Tail Incidence, Part 3 - Size

In case you forgot, the fancy calculus notationc dCL/dAlpha (pronounced " ...... Robert E., John Wiley and Sons, Inc., New. York, 1949. Airplane Aerodynamics ...
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Parts I really enjoyed meeting the many EAA members at Oshkosh '90 who were kind enough to come up and tell me they were having fun playing with the spreadsheets from these articles. I also got some really good questions at my forums, and I'll try to answer these before this series of articles ends. As you discovered, you had a lot of freedom in designing the packaging for your people, engines, and so on. You also had a lot of freedom to play with wing sizes and shapes. You were also pretty much free to choose some combination of tail areas and their lever arms which met the stability requirements and made you happy. The price you're paying for all this initial freedom is the amount of work you're going through to make all of the above balance. As the saying goes, "It all comes out in the wash." The incidence of your horizontal tail is the wash! WHERE ARE WE?

So far we've accomplished the following steps: • We calculated the pitching moments caused by the lift and drag of the wing times the distance from the wing's aerodynamic center to the airplane's center of gravity. • We calculated the pitching moments caused by the wing airfoil shape itself. • We calculated the pitching moments caused by the interaction of the wing and the fuselage. • We calculated the pitching moments caused by the propeller's lift (normal force) times its distance to the center of gravity. • We calculated the pitching moments caused by the propeller's thrust times the distance from the propeller's thrust line to the center of gravity. Those 5 steps let us know the locations of some of the forces which are acting on our airplane/see-saw which we are trying to balance. Some of those forces are weights and others are lifting balloons tied to the teetertotter. • We calculated the angle of downwash over the horizontal tail.

by JOHN G. RONCZ, EAA 112811 15450 Hunting Ridge Tr. Granger, IN 46530-9093

• We calculated the dynamic pressure of the air over the horizontal tail caused by the propeller thrust. These last two steps looked at the local conditions at the horizontal tail, so that we know how much air our tail has to work with, and what direction that air is coming from. LIFT-CURVE SLOPE OF THE HORIZONTAL TAIL

We can't possibly set the tail incidence without knowing how much lift we get for every degree of incidence. We already addressed this issue when we did our wing incidence. You'll recall that we needed the aspect ratio, the sweep of the maximum thickness line, and the Mach number in order to do this. Rather than make you copy all the formulas again into our tail spreadsheet, I'm going to ask you to go back to the spreadsheet published in April 1990. You'll use the same speed and altitude that you used for setting the wing incidence. But you'll change the wing area, wing span, and sweep (in degrees) to the tail's values rather than the wing's values. As an example, for my homebuilt I set the speed at 213.3 mph, the altitude at 7500 feet, and the weight at 1875 pounds. My tail's area is 19.55 square feet, its span is 11.333 feet, and the sweep of the fattest part of the airfoil shape is 3.5 degrees. The April spreadsheet reported dCL/dAlpha as .08364 per degree. Technically, this is wrong. The reason is that the propeller has speeded up the air over the horizontal tail somewhat, which in turn increases its Mach number. For those of you who bought F117A plans at Oshkosh, you can take the square root of

spreadsheet when you calculate the liftcurve slope of your tail. For my homebuilt, the difference was less than 2 tenths of one percent, so it isn't worth the bother. In case you forgot, the fancy calculus notationc dCL/dAlpha (pronounced "Dee See-Ell Dee-Alfa") simply tells you how much lift coefficient a wing will make for each degree angle of attack it has. But if you rotate the airplane to a one degree higher angle of attack, the wing will make more lift. As you recall, the extra lift comes from throwing more air at the ground. This, of course, makes more downwash over the tail. From the tail's point of view, part of that extra degree angle of attack disappears due to the extra wing downwash! How much disappears? If you remember the August spreadsheet, you already figured out what the angle of downwash was over the tail. In that spreadsheet, in cell A42, I fiendishly printed out something called dEPSILON/dALPHA without explaining what it was. The shorthand for the downwash angle at the tail is K (the Greek letter epsilon). The quantity dEpsilon/dAlpha tells you what the downwash at the tail is for each degree of wing angle of attack. In plain English, this is the amount of angle of attack that disappears due to the wing's downwash. For my bird, this number was .45886 degrees. So if I fly my homebuilt and increase its angle of attack by one degree, the angle of attack at the tail will go up by (1 - .45886) degrees, or .54114 degree. This means that the tail on my homebuilt sees only 54.114% of the angle of attack of the rest of the airplane. The missing 45.886% disappears in the extra wing downwash. For this reason, you'll keep running into the quantity "1 - de/da" in the equations for stability and trim. This is the correction for the tail's angle of attack due to wing downwash. THE NEUTRAL POINT

Most pilots have noticed that the amount

the number in cell D104 from last month's

of elevator it takes to flare the airplane on

spreadsheet, and multiply your airplane's speed by that amount in cell B5 of April's

landing depends very much on the center of gravity location. If the center of gravity is far SPORT AVIATION 45

forward, you have to really pull hard on the wheel or stick to flare the airplane. If you don't pull hard enough, you'll land nosewheel first. On the other hand, if you fly the same plane with the center of gravity near its aft limit, you'll notice that you can flare the plane with very little pull on the stick. Assume that you had a gauge in the cockpit that told you what the elevator setting was. At cruise speed, say it read -1 degree. At 5 knots above stall speed say it read -14 degrees. Chances are good that you're near the forward center of gravity limit. Now bolt some lead to the tail to move the center of gravity towards its rear limit. In cruise, perhaps the elevator position might be +1 degree. Five knots above the stall, it may read -8 degrees. It's taking less elevator to trim out the airplane at different speeds. Imagine that you add even more lead to the tail, and repeat the flight test. This time the elevator position in cruise was + 1.6 degrees, and when you finally got the beast stabilized at five knots above stall speed the elevator position was also +1.6 degrees. You also noticed that the airplane was extremely difficult to fly. If you can achieve any angle of attack with the same final elevator position, then the airplane's center of gravity is at the neutral point. This is the fuselage station of the center of gravity at which the airplane has neutral stability. You definitely don't want to fly there! (Modern fighter aircraft do fly there, and the X-29 you saw at Oshkosh '90 actually flies with the center of gravity even further aft - but they use sophisticated computers, sensors and software to move the elevators to keep the airplane under control. The reason they do this is that airplanes without stability are easier to toss around the sky in a dogfight.)

• cruise C

Meg

0.0 att eg

forward eg

-faster

-

£.

+

FIGURE 1

(*

•£

Meg

WHAT IS STABILITY, ANYWAY?

0.0

Stability is the tendency of your homebuilt to return to its trimmed airspeed. This means if you're trimmed for cruising flight, and push

the stick over to dive the plane, when you let go of the stick the plane should pitch nose up, slow itself down, and eventually after a few oscillations arrive back at the airspeed you trimmed it for. Likewise, if you are trimmed for cruising flight and pull back on the yoke, slowing the plane, then when you let go the airplane should dive a bit to recover its missing airspeed. You can see from these examples that stability has something to do with what happens when you change the airplane's angle of attack. Since changing the angle of attack also changes the lift coefficient, you can also say that stability has something to do with what

happens when you change the airplane's lift coefficient.

If your plane is trimmed for cruising flight,

it means that you've got the right amount of

lift on your tail to balance the teeter-totter. As we saw before, this means that the moments about the center of gravity are zero. Pulling or pushing on the control yoke will change the lift on the tail. The airplane will climb or dive until it finds the speed at which it's back in trim based on its new tail lift, or until it stalls or the wings come off, whichever comes first.

Let's figure out what we need. If I pull back

on the yoke, the airplane slows down. To be

stable, I want the plane to lower its nose 46 OCTOBER 1990

slower

nose-up trim

(*

^7

nose-down trim

faster

-

Ci

+

slower

FIGURE 2

when I let go. In other words, at higher lift coefficients (lower speeds) I want a nosedown pitching moment. If I push on the yoke, the airplane speeds up. To be stable, I want

also a horizontal line showing zero pitching moments. Let's start with the line marked "forward eg". As you can see, this line crosses the zero pitching moment line at the

this case, at lower lift coefficients (higher speeds) I want a nose-up pitching moment. This situation is illustrated in Figure 1. The

in balance at cruise speed. Let's say the pilot now pushes on the stick, and speeds up the

the plane to raise its nose when I let go. In

cruise condition. The pilot has trimmed the elevator for cruise, so that the teeter-totter is

vertical axis shows the moments about the center of gravity. Towards the top of the page gives a positive pitching moment,

plane in a mild dive. The lift coefficient reduces because of the lower angle of attack. Looking at the graph, we see that to the left of the cruise CL the moments about the center of gravity become positive. This

tive pitching moment, which would rotate the plane nose-down. The horizontal axis shows the airplane's lift coefficient. Faster speeds are to the left, which would require a lower lift coefficient. Slower speeds are to the right, which require higher lift coefficients. The CL corresponding to cruising flight is indicated by a dashed vertical line. There is

means that the airplane would respond by

which would rotate the plane nose-up. Towards the bottom of the page gives a nega-

raising its nose. Exactly what we wanted.

Now say that the pilot pulls back on the

stick, and slows down the plane. The higher angle of attack increases the lift coefficient. To the right of the cruise CL the moments

about the c of g become negative. This

means that the airplane would respond by

lowering its nose. Again, it's exactly what we wanted. This is a stable airplane! If the graph of moments versus CL slopes down towards the right, the airplane is stable. If you move the center of gravity further aft, the airplane is still stable, as the graph shows. However, the amount of moments generated to push the nose back where it belongs are much lower. If you moved the c

of g further back, eventually the line would no longer slope down to the right, but would be absolutely horizontal - it would lie right on top of the zero moment line. This means that

the airplane would have neutral stability. The center of gravity position which gave us this

condition would be called the neutral point. In effect, the steeper the slope of the moments about the c of g versus CL line, the harder it is to change the airplane's angle of attack. That's why it's a chore to flare the plane at the forward c of g limit, and easy to overcontrol at the aft c of g limit. The steep-

ness of the slope of this line is directly controlled by how far the center of gravity is from the neutral point. In calculating the stability and control of your plane, it helps to know how much each part of the plane that's contributing moments about the center of gravity

affects the slope of this line. That's why I keep printing out the values of dC^/d^ (shorthand for the slope of this line) for each

contributor. If the value is positive, it means

that it's making the line's slope more shallow. This is destabilizing, because it's the same as moving the center of gravity rearwards. If the dCMcg/dCL contribution of some element is negative, then it's stabilizing influence, because it has the effect of moving the center of gravity forward by steepening the line. You're probably wondering what effect the elevator setting has on this stability. Figure 2 shows the answer. If you want to trim the airplane for climb, you reach down and crank in nose-up trim. This shifts the whole line to the right, but does not change its slope. Simi-

larly, if you want a fast descent, you pull off

the power and crank in nose-down trim. This would shift the line to the left, but not change

its slope. So the effect of elevator deflection is to move the trim point (the CL or speed at

which the airplane is trimmed to zero pitching moments) which doesn't affect stability. What if the moments line sloped up to the right instead of down, which it would do if

you moved the center of gravity even further

aft? In this case, if you pulled back on the stick, positive moments would be generated which would raise the nose even further. Or, if you pushed on the stick, negative moments would be generated which would cause the

plane to dive even more. In effect, the

airplane would amplify any little stick movement or gust-induced pitch change, and you would have your hands full just keeping it level! This is not good.

So you can see that the line describing the

moments versus the lift coefficient has to slope downwards. As you've seen, the calculus shorthand for the slope of that line is dCMcg/dCL, and it must have a negative value in order to make a stable airplane. Perhaps you don't care about all this theory, but just want to know how to make a stable airplane. O. K. That's easy! Always put the center of gravity forward of the neutral point, and the line will slope down like it should and your plane will be stable. It would help a lot to know where this neutral point is

A " POWER O F F NEUTRAL POINT '

106

"

108 HORIZONTAL TAIL dCL/dALPHA; 109 FAIL AREA: 110 FUSELAGE STATION OF TAIL AERODYNAMIC CENTER: J OWER OFF DYNAMIC PRESSURE RATIO AT TAIL: 111

112 113 114 1 15 116 117 118 119 120 121 122 123 124 125 | 126 127 128 129 130 131

] "• i 1

E GET FROM APRIL 1990 SPREADSHEET

1 • •"•"

i

„..'SQUARE FEET

;* * « •

NCHES

— jrrrs

4.--.-.1INCHES~ ~ —— — -

UNCORRECTED POWER-OFF NEUTRAL POINT BAT FST" CORRECTION FOR FUSELAGE: 'OWER OFF NEUTRAL POINT WTTH FUSELAGE: 1 ........... ______

{| * ' ') NCHES

~

_

! * " t NCHES AFT OF OATlJM t 1

"PROPELLER INDUCED DOWNWASH'" THRUST COEFFICIENT: VALUE OF RIBNER CURVE "A":

t

VALUE OF RIBNER CURVE -fl": dCNp/dALPHA FOR ZERO THRUST: dEPSILON.prop/dALPHA : dCMttCL DUE TO PROPELLER DOWNWASH: PROPELLER DOWNWASH MOVES NEUTRAL PCHNT_:______

'"PROPWASH OVER TAIL CONTRIBUTION'" EXTRA Q OVER TAIL MOVES NEUTRAL PONT: — • POWER ON NEUTRAL POINT ""

• •* •"• • • . .. —— TTT —— ————

NONES_____________ ____

—— ———

•"NORMAL FORCE CONTRIBUTION"' dCM/dCL DUE TO NORMAL FORCE: PROPELLER NORMAL FORCE MOVES NEUTRAL PONT :

132 —THRUST LINE OFFSET CONTRIBUTION'" 133 134 PERKINS «HAGE 'K' FACTOR: 135 dCMAiCL DUE TO THRUST LINE OFFSET: 136 THRUST LINE OFFSET MOVES NEUTRAL POINT: 137 138 139 140 141 142 143 144

IBICI D j .

i

107

I i 1 T

_,_{.. ———,

—pr. ! * * ' ! NCHES, _____________ __ . _ .__ ____

^±^ • } [•' 'INCHES H -. —— "T" "

i

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POWER OFF NEUTRAL POINT: NCHES 145 CORRECTED FOR PROPWASH: INCHES 1 4 6 CORRECTED FOR NORMAL FORCE: INCHES ! ! ! • • ' INCHES 147 CORRECTED FOR THRUST LNE OFFSET: 148 CORRECTED FOR TAl DYNAMIC PRESSURE DUE TO PROPri ' ?'" INCHES ; 149 INCHES AFT OF DATUM 150 FINAL POWER ON NEUTRAL POINT IS AT FS . —(i-i . .INCHES 151 LEADING EDGE OF MEAN AERODYNAMIC CHORD IS AT FS: __ i... % OF THE MEAN AEROOYNAMiC : CHORD 152 NEUTRAL POINT IS AT % OF THE MEAN AEROOYNAMIC CHORD 153 CENTER OF GRAVITY IS AT STATIC MARGIN IS .-.— — — — — —— "%OF THE MEAN AERODYNAMIC CHORD 154

-ifi- ..;.;:

"" _ _ir...- •

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FIGURE 3

on your airplane, so let's go do it. FINDING THE NEUTRAL POINT

cient, I told you I was picking a number that would give me a bigger tail because of the shape of my fuselage. A couple of people

We have now calculated everything we

wrote asking what I meant. The very concave rear end on my bird was carefully

your homebuilt. It's interesting to note that this point is independent of the actual

slow the air down a lot. Slowing the air down in a controlled way is good for drag. The

needed to find the power-off neutral point on

elevator position. While the elevator position will determine whether or not the airplane is trimmed, it doesn't have anything to do with the neutral point. Let's compute the power-off neutral point for my homebuilt. To do this, we need the following information, all of which came from

the spreadsheet:

Wing area = 98.12. Tail area = 19.55. Lift slope of wing = .0909 per degree. Lift

slope of tail

=

.08364 per degree.

Aerodynamic center of tail = FS 258.385. Center of gravity = FS 106.75. dr./da = .45886. Mean Aero. Chord =43.57 in. q (dynamic pressure) at cruise = 104.53 We saw that when the propeller is running in cruise, the dynamic pressure (q) at the tail

increases, because of the propwash. The tail

shaped using three-dimensional software to

price I pay over a conventionally shaped fuselage is that the air going over the horizontal tail is also slowed down, making the tail seem smaller than it is. For this reason, and

because I'm generally a pessimist in these

things, I'll use a tail dynamic pressure ratio of .80, to be safe. If your rear end is shaped more normally, you would use a higher number, like .90.

First, we'll look at the tail's part in this: (1-dK/du)' dCL/da, tail" tail's dynamic pressure * tail area ' lever arm

For my bird that's (1-.45886) * .08364 *

(.80 ' 104.53) ' 19.55 * (258.385 - 106.75),

which gives 11220.206. This equation can be explained simply: First, (1-de/du) figures

out how much angle of attack the tail has

turns this extra q into extra lift, making the tail effectively bigger than it really is. The question, then, is what the dynamic pressure ratio at the tail is without power. Perkins & Hage simply says it's less than one, meaning that the dynamic pressure at the tail is smaller than for the rest of the plane. Not much help. Dommasch says that it varies from .65 to .95 with no power. That's a very large

after subtracting out the wing downwash; dCL/da is how much lift coefficient the tail makes for each degree angle of attack. As you recall, the lift coefficient is the lift in pounds that each square foot of tail makes at a dynamic pressure of one pound per square foot. So to figure out how much lift we've got, we multiply the CL by the tail's dynamic pressure and then by the tail's area. To get the tail's moments, we multiply this

safe assumption (his words) to use .90. Back when we were selecting a tail volume coeffi-

The wing's part is dCL/d« * dynamic pressure * wing area. For my bird that's .0909 *

range Roskam says that it is frequently a

lift in pounds by the tail's lever arm.

SPORT AVIATION 47

106 107

108

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