Principles of Flight

Principles of Flight – Modular ATPL(A) Course

PRINCIPLES OF FLIGHT Contents: • Review of subsonic aerodynamics • Transonic aerodynamics • Supersonic aerodynamics • Airplane performance • Airplane stability Literature: Richard Bowyer: AERODYNAMICS FOR THE PROFESSIONAL PILOT Charles E. Dole: FLIGHT THEORY FOR PILOTS A.C. Kermode: MECHANICS OF FLIGHT, revised by R.H. Barnard, D.R. Philpot R.H. Barnard, D.R. Philpot: AIRPLANE FLIGHT D. Stinton: THE DESIGN OF THE AEROPLANE J.D. Anderson: FUNDAMENTALS OF AERODYNAMICS W.N. Hubin: THE SCIENCE OF FLIGHT H.C. Smith: THE ILLUSTRATED GUIDE TO AERODYNAMICS =5HQGXOLþMEHANIKA LETA

1

Principles of Flight – Modular ATPL(A) Course

Review of Subsonic Aerodynamics Properties of fluid State variables: • Temperature

T

[°C, °F, K]

• Pressure

p

[N/m2 = Pa, bar, atm]

• Density

ρ

[kg/m3]

Equation of state for perfect gas: p = ρRT

R = 287 J/kgK

pV = const. T

Properties: • Clasification: fluid – liquid \ gas • Continuum • Speed of sound – a longitudinal wave motion a = κRT = κ

p ρ

κ=

cp cv

= 1.4

R = c p − cv

a0 = 340 m/s = 1225 km/h = 1117 ft/s = 661 kts = 761 mph 2

Principles of Flight – Modular ATPL(A) Course

Properties of fluid: • Viscosity η

dynamic viscosity τ=η

dv dy

η = η(T)

insensitive to changes in pressure

η0 ≈ 1.8⋅10-5 Pa⋅s

air

η0 ≈ 1.1⋅10-3 Pa⋅s

water ν

kinematic viscosity ν=

η ρ

ν0 ≈ 1.46⋅10-5 m2/s

air

ν0 ≈ 1.14⋅10-6 m2/s

water

• Compressibility χ χ=− χ=

1 dυ υ dp

υ=

1 specific volume ρ

1 dρ ρ dp

dρ = ρχdp

change in pressure dp results in change of density dρ p

p + dp v

v + dv

3

Principles of Flight – Modular ATPL(A) Course

Fluid mechanics Buoyancy: The principle of Archimedes

Continuity equation: Physical principle: Mass can be neither created nor destroyed m& = ρVn A = const. along a streamtube

Momentum equation: Physical principle: Force = time rate of change of momentum Momentum equations for a viscous flow: Navier–Stokes equations Momentum equations for an inviscid flow: Euler equations

After integration of Euler equations along a streamline for the inviscid and incompressible flow Bernoulli equation can be derived 1 p + ρV 2 + ρgz = const. 2

Energy equation: Physical principle: Energy can be neither created nor destroyed; it can only change in form

Types of flow: • laminar flow • turbulent flow Reynolds number

Re =

Vl ν

4

Principles of Flight – Modular ATPL(A) Course

Basic (two dimensional) airfoil theory • Airfoil terminology

• Lift generation • Kutta-Joukowski condition • Pressure distribution Resultant aerodynamic force Center of pressure Aerodynamic center

• Airfoil stall Thin airfoil stall Leading edge stall Rear stall

• Effect of Re, airfoil thickness, chamber • High lift devices Trailing edge flap: flap Leading edge flap: slat

5

Principles of Flight – Modular ATPL(A) Course

Wing • 3-dimensional flow Induced drag Downwash Lift distribution along span Effect of aspect ratio on lift and drag characteristic Effect of aspect ratio, sweep and twist on lift distribution along span Winglets

Airplane • Arrangement of surfaces Tailless airplane Conventional Tandem Canard (tail first)

• Lift and drag characteristics • Propulsion

6

Principles of Flight – Modular ATPL(A) Course

Wake turbulence

7

Principles of Flight – Modular ATPL(A) Course

Transonic Aerodynamics • Speed of sound – a a = κRT = κ

p ρ

a0 = 340 m/s = 1225 km/h = 661 kts at 15°C

Average molecular velocity =

8 RT ≈ 460 m/s = 1650 km/h = 890 kts = 1025 mph π

Influence of temperature and altitude T [K]

a [m/s]

a/a0 [%]

0

288

340

100

1000

281.5

336

99

2000

275

332

98

3000

268.5

328

97

4000

262

324

95

5000

255.5

320

94

10000

223

299

88

11000

216.5

295

87

20000

216.5

295

87

340 330

a [m/s]

H [m]

320 310 300 290 0

5000

• Mach number Flight Mach number vTAS a - local speed of sound a Local Mach number Ma =

Ma L =

vL aL

aL, vL - speed of sound and speed of flow at point

8

10000

Principles of Flight – Modular ATPL(A) Course const. FL and VCAS varying T

} no change in Ma

given Ma varying altitude

}V

TAS

= Ma⋅a

Variation of Ma at varying altitude in the standard atmosphere with constant VCAS and VTAS VCAS = 100 m/s ρ [kg/m3] 1.2259 1.1123 0.7362 0.4124 0.3636 0.1934 0.0878

1.40

0.34

1.20 = 1 0 0 m /s)

0.35

0.33 0.32 0.31 0.30

ρ/ρ0 1 0.907 0.601 0.336 0.297 0.158 0.072

VTAS 100 105 129 172 184 252 374

VTAS = 100 m/s

Ma 0.294 0.312 0.403 0.576 0.623 0.854 1.267

490 420 Ma

1.00

350

VTAS

0.80

280

0.60

210

0.40

140

0.20

70

0.00

0.29 0

5000

10000

15000

0

20000

5000

10000 H [m]

H [m]

9

Ma 0.294 0.297 0.312 0.334 0.339 0.339 0.339

15000

0 20000

= 1 0 0 m /s)

p [Pa] 101325 89863 53983 26397 22594 12015 5456

(V

a [m/s] 340 336 320 299 295 295 295

V

T [K] 288 281.5 255.5 223 216.5 216.5 216.5

M a (V

Ma (V

= 100 m/S)

Stratosphere

Troposphere

H [m] 0 1000 5000 10000 11000 15000 20000

Principles of Flight – Modular ATPL(A) Course • Compressibility χ χ=−

1 dυ υ dp

dρ = ρχdp

υ=

1 1 dρ specific volume ⇒ χ = ρ ρ dp

change in pressure dp results in change of density dρ

Isentropic variation of density, pressure and temperature with Mach number Ma = 1 ρ  κ −1  = 1 + Ma 2  ρ0  2 

−

1 κ −1

ρ∗ = 0.634 ρ0

−

κ κ −1

p∗ = 0.528 p0

p  κ −1  = 1 + Ma 2  p0  2 

T  κ −1  Ma 2  = 1 + 2 T0  

−1

T∗ = 0.833 T0

Isentropic variation of density Mach number 1

5% variation

0.8

ρ/ρ0

0.6 0.4 0.2 0 0

0.2

0.4

0.6

Ma

10

0.8

1

Principles of Flight – Modular ATPL(A) Course • Subdivision of aerodynamic flow – distinction based on the Mach number Subsonic (Ma < 0.8) – the airflow around the airplane is completely below the speed of sound Transonic (0.8 < Ma < 1.2) – the airflow around the airplane is partially subsonic and partially supersonic Supersonic (Ma > 1.2) – the airflow around the airplane is completely above the speed of sound but below hypersonic speed Hypersonic (Ma > 5) – the airflow around the airplane is at very high supersonic speeds, leading to stronger shock waves and high temperatures behind it – viscous interactions and/or chemically reacting effects begin to dominate the flow

0

Kinetic heating effects important

SUPERSONIC

Shock system fully developed

Shock wave appear

Density changes important

Density changes unimportant

SUBSONIC TRANSONIC

1 2 Mach number (Ma) 11

3

Principles of Flight – Modular ATPL(A) Course

• Propagation of pressure waves

a)

b)

at

at

Vt

c)

d)

shock wave

shock wave

at

θ

zone of silence

Vt = at

zone of action

Vt

a) body hardly moving Ma ≈ 0; b) Speed about Ma = 0.5; c) Speed Ma = 1.0 Body has caught up with its pressure waves; d) Body moving about Ma = 1.9 Angle θ related to Ma by Ma =

1 = cosec θ sin θ

12

Principles of Flight – Modular ATPL(A) Course • Shock wave formation on wings increasing flight Ma – – – –

transition point flow breakaway local Mach number MaL = 1.0 incipient shock wave – usually near the point of maximum chamber (max. speed) – approximately normal to the surface – pressure and temperature rise, decrease of speed of flow – tendency for a breakaway and turbulent wake

• Observation of shock waves – light travels more slowly through denser air – rays bending towards higher density – „schlieren method“

schlierennem = streaking, striationang QDUHGLWLSURJHþUWH(UQVW0$&+

• Critical Mach number Macr various definition – flight Mach number at which the local airflow at some point reaches the speed of sound – flight Mach number at which the first shock wave is formed – flight Mach number at which severe buffeting begins (buffet boundary) – flight Mach number at which the drag coefficient begins to rise – flight Mach number at which the pilot loses the control Once Macr is exceeded, the airplane is flying in the transonic speed range.

13

Principles of Flight – Modular ATPL(A) Course

Normal shock waves 1

Ma 22 =

2 Ma2 < 1 shock wave

Ma1 > 1 p1 ρ1 T1 V1

1 + [(κ − 1) / 2]Ma 12 κ Ma12 − (κ − 1) / 2

p2 2κ = 1+ Ma 12 − 1 p1 κ +1

(

p2 > p1 ρ2 > ρ1 T 2 > T1

)

(κ + 1) Ma12 ρ2 = ρ1 2 + (κ − 1) Ma 12

V2 < V1

2 2κ T2   2 + (κ − 1) Ma 1 Ma12 − 1  = 1 + 2 T1  κ + 1  (κ + 1) Ma 1

(

p2/p1

ρ2/ρ1

T2/T1

1

1

1

1

1

2

0.58

4.5

2.67

1.69

3

0.48

10.3

3.86

2.68

4

0.43

18.5

4.57

4.05

1

20

Ma2

0.9

18

p2/p1 r2/r1

0.8

16

T2/T1

0.7

14

0.6

12

0.5

10

0.4

8

0.3

6

5

0.42

29.0

5.00

5.80

6

0.40

41.8

5.27

7.94

7

0.40

57.0

5.44

10.47

0.2

4

8

0.39

74.5

5.57

13.39

0.1

2

9

0.39

94.3

5.65

16.69

0

10

0.39

116.5

5.71

20.39

0 1

2

3

4

5

6

Ma1

14

7

8

9

10

p2/p1, r2/r1, T2/T1

Ma2

Ma2

Ma1

)

Principles of Flight – Modular ATPL(A) Course

Effects of shock waves Shock wave is an extremely thin region (order of 10-4 mm) across which the flow properties can change drastically. Shock wave is an almost explosive compression process. At the normal shock wave there is • a great rise in pressure • a considerable rise in temperature • a rise in density • a decrease in speed • V2 is always subsonic • breakaway of the flow from the surface This all adds up to a: • sudden increase in drag (up to 10×) • loss of lift of an airfoil • change in position of center of pressure • change in pitching moment

}

SHOCK STALL

• severe buffeting behind the shock wave Shock drag • energy dissipated in the shock wave – wave drag • increase in profile drag due to breakaway of the flow – boundary layer drag 15

Principles of Flight – Modular ATPL(A) Course

Behavior of Airplane at shock stall - high incidence stall - shock stall • compressibility correction factor

1 1 − Ma 2

• considerable changes in longitudinal trim (usually nose heavy – Tuckunder) • large control forces • buffeting • aileron buzz • loss of control • stability problems: - snaking (yaw) - porpoising (pitch) - Dutch roll

Measures: • machmeter • regions of higher temperature • slow down or accelerate • power controls • air brake

16

Principles of Flight – Modular ATPL(A) Course

Height & speed range • speed limitations: - high incidence buffet boundary - shock stall boundary • variations of speed limitations with height and weight High incidence buffet boundary difference between VEAS and VCAS

• “coffin corner” – coffin ang NUVWDSRORåLWLYNUVWR

Raising the Critical Mach Number •supercritical wing section (Whitcomb) ◊ higher Macr ⇒ higher Madiv (-1965, NACA 64 series) ◊ increment between Macr and Madiv ⇒ supercritical airfoils + relatively flat top – lover MaL + weaker shock wave - flat top – forward 60% of airfoil has negative chamber ⇒ lowers lift extreme positive chamber on the rearward 30% - high Cm a.c. 17

Principles of Flight – Modular ATPL(A) Course

•slimness ◊ smaller increase of local airflow velocity + + + + + -

formation of shock wave is delayed– increasing Macr reduced intensity of shock wave reduced boundary layer separation reduced drag improved longitudinal handling and stability reduced total lift structural problems

•sweepback ◊ component of velocity along span has no effect on the flow across the wing ◊ only the component of the velocity across the cord of the wing is responsible for the pressure distribution and so for causing the shock wave (shock wave lies parallel to the span of the wing) + higher Macr + lower drag slope and peak drag - swept wing has lower CL comparing to straight wing of same chord and α -

tip stall, pitch-up and high induced drag high minimum drag speed additional wing torsion due to lift aeroelastic effects 18

Principles of Flight – Modular ATPL(A) Course

•area rule (Whitcomb) ◊ the area of cross-section should increase gradually to maximum and then decrease gradually

•vortex generators ◊ make the boundary layer turbulent + reduced boundary layer drag + weaken the shock wave and reduce shock drag + vorticity can prevent buffeting

19

Principles of Flight – Modular ATPL(A) Course

Supersonic Aerodynamics shock wave

Mach angle

θ sin θ =

a 1 = V Ma

at

direction of flight

Vt

• the greater the Mach number, more acute the angle θ • compressible flow through convergent-divergent nozzle (Laval nozzle)

Subsonic Flow

In a Contracting Duct

In an Expanding Duct

Flow accelerates Air rarefies slightly Pressure falls

Flow decelerates Air is compressed slightly Pressure rises

Flow decelerates Supersonic Flow Air is compressed Pressure rises 20

Flow accelerates Air is rarefied Pressure falls

Principles of Flight – Modular ATPL(A) Course • supersonic flow over wedge – compressive flow - shock wave angle - change of direction and speed of flow - effect of change of Ma - effect of change of wedge angle • supersonic flow over convex corner – expansive flow

V2 V1

w1

u1

V2 Ma2 θ

V1 Ma1

u2

w2

β

Oblique shock geometry 21

w1 = w2

Principles of Flight – Modular ATPL(A) Course • supersonic flow over airfoil • boundary layer and supersonic flow - boundary layer is relatively unimportant in supersonic flow - supersonic flow can turn sharp corners • relation between supersonic flow over wedge and cone • supersonic wing shapes – plan form - at subsonic speeds the airfoil is more important than the plan form of the wing - but at supersonic speeds the plan form of the wing is more important - sweepback increases Macr - leading edge of the wing lies inside the Mach cone - structural disadvantages of sweepback - tip stalling - rectangular wing at high Ma • supersonic airfoil sections • control surfaces • supersonic engine inlets • aerodynamic (kinetic) heating

22

Principles of Flight – Modular ATPL(A) Course

Airplane Stability Definitions: Equilibrium A body is in static equilibrium when it is in a state of rest of uniform motion in a straight line and the forces acting on it are balanced out. The definition can be extended to cover those bodies in uniform motion in a curved path. There is, in these cases, a resultant force and an acceleration towards the centre of the curved path, but they can be considered as cases of dynamic equilibrium. Stability is property of the equilibrium state and there are two types of stability to consider, static stability and dynamic stability.

Static stability Static stability is concerned with the forces and moments produced by a small disturbance from the condition of equilibrium. It determines whether or not the body will initially tend to return, of its own accord, towards the equilibrium condition, once the disturbance is removed. • a body is statically stable when it tends to return to the equilibrium position • a body is statically unstable when it tends to diverge further away from the equilibrium position • a body possesses neutral static stability when it remains in the disturbed position Degree of static stability possessed by a body:

Restoring effect produced as a result of the disturbance Magnitude of the disturbance 23

Principles of Flight – Modular ATPL(A) Course

Dynamic stability Dynamic stability is concerned with the subsequent behaviour of a body which possesses static stability. The motion consists of either oscillations about the equilibrium position or aperiodic motion. There are once again three possibilities: • a body is dynamically stable when the amplitude reduces with time • a body is statically unstable when the amplitude increases with time • a body possesses neutral when the amplitude remains constant

Airplane stability • airplane is designed mainly from performance considerations, but it must also posses acceptable handling characteristics, if necessary achieved by artificial methods • motion of rigid airplane can be represented as translation along and rotation about three mutually perpendicular axes • airplane must be controllable • stability and control are closely related Assumptions - rigid airplane - conventional arrangement of surfaces

24

Principles of Flight – Modular ATPL(A) Course

System of axes x, X, u

L, P

C.G.

y, Y, v M, Q N, R

z, Z, w

vrtenje okrog: Y]GROåQHRVLvaljanje (ang. roll; nem. rollen) RNURJQDYSLþQHRVLsukanje (ang. yaw; nem. gieren) SUHþQHRVL"DQJSLWFKQHPQLFNHQ

axis

Linear velocities

Aerodynamic forces

Angular velocities

Aerodynamic moments

Moment of inertia

Angular displacement s

Ox

u

X

p

L

Ix

φ

Oy

v

Y

q

M

Iy

θ

Oz

w

Z

r

N

Iz

ψ

25

Principles of Flight – Modular ATPL(A) Course

Stability and control are analysed in three planes: MOTION

STABILITY

Pitch

Longitudinal

Yaw

Directional

Roll

Lateral

Airplane longitudinal static stability • pitch motion Cm

Cm

Balanced and stable

b

Cm0 A

B

0

α

d

Cm0

α

0

c Balanced but unstable

Unbalanced and unstable

C Unbalanced and stable

a

26

e

Principles of Flight – Modular ATPL(A) Course

Possible arrangement of wing and tail surfaces

pozitivna ukrivljenost

simetriþQLSURILO

negativna ukrivljenost

Cm0 < 0

Cm0 = 0

Cm0 > 0

a) MS

Krilo s pozitivno ukrivljenostjo pri CZ=0

Višinski rep s CZ0

Wing contribution


aerodinami center SAT



Zk

M0k Xk

lSAT

srednja aerodinami tetiva krila – SAT

27




αk

Principles of Flight – Modular ATPL(A) Course Zk





Aerodinami center SAT 

Xk

M0k

MS

hlSAT

V

lSAT

sinαk ≅ αk , cosαk ≅ 1

Cmk = Cm a.c. + CZ (h − ha.c. )

Cmk = Cm a.c. + αa(h − ha.c. )

Fuselage contribution a)

b)

Vsinα

28



Srednja aerodinami tetiva krila

hnk lSAT



αk



zlSAT

Principles of Flight – Modular ATPL(A) Course

Tail contribution xh Zh 

αkt-ε



MS Xh





Srednja aerodinami tetiva višinskega repa 

Aerodinami srednje aerodinami tetive višinskega repa













V’

V

Mach

ε

V



αh

αkt

ih



zh





Srednja aerodinami tetiva krila (SAT)

Cmh = −ηVh ah α h = −ηVh ah (α − ik − ε + ih )

Pitch moment of complete airplane

C m = C m fus + C m a.c. + aα(h − ha.c. ) − ηVh a h (α − ik − ε + ih ) + C m F + C m D

Balance or equilibrium:

Cm = 0

Static stability:

∂Cm ∂Cm < 0 or hn

masno središþH zadaj

h = hn

Cm0

Cm = Cm0 + a(h-hn)α 0

Pitch control

α

h < hn

masno središþHVSUHGDM

9SOLYOHJHPDVQHJDVUHGLãþDQDJUDGLHQWNROLþQLNDPRPHQWD

Višinski stabilizator

A

a) Šarnirna os krmila

lb

Višinsko krmilo

lhk

Trimer

yh

Šarnirna os trimerja

A

Šarnirna os krmila

b) lb lh

30

lhk

Šarnirna os trimerja

Principles of Flight – Modular ATPL(A) Course

višinski stabilizator

a) δh višinsko krmilo

Cm δh = 0 0

 





α 















kon α uravnote











za α uravnote



b)

δh > 0

CZ δh > 0 





 





 

















 









kon RT

za to



δh = 0

c)

∆CZ

α

0

Vpliv odklona višinskega krmila na Cm in CZ: a) pozitiven odklon krmila, b) diagram Cm - α, c) diagram CZ - α

31

Principles of Flight – Modular ATPL(A) Course

a)

α V

višinski stabilizator

šarnirna os krmila

višinsko krmilo

b) V

δh

Porazdelitev normalne sile na višinskem repu pri: a) spremembi vpadnega kota α ob δh = 0; b) odklonu krmila δh ob α = 0

δleb αh V

Floating elevator 32

Principles of Flight – Modular ATPL(A) Course

Longitudinal manoeuvring stability Effect of thrust on Effect of elasticity of structure on longitudinal stability Lt

∆αh = -kZh

Sprememba vpadnega kota višinskega repa pri deformaciji trupa

The aft C.G. limit The permissible aft C.G. limit is determined by the stability considerations. It is based on the location of the stick-free neutral point h’n when manual controls are employed, and on the stick-fixed neutral point hn if the elevator control is irreversible. Conservative practice is to keep the aft limit a small distance forward of the computed relevant neutral point due to the effects of wing flaps, the propulsive system, aeroelastic deformation and to provide safe handling characteristic.

33

Principles of Flight – Modular ATPL(A) Course

The forward C.G. limit As the C.G. moves forward, the stability of the airplane increases and larger control movements and forces are required to maneuver the airplane. The forward C.G. limit is therefore based on the control considerations and may be determined by one of the following requirements: 1. the stick-force per g should not exceed a specific value, 2. the stick-force gradient at trim, dP/dV, shall not exceed a specified value, 3. the stick-force required to land, from a trim at the approach speed, shall not exceed a specified value and 4. the elevator angle required to land shall not exceed maximum up elevator.

Airplane directional static stability Sideslip

x

β

V y

N

∂C n >0 ∂β

34

Principles of Flight – Modular ATPL(A) Course

Airplane lateral static stability Vzgon φ

MS

L

z

γ G

Sile na letalo v nagibu Ravnina tetive krila

y

Vn Vx

β

γ Komponente

Vy

hitrosti letala

Vz x z

Vpliv diedra oz. V-loma krila na vpadni kot krila

35

y

Principles of Flight – Modular ATPL(A) Course Visokokrilnik

Vβ

Nizkokrilnik

9SOLYWUXSDQDXþLQHNGLHGUDClβ

β V Vn

V

Vn Λ

%

$&

$!

#

!

aerodinami

"

9SOLYSXãþLFHNULODQDXþLQHNGLHGUD

smernega repa

V

zv MS

Vpliv smernega repa na Clβ

36

V

Principles of Flight – Modular ATPL(A) Course

Rigid Airplane Dynamic Stability Equations of motion for rigid airplane (6 DOF) • for inertial reference frame v v dvc F =m dt

v v dh G= dt

• for airplane-fixed reference frame ω

x P

v i

z

v di v v v = vP = ω × i dt

v k

v j

y

ω

v v dh v v G= + ω× h dt

v v δvc v v F =m + mω × vc δt Symmetrical airplane assumption • longitudinal dynamic stability (pitch)

• lateral-directional dynamic stability (roll-yaw)

37

Principles of Flight – Modular ATPL(A) Course

Small disturbance theory  ∂F   ∂F   ∂F   ∂F  & ∆F =   u+  u + L +  &  ∆δ& v +  &&  ∆&δ&v  ∂u 0  ∂u& 0  ∂∆δv 0  ∂∆δv 0

Stability derivatives Xu =

1  ∂X    m  ∂u 0

Lp =

1 Ix

 ∂L     ∂p 0

K

Yy =

K Mw =

1  ∂Y    m  ∂v 0 1 Iy

 ∂M     ∂w 0

K

Zw =

K Nr =

1  ∂Z    m  ∂w 0

K

 ∂N     ∂r 0

K

1 Iz

Linearised system of equations: Aperiodic motion • first order linear differential equation

Amplitude

• eigenvalues, eigenvectors

Oscillatory motion • second order linear differential

Time

&x& +

d k x& + x = 0 m m

&x& + 2δω0 x& + ω02 x = 0

Amplitude

equation

• PIO Time

38

Principles of Flight – Modular ATPL(A) Course

Airplane Longitudinal Dynamic Stability – 2 oscillatory modes Phugoid mode Im α1 ≈ 0.02 θ1 (ni viden) u1 ≈ 0.85 θ1 Re ω

θ1

Vector diagram of phugoid mode a)

x x’

x

b) x'– u0t

Phugoid motion path in (a) fixed reference frame (b) moving reference frame 39

Principles of Flight – Modular ATPL(A) Course

Phugoid mode • change of angle of attack is negligible (∆α ≈ 0) – velocity of airplane is approximately tangent to the path • the motion is approximately one of constant total energy, the raising and falling corresponding to an exchange between the kinetic and the potential energy • long period (T ≈ 2min) and lightly damped mode (Nhalf = 2)

Short-period mode Im

α2

Re u2

(ni viden)

θ2

ω2

Vector diagram of short-period mode • negligible speed variation (∆u ≈ 0) • the motion is approximately pure oscillatory pitch motion of the airplane • short period (T ≈ 3sec) and highly damped mode (Nhalf = 0.2)

40

Principles of Flight – Modular ATPL(A) Course

Short-period motion path

A

>

@

B

=

:

?