Cosine Functions in Kite Physics

Trigonometry is basic to kite physics. When a kite line is swept thru space like a compass, it defines a triangle whose properties reflect performance parameters. “Cosine Loss” was identified in AWE as the relative loss of crosswind power at the edge of the kite window, away from the power zone. The equal and opposite “Cosine Gain” is a proportional increase in potential energy by altitude and/or upwind position in the kite window.

Another key application of Cosine relations is in the distribution of forces in a Tri-Tether junction, aka the Rigger’s Triangle. By setting the angles in particular ways, forces can be amplified by mechanical advantage, like how a bowstring is drawn, or the way a kite is bridled. The Tri-Tether is the simplest possible variable mechanical transmission, and has huge applicability in kite design, under Cosine math.

http://mathworld.wolfram.com/Cosine.html

I find these definitions quite confusing and contrary to my current understanding of cosine loss. The term cosine gain makes no sense to me.

To define the term cosine loss I would say something like:

Cosine loss is the loss of power dependent on the angle between the downwind vector and the angle of the tether. Though not strictly a loss, cosine loss describes producing less power than would be possible directly downwind.

(the tether is considered straight in this context).

If the angle described above is \alpha, the cosine loss near the power zone may be approximated by multiplying the C_L by a factor \cos{\alpha}

A typical method of finding \alpha would be using the dot product of said vectors.

Another factor not taken into account here is the fact that in the power zone, yaw direction of the kite has no importance. When you travel away from the power zone, yaw will make a (sometimes slight) difference. Whether this effect is part of the term “cosine loss” I don’t know.

At the edge of the wind window, the cosine loss would be either complete or very big.

The term cosine loss seems to be quite loosely defined, and the use of the term is just to state that the kite is not working at full power directly downwind, rather at an angle, and thus full power is not available for harvesting. The bigger the angle, the more loss relative to working directly downwind.

I would not use the term in a mathematical exact sense.

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“Cosine Gain” is simply the inverse of Cosine Loss, according to whatever parameters chosen. For example, the kite at “zenith” has a Cosine Gain of max potential energy of mass by altitude, but a Cosine Loss relative to max kinetic energy of sweeping in the power zone.

Yaw angle is properly disregarded as extraneous here, but has its own interesting cosine complexities, with regard to the gravity vector, and position in the kite window. Pitch angle has greater kite-line cosine dependencies than yaw. The wind velocity gradient in the kite window is a further source of cosine complication.

The primary intent here is to expand our cosine analytics in kite physics, especially to tri-tether forces. However complex our real-world cases, we refer back to the exact mathematical idealization, not some vague approximation.

Please cite your reference.

1st Springer AWE Textbook, TUDelft Circle.

Please reveal your identity, as long requested.


Which chapter and page?

I donated kPower’s heavily-annotated copy to the American Wind Power Center Museum. Thanks to ChristofB for kindly providing the book. RolandS can best attribute the idea, and point to the page.

Never yet had access to the 2nd Springer Textbook, either in print or behind its paywall. For all the public funding in play, paywall AWE content is not Open-AWE.

The word “cosine” is usually associated with the word “loss”, said loss increasing by the increase of the elevation angle as an example. If “a proportional increase in potential energy by altitude” is identified for some advantageous AWE use, “sine gain” would perhaps be a more suitable expression.

We use “gain” or “loss” in context, as needed. This topic is about extending prior usage. Its simply not true that only “loss” applies.

For example, Cosine Gain occurs when a power kite pops above surface inversion calm into an LLJ.

“cosine loss” is mentioned 11 times in the 2014 book, all between pages 14 and 20 in 1.4 Fundamental Physical Limits of AirborneWind Energy.

A simple conclusion from the lemma, that gives an upper bound on the usable
power, is that no device can extract power from a constant wind field if it does not
exert a horizontal force component against this wind. Most AWE devices have some
losses, and most exert a force on the ground anchor point that is not parallel to the
wind direction. In analogy to a similar expression in solar power, we might call the
loss that is due to the fact that the total aerodynamic force is not perfectly in line
with the wind direction the cosine loss.
In fact, all tethered systems need some elevation angle that the tether forms with
the horizontal in order to reach some altitude. Fortunately, for moderate angles, the
cosine is still close to one, for example the cosine loss is less than 30% even if the
tether goes upwards with an angle of 45 degrees. Optimized AWE systems typically
fly at even lower elevation angles, and for e.g. 20 degrees we have nearly negligible
cosine losses, of only 6%.

The lemma forms a strong obstacle for all airborne wind energy concepts whose
tether rises almost vertically into the air, cf. the artistic vision at the left of Fig. 1.5.
Such systems use a large portion of the aerodynamic force to just pull the tether
upwards, without extracting the corresponding amount of power from the wind field:
they have very large cosine losses.

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Elsewhere CristinaA credits WayneG for identifying the LLJ factor, which allows for Cosine Gain, a state clearly overlooked by the TUDleft circle.

Referencing capturing higher winds above by “cosine gain” obfuscates rather than clarifies.

I won’t be able to prove it, but I also doubt that going for higher winds has any effect on the line angle, except to lower it. So the connection with cosine breaks down there too.

That’s just potential energy then, not cosine gain?

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Its already proven that the LLJ bump is a common feature of the wind gradient, as Pocock described almost two centuries ago, and CristinaA’s modern geophysical data-sets amply reconfirm today. Let anyone prove otherwise.

“Gain” here is simply any parameter that grows in value as kite angle grows. Greater potential energy of mass, or higher wind velocity at higher altitude are just starting Cosine Gain examples. Consider antenna cosine gain, for kite based ham radio using an antenna lifter kite that flies at a greater angle, therefore higher, on the same length tether; or flying a power kite just above obstructing trees or terrain, when it would not fly at all any lower. Cosine Gain.

There is also the crosswind power vector cosine relation. Hauling a PTO line crosswind, after USP3987987fig5, LeoG, JoeH, or kPower, is the cosine gain mode. Reeling upwind and downwind, even with embedded crosswind sweep, involves an inherent cosine loss component.

AWE’s evolving engineering language is a tool-set for the toolmakers to extend as needed.

By your example of reaching LLJ (potential energy being yet another concern), the “gain cosine” has nothing to do with the elevation angle since the same altitude could be reached by different elevation angles. So the “gain cosine” or even the “gain sine” I tried to evoke above do not apply. Perhaps the wind gradient could be used but as a variable. So changing words sense is likely a not good idea, above all in trigonometry.
Tallak’s definition above is suitable Cosine Functions in Kite Physics .

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Pierre,

Its long been natural for kite experts to choose a kite with the highest possible flying angle as needed, especially to operate in smaller fields. They need not call this “cosine gain”, but its the same idea here.

Rod,

We agree that “Cosine Gain” as discussed here, in a kite geometry context, by the lifted-antenna example, is not to be confused with standard antenna-design parameters, like cosine aperture gain, cosine filters, and other common “cosine” terms. We could consider specific antenna gain cases by elevation angle, but that’s not the topic here.

“Cosine gain” itself is occasionally and quite properly used across many science and engineering contexts, but here we are specifically dealing with established geometric kite physics of cosine loss, and its necessary inverse meaning, cosine gain.

The tri-tether-cosine and crosswind-cosine ideas are very powerful analytically.

Ochd give up please… yawn. Look even with years as a satcomms engineer I’d never heard of either of these uncommon terms until after you tried to invent the cosine gain term in respects to kite power.
All hail wonder Dave. Fine sure. Great job Dave! Wish we were all as awesome.

Just shouldn’t be or at most, really doesn’t need to be discussed here

Rod,

The terms “cosine loss” and “cosine gain” already existed before AWE usage. I did not invent them, nor was I first to use “cosine loss” in AWE.

This topic is about cosine functions generally in various kite geometry instances. How can I reasonably abandon hope you can understand or at least tolerate “cosine gain”, along with “cosine loss”, as acceptable terms in AWE geometric discussion?

We adopt, adapt, or coin new terms all the time in AWE, including “AWE” itself.

Could you point to a orevious use of «cosine gain»?

You could always define the language to your needs. But better if course if the AWE community is on the same page for such terms. If «cosine gain» is important, you may well be the first person to introduce it. But right now I dont see the usefullness of the term, and also the way it stands relative to the well known term «cosine loss» seems very confusing to me.

Tallak,

In kite geometry, each of these statements is true, in context-

“There is a cosine loss when a power kite flies outside of its power zone.”

“There is a cosine gain when a KAP kite flies at a higher elevation angle”

Out of thousands of prior usage results in search, mostly in antenna design, I chose this neurological case because neuronal topology and geometry is rather kite-like, and they even have a “cosine gain rule”.

(https://core.ac.uk/display/10510599)

“These spatiotemporal response properties were quantified in two independent ways. First, the data were evaluated on the basis of the traditional one-dimensional principle governed by the " cosine gain rule” and constant response phase at different stimulus orientations."

Once again, Windy Skies, on no technical basis, as an “anonymous authority”, is abusing moderation power by relegating Kite Cosine Functions, including tri-tether and crosswind load motion, to the Scrapyard. He was not even convinced this is a valid topic by his own review of Cosine discussion in AWE literature.