Re: "Exponential" differential via mechanics
David J. Gall
Larry,
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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 nonexistent. 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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