Category AERODYNAMICS, AERONAUTICS, AND FLIGHT MECHANICS

NOMENCLATURE AND ABBREVIATIONS

This listing includes only the symbols used throughout the text or a chapter. Symbols limited to a few pages are defined when used and are not listed here.

ENGLISH NOMENCLATURE

a speed of sound; slope of lift curve, dCJda

a„ acceleration normal to flight path

a0 speed of sound at sea level; slope of lift curve, dCJda

a, slope of horizontal tail lift curve, dCLJda

av slope of vertical tail lift curve, dCLJda

a average acceleration

A projected frontal area; reference area; aspect ratio, b2/S; disc area

А в cooling baffle area

b wingspan; propeller section chord in Figure 6.11

b constant (Equation 8.32)

b2 constant (Equation 8.32)

b3 constant (Equation 8.37)

b’ span between rolled-up vortices

В (M2- 1),/2; number of propeller blades

c chord length

Cf flap chord

c0 midspan chord

c, tip chord

c mean aerodynamic chord or geometric mean chord

Cd section drag coefficient = D/qc

Са average section drag coefficient for propeller

CD drag coefficient for finite wing or airplane = DlqS

CDi induced drag coefficient

Cdj, parasite drag coefficient = CD for C, = 0

CDv drag coefficient for body based on volume to the two-thirds power

Cf skin friction coefficient = DlqSw

CF average skin friction coefficient for airplane

Ch hinge moment coefficient = HjqSc (S and c for control surface)

Ci section lift coefficient = Llqc rolling moment coefficient = L/qSb

Ci average section lift coefficient for propeller

CL wing or airplane lift coefficient = LlaS

CLa airplane lift coefficient at approach speed

С/ч| trim CL; CL for zero angle of attack

Cm section pitching moment coefficient = M/qc2

CM airplane pitching moment coefficient = MlqSc

Смя дСмІ dq

C„ normal force coefficient = FJqc

CN yawing moment coefficient = N/qSb

Cp pressure coefficient = (p – p«,)lq; specific heat at constant pressure

CP propeller power coefficient = Plpn3D5

Cp. induced power coefficient

Cs propeller speed-power coefficient = (pV5IPn2)il$

Ct propeller thrust coefficient = Tlpn2D4 Cv specific heat at constant volume Cx X force coefficient = XIqS CY side force coefficient = YlqS

Cz Z force coefficient = ZlqS

Cp jet momentum coefficient = mlqc or mlqS

D drag; propeller diameter; body diameter

Dc cooling drag

Df skin friction drag

Д induced drag

e Oswald’s wing efficiency factor (Equation 4.32); voltage

/ frequency, Hz; equivalent flat-plate area

fs rate of change of total energy with fuel weight, dhJdWf

F force, jet engine thrust; Prandtl’s tip loss factor (Equation 6.37)

Fe free elevator factor (Equations 8.42 and 8.46)

Fg gross jet engine thrust

Fn force normal to chord; net jet engine thrust

F0 turbojet static thrust

g acceleration due to gravity

G gain of operational amplifier; gearing (Equations 8.27 and 8.30)

h height; location of center of gravity aft of leading edge of c as a

fraction of c, height defined in Figure 2.16; propeller section thick­ness in Figure 6.11 /labs absolute ceiling

642 NOMENCLA TURE AND ABBREVIA TIONS he total energy of airplane per unit weight

hi location of horizontal tail aerodynamic center aft of leading edge of

c as a fraction of c

hn location of airplane neutral point aft of leading edge of c as a

fraction of c

hnw location of wing neutral point aft of leading edge of c as a fraction of c

h0 distance defined in Figure 2.2

H hinge moment

і current

ihs incidence angle of horizontal stabilizer, positive nose down

i, incidence angle of horizontal tail, positive nose down

ix dimensionless mass moment of inertia about x-axis (Equation 10.16)

ixz dimensionless mass product of inertia about x-z axes (Equation

10.16)

iy dimensionless mass moment of inertia about у-axis (Equation 10.16)

іг dimensionless mass moment of inertia about z-axis (Equation 10.16)

Ix mass moment of inertia about x-axis

Ixz mass product of inertia about x-z axes

Iy mass moment of inertia about у-axis

Iz mass moment of inertia about z-axis

J propeller advance ratio = VlnD

ke constant (Equation 8.33)

Kc correction factor (Figure 6.34a)

Kp see Equation 5.97 and Figure 5.38

Kv see Equation 5.97 and Figure 5.38

/ reference length

/ac distance defined in Figure 8.5

/, distance from center of gravity to horizontal tail aerodynamic center

l„ distance from center of gravity to vertical tail aerodynamic center

L lift; rolling moment

m airplane mass; mass flow rate; doublet strength

M pitching moment; Mach number

Mcr critical Mach number

n rotational speed, rps; load factor, LIW

N yawing moment, rpm

Np propeller yawing moment

Nj jet normal force

Nt low-pressure rotor rpm (forward compressor, aft turbine)

N2 high-pressure rotor rpm (aft compressor, forward turbine)

p static pressure; distance from leading edge of airfoil to Zmax; pro­

peller pitch

p,2 total pressure at compressor inlet

ph total pressure in turbine exhaust

p dimensionless roll rate, pb/2V

рж free-stream static pressure

pa atmospheric pressure

p0 reservoir pressure; sea level atmosphere pressure

P stick force; power; roll rate

Pa power available

PB static pressure just ahead of cooling baffle

PE static pressure at engine cowling exit

Pi propeller-induced power

Pia static propeller-induced power

PN propeller normal force

Pr power required

ps excess specific power, dhjdt (Equation 7.61)

Puse useful power

Pxs excess power (Pa – Pr)

q dynamic pressure, pV2l2; source strength, point or distributed

q dimensionless pitch rate, Qcl2 V

Q volume flux; source strength; pitch rate; propeller torque

r pressure ratio for the Brayton cycle (Equation 6.69)

r0 sea level value of r

r dimensionless yaw rate, Rbl2V; radius vector from point P to

vortex element in Biot-Savart law

R universal gas constant, plpT; Reynolds number, VIIv; yaw rate;

electrical resistance; radius; radius of curvature; range Rx Reynold’s number based on x distance from leading edge

R/C rate of climb

s stick travel

sa airborne distance for takeoff or landing

S planform area; Strouhal number, fDIV; distance, reference area

Sw wetted area

t time; temperature, °С; airfoil thickness (sometimes denotes maximum

value or ratio of maximum value to c) te endurance time

t* air seconds (Equations 9.28 and 10.14)

T thrust, absolute temperature

TB temperature just ahead of cooling baffle

Tc thrust coefficient (Equation 6.22)

T0 static thrust

и x component of velocity; increment in same component above U0

й dimensionless perturbation velocity in x direction, u/U0

U0 trimmed airplane velocity; free-stream velocity

v у component of velocity

vr radial component of velocity

v„ tangential component of velocity

V local velocity; free-stream velocity

VA approach velocity

Vc velocity of climb

Vcai calibrated airspeed

Ve equivalent airspeed; resultant velocity (Figure 6.9)

VE velocity at engine cowling exit

Va ground speed

VH horizontal tail volume

V, indicated airspeed

Vlof liftoff speed

Vm volume

Vmc minimum control speed

V™ minimum unstick speed

V0 free-stream velocity

Vr resultant velocity (Figure 6.9)

VR takeoff rotation speed

Vs stalling speed

VS| stalling speed, one engine out

Vr propeller tip speed, wR

Vtr trim speed

V„ vertical tail volume

Vw wind velocity

V, critical engine failure calibrated airspeed

V2 airspeed over takeoff obstacle

V» free-stream velocity

w z component of velocity; downwash; propeller-induced velocity

wa axial component of propeller-induced velocity

w0 static value of propeller-induced velocity; impact velocity (Equation

6.60)

w, tangential component of propeller-induced velocity

W airplane weight

Wa airflow

WE airplane empty weight

Wf fuel flow rate

WP total fuel weight

Wi initial airplane weight

x cartesian coordinate, directed forward; relative radius along pro­

peller blade, r/R

xh relative hub radius for propeller, rhIR

Xj distance of center of gravity aft of inlet (Figure 8.23)

x„ x-coordinate of point at which induced velocity is to be calculated

X resultant aerodynamic force on airplane in x direction

у right-handed, orthogonal coordinate directed to the right, spanwise

distance to right of airplane centerline yp у location of point at which induced velocity is to be calculated

Y resultant aerodynamic force on airplane in у direction

2 cartesian coordinate directed downward; airfoil camber (sometimes

denotes maximum value or ratio of maximum value to c) zp z location of point at which induced velocity is to be calculated

Z resultant aerodynamic force on airplane in z direction

Zj distance of jet thrust line below center of gravity

Zp distance of propeller thrust line above center of gravity

GREEK NOMENCLATURE

a angle of attack; angle defined in Biot-Savart law (Figure 2.16)

a, induced angle of attack

a0i angle of zero lift

/8 sideslip angle, angle defined in Biot-Savart law (Figure 2.16);

Prandtl-Glauert compressibility correction factor (1 — M2)l/2; blade section pitch angle

у ratio of specific heats (1.4 for air); vortex strength (point or dis­

tributed)

Г wing dihedral angle; vortex strength

5 ratio of ambient pressure to sea level ambient pressure; flow

deflection angle that causes oblique shock wave; fractional increase in CD. above elliptic case; boundary layer thickness 8a aileron deflection, sum of left and right aileron angles, positive for

right aileron down, left aileron up <5e elevator angle, positive for down elevator

Sf flap angle, positive down

Sr rudder angle, positive to the left

8, trim tab angle, positive down

8* displacement thickness (Figure 4.2 and Equation 4.3)

Д denotes an increment

e wing twist (positive nose up); downwash angle; apex angle for delta

wing; drag-to-lift ratio eT wing twist at tip

ea де/da = rate of change of downwash angle with a

€p deldfi = rate of change of sidewash angle with /3

£ damping ratio

7j propeller efficiency = TV/P; correction to т (Figure 3.33)

t), ideal propeller efficiency

r), ratio of dynamic pressure at tail to free-stream g

в pitch angle; oblique shock wave angle; angle between thrust line and

horizontal; ratio of absolute ambient temperature to sea level value вс climb angle between velocity vector and horizontal

eD descent angle between velocity vector and horizontal

A taper ratio, c,/c0; also function in Figure 5.42; propeller advance

ratio, VIwR

A angle of sweepback

H coefficient of viscosity; Mach wave angle; dimensionless airplane

mass (Equations 9.31 and 10.16); coefficient of braking or rolling friction

v kinematic viscosity

p mass density

<7 ratio of mass density to sea level value; root of characteristic

equation; propeller section solidity, Bc/ttR t flap effectiveness factor (Equation 3.48 and Figure 3.32); dimension­

less time, tit*; time constant (time for a damped system to reach 1/e of its initial displacement)

ф roll angle; velocity potential; resultant flow angle for propeller blade

section (Figure 6.9)

фс compressible velocity potential

фі incompressible velocity potential

фт helix angle at tip of propeller trailing vortex system

ф stream function; yaw angle

a) angular velocity or circular frequency, radians per second

at curl of velocity vector, V x V

шп undamped natural frequency, radians per second

SUBSCRIPTS

a ailerons

ac aerodynamic center

am ambient

В base; cooling baffle

c compressible; corrected

CLB climb

CR cruise

DES descent

e elevator

/ flap

HLD hold

і incompressible; index, induced, initial

jet; index

index

lower

maximum

minimum

reservoir, sea level, free-stream

optimum

horizontal tail

horizontal tail

vertical tail

winglet

wing

Any quantity may indicate differentiation with respect to quantity, for example, Cia = dCJda. quarter chord midchord

just upstream of shock wave

just downstream of shock wave

conditions far removed from body (free stream)

SUPERSCRIPTS

throat conditions where M = 1

derivative with respect to time, f, or dimensionless time, т

ABBREVIATIONS

aerodynamic center brake horsepower bypass ratio

brake specific fuel consumption |

center of pressure

center of gravity

cylinder head temperature

exhaust gas temperature

engine pressure ratio, p,7lp,2

effective shaft horsepower

Federal Aviation Administration

Federal Air Regulations

gallons per hour

in-ground effect

kt

knots

LE

leading edge

In

natural logarithm

MAP

manifold absolute pressure

NACA

National Advisory Committee for Aeronautics

NASA

National Aeronautics and Space Administration

OGE

out-of-ground effect

rpm

revolutions per minute

rps

revolutions per second

SFC

specific fuel consumption

shp

shaft horsepower

ТЕ

trailing edge

thp

thrust horsepower

TIT

turbine inlet temperature

TO

takeoff

TSFC

thrust specific fuel consumption

[1] L=f (pi —pu) dx (3.5)

J о

The moment about the leading edge, defined positive nose up, will be

MLE = -| x(pi – pu) dx (3.6)

Jo

In accord with Equation 2.12, the lift and moment can be expressed in terms of dimensionless coefficients.

[2] = 0.6328

Pi

[3] Taxi and takeoff allowance.

• Climb from sea level to 41,000 ft.

• Cruise at maximum cruise thrust.

• Descent to sea level.

• Land with 45-min reserve fuel.

AIRPLANE DATA

The following samples of airplane data (Figures A.3.1 to A.3.20 and Table A.3.1) are taken from Aircraft Handling Qualities Data, by Robert K. Heffley and Wayne F. Jewell, NASACR-2144, December 1972. The figures include:

• Flight envelope.

• CL and CD as a function of M and altitude.

• Weight and moments of inertia.

• Three-view sketches to scale.

In addition, stability derivatives are tabulated for the Convair 880… The following definitions apply to notations on the sketches.

0.25c F. S. 205.5

F. S. = fuselage station = inches aft of reference transverse plane near nose.

W. L. = waterline = inches above horizontal reference plane.

B. L. = buttline = inches to right of vertical plane of symmetry.

Nominal Configuration

1.4 Vs

W= 11,8001b eg at 0.260c, W. L. 100 Ix = 12,700 slug-ft’

Iy = 20,700 slug-ft’

Iz = 32,000 slug-ft‘

/„ = 480 slug-ft2

Figure A.3.4 X-15; a general arrangement.

Nominal Configuration

Zero fuel Lower ventral on Speed brakes retracted W= 15,5601b eg at 0.22c ix = 3650 slug-ft2 I, = 80,000 slug-ft2 Iz = 82,000 slug-ft2 Іхг = 590 slug-ft2

1.0

0.5

0.2

0.1

0.05

0.02

0.01

Figure A.3.6 X-15; 15,560 lb.

S = 6297.8 ft2 b = 105 ft c = 78.53 ft

^—————– r-TT-n———–

00 00 00 О

Figure A.3.10 XB-70A; a general arrangement.

Nominal Configuration

W = 300,000 lb eg at 0.235 c lx = 1.45 x 106 slug-ft2 /,. = 16 x 106 slug-ft2 U = 17.2 X 106 slug-ft2 /„ = -0.6 x 106 slug-ft2

________ SL

——— 20,000 ft

——– 40,000 ft

——— 60,000- ft

Table A.3.1 CV-880 M Longitudinal Nondimensional Stability Derivatives

Flight condition

1

2

3

4

5

6

7

Configuration

L

PA

Speed

134 KTAS

165 KTAS

0.6M

0.86M

0.7M

0.8M

0.86М

Altitude

SL

SL

23 К

23 К

35 К

35 К

35 К

<*o (deg)

5.2

4.3

5.3

2.8

8.3

4.7

4.0

CL

1.03

0.68

0.36

0.175

0.454

0.347

0.301

CD

0.154

0.080

0.022

0.019

0.025

0.024

0.023

Си, (1/rad)

4.66

4.52

4.28

4.41

4.62

4.8

4.9

CD, (1/rad)

0.43

0.27

0.14

0.07

0.18

0.15

0.13

Cm„ (1/rad)

-0.381

-0.903

-0.522

-0.572

-0.568

-0.65

-0.74

CL, (1/rad)

2.7

2.7

2.44

2.5

2.75

2.75

2.9

CLe (1/rad)

7.92

7.72

6.76

6.37

7.51

7.5

7.62

Cm, (1/rad)

-4.17

-4.13

-4.16

-4.66

-4.4

-4.5

-4.6

C„a (1/rad)

-12.2

-12.1

-11.5

-11.8

-12.0

-12.0

-12.0

CLs (1/rad)

0.22

0.213

0.193

0.141

0.203

0.190

0.180

Cmh (1/rad)

-0.657

-0.637

-0.586

-0.438

-0.618

-0.57

-0.532

Chs (1/rad)

-10.326

-0.328

-0.336

-0.278

-0.342

-0.31

-0.285

Cl, (1/rad)

0.055

0.0532

0.482

0.0352

0.0508

0.047

0.0450

Cm„; (1/rad)

-0.164

-0.159

-0.146

-0.11

-0.155

-0.14

-0.134

Ch,’ (1/rad)

-0.287

-0.285

-0.297

-0.343

-0.312

-0.335

-0.352

S = 160 ft2 b= 13.6 ft T= 21.17 ft

Nominal Configuration

Zero fuel (burnout)

Gear up

Transonic or subsonic configuration, depending on flight condition W = 6466 lb

eg at 0.517c, W. L. Ix = 1353 slug-ft2 Iy = 6413 slug-ft2 J2 = 7407 slug-ft2 /„ = 399 slug-ft2

Figure A.3.19 HL-10; flight envelope; nominal envelope extremes.

Configuration

Speed Brakes

Elevon Flaps

Tip-Fin Flaps

Subsonic

Zero

Zero

Zero

Transonic

30°

30.5°/32.5°

STANDARD ATMOSPHERE

English Units


Altitude Temperature Pressure P, Z* ft T, °R lb/ft2

Speed of Kinematic Density p, sound, viscosity,

lb sec2/ft4 ft/sec ft2/sec

0

518.69

2116.2

1,006

515.12

2040.9

2,000

511.56

1967.7

3,000

507.99

1896.7

4,000

504.43

1827.7

5,000

500.86

1760.9

6,000

497.30

1696.0

7,000

493.73

1633.1

8,000

490.17

1572.1

9,000

486.61

1512.9

10,000

483.04

1455.6

11,000

479.48

1400.0

12,000

475.92

1346.2

13,000

472.36

1294.1

14,000

468.80

1243.6

15,000

465.23

1194.8

16,000

461.67

1147.5

17,000

458.11

1101.7

18,000

454.55

1057.5

19,000

450.99

1014.7

20,000

447.43

973.27

21,000

443.87

933.26

22,000

440.32

894.59

23,000

436.76

857.24

24,000

433.20

821.16

25,000

429.64

786.33

2.3769-3

1116.4

І.5723-4

2.3081

1112.6

1.6105

2.2409

1108.7

1.6499

2.1752

1104.9

1.6905

2.1110

1101.0

1.7324

2.0482 .

1097.1

1.7755

1.9869 3

1093.2

1.820Г4

1.9270

1089.3

1.8661

1.8685

1085.3

1.9136

1.8113

1081.4

1.9626

1.7556

1077.4

2.0132

1.7011 3

1073.4

2.0655"4

1.6480

1069.4

2.11%

1.5961

1065.4

2.1754

1.5455

1061.4

2.2331

1.4962

1057.4

2.2927

1.4480 3

1053.3

‘ 2.3544-4

1.4011

1049.2

2.4183

1.3553

1045.1

2.4843

1.3107

1041.0

2.5526

1.2673

1036.9

2.6234

1.2249^3

1032.8

2.6966~4

1.1836

1028.6

2.7724

1.1435

1024.5

2.8510

1.1043

1020.3

2.9324

1.0663

1016.1

3.0168

English Units

Altitude Z* ft

Temperature T,° R

Pressure P, lb/ft2

Density p, lb sec2/ft4

Speed of sound, ft/sec

Kinematic

viscosity,

ft2/sec

26,000

426.08

752.71

1.0292’3

1011.9

3.1044-

27,000

422.53

720.26

9.931 Г4

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

8.584Г4

990.54

3.5937-

32,000

404.75

574.58

8.2704

986.22

3.7030

33,000

401.19

548.54

1.9656

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—

968.75

4.1852-

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

4.81%

40,000

389.99

393.12

5.8727

968.08

5.0560

41,000

389.99

374.75

5.5982 4

968.08

5.3039-

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-

968.08

6.7380-

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-

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-

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

English Units

Altitude

Z,*ft

Temperature T, °R

Pressure P, lb/ft2

Density p, lb sec2/ft4

Speed of sound, ft/sec

Kinematic

viscosity,

ft2/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

1.3341 4

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

SI Units

Altitude Z, m

Tempera­

ture

T, K

Pressure P N/m2,

Density. p kg/m3

Speed of sound m/s

Kinematic

Viscosity,

m2/s

0

288.16

1.01325 + 5

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

SI Units

Tempera – Speed of Kinematic

Altitude ture Pressure Density sound Viscosity,

Z, m Г, К P N/m2 p kg/m3 m/s m2/s

SI Units

Altitude Z, m

Tempera­ture T, К

Pressure P N/m2

Density p kg/m3

Speed of sound

m/s

Kinematic

Viscosity,

m2/s

16,500

215.66

9.5717 + 3

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

7.9295 + 3

1.2751-1

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

THE SI SYSTEM

SI is an abbreviation for Systeme International and is a metric system of units that is being adopted internationally for all scientific and engineering work. This brief presentation presents only the aspects of the system per­tinent to this text.

Names of International Units

Physical Quanity

Name of Unit

Symbol

Length

Meter

m

Mass

Kilogram

kg

Time

Second

s

Electric current

Ampere

A

Temperature

Kelvin

К

DERIVED UNITS

Area

Square meter

rrr

Volume

Cubic meter

m3

Frequency

Hertz

Hz

(s")

Density

Kilogram per cubic meter

kg/m3

Velocity

Meter per second

m/s

Angular velocity

Radian per second

rad/s

Acceleration

Meter per second squared

m/s2

Angular acceleration

Radian per second squared

rad/s2

Force

Newton

N

(kg-m/s2)

Pressure

Newton per square metera

N/m2

Kinematic viscosity

Square meter per second

m2/s

Dynamic viscosity

Newton-second per square meter N-s/m2

Work, energy, quantity of heat

Joule

J

(N-m)

Power

Watt

W

(J/s)

Entropy

Joule per kelvin

J/K

Specific heat

Joule per kilogram kelvin

Jkg-‘K 1

Thermal conductivity

Watt per meter kelvin

Win ‘K 1

“I N/m2= I Pascal, denoted by Pa.

DEFINITIONS

newton (N) The newton is the force that gives a mass of 1 kg an acceleration of 1 m/s/s.

joule (J) The joule is the work done when the point of application of 1 N is displaced a distance of 1 m in the direction of force.

watt (W) The watt is the power that gives rise to the production of energy at the rate of 1 J/s.

Conversion Factors

Multiply By To Get

Newtons (N)

Meters (m)

Kilograms (kg)

Kilograms per cubic meter (kg/m3) Kilowatts (kW)

Pascals (Pa)

Pascals (Pa)

Meters per second (m/s)

Meters per second (m/s)

MISCELLANEOUS

g = 32.2 ft/sec2 = 9.81 m/s2 at sea level °K = °С+ 273.19 °R = °F +459.7 °С = (°F – 32)(5/9)

Universal Gas Constant (p = pRT)

R = 1716 ft2/s2/°R = 287.0 m2/s2/°K

PREFIXES

The names of multiples and submultiples of SI units may be formed by application of the following prefixes.

Factor by which unit is multiplied

Prefix

Symbol

1012

tera

T

109

giga

G

106

mega

M

103

kilo

к

102

hecto

h

10

deka

da

10~‘

deci

d

10~2

centi

c

10~3

milli

m

10~6

micro

p

io-9

nano

n

10~12

pico

p

10’15

femto

f

10 18

atto

a

Spiral Stability

The use of Table 10.3 should be obvious and needs no explanation. For the Cherokee example, the “time to double” of 30.2 sec is seen to exceed all of the minimum times specified in this table.

SPINNING

When stalled an airplane has a tendency to “drop off” into a spin, particularly if the stall is asymmetrical or entered with power. A pilot can purposefully initiate a spin by kicking the rudder hard to one direction at the top of the stall.

In a spin, an airplane rotates around a vertical spin axis as it is descend­ing rapidly in a nose-down attitude. The path of its center of gravity pres­cribes a helix around the spin axis. The airplane’s motion and attitude result in a high angle of attack on the order of 45° or more.

The stall/spin is one of the major causes of light plane accidents today. In my opinion, the blame for this can be placed on pilot training instead of current airplane designs. On a landing approach, too many pilots fly too slow. Rarely is there a valid reason for dragging a light airplane in under power at a speed just above the stalling speed. Under such a condition, an unexpected gust or a maneuver on the part of the pilot can produce an asymmetrical stall. If this occurs on the approach, the altitude may be insufficient to recover from either the stall or ensuing spin, regardless of how well the airplane is designed.

FAR Part 23 requires, for the normal category, that a single-engine airplane be able to recover from a one-turn spin in not more than one additional turn, with the controls applied in the manner normally used for recovery. Normally, to recover from a spin, one uses forward stick and rudder opposite to the direction of the spin. Ailerons are generally ineffective for spin recovery, since the wing is fully stalled. FAR Part 23 also requires, for the normal category, that the positive limit maneuvering load factor and applicable airspeed limit not be exceeded in a spin. In addition, there may be no excessive back pressure (on the stick) during the spin or recovery, and it must be impossible to obtain uncontrollable spins with any use of the controls.

The analysis of spinning, in order to design for good spin-recovery characteristics, is difficult because of the nonlinear nature of the problem. However, one can understand the principal factors influencing spin recovery by reference to Figure Ю.6. A side view of the spinning airplane is shown in Figure 10.6a. As the airplane descends, the aerodynamic forces on the aft fuselage and tail, Ff and FT, tend to nose the airplane downward. However, in a nose-down attitude, because of the rotation, centrifugal forces are developed on the airplane’s mass to either side of the center of gravity. These create a pitching moment about’ the center of gravity that opposes the nose-down aerodynamic moment.

The wing is completely stalled in a spin so that its resultant force, F„, is approximately normal to the wing. In a nose-down altitude, the horizontal component of F„ points in toward the spin axis and balances the centrifugal force

while the vertical component of F„. balances the weight. If 0 is the angle of the nose up from the vertical, as shown in figure 10.6 (a), then the angular velocity about the spin axis, Д the spin radius, R,, and 0 are related by,

R, = g/П2 tan 0

For a typical light aircraft Rs, for a steep spin, is of the order of 0.2 of the wing span and decreases to.06 b for a flat spin. Corresponding 0 values equal approximately 45° and 60°.

(a)

Figure 10.6 Aerodynamic and inertia forces influencing the spin behavior of an airplane (a) Side view. (b) Top view, (c) Autorotative forces.

Normally, the angle of roll, ф, is small in a spin. Using equations 10.&d, e, and/, it can be shown that the inertia moments about the spin axis and about the airplane’sy-axis are proportional to fl2 and (1-І,). Thus, the spinning behavior of an airplane is determined as much, possibly more so, by its mass distribution as by its aerodynamic shape.

The pro-spin, autorotative forces can be produced by both the wing and the fuselage. Referring to figure 10.6 (b), consider the resultant velocities in a spin at each wing tip and at a typical fuselage section aft of the eg. Assume Rs to be small as in the case of a flat spin. The velocities at each of these locations combine vectorially with the descent velocity, VD as shown in figure 10.6 (c) to produce extremely high angles of attack for the wing sections and a nearly vertical flow from beneath the fuselage. For the wing, particularly one with a well-rounded leading edge, a net moment in the direction of rotation results from the higher

angle of attack on the left side compared to the right side. The aft fuselage, depending upon its geometry, can develop a force also in the direction of rotation.

Rudder control is the principal means of recovering from a spin. There­fore, in designing the empennage, the placement of the horizontal tail relative to the rudder is important. If the horizontal tail is too far forward, in a spin its wake will blanket the rudder, making it ineffective.

An attempt to quantify the blanketing of the vertical tail by the horizontal tail is presented in Figure 10.7 (taken from Ref. 10.1). Referring to this figure, a term called the tail damping power factor (TDPF) is defined by

TDPF

mb2

Figure 10.9 Criteria for spin recovery.

Inertia yawing moment parameter, Ux — !y) / mb2 X 104

Figure 10.10 Spin recovery for a light airplane with different tail configurations.

shape of this area must certainly influence the damping effectiveness of this area.

According to Reference 10.1, the TDPF for satisfactory spin recovery is a function of an airplane’s mass distribution. This is shown in Figure 10.8 (taken from Ref. 10.1). Boundaries are suggested on this figure that divide regions of satisfactory spin recovery characteristics from unsatisfactory regions. However, from the points included on the figure, it is obvious that these boundaries are not too well defined. Indeed, there are several unsatis­factory points lying well within the region denoted as being satisfactory. Similar graphs for ц values as high as 70 can be found in the reference.

The scatter and overlapping of the points in Reference 10.1 appear to rule out any valid definition of the criteria as a function of /a. Instead, the graph of Figure 10.9 is offered as representative of any /л value. In the region labeled “satisfactory,” there were no unsatisfactory points to be found in the reference. For the region labeled “possibly satisfactory,” there were ap­proximately an equal number of satisfactory and unsatisfactory points. In the region labeled “probably unsatisfactory,” the data points were predominan – tely unsatisfactory.

Despite the uncertainty associated with this figure, one point is obvious. For satisfactory spin recovery characteristics, the moments of inertia about the pitching and rolling axes should be significantly different.

Figure 10.10 is taken from Reference 10.3. Obviously, neither the shape of the curve dividing the satisfactory region from the unsatisfactory region nor the values of TDPF for satisfactory recovery agree with Figure 10.9. Tails 1,3, and 4 were found to be unsatisfactory for spin recovery with ailerons neutral. With ailerons deflected, tails 2 and 7 were also unsatisfactory. This reference concludes that TDPF cannot be used to predict spin recovery. However, it is important to provide damping to the spin and very important to provide exposed rudder area for spin recovery. In modern aircraft, the T-tail is becoming very popular. The reason for this is twofold. First, from Figure

10.10, such a tail configuration provides excellent spin recovery charac­teristics. Second, the horizontal tail is removed from the wing wake, thereby minimizing downwash effects. For the same tail effectiveness, the T-tail will allow a smaller horizontal tail, thereby saving on weight and drag.

Roll Mode

Table 10.1 presents the maximum roll mode time constants that should not be exceeded in order to achieve a given flying quality level. In order to interpret this table, one needs to understand the meaning of the term “time constant.”

A first-order, linear system x(t) obeys a differential equation, which can be written as

x + — x=f(t) (10.36)

T

where r is defined as the time constant. For the homogeneous solution, x will be of the form

x = ХоЄ^’

so Equation 10.36 becomes

(<r + i)x = °

or

v = – (10.37)

The homogeneous solution for x can therefore be written as

X = x0e~tlr (10.38)

Thus, the time constant, r, is a measure of the damping in a first-order system.

The more heavily damped a system, the smaller will be its time constant. Given an initial displacement and released, the system will damp to Це or 0.368 of its initial displacement in a time equal to the time constant. The time to halve amplitude and the time constant, r, are related by

Tm = 0.693 t

For the Cherokee 180 example, for the roll mode, cr = —2.79, so

t = 0.350 air sec

or, in real time,

t = 0.033 sec

This value is well within the level 1 criteria for the class I airplane for all flight phases.

Dutch Roll Mode

Table 10.2 presents criteria for the frequency and damping ratio for the Dutch roll mode. Note that minimum values are specified for £, ш„, and for the product £шп. The minimum value for is determined from the column labeled o)„. However, the governing damping requirement equals the largest value of £ obtained from either of the two columns labeled £ and £<a„.

For the Cherokee 180 example, in real time,

tr = -0.601 ± 3.03/

Thus, from Equation 9.51,

£ = 0.194 a>„ = 3.09 rad/sec

With reference to Table 10.2, the Cherokee’s damping in this mode is governed by the column labeled £ and is seen to be nearly equal to the minimum value prescribed for a flying quality level of 1 in the flight phase category A. Thus one would not expect to encounter any problems from the Dutch roll mode with this airplane.

LATERAL DIRECTIONAL FLYING QUALITIES

Requirements on the three lateral-directional modes to assure a desired level of flying quality can be found in Reference 9.1. Although specific to military aircraft, these criteria can obviously prove of use in the evaluation or design of civil aircraft. Tables 10.1, 10.2, and 10.3 present criteria for the three

Table 10.1 Maximum Roll Mode Constant

Right Phase Category

Class"

Level

1

2 3

A

I, IV

1.0

1.4

II, III

1.4

3.0

В

All

1.4

3.0 10

C

I, II-C, IV

1.0

1.4

II-L, III

1.4

3.0

“ C and L refer to carrier and land operations.

Table 10.2 Minimum Dutch Roll Frequency and Damping

Level

Right Phase Category

Class

Min £

Min £w„, rad/sec

Min 0)„, rad/sec

A

I, IV

0.19

0.35

1.0

II, III

0.19

0.35

0.4

1

В

All

0.08

0.15

0.4

C

I, II-C

IV

0.08

0.15

1.0

II-L, III

0.08

0.15

0.4

2

All

All

0.02

0.05

0.4

3

All

All

0.02

0.4

Table 10.3 Spiral Stability—Minimum Time-to-Double Amplitude

Class

Flight Phase Category

Level 1, sec

Level 2,

sec

Level 3, sec

I and IV

A

12

12

4

В and C

20

12

4

II and III

All

20

12

4

modes as a function of the Cooper-Harper level, flight phase category, and class of airplane. Airplane class is defined according to:

Class I —Small, light airplanes.

Class II —Medium weight, low-to-medium maneuverability.

Class III—Large, heavy, low-to-medium maneuverability.

Class IV—High maneuverability.

Oscillatory or “Dutch Roll" Mode

The pair of complex roots of the characteristic equation for the lateral – directional motion of the Cherokee was given previously as -0.055±0.277i. When substituted into Equation 10.29 (using +0.277/) and Equation 10.30, the following results are obtained.

-0.192 + 0.864/

or

0.0392-0.263/

Replacing r by сгф gives

j|= -0.940 + 0.045/ or

j|= 0.941** (10.34)

where у = 177.3°.

Relating ф to <f>,

^7 = 1.06** (10.35)

Ф

where у = 74.8°.

This mode is a damped oscillation where the three angles ф, /3, and ф have approximately the same magnitudes. In the case of the Cherokee 180, the time to damp to half-amplitude equals 1.15 sec with a period of 2.07 sec.

The motion of the airplane for this mode can be described as follows. Imagine that the airplane begins to yaw to the right. As it does so, it slips to
the left, so that its path remains nearly a straight line. As it yaws to the right, it begins to roll in that direction. While still rolled to the right, the airplane begins to yaw to the left and slip to the right. This turning and rolling motion is somewhat mindful of the weaving and twisting that an ice skater undergoes in skating along the ice. Hence the mode has come to be called the “Dutch roll” after the country well known for its ice skating.

This mode, for the Cherokee 180, is barely discernable to the pilot. It can be excited by a step input to the rudder and is observable. However, as predicted, it damps rapidly. For some airplanes the Dutch roll mode is lightly damped, making them somewhat unpleasant to fly. One gets the feeling that the rear end is trying to pass the front end.

Mode Shapes

The significance of these roots and the instability exhibited by the positive real root can be examined by looking at the shapes of each mode in the same manner that was followed for the longitudinal motion.

For the transient solution, 8„ and 8r are zero. Using Equation 10.27a and 10.27b, we can eliminate r to solve for the ratio of ф to /3. This result, which holds for any a, is

ф _ a + 0.513

J~~ 0.766o-2 + 2.150- – 0.0180

rlj3 can then be obtained from any one of the equations. Using Equation 10.27a

^ = -(1.007a – + 0.0131) – & (0.00129a – – 0.0180) (10.30)

Г* г*

Roll Mode

For cr = -2.79, фіі8 and rip become

f=-42.3
г188

Since r — ф, we can replace r by стф, so that

£–0.674

or

For this mode, it is seen that both /3 and ф are small compared to ф. Thus, this mode is predominantly a damped rolling motion. Indeed, if one neglects all but the ф terms in Equation 10.27b, a value for <r of —2.80 that is very close to the exact value is obtained immediately.

The time for the roll rate to damp to half of its initial value can be found from Equation 9.72 to be

For this mode, the motion is seen to be predominantly a heading change with a small roll angle and sideslip angle. With o – being positive, these angles increase with time, so the mode is actually unstable! With ф increasing exponentially with time, the flight path of the airplane describes a spiral. Thus, this mode is referred to as the spiral mode. If it is unstable, as in this case, the motion is referred to as spiral divergence; otherwise, it is referred to as spiral convergence.

The time to double amplitude is found from

_ln 2

Tdbl —

<T

= 330 air sec

or

Tdbi = 30.2 sec

This time is characteristic of many aircraft and is sufficiently long so that the pilot compensates for the divergence without realizing it. Although spiral divergence cannot be described as unsafe, it can result in extreme attitudes if the pilot should be studying a chart and forgets to fly the airplane for a few moments. It can prove catastrophic for the noninstrument-rated pilot who finds herself or himself in instrument conditions.

The root for the spiral mode is normally small, so it can be closely approximated by the constant term in the characteristic equation divided by the coefficient of a to the first power. From Equation 10.17, the determinant defining the characteristic equation is:

When this determinant is expanded for typical values of the stability deriva­tives, one obtains approximately

2n(Cl0CN. – СщСч)<т = C^Cn’Q, – ClfCNf) or

Equation 10.31 neglects terms in a of order higher than the first. It also neglects some first-order terms in a that are typically small. For the Cherokee, this approximation for the root of the spiral mode gives a value of 0.00227 that is 8% higher than the exact value.

The denominator of Equation 10.31 is usually positive, so the com­bination of terms in the numerator governs whether or not the spiral mode

is stable. For spiral convergence,

Сц0Сір < C^Cn. (10.32)

Since most of the contribution to Ctf results from the wing, this derivative is not too easily adjusted. Varying the vertical tail size will change and CNf approximately in the same proportion. Also, the vertical tail size is normally fixed by other considerations. Hence the primary control on the spiral mode is exercised through Ch, the dihedral. Increasing the dihedral effect will tend to make the spiral mode more stable. However, as stated previously, too much dihedral leads to an unpleasant feel to the airplane.

LATERAL-DIRECTIONAL EQUATIONS FOR THE CHEROKEE 180

The three equations governing the lateral-directional motion and control of the Cherokee 180 are obtained by substituting the calculated stability and control derivatives into Equation 10.17. This reduces to:

30.4/3 + 0.396/8 + 0.039<b – 0.543$ + 30.2r = 0.117 Sr (10.27a)

0.0993/3 + 0.153$ + 0.429$ – 0.198r = 0.0531 Sa + 0.0105 8r (10.27b)

-0.0672/8 + 0.0735$ + 1.18f + 0.0873r = -0.0509 8r (10.27c)

The characteristic equation for this set of simultaneous, linear differential equations is obtained from the determinant.

+(0.039a-0.543) 30.2

(0.153a-2 + 0.429a) -.198 + 0.0735a – (1.18a-+0.08.73)

This determinant reduces to the following.

a4 + 2.90a3 + 0.381a2 + 0.215a – 0.000454 = 0

Since the constant is negative, it is obvious that Equation 10.28 will have a positive real root. Thus the Cherokee is predicted to possess at least one mode of the lateral-directional motion that is unstable.

Typical of lateral-directional motion, Equation 10.28 has two real roots and a pair of complex roots. The real roots can be found from trial and error,

by graphical means, or otherwise to equal

<r, = -2.79 0-2 = 0.00210

(cr — cr,) can be divided into Equation 10.28 (simply follow the same procedure as is done in a long division problem) to obtain a cubic. The cubic is then divided by (cr — cr2) to obtain a quadratic, which can be solved for the pair of complex roots

0-3 =-0.055 + 0.277/ o-4= -0.055 -0.277 і