Tail Incidence

If you set the incidence too high, you will ... it can be changed in flight test. ... much lift the tail needs to produce to balance ...... •b"y*i.46.7/j(p78p'p7p/2^. _.
7MB taille 9 téléchargements 339 vues
Parti IN reading through June's article, I realized that I screwed up an explanation. We were talking about Reynolds numbers, and how they get smaller with higher viscosity and higher with lower viscosity. To get higher Reynolds numbers, I said that you need longer chord lengths, or could fly faster or at lower altitudes. That was all true. Then I told you that to lower the viscosity, you need to add gobs of heat. That was true only if you are flying a submarine! Fluids like oil get less "sticky" if you get them hot. However, air is a gas, and gasses get less "sticky" when you make them cold. One practical example of this principle is NASA Langley's wind tunnel called the National Transonic Facility (NTF) which uses cryogenics to make the air very cold indeed. Chilling the air both increases its density and reduces its viscosity. These effects increase the Reynolds numbers so that the scale model behaves more like the full size thing. I've seen the NTF, and it is awesome. So I should have told you that to increase Reynolds numbers by decreasing viscosity, you need to fly in supercold air. Those EAA members who have flight-planned a trip to the equator in search of higher Reynolds numbers should immediately amend their destinations to Siberia. Sorry.

In doing my homework for writing this arti-

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

of incidence you needed to lift your airplane at the design condition. With the tail incidence, you proceed in the same way. However, figuring the tail CL required isn't so simple. For starters, let's list some things that you'll have to consider:

• You have to have enough elevator power to stall the plane in ground effect with flaps deflected at the forward C of G position. • You have to balance the pitching moment of the wing. • You have to balance the pitching moment of the fuselage. • You have to balance the C of G position. • You have to compensate for the effects of the powered propeller. The tail lives in a horrible place. Since the wing makes its living by throwing air at the

ground, the air is curved downward behind the wing, and the tail feels this downwash. Then it is assaulted by the propeller wash. Finally, you can add the interference caused by the vertical tail and the fuselage. It's a miracle that the horizontal tail doesn't go on strike for higher wages.

cle, I visited the Notre Dame Library and looked up tail incidence in several airplane design books. None of them provided a procedure for determining tail incidence. Many simply gave a suggested range, or pointed out that these things are determined by wind tunnel tests. Then I talked with Burt Rutan, and asked him what he does. He does the best he can using a computer program he wrote which takes into account the wing downwash, and then added that he makes the tail (or canard) incidence variable so that it can be changed in flight test. So we are going to do the best we can, and most probably our tail incidence will be close enough not to present any serious problems. We'll break the problem of tail incidence into two areas of inquiry: what is the angle and strength of the local airflow over the horizontal tail (the downwash issue), and how much lift does the horizontal tail have to produce in order to accomplish its mission of trimming the airplane (the moments issue). An airplane can be thought of as a kind of teeter-totter or see-saw, like that in Figure 1. You remember from your childhood that if

the two people are of different weights, the larger person must sit closer to the pivot

TRIMMING YOUR AIRPLANE

A few months ago we talked about calculating the angle of incidence for your wing. If you set the incidence too high, you will cruise nose low. If you set the incidence too low, then you will cruise nose high. It's time to look at the question of the horizontal tail incidence. The penalty for getting the tail incidence wrong is that you end up carrying around a lot of elevator deflection in cruise, which adds drag and could make you run out of elevator power when landing.

o

As you saw, it was pretty easy to calculate the wing CL, then correct for the sweep,

Mach number and Aspect Ratio. With those few quantities you were able to find the angle 36 AUGUST 1990

Figure 1

Figure 2 point in order to make the thing balance level. The principle involved is that the product of each person's weight times their distance to the pivot must be the same. In engineering parlance, the sum of the moments about the pivot must be zero. Remember that a moment is simply a force times a distance. If a 180 pound person sits 4 feet from the pivot, the moments at the pivot would be 720 foot-pounds. A person who weighed 90 pounds could then balance the teeter-totter by sitting 8 feet to the other side of the pivot, thereby generating -720 foot-pounds of moment at the pivot. The moments about the pivot would sum to zero. Your airplane's pivot point is, of course, its center of gravity. However, instead of having just two people parked on the teeter-totter, there are several forces being applied at different distances from the pivot. Not only are weights piled on the teeter-totter, but we have also tethered some helium-filled balloons along its length! The balloons represent lift forces being applied along the teetertotter. Our mission is to find the unknown force to be produced by the tail in order for this peculiar teeter-totter to balance. We already know the lever arm of the tail, which is simply the distance from its aerodynamic center to the center of gravity of the airplane. What we need to find out is: 1) how much lift does the tail need to balance the airplane, and 2) what angle of incidence do we need to produce this amount of lift? So before we can decide on the tail incidence angle, we first have to know how much lift the tail needs to produce to balance the airplane. We'll find out by analyzing all the moments generated around the center of gravity.

aerodynamic center of the wing. So our first task is to locate this circle along the Mean Aerodynamic Chord (MAC) of the wing. You have already computed the buttline for the MAC using the April spreadsheet (it's in cell E20). To find out where the aerodynamic center is along the MAC, you only need 3 quantities. The first is the pitching moment about the 1/4 chord when the airfoil is producing zero lift. The second is the lift coefficient at some small angle of attack (I used 4 degrees). The last is the pitching moment coefficient at that same angle of attack (again, I used 4 degrees). You get these values from the wind tunnel data plots like those in Theory of Wing Sections or computer predictions if you don't have wind tunnel data. The formula is then .25 - (CM, positive lift - CM, zero lift)/CL, positive lift. This formula is in this month's spreadsheet for you. For

my wing, the spreadsheet gives .2419 as the answer. This means that the aerodynamic center lies at 24.19% of the chord length of the MAC, which is where I've drawn the little circle in Figure 2. The answer for most airfoils is pretty close to the 25% of chord location, so if you can't get the numbers you need for this calculation, you can use 25% of chord without being too far off. Above the MAC airfoil are two horizontal circles, which mark the forward and aft center of gravity (C of G) locations for my airplane. These positions were calculated using the spreadsheet from May. I've drawn a line from the aerodynamic center to the forward C of G position. Imagine that you tear Figure 2 out of the magazine, then poke a pin through the center of the forward C of G location. If you then pulled on the arrowhead representing the lift, the picture would rotate nose down. If you pulled on the arrowhead representing the drag, the picture would also rotate nose down. So for the forward C of G case, both lift and drag cause a negative or nose down pitching moment. If you shift the pin to the aft C of G position, and pull on the lift arrow, the plane rotates nose up, while the drag still rotates the plane nose down. Note that if this was a high-wing airplane, the wing drag would rotate the nose up. In practice, you need to pick a center of gravity location which you consider 'Typical" for your airplane. I used the mid C of G position. Using the spreadsheet, of course, you can move the C of G later, and all the calculations will be redone automatically for the new location. This is an accurate physical picture of how your plane reacts in flight as well. The lift (positive or negative) from your tail is going to have to balance these pitching moments, so we need to calculate their values. You'll need to gather some numbers for your airplane in order to do this, so while you're getting the buttline for the MAC from the April spreadsheet, also write down the slope of the lift curve from cell E13, the wing

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MOMENTS DUE TO THE CENTER OF GRAVITY POSITION

Figure 2 is a plot of my airplane. On it I've drawn the cross-section of the airfoil at the buttline location of the Mean Aerodynamic Chord. Along its chord length is a circle. For computational purposes, the lift and drag forces of the wing can be considered to originate at this circle. This location is called the

Figure 3 SPORT AVIATION 37

The wing lift, in pounds, then is

x=o

dynamic pressure * wing area * lift coefficient.

The wing drag in pounds is found by

dynamic pressure * wing area ' drag coefficient. These 2 values give you the forces which will create a moment about the center of gravity Now you need their lever arms. FS means fuselage station and WL means

t

waterline: vertical lever arm = WL of aerodynamic center - WL of C of G horizontal lever arm = FS of C of G - FS of aerodynamic center

Be careful to do the subtractions in the order given, so that the lever arms have the proper sign (positive or negative). Then the moments generated by the lift of the wing are weight * horizontal lever arm, and the moments generated by the drag

of the wing are

34.410

pounds of drag " vertical lever arm. The spreadsheet will do all this for you, once you give it the information it needs.

TRAILING EDGE TO CENTER OF PANEL

PITCHING MOMENTS DUE TO THE WING AIRFOIL

TRAILING EDGE 99.794 TO AC OF HORIZTAIL

This is an easy one. You'll need the pitching moment coefficient (about the 1 /4 chord)

for the airfoil you're using, at the airplane's

lift coefficient you've chosen as "typical cruise." Again, you get this from the wind tunnel data or the computer predictions. If you're using a single airfoil from root to tip, this is easy. If, like me, you're using different airfoils at the root and tip (I change camber and thickness from root to tip), then you can use a graph. Draw a horizontal line to scale and mark the buttlines of the wing root and tip airfoils. On the vertical scale, put a dot

Figure 4

above the root airfoil location representing

X=L

area from B9, the root and tip chords from B18 and 19, the wingspan from B10, the mean aerodynamic chord from E19, and the lift coefficient from E9. Make sure the case you're running in the April spreadsheet represents a "typical" cruise condition. There's not much point in optimizing the tail incidence for some weird weight and speed! For some reason the dummy who wrote the April spreadsheet didn't print out the dynamic pressure, called "q", anywhere. So

A I B I SPREADSHEET NUMBER 6 FROM SPORT AVIATION 6/90 JGR

ifa s s

7

9

9 10 11 12 1? 14 19 11 17 1*

while you've still got it on your screen, enter

the title "dynamic pressure, q:" into cell A13. Then enter the formula " = .5 * E8 * (B5 *

19 20

1.467) "2" into cell B13. Lotus 1-2-3 users

don't need the " =" at the beginning of the formula, and nobody should enter the quotation marks. The alternative is to get the dynamic pressure from the March spreadsheet, using the same weight, speed and altitude. You also need to estimate the drag coefficient of your wing at the cruise condition

you've picked. I used the basic airfoil drag

coefficient, then added 10% more drag, because I'm sawing slots in the wing to accom-

modate the ailerons and flaps, and these holes aren't free. In fact, 10% may not be enough. You get the drag coefficient corresponding to the cruise lift coefficient from the

wind tunnel data or computer predictions, as before. 38 AUGUST 1990

the pitching moment of the root, and above

,

22 23 24 25 26 27 28 2« 30 31 32 33 34

15

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•"CALCULATION OF THE AERODYNAMIC CENTE R ALONG THE MAC"

ZERO LFT PITCHNG MOMENT JL FOR ALPHA-4 JM FOR AIPHA.4

0 02568 0 7275 0 03158

! I !

.025.(D8-D6)/D7

AEROOYNAMC CENTER: —PITCHING MOMENTS DUE TO THE FUSELAGEWING COMBINATION"' STRIP NUMBER WIDTH (IN) STRIPS IN FRONT OF WING 1;

HEIGHT (IK )

36.111:

2l 42.7621 3\ 45.62* STRIP JUST AHEAD OF LEADING EDGE 4 47 492 STRIPS BEHIND TRAILING EDGE: 5 42.8971 C 2».64d 7 20.021] 8 13884 9 9118

10 15.0 16.000! 180 >0j

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57 25C 40 750 22589

50j 13.7

6.875

21.4 10 26.0 )0 10 28.0 r 3 0 0 >0 29828

10705 34 410 61 410 90410 120324

CHORD N FUSELAGE ; DISTANCE FROM TRAIL EDGE TO AC OF TAl: WING ROOT CHORD (N): " " WING TIP CHORD (IN): WING AREA (SO FT) WING SPAN (FT)

DISTANCE (Ml

i

______

56840 99794 59 405 22 000 98 120 30666

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151 043

DISTANCE FROM WING AC TO TAIL AC (IN):

36 DESIGN LIFT COEFFICIENT 3 7 WING LIFT CURVE SLOPE OCL/dALPHA 3 » DYNAMIC PRESSURE q AT DESIGN POINT ~39~l .................... "t 4 0 MEAN AERODYNAMIC CHORD (M) I 4 1 DOWNWASH AT TAIL (DEGREES): T 42 dEPSILONWALPHA 43 44 PITCHING MOMENTS DOE TO FUSELAGE: 45 IdCM/dCL ol lus»lao«:

0 185 0 0909 104 S3

.2/3'D3r(1.|D32/D31)»(D32;D31) A 2)/(l.(D32/D31)) -2~0''D36'(D31/D32)"0 3;(D34-2/D33)*0 725'(3'D40/D35)'0 25 -O41/D36-D37

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DIST/CHORO .dBETA/dALPHA

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GET FROM APRIL SPREADSHEET IN SPORT AVIATION i GET FROM MARCH SPREADSHEET IN SPORT AVIATION

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40

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NOTE: IF TAIL IS MORE THAN 50% OF MAC I ABOVE OR BELOW WING, CHANGE .20 TO .18 • • • • • • • • . _. FOOT-POUNDS _________

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Figure 6 the tip airfoil location put a dot to scale representing the tip moment coefficient. Connect the two pitching moment dots with a straight line. Then find the buttline for the MAC, and find the pitching moment on the line corresponding to this buttline.

The pitching moments about the aerodynamic center of your wing are then: dynamic pressure ' wing area * mean aerodynamic chord (in feet)' moment coefficient. The spreadsheet will do this for you also. THE PITCHING MOMENTS DUE TO THE FUSELAGE

I've saved the best until last. The pitching moments of the fuselage are the hardest to

get a handle on. Ahead of the wing there is upwash, which is quite strong just ahead of

the wing. Behind the wing is downwash, because the wing is throwing air at the ground.

The fuselage by itself is pretty much like an airfoil, and its center of lift would normally be located at about 25% of its length. However, plug a wing onto it and the upwash and downwash affect the fuselage strongly, and move its center of lift. While there are some simple formulas which have been offered to predict the moments of the fuselage, I've opted to use the more accurate but tedious method presented in Perkins & Hage, and also reproduced in Roskam's book. Even this method isn't completely accurate, but the only alternative I know of is an obscure

from a Pterodactyl. These are hard to find even in the Oshkosh Fly Market! So we'll do it the hard way. Figure 3 is a picture of my homebuilt. On it I've marked the aerodynamic centers of the wing and tail with circles. The buttlines locating the Mean Aerodynamic Chords of the wing and tail were computed using the April spreadsheet. The position of the aerodynamic center along the wing MAC was calculated using the formula used in this month's spreadsheet. The tail's aerodynamic center is at 25% of chord, because it is a symmetrical airfoil. As Figure 3 shows, you need the distance (in inches) from the fuselage station of the wing's aerodynamic center to that of the tail.

57

Figure 4 shows you the rest of your task. Draw lines across the fuselage at the wing leading and trailing J_ edges. Measure the distance be•PITCHING MOMENTS DUE TO WING AIRFOIL"' tween these two lines, which on Figure 3 is called the Reference 0.02394! CM OF MAC AIRFOIL AT THE DESIGN CL Chord. You'll eventually enter this 'PITCHING MOMENTS DUE TO WING: .D3 8 *033;* D4.6/121* D4'$ 'FOOT-POUNDS number into cell D28 on this —PITCHING MOMENTS DUE TO CENTER OF GRAVITY' month's spreadsheet. Ahead of the wing, divide the fuFS OF CENTER OF GRAVITY (INCHES): 1 06.742! selage into four strips, which I've ........~... WATERLINE OF CENTER OF GRAVITY (INCHES): numbered 1 to 4. Behind the wing, r FSOF AERODYNAMIC CENTER (INCHES): divide the fuselage into 5 strips, •• • WATERLINE OF AERODYNAMIC CENTER (INCHES):

58

WING DRAG COEFFICIENT:

47 48 49

52 53 54 55 56

Voodoo ritual which involves some feathers

which I've numbered 5 through 9.

59

60 61

62 63 64 65

WING DRAG, POUNDS:! ! i »D58*D38'D33! WING LIFT, POUNDS: '""""" ••--""-••|"~"——""--y••••••———— zD38.D33.D36! PITCHING MOMENTS DUE TO WING LIFT:•••••••-————!————— .D61 •(054.056): FOOT-POUNDS" PITCHING MOMENTS DUE TO WING DRAG: ' -D60'(D57-D55) FOOT-POUNDS TOTAL MOMENTS AEOUT THE C OF G DUE TO WING: """"""""""""""".050+062+063! FOOT-POUNDS

MOMENT COEFFICIENT CM.cg wing:__________'

Figure 7

=b64/(D3"8'D33'D40/12):

They don't have to be the same size. For the four strips ahead of the wing, find the distance from the wing's leading edge to the center of each panel. For the five aft strips, find the distance from the trailing edge to the center of each

panel.

SPORT AVIATION 39

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SPREADSHEET NUMBER 6 FROM SPORT AVIATION 6/90 JGR

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STRIP NUMBER WIDTH (IN) : HEIGHT (IN) DISTANCE (IN) DIST/CHORD dBETA/dALPH^CONTRIBUTKDN j STRIPS IN FRONT OF WING: . ! ; 15 .000 1j 36.111! 57.250! 1.01! 1.351 15.31? I 2; 42.7621 18 .000 40.750! 1.45] . . 0.72! 27.571 j. r 3 45.829 18 "666 22.589]. 0.40] 1.64! 35.81! i ! STRIP JUST AHEAD OF LEADING EDGE: .........................}... 13 .750! 6.875! 6.12 4 47492! 83.67! I i "1 STRIPS BEHIND TRAILING EDGE: 5 42.897: 0.06! 21 .410! 10.705! 1.32 0.19! 6! 29.640 26 .OOO' 34 410! 0.19! 2.47! 0.61! : 28 .ooo 7! 26.621; 6 1.4 1'6T 6.331 2. 16! """'" '""' t """" """""""" 30 .000 8] 13.884! 90.410-: •••••••- ""TMT" 0.49] 1.64! f 9 9.118 29 •828: 120,324]. 0.94! ...2...12; 0.65! 1 i i 1 1 1 T 56.840! CHORD IN FUSELAGE: ! i i I j • 99.794! DISTANCE FROM TRAIL. EDGE TO AC OF TAIL: ...... ——— ............. !

I

WING ROOT CHORD (IN):

59.405!

WING TIP CHORD (IN): [WING AREA (SO FT): [WING SPAN (FT) DISTANCE FROM WING AC TO TAIL AC (IN): DESIGN LIFT COEFFICIENT:

22.000: 98.120

30.666! 151.043! I

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T 0.934 NOTE: IF TAIL IS MORE THAN 50% OF MAC ABOVE OR BELOW WING. CHANGE -20 TO -18 ........................ 0.45886

DOWNWASH AT TAIL (DEGREES):

dEPSILON/dALPHA:

•"I:::....::]::: i ]

0 185 0 0909 GET FROM APRIL SPREADSHEET IN SPORT AVIATION ,...-..... ^ 1 0 4 5 3 GET FROM MARCH SPREADSHEET IN SPORT AVIATION

3 7 WING LIFT CURVE SLOPE dCL/dALPHA: 3 8 DYNAMIC PRESSURE q AT DESIGN POINT: 39 I 40 MEAN AERODYNAMIC CHORD (IN): 41 42 43 44 45

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10 AERODYNAMIC CENTER: 0.2419! CHORD 11 I 1 2 —PITCHING MOMENTS DUE TO THE FUSELAGE>WING COMBINATION'"

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6 7 8 9

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PITCHING MOMENTS DUE TO FUSELAGE:

992.49 FOOT-POUNDS 0.14408

dCM/dCL of fuselage:

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Figure 8 Now measure the width of the fuselage at the center of each panel, 1 through 9. Next, measure the height of each strip. Finally, measure the distance from the wing's trailing edge to the aerodynamic center of the tail, as shown. Enter these values into the spreadsheet, which will then spit out the pitching moments of the fuselage. The procedure calls for looking up values on a graph made by H. Multhopp at the NACA in 1941. In order to save you this grief, I fit polynomial curves to the curves in this graph. You won't thank me after you've seen the formulas you've got to type in column F of this month's spreadsheet! One of the

47 48 49 50 51 52

53 54 55 56 57 58 59 60 61 62 63 64 65

curves had to be fit in two sections, because of its weird shape, making the formulas that much worse. The upwash just ahead of the wing is very strong, and uses a different curve on e graph. This curve was easier to fit. The panels behind the wing use percentages of the downwash at the tail, so this had to be calculated as well. While the spreadsheet reports the downwash at the tail, we'll leave our discussion of downwash until next time. MAKING THE SPREADSHEET

Because of the size of this month's

A I B I C I —PITCHING MOMENTS DUE TO WING AIRFOIL"' ;

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spreadsheet, I printed it out in pieces. Start by typing in the titles, formulas, and sample

numbers in columns A-D, rows 1-45 from

Figure 5. Then continue with Figure 6, for columns E and F, rows 1 -45. The formula in cell F16 can be copied fo F17 and F18, so you don't have to type this monster 3 times! You can type the formula into cell F22, then copy it to cells F23 to F25 as well.

Finally, copy everything from Figure 7 into columns A-E, rows 47-65. If you copied everything correctly, your spreadsheet should match the one in Figures 8 and 9. Next month we will continue our quest for the mysterious tail incidence. Hope to see you at Oshkosh '90!

E

:

CM OF MAC AIRFOIL AT THE DESIGN CL PITCHING MOMENTS DUE TO WING: i

—PITCHING MOMENTS DUE TO CENTER OF GRAV TY— FS OF CENTER OF GRAVITY (INCHES): WATERLINE OF CENTER OF GRAVITY (INCHES): FS OF AERODYNAMIC CENTER (INCHES): WATERLINE OF AERODYNAMIC CENTER (If JCHES): WING DRAG COEFFICIENT: WING DRAG, POUNDSI WING LIFT, POUNDS: ! PITCHING MOMENTS DUE TO WING LIFT: PITCHING MOMENTS DUE TO WING DRAG: TOTAL MOMENTS AFJOUT THE C OF G DUE TO WING: j MOMENT COEFFICIENT CM.cg wing:

Figure 9 40 AUGUST 1990

0.02394! 891.431 FOOT-POUNDS

!

REFERENCES

V

i 06.7421 -7.520; 107.350! -16.239!

6. obss;

56.411 1897; FOOT-POUNDS -1154.19 -491.83 FOOT-POUNDS -754.59 ^COT-POUNDS -0.020265;

Airplane Performance, Stability, and Control, Parkins, Court/and D. and Hage, Robert

E., John Wiley& Sons, Inc., New York, 1949. Airplane Flight Dynamics and Automatic Flight Controls, Roskam, Jan, Roskam Aviation and Engineering Corporation, Route 4, Box 174, Ottawa, Kansas 66067.

Part 2

Last month we looked at some of the forces which create pitching moments around your airplane's center of gravity. We computed the moments caused by the wing's lift and drag, the marriage of the wing and the fuselage, and the wing airfoil's pitching moment. Hopefully you survived last month's spreadsheet. This month we are going to consider the effects of the propeller on the pitching moments. Then we will look at the local weather in the vicinity of the tail, to find out the wind speed and direction. PROPELLER NORMAL FORCE

When your plane is in cruising flight, the wing must produce lift, and in order to make lift it needs to be at some positive angle of attack. While it's true that a cambered airfoil produced lift at zero or even negative angles of attack relative to its own chord line, it's still considered to be at a positive angle of attack relative to its zero lift line. The zero lift line is the angle of attack at which the airfoil produces zero lift. A wing producing lift has upwash ahead of the leading edge, and downwash behind the trailing edge. The air moves up ahead of the wing because the wing creates a pressure wave out ahead of it, which travels at the speed of sound. This means that if the plane is supersonic, the plane gets there before the air has a chance to get out of the way. The result is a shock wave and the familiar sonic boom. Assuming that your plane is subsonic, the propeller is also at some angle of attack, because

by JOHN G. RONCZ, EAA 112811

15450 Hunting Ridge Tr. Granger, IN 46530-9093

it is in this upwash field. This makes the propeller behave like it were a small wing, generating lift. To model it we need to tether a helium-filled balloon to one end of the see-saw. This propeller lift is called the propeller normal force. The word "normal" has nothing to do with its psychological or physical health. Airplane engineers use the word "normal" to mean at right angles to something; in this case, the propeller is producing lift at a 90 degree angle to its thrust line. The propeller's lift or normal force can be very considerable, particularly on little airplanes with great big motors - the kind EAAers drool over. The worst case is when you are on short final, at low airspeed, and a group of those darn kangaroos hop onto the runway, as they so often do, and you have to apply full power for a go-around. In this case the propeller normal force can actually cause the airplane to pitch up and stall. Therefore, it is useful to know how much normal force the propeller is producing. Dommasch suggests the following equation to calculate the coefficient of the propeller's normal force: CNP = n * F T 2 / ( 2 ' V * S ) * B * b b * a *

(2*TT*pb'V/(0*R)-(1 +TC'/4*S/ (IT * R-2) ) + Cd)

That equation might send you run-

ning for cover. Yet like most things which seem a bit intimidating at first glance, this one is a piece of cake once you wade through the hieroglyphs. To solve this equation, first you make a list of the things you need to know: • The propeller diameter • The rpm you're using • The number of blades • The average width of each blade • The equivalent flat plate drag area of your airplane • The speed of your airplane • The wing area of your airplane • The lift coefficient of your airplane • The lift curve slope of the wing The rather pretty Greek letter ft (Omega) represents the rotation rate of the propeller in radians per second. Take the propeller rpm you're using, and divide by 60 to get the number of revolutions per second. I'll use 2400 rpm, so the prop is turning 2400/60 = 40 times each second. Since there are 2tr radians in each circle, we multiply 40 by 6.2831853 and get 251.33 radians per second. The diameter of my prop is 78 inches, or 6.5 feet. The radius is half that, or 3.25 feet, which is the value of the letter R in the equation. The letter V is the speed of the airplane in feet per second. My homebuilt should cruise at 213 mph; to convert to feet per second, multiply by 1.467, giving 312.47 feet per second. The letter S means wing area, which for my bird is 98.12 square feet. The letter B is the number of blades, which is 2. The b-sub-b combination bb means the width of the blade. I don't know, but SPORT AVIATION 35

"D

27.0°

GO TJ

m m

o

SPEED DUE TO PROPELLER ROTATION FIGURE 1

5 inches is a good guess. That's .417 feet, because we can't mix feet and inches in this equation! The next bit of Greek is a (alpha), which is the angle of attack of the propeller. According to Perkins & Hage, the angle of attack of the propeller is always larger than that of the wing, because the tractor propeller lives in the upwash of the wing. The wing's angle of attack is simply the airplane's lift coefficient divided by the wing's lift-curve slope (the amount of CL it produces for each degree angle of attack). If my plane's cruise CL is .185, and my wing's liftcurve slope is .0909 per degree, then my wing is at an angle of attack of .185/ .0909 or 2.04 degrees from its zero-lift line. This leaves the question of how much to add to account for the wing upwash. The spreadsheet from August fortunately (or unfortunately) made you slice your fuselage into strips. Then it calculated the upwash for each strip. Since the propeller is out ahead of the fuselage, the upwash is a bit smaller than for the first fuselage strip, say about 90% of that value. The value for my first fuselage strip was 1.35, so 90% of that is 1.215, or 21.5% more than the angle of attack of the wing. Multiplying 2.04 degrees by the upwash factor of 1.215 gives us 2.53 degrees, which I'll assume is close to the angle of attack of my propeller in cruise. This needs to be converted to radians, so we multiply by iT/180, which gives .04409 radians. This brings us to pb, which is the blade pitch setting at 75% of the length of one blade. Rather than running around with a protractor, trying to measure the blade angle, which is pretty useless on a constant-speed prop anyway, we'll cheat a little here. If my prop's

distance this airfoil travels in one second, I can multiply this circumference by the number of revolutions per second the propeller is turning. Our friend Omega O conveniently contains both 2-n and the number of revolutions per second. So all I need to do is multiply O. by 2.4375. For my plane, that's 251.33 * 2.4375, or 612.62 feet. This

means that even when the plane is standing still, with the prop spinning at 2400 rpm, the airfoil at 75% of the blade length is traveling at 417 miles per hour (612.62/1.467)! Hopefully in cruise the plane is not standing still, however, but is going 213 miles per hour. That's 312.47 feet per second (213 * 1.467). This gives us a wind triangle (Figure 1), whose base is 612.62 feet (the wind speed coming from the prop's own rotation), and whose height is 312.47 feet (the wind coming from the forward speed of the airplane). Resolving the

wind triangle, we find that the angle of the relative wind is 27 degrees. You compute that by taking the arctangent of (312.47/612.62). In order to make the propeller airfoil at that location make positive lift, we are going to add 4.5 degrees to that. This 4.5 degrees compensates for the downwash coming from the blade, and adds some positive incidence. So pb is 27 + 4.5, or 31.5 degrees. This would be the twist of the propeller at 213 mph and 2400 rpm, for the slice of blade which is 2.4375 feet from the center of the hub. This angle needs to be in radians rather than degrees, so 31.5 * -iT/180 gives .54978 radians. The next item in the equation we need to find is the quantity Tc'. Dommasch defines this as the propeller thrust/(wing area ' dynamic pressure). Well, as we all learned in Private pilot ground school, in level unaccelerated flight the thrust has to equal the drag. In the March spreadsheet, you calculated the equivalent barn door area of your homebuilt. We'll just use the drag of the plane, and assume the thrust is the same as the drag. This makes Tc' equal to airplane drag/(wing area ' dynamic pressure), which simplifies to flat plate drag/wing area. The equivalent flat plate drag of my homebuilt is (hopefully) 1.95 square feet, and dividing by 98.12 gives .0199, which is the value of Tc'. The last missing item is cd, which represents the propeller's profile drag coefficient. Profile drag is the drag which comes from skin friction and any flown separation (stalling). Dommasch suggests a value of .02 for this, so we'll go with his

IF C OF G IS ABOVE THRUST LINE, NOSE PITCHES UP WITH POWER

radius is 3.25 feet, then 75% of that is 2.4375 feet. The propeller airfoil that

lives 2.4375 feet from the hub sweeps out a circle once per revolution. Since the circumference of a circle is 2irr, the circumference at this distance from the

hub is 2 * TI * 2.4375. To get the total 36 SEPTEMBER 1990

IF C OF G IS BELOW THRUST LINE, NOSE PITCHES DOWN WITH POWER

FIGURE 2

down. If, on the other hand, the prop's thrust line is below the center of gravity, adding power is going to pitch the plane nose up. A lot of planes have the engine tilted down between 1 and 2 degrees, in order to place the thrust line above the center of gravity, in order to avoid a pitchup with power. It is simple to calculate the moments

about the center of gravity caused by the position of the thrust line. First you measure the vertical distance from the thrust line to the center of gravity, as shown for my homebuilt in Figure 3.

FIGURE 3

suggestion. Having made a list of all the unknown quantities this equation needs to munch, we can proceed to solve the beast: Start with the innermost parentheses: 2* V S becomes 2'312.47'98.12 = 61319.113 II ' R becomes 251.33 * 3.25 = 816.6225 TT * FT2 becomes 3.14159 * 3.25"2 = 33.1831

Looking at the remaining parentheses leaves: (2* T T * pb* V/816.6225* (1 + T c ' / 4 * S/33.1831) + cd)

The innermost set becomes: 1 + .0199/4*98.12/33.1831 = 1.0147

Then we do the outermost set:

(2

* 3.14159

*

.54978 * 312.47/

816.6225 * 1.0147 + .02) = 1.3612

Now we kill off the rest:

fl * R"2/61319.113 * B * b b * « * 1.3612 becomes 251.33 *3.25'2/61319.113* 2 * .417 * .04409 * 1.3612 = .002167

gravity. Since this is a positive number, the propeller's normal force will pitch the airplane nose up.

The spreadsheet for this article will do all this math for you, of course. I've taken the space to write all this out for those of you who have written asking for a bit more help because you're doing this by hand. I also want to demonstrate how you take a rather complicated equation and break it down in steps. Airplane design has a lot of these, and they are not beyond your ability or intelligence, once you overcome your fear of funnylooking symbols. THRUST LINE EFFECTS ON PITCHING MOMENTS

The propeller's axis of rotation establishes a direction for its thrust. This thrust line can be above the center of gravity or below it (Figure 2). If the thrust line is above the C of G, then adding power is going to pitch the plane nose

Find the waterline of the center of gravity, then subtract the waterline of the thrust line. If the thrust line is above the C of G, you will get a negative number, otherwise you will get a positive number. Then you multiply by the propeller thrust. If the thrust line is above the C of G, this results in negative pitching moments, meaning that the thrust will try to pull the nose down. If my homebuilt in cruise produces 203.83 pounds of thrust, the waterline of the C of G is -7.52, and the waterline of the thrust line is 0.0, then the thrust line's lever arm is - 7.52 - 0.0, which is -7.52. Converting to feet gives -.6267

feet, which times 203.83 pounds of thrust would generate -127.73 footpounds of moment. Since the moments are negative, my plane will pitch nose down when power is applied. OTHER THRUST VALUES

Sometimes it's useful to evaluate the pitching moments due to thrust at power settings which are causing the airplane to accelerate. Takeoff and climb are obvious examples of this. In this case the thrust values built into the spreadsheet are wrong, because if the airplane is climbing or accelerating, then you have more thrust than drag. Rather than leave you out in the cold, I've added a section to the spreadsheet which will estimate the thrust in pounds for you for any condition. Normally in order to do this, you

This is the coefficient of propeller normal force. To find the prop's lift in pounds, we multiply this coefficient by the dynamic pressure and by the wing area. You calculated the dynamic pressure in the March spreadsheet, and for

my bird in cruise I'll use 104.53 pounds per square foot. My wing area is 98.12 square feet. So the propeller normal force is .002167 * 104.53 * 98.12, or 22.23 pounds. Next you need the distance from the prop to the center of gravity location which you're using to set your trim. For my plane it's 75 inches, which is 6.25 feet. The moment, then, in foot-pounds is 6.25 * 22.23, or

138.9 foot-pounds, about the center of

WING CXDWNWASH PUTS TAIL AT NEGATIVE ANGLE OF ATTACK

FIGURE 4 SPORT AVIATION 37

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A '"PROPELLER NORMAL FORCE*"

B

c

PROPELLER RPM: PROP DIAMETER (INCHES) : AVERAGE BLADE WIDTH (INCHES): AIRPLANE FLAT PLATE DRAG (SQUARE FEET): AIRPLANE SPEED (MILES PER HOUR): NUMBER OF BLADES: DISTANCE FROM PROP TO C OF G (INCHES):

PROPELLER ANGLE OF ATTACK (DEG): BLADE Pit CH ANGLE AT 75% OF RADIUS: PROPELLER THRUST (POUNDS): ROTATION SPEED (RADIANS/SEC): fc PRIME: COEFFICIENT OF PROP NORMAL FORCE PROPELLER NORMAL FORCE (POUNDS): PITCHING MOMENTS DUE TO PROP: "•Pit CHING MOMENTS DUE TO PROP THRUST LINE"* WATERLINE OF THRUST LJNE AT FS OF C OF G: LEVER ARM OF THRUST UNE: PitCHING MOMENTS DUE TO THRUST:

•••PROPELLER THRUST FOR OTHER CONDltiONS*" TRUE AIRSPEED (MILES PER HOUR): HC>RSEFOWER: PRO'p'EFFiciENCY:

THRUST:

•••AIRSPEED AT THE tAIL*** TAIL DYNAMIC PRESSURE: FIGURE 5

would need to know the propeller efficiency over a large range of speeds and powers. This is difficult to get. What I've done for you is to design a constantspeed propeller for my 180 hp engine, and then calculate what its efficiency would be at various airspeeds, assuming full power. Then I took these values and fit a polynomial curve to them. If

you don't know what that means don't

worry, because you don't need to. The expression I use is:

prop efficiency = .1220655 + .01223503 ' Vkts - .00006986597 * Vkts"2 + .00000013532* Vkts"3

The symbol Vkts means your true airspeed in knots. The thrust that your

engine + propeller duces is

combination

38 SEPTEMBER 1990

DOWNWASH AT THE TAIL

The first local weather question is to find the local wind direction at the tail. This is a difficult question, because the answer depends on how the wing itself is loaded spanwise (it should be semielliptical, remember?), the shape of the fuselage, the shape of the tail, and the spacing between all these parts. Dommasch offers an empirical equation for finding the downwash angle at the centerline of the wing wake. "Empirical" means he has looked at a lot of wind tunnel tests and wrote an equation which fits the results with some degree of accuracy. Sounds OK to me! He gives this equation as: e = 20* CL* X'O.S/AR".725*

prop efficiency is .36, or only 36%. It's true, however. While there's no guarantee that your propeller's efficiency matches the one I used to write this efficiency formula, it's better than nothing and probably is pretty close. For fixedpitch propellers, you could subtract .03 or 3% from the efficiency in cruise. To use this part of the spreadsheet, just type in the speed and horsepower, and the spreadsheet will spit out the thrust in pounds for full power at that speed. You then type the thrust value into cell D79, and the pitching moments due to thrust will be recomputed for you using this new value.

pro-

Thrust (in pounds) = Horsepower * prop efficiency * 550/true airspeed in feet per second. When you try this out in the spreadsheet (or by hand calculator) you may be surprised to see that at 25 mph the

make thrust, and the wing is happily grabbing air and accelerating it gently towards the ground, which obliges by pushing the airplane away and keeping it at a constant altitude. All's right with the world. The horizontal tail is happy to be flying along with you, keeping everything in balance. Yet the conditions at the tail are different. The propeller thrust has increased the wind speed over the tail, and the wing's downwash has changed the direction from which the wind blows over the tail. The angle of attack of any wing is the angle between its chord line and the relative wind. To get zero lift on a symmetrical wing, like the horizontal tail, you line it up with the relative wind. Because of the wing's downwash, the relative wind at the tail is coming from above. So if you mount the horizontal tail at zero incidence, the tail will make negative lift, because it will have a negative angle of attack (Figure 4).

WEATHER CONDITIONS AT THE TAIL

You are cruising along above the weather, the sky is blue, the engine is humming, the propeller is happily grabbing air and accelerating it rearwards to

(3 * c/l)~.25 degrees where e, (the Greek letter epsilon) stands for the downwash angle we're looking for, CL is the lift coefficient of the wing, \ (the Greek letter lambda) is defined as the root chord divided by the tip chord (he says to use 1.5 if you have an elliptical wing, like a Spitfire replica), AR is the aspect ratio of the wing, which is the square of its span divided by its area, c stands for the Mean Aerodynamic Chord of the wing, and I is the

distance from aerodynamic center of the wing to the aerodynamic center of the tail. If I check out my homebuilt at another cruise condition, the CL is .206, the root chord is 59.402 inches and the tip chord

is 22 inches, making X 2.7, the aspect ratio AR is 9.58, the mean aerodynamic chord is 43.57 inches, and the tail lever arm is 258.39 inches. The equation

then becomes

e = 20 * .206 * 2.7-Q.3 / 9.58'.725 *

(3 * 43.57 / 258.39)'.25

67

68 69 70 71 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

100 101 102 103 104

2400

' 78.po: INCHES'""" """ s.pa INCHES"""""""" i.95 SQUARE FEET"

"213!bpiMl£SPERHOUR 2" "" 75Vda INCHES -D36/D37-0!9'F16 'DEGREES -(«Un(p73'1.467/(2'plO'069/80-Q,75'D70/24))