seen edgewise. Locate this line so that the angles of the push-pull line, hard over to hard over, are about equal with reference to it. This is because you'll probably use self aligning rod ends in

By Start Hall (EAA 10883) DIFFERENTIAL bellcrank is, of course, the kind of crank that is commonly used to operate differential ailerons. And differential ailerons travel more in one direction than the other even though the stick throw is equal, each side of neutral. An aileron having an up-travel twice that of its down-travel, for example, is operating at a differential of 2 to 1. Because of the several variables involved, laying out the geometry of a differential bellcrank can be one of your more frustrating endeavors. Such variables include the numerical value of the differential motion itself, the crank radii at both its input and output ends and the positions of the neutral points along the input and output radii. Although mathematical approaches to the problem undoubtedly exist, more often than not it is the practical, trial and error approach that prevails. The usual penalty for trial and error is, of course, frustration — and there's

enough of that already in airplane design. Having frequently faced this problem, and counting myself as one of that large majority of designers who are short on patience, I have developed a shortcut to the trial and error procedure, a graphical one which almost always works the first time. The idea isn't likely to be original with me, but I like it anyway. Here's how you can apply the technique on an aileron system. First, lay out the side view of the trailing part of the wing, where the aileron is. Draw in the aileron travel desired. Figure 1, which represents a working example of a 2 to 1 differential aileron system, shows an aileron uptravel of 30° and a down-travel of 15°. The bellcrank rotates 30° each way. Draw in whatever aileron horn radius appeals to you and spot in on this radius the push-pull connecting points for both the up and down positions. Now connect these points with a straight line which extends forward. This line represents the centerline of the push-pull. "Rotate" the aileron horn (without rotating the aileron) so that the forward end of this line is up inside the wing where you judge the bellcrank is likely to end up. Go back and spot in the neutral position on the horn radius. The line representing the 32 MARCH 1978

push-pull centerline will also be the plane of rotation of the bellcrank. Adjust a beam compass or other appropriate device to the estimated or desired push-pull length and lay off the three positions of the forward end of the push-pull on the bellcrank rotation plane; aileron up, neutral and down. Project these three positions upward at 90° to that rotation plane as three parallel lines. Now, on a separate piece of translucent drawing vellum, draw three intersecting radial lines to represent the angular throw of the bellcrank (30° each side of neutral, in the example of Figure 1) and draw a number of randomly-dimensioned radii from their point of intersection. What you've just drawn is the geometry of a bellcrank having several "test" radii drawn on it. One of these radii will give you the required differential linear output of the push-pull, obtained at the required angular rotation of the bellcrank. To find out which one it is, place the overlay over the three parallel lines you extended upward at 90° to the bellcrank rotation plane. Work the overlay around until one of your test radii intersects all three parallel lines on the drawing underneath as well as all three radial lines on the overlay. This is the radius you're looking for. Set your compass for this radius and transfer it to the drawing underneath. Before doing this, however, prick the center point through to the drawing underneath. This is the bellcrank pivot point. Don't lose it. Let's now move back to the aileron horn down in the side view. Project the three points representing the aft end of the push-pull travel upward by three parallel lines just as you did with the three points at the forward end. Draw a short line across these three lines, parallel with the bellcrank rotation plane. You now have three points of intersection, right? Each of these points represents a position of the aft end of the push-pull. Connect the forward points to the corresponding aft points,,and there's your geometry. Before doing this, however, you'll have to properly locate that short line (the one identified in the above paragraph) upon which the aft ends of the push-pull fall. This line is, of course, the rotation plane of the aileron horn,

the push-pull, and the self-alignment angle should be split, half one way and half the other. If you'll examine the sketch at the top of Figure 1 you'll see a problem; the aileron won't come all the way down because the push-pull interferes with the bellcrank at a point near the crank's pivot point. I drew it this way on purpose to illustrate a difficulty one can encounter in using the "wrong" combination of radius, angular throw and push-pull travel. As shown, trying to meet all the requirements in one "stage" doesn't always work. You'll have to change something and try again — or handle the job in two stages, half at the input end of the crank and half at the output end. For really weird arrangements you can "differential-ize" in several stages. I define the "input" end as the place on the bellcrank which picks up the push-pull from the control stick and the "output" end as that part of the crank that actually drives the aileron. Figure 2 shows the Figure 1 problem solved in two stages. Here, you'll need two overlays instead of one (or use the same overlay in two places), which are developed on the basis of the "K" factors shown in the table in Figure 2. The K factor merely helps locate the angular position of the neutral radial line with reference to the full-throw lines. And, of course, this is where you start. In the 2-stage differential the angular throw of the bellcrank about the neutral position is no longer equal as it is in the single stage differential. For the 2 to 1 differential used in the example the table shows the K factor to be 2.41. Therefore, according to the little formula shown in the table, the crank rotation angle from aileron neutral to aileron down (-e-) is total crank throw (60°) divided by 2.41, or 24.9°. I used 25°. Draw the overlay(s) just as before but this time rotate the neutral radial line 5° (in this example) toward the aileron-down side, to give the required 25°. The remaining angle is 35°, right? Now go through the Figure 1 process again at the output side of the crank, moving the overlay around until all three vertical lines, all three radial lines and the appropriate "test" radius intersect. Locate the crank center as before. Extend a line from this center point, upwards, as shown in the figure. Extend it far enough to give yourself some room for another overlay problem. Now draw three parallel lines across it, at right angles. The spacing of these three parallel lines is equal, each line representing a position of the

push-pull to the stick, as dictated by the stick movement from hard over one way, to neutral, to hard over the other way. Although the stick throws are equal, each side of neutral, they have to cause the bellcrank to rotate unequally at, in this example, 25° and 35°. Apply the overlay upon these three parallel lines and go through the moving-around process just as before until all the required lines and test

radius intersect. By the way, while you're moving the

overlay around, don't fret if the crank

center doesn't immediately fall on the line projected up from the crank center

AILERON HORN ROTATION PLANE

I

in the lower view. Merely draw in the required radius, radial lines and crank

center wherever they happen to fall — and carefully project them to the required crank center line. You now have both the input and output geometries worked out. All you need do now is combine them into one bellcrank as shown in the figure. It should be noted that the technique just described contains a small error because it assumes that

the bellcrank and the aileron horn operate in the same plane. In reality they operate in planes 90° to one another. This implies that since the horn and bellcrank are tied together with a push-pull of finite length the angular

BELLCRANK ROTATION PLANE AND PUSH-PULL CENTERLINE FIGURE 1 — EXAMPLE OF SINGLE-STAGE. 2 TO 1 AILERON DIFFERENTIAL

Drawings by Author, Stan Hall (EAA 10883), 1530 Belleville Way, Sunnyvale, CA 94087

throws of the aileron will be very slightly less — or very slightly more —

than desired because of the rise and fall of the arcs. The longer the pushpull, of course, the smaller the error. There is no simple way to account for this small error by graphical means. And, since the error is likely to be on the order of a degree or less, I for one am inclined to ignore it. In my opinion there is nothing holy about maintaining an exact, precise differential motion in ailerons. However, if you've a mind toward precision at whatever cost, Go For It. With

adequate trial and error, you'll eventually get there. A final point; as described earlier, the K factor in Figure 2 leads to proper positioning of the surface-neutral radial line on the crank geometry between the maximum-throw lines — for a crank which provides half the differential at the input end and half at the output end. The division between input and output need not be half and half; it can be anything that works for you. However, this calls for more trial and error, and that's what this article attempts to minimize. Try the half and half arrangement first. If this doesn't work, try shifting the surface-neutral radial line on the overlay(s) and forget the K factor.

OVERLAY FOR INPUT STAGE

DIFFEREN 1. to 2, 10 2. to 3, to 3. to 4. 10

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SPORT AVIATION 33