#### Re: "Exponential" differential via mechanics

David J. Gall

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

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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 asan oval orellipse; a simple diamond or even a rectangle will work.