# Category Dynamics of Flight

## Data for the Boeing 747-100

The Boeing 747 is a highly successful, large, four-engined turbofan transport aircraft. The model 100 first entered service in January 1970, and since then it has continued to be developed through a series of models and special versions. As of May 1990, only versions of the model 400 were being marketed. By the year 1994, close to 800 Boeing 747s were in operation around the world, and the aircraft was still in produc­tion.

The data for the Boeing 747-100 contained in this appendix are based on Heffley and Jewell (1972). A three-view drawing of the aircraft is given in Fig. E. l. A body axis system FB is located with origin at the CG and its x-axis along the fuselage refer­ence line (FRL). The CG is located at 0.25 c (i. e., h = 0.25), and this is the location that applies for the tabulated data. The thrust line (TL) makes an angle of 2.5° with respect to the FRL as shown.

Three flight cases are documented in the data tables. They all represent straight and level steady-state flight at a fixed altitude. Case I has the aircraft in its landing configuration with 30° flaps, landing gear down, and an airspeed 20% above the stalling speed. Cases II and III represent two cruising states with the flaps retracted and the gear up.

The data in Table E. l define the flight conditions that apply to the three cases. It should be noted that the moments and product of inertia are given relative to the body frame FB shown in Fig. E. l. Flere the weight and inertias for Case I are smaller than those for the other two cases because the amount of fuel on board during landing is less than that during the cruise. If the data are to be applied to a reference frame dif­ferent from FB (e. g., to stability axes Fs) then the given inertias will have to be trans­formed according to (B.12,3). Note that FB can be rotated into Fs by a single rotation of f about the у-axis. Values for f are contained in Table E. 1.

The dimensional derivatives corresponding to FB of Fig. E. 1 are contained in Ta­bles E.2 to E.4. Since FB can be rotated into the stability axes Fs by a single rotation of £ about the у-axis, it follows that the transformations of (B. 12,6 and B. 12,7) can be used to obtain the derivatives corresponding to Fs. Values for £ are contained in Table E. l.

Table E. l

Boeing 747-100 Data

(S = 5,500 ft2, b = 195.68 ft, c = 27.31 ft, h = 0.25)

 Case I Case II Case III Altitude (ft) 0 20,000 40,000 M 0.2 0.5 0.9 V (ft/s) 221 518 871 W(lb) 5.640 X 105 6.366 X 105 6.366 X 105 Ix (slug-ft2) 1.42 X 107 1.82 X 107 1.82 X 107 /,, (slug-ft2) 3.23 X 107 3.31 X 107 3.31 X 107 h (slug-ft2) 4.54 X 107 4.97 X 107 4.97 X 107 (Slug-ft2) 8.70 X 105 9.70 X 105 9.70 X 105 £ (degrees) -8.5 -6.8 -2.4 CD 0.263 0.040 0.043

Table E.2

Boeing 747-100 Dimensional Derivatives

Case I (M = 0.2) Longitudinal

 X(lb) Z(lb) M(fflb) и (ft/s) -3.661 x 102 -3.538 X 103 3.779 X 103 w (ft/s) 2.137 X 103 -8.969 X 103 -5.717 X 104 q (rad/s) 0 -1.090 X 105 -1.153 X 107 w (ft/s2) 0 5.851 X 102 -7.946 X 103 8,(rad) 1.680 X 104 -1.125 X 105 -1.221 X 107

Lateral

 У (lb) L(ft-lb) N(fflb) v (ft/s) -1.559 X 103 -8.612 X 104 3.975 X 104 p (rad/s) 0 -1.370 X 107 -6.688 X 106 r (rad/s) 0 4.832 X 106 -1.014 X 107 8a (rad) 0 -3.200 X 106 -1.001 X 106 8r(rad) 5.729 X 104 1.034 X 106 -6.911 X 106

Table E.3

Boeing 747-100 Dimensional Derivatives

Case II (M = 0.5) Longitudinal

 X(lb) Z(lb) M(fflb) и (ft/s) -4.883 X 10′ -1.342 X 103 8.176 X 103 w (ft/s) 1.546 X 103 -8.561 X 103 -5.627 X 104 q (rad/s) 0 -1.263 X 105 -1.394 X 107 w (ft/s2) 0 3.104 X 102 -4.138 X 103 Se (rad) 3.994 X 104 -3.341 X 105 -3.608 X 107

Lateral

 У (lb) Lift-lb) N(ft-lb) v (ft/s) -1.625 X 103 -7.281 X 104 4.404 X 104 p (rad/s) 0 -1.180 X 107 -2.852 X 106 r (rad/s) 0 6.979 X 106 -7.323 X 106 Sa (rad) 0 -2.312 X 106 -7.555 X 105 Sr(rad) 1.342 X 105 3.073 X 106 -1.958 X 107

Table E.4

Boeing 747-100 Dimensional Derivatives

Case III (M = 0.9) Longitudinal

 X(lb) Z(lb) M(ft-lb) и (ft/s) -3.954 X 102 -8.383 X 102 -2.062 X 103 w (ft/s) 3.144 X 102 -7.928 X 103 -6.289 X 104 q (rad/s) 0 -1.327 X 105 -1.327 X 107 w (ft/s2) 0 1.214 X 102 -5.296 X 103 Se (rad) 1.544 X 104 -3.677 X 105 -4.038 X 107

Lateral

 У (lb) L(ft-lb) N(ft-lb) v (ft/s) -1.198 X 103 -2.866 X 104 5.688 X 104 p (rad/s) 0 -8.357 X 106 -5.864 X 105 r (rad/s) 0 5.233 X 106 -7.279 X 106 Sa (rad) 0 -3.391 X 106 4.841 X 105 S, (rad) 7.990 X 104 2.249 X 106 -2.206 X 107

[1]An excellent account of the early history is given in the 1970 von Karman Lecture by Perkins (1970).

[2]It is also possible to speak of the stability of a transient with prescribed initial condition.

[3]For a more complete discussion, see AGARD (1959); Stevens and Lewis (1992).

[4]This word describes the position of movable elements of the airplane—for example, landing con­figuration means that landing flaps and undercarriage are down, climb configuration means that landing gear is up, and flaps are at take-off position, and so forth.

[5]When partial derivatives are taken in the following equations with respect to one of these variables, for example, ЭCJda, it is to be understood that all the others are held constant.

[6]The notation h„w indicates that the mean aerodynamic center of the wing is also the neutral point of the wing. Neutral point is defined in Sec. 2.3.

[7]Equivalent airspeed (EAS) is VE = vVpfpa where p0 is standard sea-level density.

[8]Note that a is still the angle of attack of the zero-lift line of the basic configuration, and that the lift with flap deflected is not zero at zero a.

[9] We may neglect as well all the derivatives of the symmetric forces and mo­ments with respect to the asymmetric motion variables.

[10] We may neglect all derivatives with respect to rates of change of motion vari­ables except for ZH, and Mlv.

[11]For two-dimensional incompressible flow, the area 5(f) diverges as t —» °°. That is, the derivative concept is definitely not applicable to that case.

[12]Exactly for supersonic wings, and approximately for subsonic wings.

[13]See bibliography.

[14]Rodden and Giesing (1970) have extended and generalized this method. In particular they give re­sults for finite wings.

[15]For the effect of the wing at low speeds, see Campbell and McKinney (1952).

[16]Exactly, in an isothermal atmosphere of uniform composition; approximately, in the real atmos­phere.

[17]The De Havilland Buffalo airplane.

[18]Program CC, see Appendix A.7.

[19]Based on the Piper Cherokee. The control derivatives were taken from McCormick (1979). We es­timated the stability derivatives. The numerical values used may not truly represent this airplane.

[20]Neglecting the fact that the pilot and indicator are not right at the CG.

[21]The term root locus is used throughout this chapter with the meaning ordinarily ascribed to it in the

control theory literature.

[24]

## The Standard Atmosphere and Other Data

The Standard Atmosphere

The tables that follow are derived from The ARDC Model Atmosphere, 1959, by Minzner, R. A., Champion, K. S. W., and Pond, H. L. Air Force Cambridge Research Center Report No. TR-59-267, U. S. Air Force, Bedford, MA, 1959. The values in the tables are the same, for most engineering purposes, as those derived from U. S. STAN­DARD ATMOSPHERE, 1976. Prepared by the USAF, NASA, and the NOAA.

English Units"

 Altitude h. ft Temperature T, °R Pressure P, lb/jp Density p, lb seU/ff Speed of sound, ft/sec Kinematic viscosity, fd/sec 0 518.69 2116.2 2.3769~3 1116.4 1.5723~4 1,000 515.12 2040.9 2.3081 1112.6 1.6105 2,000 511.56 1967.7 2.2409 1108.7 1.6499 3,000 507.99 1896.7 2.1752 1104.9 1.6905 4,000 504.43 1827.7 2.1110 1101.0 1.7324 5,000 500.86 1760.9 2.0482 1097.1 1.7755 6,000 497.30 1696.0 1.9869’3 1093.2 1.820Г4 7,000 493.73 1633.1 1.9270 1089.3 1.8661 8,000 490.17 1572.1 1.8685 1085.3 1.9136 9,000 486.61 1512.9 1.8113 1081.4 1.9626 10,000 483.04 1455.6 1.7556 1077.4 2.0132 11,000 479.48 1400.0 1.7011 3 1073.4 2.0655 ~4 12,000 475.92 1346.2 1.6480 1069.4 2.1196 13,000 472.36 1294.1 1.5961 1065.4 2.1754 14,000 468.80 1243.6 1.5455 1061.4 2.2331 15,000 465.23 1194.8 1.4962 1057.4 2.2927 16,000 461.67 1147.5 1.4480’3 1053.3 2.3544-4 17,000 458.11 1101.7 1.4011 1049.2 2.4183 18,000 454.55 1057.5 1.3553 1045.1 2.4843 19,000 450.99 1014.7 1.3107 1041.0 2.5526 20,000 447.43 973.27 1.2673 1036.9 2.6234

 Altitude h. ft Temperature T, °R Pressure P, Ib/ft2 Density p, lb sec^/ft4 Speed of sound, ft/sec Kinematic viscosity, ff’/sec 21,000 443.87 933.26 I.2249-3 1032.8 2.6966~4 22,000 440.32 894.59 1.1836 1028.6 2.7724 23,000 436.76 857.24 1.1435 1024.5 2.8510 24,000 433.20 821.16 1.1043 1020.3 2.9324 25,000 429.64 786.33 1.0663 1016.1 3.0168 26,000 426.08 752.71 1.0292~3 1011.9 3.1044~4 27,000 422.53 720.26 9.93 ІГ4 1007.7 3.1951 28,000 418.97 688.96 9.5801 1003.4 3.2893 29,000 415.41 658.77 9.2387 000.13 3.3870 30,000 411.86 629.66 8.9068 994.85 3.4884 31,000 408.30 601.61 8.584 Г 4 990.54 З.59З7-4 32,000 404.75 574.58 8.2704 986.22 3.7030 33,000 401.19 548.54 7.9656 981.88 3.8167 34,000 397.64 523.47 7.6696 977.52 3.9348 35,000 394.08 499.34 7.3820 973.14 4.0575 36,000 390.53 476.12 7.1028~4 968.75 4.1852 4 37,000 389.99 453.86 6.7800 968.08 4.3794 38,000 389.99 432.63 6.4629 968.08 4.5942 39,000 389.99 412.41 6.1608 968.08 4.8196 40,000 389.99 393.12 5.8727 968.08 5.0560 41,000 389.99 374.75 5.5982~4 968.08 5.ЗОЗ9-4 42,000 389.99 357.23 5.3365 968.08 5.5640 43,000 389.99 340.53 5.0871 968.08 5.8368 44,000 389.99 324.62 4.8493 968.08 6.1230 45,000 389.99 309.45 4.6227 968.08 6.4231 46,000 389.99 294.99 4.4067 ~4 968.08 6.7380 4 47,000 389.99 281.20 4.2008 968.08 7.0682 48,000 389.99 268.07 4.0045 968.08 7.4146 49,000 389.99 255.54 3.8175 968.08 7.7780 50,000 389.99 243.61 3.6391 968.08 8.1591 51,000 389.99 232.23 3.4692~4 968.08 8.5588~4 52,000 389.99 221.38 3.3072 968.08 8.9781 53,000 389.99 211.05 3.1527 968.08 9.4179 54,000 389.99 201.19 3.0055 968.08 9.8792 55,000 389.99 191.80 2.8652 968.08 1.0363-3 56,000 389.99 182.84 2.7314~4 968.08 1.0871 3 57,000 389.99 174.31 2.6039 968.08 1.1403 58,000 389.99 166.17 2.4824 968.08 1.1961 59,000 389.99 158.42 2.3665 968.08 1.2547 60,000 389.99 151.03 2.2561 968.08 1.3161

 Altitude h. ft Temperature T, °R Pressure P, Ib/ft2 Density p, lb sec2/ft4 Speed of sound, ft/sec Kinematic viscosity, ff/sec 61,000 389.99 143.98 2.1508“4 968.08 1.3805-3 62,000 389.99 137.26 2.0505 968.08 1.4481 63,000 389.99 130.86 1.9548 968.08 1.5189 64,000 389.99 124.75 1.8636 968.08 1.5932 65,000 389.99 118.93 1.7767 968.08 1.6712 66,000 389.99 113.39 1.6938 4 968.08 1.7530-3 67,000 389.99 108.10 1.6148 968.08 1.8387 68,000 389.99 102.06 1.5395 968.08 1.9286 69,000 389.99 98.253 1.4678 968.08 2.0230 70,000 389.99 93.672 1.3993 968.08 2.1219 71,000 389.99 89.305 1.334Г4 968.08 2.2257 “3 72,000 389.99 85.142 1.2719 968.08 2.3345 73,000 389.99 81.174 1.2126 968.08 2.4486 74,000 389.99 77.390 1.1561 968.08 2.5683 75,000 389.99 73.784 1.1022 968.08 2.6938
 “Note: the notation xxx " means xxx X 10 ".

SI Units

 Altitude h, m Tempera­ture T, К Pressure PN/m2 Density p kg/m3 Speed of Sound m/s Kinematic Viscosity, m2/s 0 288.16 1.0132545 1.2250 340.29 1.4607-5 300 286.21 9.7773+4 1.1901 339.14 1.4956 600 284.26 9.4322 1.1560 337.98 1.5316 900 282.31 9.0971 1.1226 336.82 1.5687 1,200 280.36 8.7718 1.0900 335.66 1.6069 1,500 278.41 8.4560 1.0581 334.49 1.6463 1,800 276.46 8.1494 1.0269 333.32 1.6869 2,100 274.51 7.8520 9.9649 1 332.14 1.7289 2,400 272.57 7.5634 9.6673 330.96 1.7721 2,700 270.62 7.2835 9.3765 329.77 1.8167 3,000 268.67 7.0121 9.0926 328.58 1.8628 3,300 266.72 6.7489 8.8153 327.39 1.9104 3,600 264.77 6.4939 8.5445 326.19 1.9595 3,900 262.83 6.2467 8.2802 324.99 2.0102 4,200 260.88 6.0072 8.0222 323.78 2.0626 4,500 258.93 5.7752 7.7704 322.57 2.1167 4,800 256.98 5.5506 7.5247 321.36 2.1727 5,100 255.04 5.3331 7.2851 320.14 2.2305 5,400 253.09 5.1226 7.0513 318.91 2.2903 5,700 251.14 4.918844 6.8234“1 317.69 2.3522-5

 Altitude h, m Tempera­ture T, К Pressure PN/m2 Density p kg/m3 Speed of Sound m/s Kinematic Viscosity, m2/s 6,000 249.20 4.7217 6.6011 316.45 2.4161 6,300 247.25 4.5311 6.3845 315.21 2.4824 6,600 245.30 4.3468 6.1733 313.97 2.5509 6,900 243.36 4.1686 5.9676 312.72 2.6218 7,200 241.41 3.9963 5.7671 311.47 2.6953 7,500 239.47 3.8299 5.5719 310.21 2.7714 7,800 237.52 3.6692 5.3818 308.95 2.8503 8,100 235.58 3.5140 5.1967 307.68 2.9320 8,400 233.63 3.3642 5.0165 306.41 3.0167 8,700 231.69 3.2196 4.8412 305.13 3.1046 9,000 229.74 3.0800 4.6706 303.85 3.1957 9,300 227.80 2.9455 4.5047 302.56 3.2903 9,600 225.85 2.8157 4.3433 301.27 3.3884 9,900 223.91 2.6906 4.1864 299.97 3.4903 10,200 221.97 2.5701 4.0339 298.66 3.5961 10,500 220.02 2.4540 3.8857 297.35 3.7060 10,800 218.08 2.3422 3.7417 296.03 3.8202 11,100 216.66 2.2346 3.5932 295.07 3.9564 11,400 216.66 2.1317 3.4277 295.07 4.1474 11,700 216.66 2.033544 3.2699 295.07 4.З475-3 12,000 216.66 1.9399 3.1194 295.07 4.5574 12,300 216.66 1.8506 2.9758 295.07 4.7773 12,600 216.66 1.7654 2.8388 295.07 5.0078 12,900 216.66 1.6842 2.7081 295.07 5.2494 13,200 216.66 1.6067 2.5835 295.07 5.5026 13,500 216.66 1.5327 2.4646 295.07 5.7680 13,800 216.66 1.4622 2.3512 295.07 6.0462 14,100 216.66 1.3950 2.2430 295.07 6.3378 14,400 216.66 1.3308 2.1399 295.07 6.6434 14,700 216.66 1.2696 2.0414 295.07 6.9637 15,000 216.66 1.2112 1.9475 295.07 7.2995 15,300 216.66 1.1555 1.8580 295.07 7.6514 15,600 216.66 1.1023 1.7725 295.07 8.0202 15,900 216.66 1.0516 1.6910 295.07 8.4068 16,200 216.66 1.0033 1.6133 295.07 8.8119 16,500 216.66 9.571743 1.5391 295.07 9.2366 16,800 216.66 9.1317 1.4683 295.07 9.6816 17,100 216.66 8.7119 1.4009 295.07 1.0148 4 17,400 216.66 8.3115 1.3365 295.07 1.0637 17,700 216.66 7.929543 1.2751 [24] 295.07 1.1149 4 18,000 216.66 7.5652 1.2165 295.07 1.1686 18,300 216.66 7.2175 1.1606 295.07 1.2249 18,600 216.66 6.8859 1.1072 295.07 1.2839 18,900 216.66 6.5696 1.0564 295.07 1.3457

Other Data

Conversion Factors

 Multiply By To Get Pounds (lb) 4.448 Newtons (N) Feet (ft) 0.3048 Meters (m) Slugs 14.59 Kilograms (kg) Slugs per cubic foot (slugs/ft3) 515.4 Kilograms per cubic meter (kg/m3) Miles per hour (mph) 0.4471 Meters per second (m/s) Knots (kt) 0.5151 Meters per second (m/s) Knots (kt) 1.152 Miles per hour (mph)

## M. a. Chord and m. a. Center for Swept and Tapered. Wings (Subsonic)

The ratio c/cr is plotted against A in Fig. C.2 for straight tapered wings with stream – wise tips. The spanwise position of the m. a. center of the half-wing (or the center of pressure of the additional load) for uniform spanwise loading is also given in Fig. C.2. These functions are given in Table C. l.

The m. a. chord is located by means of the distance x of the leading edge of the m. a. chord aft of the wing apex:

where A0 = sweepback of wing leading edge, degrees.

The sweepback of the leading edge is related to the sweep of the nth-chord line A„ by the relation

Using (C.3,2) and the expression for c/cn x can be obtained in terms of c and An from

The fractional distance of the m. a. center aft of the leading edge of the m. a. chord, hnw, is given for swept and tapered wings at low speeds and small incidences in Fig. C.3. The dotted lines show the aerodynamic-center position for wings with unswept trailing edges. The curves have been obtained from theoretical and experi­mental data. The curves apply only within the linear range of the curve of wing lift against pitching moment, provided that the flow is subsonic over the entire wing. The probable error of h„n given by the curves is within 3%.

The total load on each section of a wing has three parts as illustrated by Fig. CAa. The resultant of the local additional lift la, is the lift La acting through the m. a. center (Fig. CAb).

The resultant of the distribution of the local basic lift lh is a pitching couple whenever the line of aerodynamic centers is not straight and perpendicular to x. This couple is given by

rbf2

(x — x)lh dy = 2 J xlh dy о Jo

Figure C.3 Chordwise position of the mean aerodynamic center of swept and tapered wings at low speeds expressed as a fraction of the mean aerodynamic chord. (From Royal Aeronautical Data Sheet Wings 08.01.01.)

С.4 С,„

The total pitching-moment coefficient about the m. a. center is then

= Cm] + Cm2 = const (C.4,3)

If Cmar is constant across the span, and equals Cm2, then (C.4,2) also becomes the defining equation for c.

## Comparison of m. a. Chord and m. a. Center for. Basic Planforms and Loading Distributions

In Table C. l taken from (Yates, 1952), values of m. a. chord and у are given for some basic planforms and loading distributions.

In the general case the additional loading distribution and the spanwise center-of – pressure position can be obtained by methods such as those of De Young and Harper (1948), Weissinger (1947), and Stanton-Jones (1950). Fora trapezoidal wing with the local aerodynamic centers on the nth-chord line, the chordwise location of the mean aerodynamic center from the leading edge of the m. a. chord expressed as a fraction of the m. a. chord /i„ r is given by

Table C. l

 Planform Additional Loading Distribution М. Л.С. c у Constant taper and sweep Any 2cr 1 + A + Л2 r?, „ ‘ b (trapezoidal) 3 1 + A 2 Constant taper and sweep Proportional 2 cr 1 + A + A2 b 1 + 2k (trapezoidal) to wing chord (uniform C, J 3 1 + A 2 3(1 + A) Constant taper and sweep Elliptic 2 cr 1 + Л + A2 b 4 (trapezoidal) 3 1 + A 2 3 7T Elliptic (with straight Any c, 8 VcP’ b sweep of line of local a. c.) 3 7Г ~~2 Elliptic (with straight Elliptic (uniform cr 8 b 4 sweep of line of local a. c.) c,„) 3 7Г 2 3 7Г Any (with straight sweep of Elliptic 2 fW2 b 4 line of local a. c.) T dy S J о У ’ Зтг

where A = aspect ratio, b2/S

A = taper ratio, cjcr c, = wing-tip chord

The length of the chord through the centroid of area of a trapezoidal half-wing is equal to c. For the same wing with uniform spanwise lift distribution (i. e., Cla = const) and local aerodynamic centers on the nth-chord line, the m. a. center also lies on the chord through the centroid of area. The chord through the centroid of area of a wing having an elliptic planform is not the same as c, but the m. a. center for elliptic loading and the centroid of area both lie on the same chord (see Yates, 1952).

## Mean Aerodynamic Chord, Mean Aerodynamic Center, and Cm

”Lac

C. l Basic Definitions

In the normal flight range, the resultant aerodynamic forces acting on any lifting sur­face can be represented as a lift and drag acting at the mean aerodynamic center (x, y, z), together with a pitching couple C„,ut which is independent of angle of attack (see Fig. 2.8).

The pitching moment of a wing is nondimensionalized by the use of the mean aerodynamic chord c.

Both the m. a. center and the m. a. chord lie in the plane of symmetry of the wing. However, in determining them it is convenient to work with the half-wing.

These quantities are defined by (see Fig. C. l)1

2 r/7/2

c=- C2dy (C.1,1)

S Jo

2 (Ы 2

x==-— C, cxdy (C-1,2)

CLS Jo

2 rw2 b

У = 777 I C, cydy = Пер – z (C-1,3)

CLS Jo I

2 pa

Z=yzz Ctczdy (C.1,4)

CLS Jo

where b = wing span

c = local chord CL = total lift coefficient

Ci = local additional lift coefficient, proportional to CL Clh = local basic lift coefficient, independent of C,

C, = C, b + C, a = total local lift coefficient

‘The coordinate system used applies only to this appendix.

mac = pitching moment, per unit span, about aerodynamic center (Fig. C.4)

S = wing area

у = spanwise coordinate of local aerodynamic center measured from axis of symmetry

x = chordwise coordinate of local aerodynamic center measured aft of wing apex

г = vertical coordinate of local aerodynamic center measured from xy plane VcP = lateral position of the center of pressure of the additional load on the half­wing as a fraction of the semispan

The coordinates of the m. a. center depend on the additional load distribution; hence the position of the true m. a. center will vary with wing angle of attack if the form of the additional loading varies with angle of attack. For a wing that has no aerodynamic twist, the m. a. center of the half-wing is also the center of pressure of the half-wing. If there is a basic loading (i. e., at zero overall lift, due to wing twist), then (x, y, z) is the center of pressure of the additional loading.

The height and spanwise position of the local aerodynamic centers may be as­sumed known, and hence у and z for the half-wing can be calculated once the addi­tional spanwise loading distribution is known. However, in order to calculate x, the fore-and-aft position of each local aerodynamic center must be known first. If all the local aerodynamic centers are assumed to lie on the nth-chord line (assumed to be straight), then

x = ncr + у tan A„

where cr = wing root chord

An = sweepback of nth-chord line, degrees

Ideal two-dimensional flow theory gives n = for subsonic speeds and n = | for su­personic speeds.

The m. a. chord is located relative to the wing by the following procedure:

1. In (С. 1,2) replace Cla by CL, and for x use the coordinates of the і-chord line.

2. The value of x so obtained (the mean quarter-chord point) is the j-point of the m. a. chord.

The above procedure and the definition of c (see С. 1,1) are used for all wings.

## TRANSFORMATION OF INERTIAS

The inertia matrix I connects angular momentum with angular velocity [see (4.3,4) and (4.3,5)] via

h = ltd

and hence belongs to the class of matrices covered by (A.4,26). It follows that for two sets of body axes, denoted FBl and Fb 2 connected by the transformation L12, the inertias in frame Fb2 can be obtained from those in FBl by

I2 = L2,I, L12 (В.12,1)

If the two frames are two sets of body axes such that xBl is rotated about ySl through angle £ to bring it to xB?, then (see Appendix A.4)

(B.12,2)

The inertias in frame FB2, denoted by an asterisk, are then obtained from those in FBl, with the usual assumption of symmetry about the xz plane, by the relations

/* = lx cos2 £ + L sin2 £ + 1^ sin 2£

/* = lx sin2 £ + lz cos2 £ – Izx sin 2£ (B.12,3)

= Wx – lz) sin 2£ + /«(sin2 £ – cos2 0

TRANSFORMATION OF STABILITY DERIVATIVES

All of the stability derivatives with respect to linear and angular velocities and veloc­ity derivatives can be expressed as sums of expressions of the form of (A.4,23). That is, with the usual assumptions about separation of longitudinal and lateral motion, we can write

 AX ‘xu 0 Xw~ Au "0 xq 0" V "0 0 0 ‘ Дм AT = 0 0 V + 0 <7 + 0 0 0 V A Z. _ZH 0 _ A w _ . 0 Z, 0 _ _ r _0 0 _ Avv_

(B.12,4)

 A L ‘ " 0 Lv 0 ‘ A и Л 0 к ~P~ "0 0 0 ‘ Ай AM = Mu 0 Mw V + 0 Mq 0 q + 0 0 0 V AN _ _ 0 N0 0 . _ A w_ Л 0 Nr. r .0 0
 (B.12,5)

Each of the six matrices of derivatives above transforms according to the rule (A.4,26). When L is given by (B.12,2) we have the transformation from an initial set of body axes (unprimed) to a second set (primed) as follows:

Longitudinal

(Xu)’ = Xu cos3 f – (Xw + ZJ sin £cos f + Zw sin2 £ (Xwy = Xw cos2 £ + (Xu – ZJ sin f cos £ – Z„ sin2 £ (*,)’ = X„ cos £ – Zq sin f (*«)’ = Z* sin2 £ (1)

(X*)’ = – Z* sin £cos І (1)

(zu)’ = zu cos2 f – (Zw – XJ sin £cos ^ Xw sin2 £ (ZJ’ = Zw cos2 £ + (Z„ + XJ sin £ cos £ + XH sin2 f (ZJ’ = Zq cos £ + Xq sin £

(ZJ’ = – Z* sin £ cos £ (1)

(ZJ’ = Zlv cos2 £

(M„)’ = Mu cos ij – Mw sin £

(Mw)’ — Mw cos £; + Mu sin £

(Mqy = Mq

(МйУ = —M# sin £ (1)

(MJ’= M* cos £ (1)

Lateral

0J’ =

(Ур)’ = Ур cos £ – Уг sin £

(Yr)’ = Уг cos £ + Yp sin £

(Lp)’ = Lp cos I – Nv sin £

(Lp)’ = Lp cos2 £ – (Lr + AJ sin £ cos £ + 7Vr sin2 £ (L,.)’ = Lr cos2 £ — (Nr — Lp) sin £cos £ — Np sin2 £ (yVp)’ = Nv cos ^ + Lp sin £

(Np)’ = /Vp cos2 £ – (Nr — Lp) sin £ cos £, — L, sin2 £ (AJ’ = iVr cos2 % + (Lr + Np) sin £ cos £ + Lp sin2 £

(1) For consistency of assumptions, the derivatives with respect to и and (X*)’ are usually ignored.

## Changes in Inertias and Stability Derivatives with. Change of Body Axes

A matrix A that connects two vectors u and v as in (A.4,23) transforms between two reference frames as in (A.4,26). We now apply this rule to the inertia matrix and to matrices of stability derivatives.

 Aspect ratio, A

## Wing Yawing Derivatives C/r, Cnr

The following methods are simplified versions of those given in USAF Datcom. They apply to rigid straight-tapered wings in subsonic flow at low values of CL.

The derivative C,:

*r

where

C,

—1 is the slope of the rolling moment due to yawing at zero lift given by

CL jcL =o

M

(B.11,2)

where

В is given by (B.8,3).

(—- ‘j is the slope of the low-speed rolling moment due to yawing at zero ‘ ‘ck, Zо lift, obtained from Fig. B. l 1,1 as a function of aspect ratio, sweep of

the quarter-chord, and taper ratio. (B.11,2) modifies the low-speed value by means of the Prandtl-Glauert rule to yield approximate cor­rections for the first-order three-dimensional effects of compressible flow up to the critical Mach number.

is the increment in Clr due to dihedral, given by
ДС, 1 ttA sin Ac/4

— =——————— — (B.11,3)

Г 12 A + 4 cos A(V4

is the geometric dihedral angle in radians, positive for the wing tip above the plane of the root chord.

is the increment in Clr due to wing twist obtained from Fig. B. l 1,2.

Q is the wing twist between the root and tip sections in degrees, nega­

tive for washout (see Fig. B. l 1,2).

The derivative C„ :

where

CL is the wing lift coefficient.

Cnr

~—f is the low-speed drag-due-to-lift yaw-damping parameter obtained from L Fig. B. l 1,3 as a function of wing aspect ratio, taper ratio, sweepback, and CG position.

Cn

—— is the low-speed profile-drag yaw-damping parameter obtained from Fig. Do B. l 1,4 as a function of the wing aspect ratio, sweep-back, and CG posi­tion.

CDo is the wing profile drag coefficient evaluated at the appropriate Mach number. For this application Co,, is assumed to be the profile drag associ­ated with the theoretical ideal drag due to lift and is given by

C-Dn Cd

where CD is the total drag coefficient at a given lift coefficient.

 Figure B.11,3 Low-speed drag-due-to-lift yaw-damping parameter.

## Wing Rolling Derivatives Clp, Cnp

The following methods are simplified versions of those given in USAF Datcom. They apply to rigid straight-tapered wings in subsonic flow, in the linear range of CL vs. a. The derivative C,:

lP

where

j is the roll-damping parameter at zero lift, obtained from Fig. ‘Cl=о в.10,1 as a function of Af3 and /3A/к.

The parameter к is the ratio of the two-dimensional lift-curve slope at the appropriate Mach number to 2tt//3 that is, (C, JM/(2 77//З). The two-dimensional lift-curve slope is obtained from Sec. B. l. For wings with airfoil sections varying in a reasonably linear man­ner with span, the average value of the lift-curve slopes of the root and tip sections is adequate.

The parameter is the compressible sweep parameter given as

. , / tan Лс/4 .———

Ap = tan 1 f—- —— j, where /3 = Vl — M2.

and Лс/4 is the sweepback angle of the wing chord line.

f

— is the dihedral-effect parameter given by

(C/p) r=o

where

Г is the geometric dihedral angle, positive for the wing tip above the plane of the root chord.

 (6) X. = 0.50

 Н=ї.0 PA

z is the vertical distance between the CG and the wing root chord, positive for the CG above the root chord. b is the wing span.

(AC, p)drag is the increment in the roll-damping derivative due to drag, given by

(AQ„)drag = Cl-j CDo (B. 10,3)

where (C )

—lp is the drag-due-to-lift roll-damping parameter obtained from Fig.

<~’L B.10,2 as a function of A and Лс/4.

CL is the wing lift coefficient below the stall.

CDo is the profile or total zero-lift drag coefficient.

The derivative Cn

is the roll-damping derivative at the appropriate Mach number esti­mated above is the angle of attack, is the lift coefficient.

I is the slope of the yawing moment due to rolling at zero lift given by

CL= 0

M

(B.10,6)

is the effect of linear wing twist obtained from Fig. B.10,3.

is the wing twist between the root and tip stations in degrees, nega­tive for washout (see Fig. B.10,3).

## LIFT DUE TO SLIPSTREAM

The method of Smelt and Davies (1937)[23] can be used to estimate the added wing lift due to the slipstream. It is given by

ACL = s(XCLo – 0.6а0в)

where

Dx = diameter of slipstream at the wing C. P.

= Щ1 + a)/( 1 + s)]1/2 c = wing chord on center line of slipstream

Figure B.7,2 Variation of Ск^ with blade angle. (From NACA Wartime Rept. L-25, 1944, by H. S. Ribner.)

Side-force factor, SFF

Figure B.7,3 Ratio of normal force derivatives. (From NACA Wartime Rept. L-25, 1944, by H. S. Ribner.)

Distance behind root quarter-chord point, root chords

Figure B.7,4 Value of 1 — de/da on longitudinal axis of elliptic wing for aspect ratios 6, 9, and 12. (From NACA Wartime Rept. L-25, 1944, by H. S. Ribner.)

S = wing area s = a + ax/(D2l4 + x2)l/2 D = propeller diameter a = -* + 1(1 + STJtt)1’2 x = distance of wing C. P. behind propeller CLq = lift coefficient at section on slipstream center line, in absence of the slip­stream

a0 = two-dimensional lift-curve slope of wing section

0 = angle of downwash of slipstream at wing C. P. calculated from the equa­tion

1/008 = 0.016jc/£> + 1/0O08

where

0O = аф/( l + a)

ф = angle between propellor axis and direction of motion. A is an empirical constant given in Fig. B.7,5.

B.8 Wing Pitching Derivative Cmq

The method of USAF Datcom for estimating this derivative for a rigid wing in sub­sonic flow is as follows. The low-speed value (M = 0.2) of Cmq is given by

(B.8,1)

where

C, is the wing section lift curve slope from Sec. В. 1 (per rad). Ac/4 is the sweepback angle of the wing chord line.

 A3 tan2 Л..М 3 ——————– 1- — AB + 6 cos Лс/4 В A3 tan2 A£./4 ———— — + 3 A + 6 cos Лс/4

For higher subsonic speeds the derivative is obtained by applying an approximate compressibility correction.

where A is aspect ratio, and

В = V1 — M2 cos2 Лс/4 (В.8,3)

В. 9 Wing Sideslip Derivatives Clp, Cnp

The methods that follow are simplified versions of those given in USAF Datcom. They apply to rigid straight-tapered wings in subsonic flow.

The derivative Clp.

For A > 1.0:

For A < 1.0:

where

(^7/i j is the wing-sweep contribution obtained from Fig. B.9,1.

CL / Л.-/2

KMa is the compressibility correction to the sweep contribution, obtained

from Fig. B.9,2.

/ is the aspect-ratio contribution, including taper-ratio effects, ob-

CL JA tained from Fig. B.9,3.

C. ■

—— is the dihedral effect for uniform geometric dihedral, obtained from

Г Fig. B.9,4.

Г is the dihedral angle in degrees.

KM is the compressibility correction factor to the uniform-geometric-di­

hedral effect, obtained from Fig. B.9,5.

is the wing-twist correction factor, obtained from Fig. B.9,6.

is the wing-twist between the root and tip stations, negative for washout (see Fig. B.9,6). is the sweepback angle of the midchord line, is the sweepback angle of the chord line.

(СпЛ _ ( A + 4 cos Ac/4 IA2B2 + 4AB cos Лс/4 – 8 cos2 Лс/4 у СП0

СІ /м ЛВ + 4 cos Лс/4 Д А2 + 4А cos Лс/4 – 8 cos2 Лс/4 j С )low speed

(B.9,4)

where В is given by (B.8,3).