Hi. I wanted to present an idea for a calculation of TRPT (torsional ring power transfer?), or I guess any torsional transfer rig, where the shaft must be maintained at a certain diameter. The starting point is that the kite flies at a speed relative to the wind speed, glide number G_e and the amount of force in drag direction (drag mode) applied by the torsional shaft, it’s orientation relative to the wind, sustaining it’s own mass and such.
It all is quite similar to any drag-mode kite, such as Makani’s setup. The value of \gamma_0 would represent the optimum drag to maximize power, and thus may be assumed constant.
You could choose any number of kites and layering of kites. For most simplicity I will only assume one kite and one tether. We will assume that the rig could still fly in circles with a hollow shaft. The hollow shaft is assumed to consist of a number of tethers going from ground station to kite, with a compressive spacer structure keeping them apart. I guess this would probably be a part built with carbon tubes in practice.
What I want to investigate here is what happens when the diameter (or radius R) changes along the length of the shaft.
We know that no energy is lost, produced or stored in the shaft. This would be physically impossible. This means that we may assume that the power transfer is constant throughout the shaft, at any given distance from the kite’s looping path/plane.
If the rig is rotating at a rotational speed \omega, the speed of the kite would be v = \omega R_0, and the power produced by the kite would be P_0 = T \sin{\gamma} \, \omega R_0. Since the power in the shaft is constant, we may assume:
P_0 = P_1
\frac{\sin{\gamma_1}}{\sin{\gamma_0}} = \frac{R_0}{R_1}
\gamma_1 R_1 = \gamma_0 R_0
Physically, the tether will spiral tighter for a lower radius. The tension T of the tether and rotational speed \omega must remain constant.
From a similar discussion at Medium scales for torque transfer systems onshore and offshore we (I) arrived at:
\frac{d C}{d s} = \frac{T}{R} \sin^2{\gamma}
This equation represents the force per length of tether that must be applied outwards to maintain the hollow torsional shaft shape.
If we scale the radius of the shaft by a factor R_1 = \frac{1}{x} R_0 we get:
\frac{d C_1}{d s} = \frac{T}{\frac{1}{x} R_0} \sin^2{x \gamma_0}
If we assume that \gamma values are small, we get \sin^2{x \gamma} \approx x^2 \sin^2{\gamma} and:
\frac{d C_1}{d s} = x^3 \frac{d C_0}{d s}
To see how the weight of the supporting structures would scale with a reduction in R, we look at the buckling load of the carbon rods, which not must be reduced in length such that L_0 = x L_1 and F_1 = x^3 F_0. The same type of calculations were done in the other thread, so I’ll just “gloss” over most of the calculation, ending up with:
The new radius of the rods r_1 = x r_0. Thus the mass of the rods will be
m_1 = \left( \frac{1}{x} L \right) \pi (x r_0)^2 = x m_0
So if you reduce the diameter of the shaft by a factor of x, you would have to increase the strength of each ring such that the weight increases by a factor x.
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The tether drag of the shaft is determined by aerodynamic drag proportional to tether speed (approximately) D_{tether} \propto R^2 \omega^2. Furthermore, tether drag increases with the number of tethers in the shaft n by a factor of \sqrt{n} (I will assume this is prior knowledge as it is material for a quite long description by itself).
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At this point, it should be possible to optimize the tether drag against the mass of the hollow shaft to obtain a most optimal solution/construction of the rig.
This is what I would expect: The mass of the shaft is quite significant but so is the tether drag if you want to go high (compared to a traditional single kite rig). By reducing the shaft diameter you are incurring a cost in mass by the scale x while the reduction in tether drag is scaled by \frac{1}{x^2}. Presumably the optimum shaft diameter is somewhat smaller than the initial looping diameter of the kite (eg. 6 times wingspan), and thus the \sqrt{n} increase of tether drag will be compensated for quite easily. It is difficult to say which of the two kinds of rig could go higher…
It seems the fewer number of tethers n would not increase tether drag much, and also provide an easier mechanical build, as rods will be mostly pointing inwards in a triangle compared to “rings” with more than three edges. The next logical step for many n would be to have the rings built as stars rather than triangles. These detail design questions will greatly effect the mass of the rings.
I guess even a helix (n = 2) is possible and perhaps even a good solution.
