make the stiffness of the blades and blade supports large enough to resist the tendency to assume a troposkein shape. Don’t get me wrong… I think the H Rotor is a good idea for small machines that will run the meter backwards at somebody’s home. But I just want to point out the irony in this approach, for if you go in the opposite direction, the problem goes away! To see this, simply imagine an H Rotor with a diameter of 10 miles. Now the velocity of the rotor blades is essentially linear, and there is no centrifugal force to bow the blades! (A little thought will reveal that the centrifugal force disappears because the airfoil velocity is always the same, regardless of the rotor diameter. This is true because the blade velocity is determined by wind speed and by the desired tip speed ratio. So if the wind speed is 20 mph and the desired tip speed ratio is 6, then the airfoil will travel at 120 mph regardless of the rotor diameter.) Of course, the aerodynamic forces also cause the blades to bend. In any case, it would be nice if there were an alternative technique, and there is! It is the mysterious and enchanting Eye of the Cat, and it eliminates the tendency of the blades to bow, whether this tendency is caused by aerodynamic forces or by centrifugal forces!

Eye of the Cat Darrieus

The aerodynamics of this rotor are close to ideal because we can select any blade shape that has the symmetrical force-balancing properties depicted above. (Actually the centrifugal force on the outer blade is greater than the centrifugal force on the inner blade, but they are approximately equal. If necessary, weight can be added to the inner blade to balance the opposing forces.)

I’m not sure what to think of the Cat’s Eye Turbine. It seems to me that it will work as described, but it does have some strange properties. For one thing, it is unstable. To see this, imagine the blades are supported by cables instead of beams. We have carefully added weight to the inner blades so that the vertical forces tending to increase the distance between the ends of the cables exactly balance the vertical forces that draw them together. However, any perturbation that slightly increases blade curvature also increases the centrifugal force on the outer blades, and decreases the centrifugal force on the inner blades. In this case, if the blades are sufficiently flexible, they will subsequently increase in curvature until the outer blade and the supporting cables form an approximately troposkein shape, while the inner blade mirrors the shape of the outer blade in the opposite direction. Furthermore, any perturbation that slightly straightens the blades will also increase the centrifugal force on the inner blades, and decrease the centrifugal force on the outer blades. In this case, both blades will eventually become perfectly straight, and then both blades will bow all the way out together until the whole structure, cable and blades, assumes a troposkein shape.

But on the positive side, consider how even the aerodynamic forces seem to resolve into approximately equal and opposite vertical forces on the ends of the supporting beams! Meow!!

It guess the feasibility of this machine boils down to whether the beams that support the blades can have stiffness sufficient for keeping the rotor from becoming unstable, and also whether the designers can put the rotor’s natural frequencies into an appropriate range.

Rotating Guy Wire Variation

Cat's Eye Darrieus, Rotating Guy Wire Variation

Traditionally Guyed Variation

Cat's Eye Darrieus, Traditionally Guyed Variation

Non-Vertical Axis Applications

The Eye of the Cat can also be applied to the Direct Drive Linear Turbine With Yawing Oblong Track, Radially Displaced HAWT Rotor, the HAWT With No Shaft.

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