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When trying to calculate, from first principles, the lift produced by the rotor of an autogyro, I have tried imagining the rotor as a circular flat wing set at an angle to the relative wind. The mass flow rate would be the volume times density of the swept 'air tube' deflected down per unit time. That mass flow rate, times the airspeed and the sine of the deflection, will give the lift... And if the gyro is flying s/l, you'll know that lift = weight, so that the deflection angle can be calculated.

But I don't know how to estimate the area of the base of that 'air tube'. In some papers, I've seen that it's taken as a circle with a diameter equal to the wingspan, but I find that somewhat arbitrary...

Is there a rational criterion for that area...?

xxavier
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  • If you use that circle with diameter = wingspan, the induced drag will be that of an elliptic wing. If you have less than optimal lift distribution, the circle will become a bit smaller. – Peter Kämpf Sep 17 '20 at 10:17
  • I see... Well, the projection on the horizontal of a circular wing is an ellipse... Any reference for the derivation...? No problem if in German... – xxavier Sep 17 '20 at 10:53
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    I did explain it here. The disk is perpendicular to the direction of flight, so there is no ellipse but really an idealized circular tube. Similar to the Froude hypothesis for propellers. – Peter Kämpf Sep 17 '20 at 12:08
  • @PeterKämpf Thinking in your explanations, I have a doubt concerning the statement that for a wing with a less-than-optimal lift distribution, the section of the associated 'air tube' would be smaller. I think that, for those less-than-optimal wings, the section would be larger, as the vector sum of lift and drag will be larger for those sub-optimal wings, hence the total force on them will be higher, and the product mass flow rate x airspeed should be also higher too, something that –for all other things equal– would mean a larger tube section... – xxavier Sep 18 '20 at 05:30
  • Less than optimum means that some sections deflect the air by more than others, and this unequal distribution is less efficient than a constant one. Using a smaller tube is only a mathematical trick to simulate wings with lightly loaded tips and I agree, this is not universally applicable. But making the tube bigger means that more air is deflected by less, creating lower drag. This only would be justified with a larger wing span. The simplification of a stream tube uses constant deflection over span which is not realistic for non-elliptical wings. – Peter Kämpf Sep 18 '20 at 10:37
  • @PeterKämpf Thanks. I'm trying to adjust my 'paper results' to the real world of small gyro performance, and the area of the base of the 'stream tube' seemed to me a potentially modifiable parameter, that (in order that paper and real world might fit) should be corrected with a coefficient of approx 1,2... – xxavier Sep 18 '20 at 10:46

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