http://www.gizmag.com/steve-durnin-d...-geared/15088/

It's pretty obvious why this works at this scale. Do you have any
insight into how this might work at a useful scale?
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The reason it's hard to understand as explained is because it is
explained verbally with no diagrams or drawings. It is, in essence,
two planetary gear drives in tandem arranged to comprise a
differential.

The gear reduction system in a $49.95 winch from Harbor Freight is a
rather similar arrangement, except that in these the ring gears have
zero speed -- actually a locked-to-frame "barrel gear" that engages
both sets of sun gears. One set of sun gears has one tooth less, so
differential action takes place with high ratio like 157 : 1. The
reason they're so ungodly noisy is because they don't alter the tooth
profiles or pitch diameters even though they change the number of
teeth. Whaddya expect for 50 bux? Matter of fact, I used a Warne
winch found in Mike's garage of design the Chinese copied. The Warne
was vintage good stuff, cost lots. Probably made in Taiwan...

I electrified my boat lift with that Warne winch. It is audible 1/4
mile away. But it works great and nobody complains though I think
Mary is still slightly embarrassed. But enough about that...

I'm pretty sure Mr. D's diffy uses identical ring, planet and sun
gears, with the ring gears are not locked together but able to rotate
at differing speeds. In any case, they declare that the assembly is a
differential.

A differential produces an output speed that is proportional to the
difference of two input speeds. Let's consider one input as drive
and the other as control, where the speed of the control input can be
conveniently, easily and efficiently varied from 0 to +/- the drive
speed, which suggests the possibility of reverse at 65 mph in the US
and far more in Europe. Toyota thought it had problems with a sticky
gas pedal!

After devining or being told which input is "drive" and which is
"control", a key question then becomes: how much power must the
control input have? Mr. D feels that it would be quite small,
"order of magnitude" smaller than drive. Dayum, where's my checkbook
and how do I invest?

Let's consider a perfectly efficient system with two inputs and one
output where the output speed is equal to the difference between the
input speeds as Sout = S1 - S2.
Now let's look at power. Pout = P1 + P2 by whichever law of
thermogodammics makes me right. P2 could be negative, which doesn't
violate the perfectly efficient system assumption if it somehow routes
its power back to the prime mover. Easy peasy with electric motors
that can also generate, but then we don' need no steenkin' variable
speed tranny any more than we do in a 3000 HP diesel-electric
locomotive.

Let's assume this gizmo is arranged to make both input powers positive
in "forward gear" since the definition of "forward" is arbitrary.

Power is proportional to torque * speed, P = T*S

Pout = P1 + P2 = S1*T1 + S2*T2 = Sout*Tout = (S1 - S2) * Tout
but Tout= (S1*T1 + S2*T2)/(S1-S2) so T2 is inversely proportional to
S2 for given Pout.

I've skipped a few steps here but I think the path to conclusion is
clear.

As output speed approaches zero the input powers approach equality
because their speeds approach equality. It can git down and grunt in
the mud, deliver "ratio" even though no gear ratios have changed,
provided that the "control" input can deliver nearly half of the
power. That would be the variable speed control input.
Non-electric, since we don' need no steenkin' variable-speed tranny in
an electric system. Wonder how that works... :)