Larry,
One does not need a "smoothly increasing radius" to get a smoothly
increasing differential control effect. Nor do we need a "smoothly"
increasing differential effect, just one that is not discontinuous or too
abrupt (no sudden "shifting gears" to unnerve the pilot). The diamond and
rectangle each meet this criterion. Consider:
The effect of your oval cam comes from the increasing arm length
perpendicular to the cable as the angular deflection moves away from
neutral. Rhetorical question: Were we to use your "oval" as a mathematical
ellipse, what aspect ratio would you advise? In the limit, the aspect ratio
could go to zero (minor axis length divided by major axis length) and we
would have a "bar" oriented parallel to the rudder cables, with said rudder
cables attached at the fore end (farthest from the rudder). As the belcrank rotates this "bar," initially the infinitesimal motion
transmitted to the tailwheel belhorn is zero (yes, that's a problem we'll
deal with in just a moment). Then the aft end of the ellipse ("bar") "picks
up" the cable and starts to move it laterally away from the belcrank pivot,
giving an increasing arm perpendicular to the cable and starting to pull on
the cable. You'll notice that the effective arm length increases gradually
with rotation of the belcrank, not suddenly, so it gives a progressive
increase in effectiveness, just like your ellipse would give; it IS an
ellipse (okay, a degenerate ellipse if you must). Hence, the "bar" is
equivalent to the ellipse in providing a progressive differential at
increasing deflections from neutral. Using the "bar" with the rudder cables
attached at the fore end, the opposite cable moves with the fore end of the
bar giving just enough slack to let the tailwheel belhorn pivot without
letting the cables actually go slack, just like your ellipse.
What you achieve with your ellipse is that you control the "minimum" ratio
between belcrank and belhorn by choosing a minor axis length of the ellipse
that is greater than zero. The "bar" version of the ellipse has the
disadvantage that control near neutral is non-existent. In both cases, the
major axis of the ellipse/length of the bar sets the maximum ratio of
belcrank to belhorn. (The amount of differential is the ratio between the
minimum and maximum described above.)
So, the drawback to the "bar" is that it is not wide enough near neutral,
resulting in not enough control deflection, so the remedy is to make the bar
wider. Whether the long end of the bar "picks up" the cable in a perfectly
elliptical manner or not is such a minor difference that my fat feet will
never notice it. Make the "bar" wider by making it a rectangle and the
differential effect will start immediately on deflection away from neutral;
make the bar a diamond and you can enforce a small region near neutral where
the ratio stays low, then increases after the aft portion of the diamond
"picks up" the cable and starts to move it laterally, mimicking your perfect
ellipse with much simpler manufacturing effort. The only real limitation to
the shape of the ellipse/bar/diamond/rectangle cam is that it must force the
cables into convex symmetry about the forward part of the device at all
anticipated deflections so that the cables don't go slack.
Work it out in your favorite modelling software, or go prototype it in
cardboard and thumbtacks and string and convince yourself that it works just
as well with less fabrication effort than machining an elliptical plate with
a groove along its edge (that would be a pricey part indeed!)
David J. Gall

-----Original Message-----
From: Q-LIST@... [mailto:Q-LIST@...] On Behalf Of Larry Hamm
Sent: Sunday, October 22, 2006 8:57 PM
To: Q-LIST@...
Subject: Re: [Q-LIST] "Exponential" differential via mechanics

David,
So, how does one achieve a smoothly increasing radius, and hence the exponential control effect, with a diamond or a rectangle?? I'm not real clear on that!
Larry Hamm

David J. Gall wrote:

P.S. Larry's suggestion does not have to be fabricated as

an oval or

ellipse; a simple diamond or even a rectangle will work.

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