Post on 15-Oct-2021
Principles of Flight – Modular ATPL(A) Course
1
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
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Principles of Flight – Modular ATPL(A) Course
2
Review of Subsonic Aerodynamics
Properties of fluidState 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
const.=TpV
Properties:• Clasification: fluid – liquid
\ gas
• Continuum
• Speed of sound – alongitudinal wave motion
ρκ=κ= p
RTa vpv
p ccRc
c−===κ 1.4
a0 = 340 m/s = 1225 km/h = 1117 ft/s = 661 kts = 761 mph
Principles of Flight – Modular ATPL(A) Course
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Properties of fluid:• Viscosity
dynamic viscosity η
dydvη=τ
η = η(T) insensitive to changes in pressure
η0 ≈ 1.8⋅10-5 Pa⋅s air
η0 ≈ 1.1⋅10-3 Pa⋅s water
kinematic viscosity ν
ρη=ν
ν0 ≈ 1.46⋅10-5 m2/s air
ν0 ≈ 1.14⋅10-6 m2/s water
• Compressibility χ
volumespecific 1
1
ρ=υυ
υ−=χ
dpd
dpdρ
ρ=χ 1
dpd ρχ=ρ change in pressure dp results in change of density dρ
p
v + dv v
p + dp
Principles of Flight – Modular ATPL(A) Course
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Fluid mechanicsBuoyancy:The principle of Archimedes
Continuity equation:Physical principle: Mass can be neither created nor destroyed
streamtube a along const. =ρ= AVm n&
Momentum equation:Physical principle: Force = time rate of change of momentumMomentum 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 andincompressible flow Bernoulli equation can be derived
const.21 2 =ρ+ρ+ gzVp
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 ν
= lV Re
Principles of Flight – Modular ATPL(A) Course
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Basic (two dimensional) airfoil theory• Airfoil terminology
• Lift generation
• Kutta-Joukowski condition
• Pressure distributionResultant aerodynamic forceCenter of pressureAerodynamic center
• Airfoil stallThin airfoil stallLeading edge stallRear stall
• Effect of Re, airfoil thickness, chamber
• High lift devicesTrailing edge flap: flapLeading edge flap: slat
Principles of Flight – Modular ATPL(A) Course
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Wing• 3-dimensional flow
Induced dragDownwashLift distribution along spanEffect of aspect ratio on lift and drag characteristicEffect of aspect ratio, sweep and twist on lift distribution along spanWinglets
Airplane
• Arrangement of surfacesTailless airplaneConventionalTandemCanard (tail first)
• Lift and drag characteristics
• Propulsion
Principles of Flight – Modular ATPL(A) Course
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Wake turbulence
Principles of Flight – Modular ATPL(A) Course
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Transonic Aerodynamics
• Speed of sound – a
ρκ=κ= p
RTa a0 = 340 m/s = 1225 km/h = 661 kts at 15°C
Average molecular velocity = RTπ8 ≈ 460 m/s = 1650 km/h = 890 kts = 1025 mph
Influence of temperature and altitudeH [m] 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
• Mach numberFlight Mach number
avTAS=Ma a - local speed of sound
Local Mach number
L
LL a
v=Ma aL, vL - speed of sound and speed of flow at point
290
300
310
320
330
340
0 5000 10000
a [m
/s]
Principles of Flight – Modular ATPL(A) Course
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const. FL and VCAS
varying T } no change in Ma
given Mavarying altitude } VTAS = Ma⋅a
Variation of Ma at varying altitude in the standard atmosphere with constant VCAS and VTAS
VCAS = 100 m/s VTAS = 100 m/sH [m] T [K] a [m/s] p [Pa] ρ [kg/m3] ρ/ρ0 VTAS Ma Ma
0 288 340 101325 1.2259 1 100 0.294 0.2941000 281.5 336 89863 1.1123 0.907 105 0.312 0.2975000 255.5 320 53983 0.7362 0.601 129 0.403 0.312
10000 223 299 26397 0.4124 0.336 172 0.576 0.334
Tro
posp
here
11000 216.5 295 22594 0.3636 0.297 184 0.623 0.33915000 216.5 295 12015 0.1934 0.158 252 0.854 0.33920000 216.5 295 5456 0.0878 0.072 374 1.267 0.339
Stra
tosp
here
0.29
0.30
0.31
0.32
0.33
0.34
0.35
0 5000 10000 15000 20000
H [m]
Ma
(V
= 1
00 m
/S)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 5000 10000 15000 20000
H [m]
Ma
(V
= 1
00 m
/s)
0
70
140
210
280
350
420
490
V (
V =
100
m/s
)Ma
VTAS
Principles of Flight – Modular ATPL(A) Course
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• Compressibility χ
volumespecific 1
1
ρ=υυ
υ−=χ
dpd ⇒
dpdρ
ρ=χ 1
dpd ρχ=ρ change in pressure dp results in change of density dρ
Isentropic variation of density, pressure and temperature with Mach number
Ma = 1
11
2
0
Ma2
11
−κ−
−κ+=
ρρ 634.0
0
=ρρ∗
12
0
Ma2
11
−κκ−
−κ+=
pp 528.0
0
=∗
pp
12
0
Ma2
11
−
−κ+=
TT 833.0
0
=∗
TT
Isentropic variation of density Mach number
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Ma
ρ/ρ0
5% variation
Principles of Flight – Modular ATPL(A) Course
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• Subdivision of aerodynamic flow – distinction based on the Mach numberSubsonic (Ma < 0.8) – the airflow around the airplane is completely below the
speed of soundTransonic (0.8 < Ma < 1.2) – the airflow around the airplane is partially subsonic
and partially supersonicSupersonic (Ma > 1.2) – the airflow around the airplane is completely above the
speed of sound but below hypersonic speedHypersonic (Ma > 5) – the airflow around the airplane is at very high supersonic
speeds, leading to stronger shock waves and high temperaturesbehind it – viscous interactions and/or chemically reacting effectsbegin to dominate the flow
0 1 2 3Mach number (Ma)
Den
sity
cha
nges
unim
porta
nt
Den
sity
cha
nges
impo
rtant
Shoc
k w
ave
appe
ar
Shoc
k sy
stem
fully
deve
lope
d
Kin
etic
hea
ting
effe
cts
impo
rtant
SUBSONIC TRANSONIC SUPERSONIC
Principles of Flight – Modular ATPL(A) Course
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• Propagation of pressure waves
at at
at
Vt
Vt Vt = at
θ
a) b)
c) d)shockwave
shockwave
zone ofsilence
zone ofaction
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 θ=θ
= cosecsin
1Ma
Principles of Flight – Modular ATPL(A) Course
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• Shock wave formation on wingsincreasing 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� �QDUHGLWL�SURJH��þUWH���(UQVW�0$&+
• 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.
Principles of Flight – Modular ATPL(A) Course
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Normal shock waves
1 2
Ma1 > 1 Ma2 < 1
p1
ρ1
T1
p2 > p1
ρ2 > ρ1
T2 > T1
V1 V2 < V1
sho
ck w
ave
( )[ ]( ) 2/1Ma
Ma2/11Ma 2
1
212
2 −κ−κ−κ+=
( )1Ma1
21 2
11
2 −+κκ+=
pp
( )( ) 2
1
21
1
2
Ma12Ma1
−κ++κ=
ρρ
( ) ( )( ) 2
1
212
11
2
Ma1Ma12
1Ma1
21
+κ−κ+
−
+κκ+=
TT
Ma1 Ma2 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
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
8 0.39 74.5 5.57 13.39
9 0.39 94.3 5.65 16.69
10 0.39 116.5 5.71 20.39
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 2 3 4 5 6 7 8 9 10
Ma1
Ma2
0
2
4
6
8
10
12
14
16
18
20
p2/
p1, r
2/r1
, T2/
T1
Ma2
p2/p1
r2/r1
T2/T1
Principles of Flight – Modular ATPL(A) Course
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Effects of shock wavesShock wave is an extremely thin region (order of 10-4 mm) across whichthe 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
• 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
} SHOCK STALL
Principles of Flight – Modular ATPL(A) Course
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Behavior of Airplane at shock stall- high incidence stall- shock stall
• compressibility correction factor 2Ma1
1
−
• 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
Principles of Flight – Modular ATPL(A) Course
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Height & speed range• speed limitations: - high incidence buffet boundary
- shock stall boundary
• variations of speed limitations with height and weight
High incidence buffet boundarydifference between VEAS and VCAS
• “coffin corner” – coffin ang� �NUVWD��SRORåLWL�Y�NUVWR
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.
Principles of Flight – Modular ATPL(A) Course
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• 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 wavelies 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
Principles of Flight – Modular ATPL(A) Course
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• 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
Principles of Flight – Modular ATPL(A) Course
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Supersonic Aerodynamics
Mach angle
Ma1
sin ==θVa
• the greater the Mach number, more acute the angle θ• compressible flow through convergent-divergent nozzle (Laval nozzle)
In a Contracting Duct In an Expanding Duct
Subsonic FlowFlow acceleratesAir rarefies slightlyPressure falls
Flow deceleratesAir is compressed slightlyPressure rises
Supersonic FlowFlow deceleratesAir is compressedPressure rises
Flow acceleratesAir is rarefiedPressure falls
at
Vt
θ
shockwave
directionof flight
Principles of Flight – Modular ATPL(A) Course
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• 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
V1 Ma1
V2 Ma2
w2
w1u1
u2
β
θ
V1
V2
w1 = w2
Oblique shock geometry
Principles of Flight – Modular ATPL(A) Course
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• 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 thewing- 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
Principles of Flight – Modular ATPL(A) Course
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Airplane StabilityDefinitions:EquilibriumA body is in static equilibrium when it is in a state of rest of uniform motion in a straightline 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 thecurved 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 stabilityStatic stability is concerned with the forces and moments produced by a small disturbancefrom the condition of equilibrium. It determines whether or not the body will initially tendto return, of its own accord, towards the equilibrium condition, once the disturbance isremoved.
• 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:
edisturbanc theof Magnitudeedisturbanc theofresult a as producedeffect Restoring
Principles of Flight – Modular ATPL(A) Course
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Dynamic stabilityDynamic stability is concerned with the subsequent behaviour of a body which possessesstatic stability. The motion consists of either oscillations about the equilibrium position oraperiodic 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 threemutually perpendicular axes
• airplane must be controllable
• stability and control are closely related
Assumptions- rigid airplane- conventional arrangement of surfaces
Principles of Flight – Modular ATPL(A) Course
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System of axes
C.G.
N, R
M, Q
L, P
z, Z, w
y, Y, v
x, X, u
vrtenje okrog:Y]GROåQH�RVL�valjanje (ang. roll; nem. rollen)RNURJ�QDYSLþQH�RVL�sukanje (ang. yaw; nem. gieren)SUHþQH�RVL�����"������DQJ��SLWFK��QHP��QLFNHQ�
axis Linearvelocities
Aerodynamicforces
Angularvelocities
Aerodynamicmoments
Moment ofinertia
Angulardisplacement
s
Ox u X p L Ix φ
Oy v Y q M Iy θ
Oz w Z r N Iz ψ
Principles of Flight – Modular ATPL(A) Course
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Stability and control are analysed in three planes:
MOTION STABILITY
Pitch Longitudinal
Yaw Directional
Roll Lateral
Airplane longitudinal static stability• pitch motion
e
d
Cm0
Cm
α 0
Unbalanced andunstable
Unbalanced andstablea
bCm0
Cm
c0
A B
C
α
Balanced andstable
Balanced butunstable
Principles of Flight – Modular ATPL(A) Course
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Possible arrangement of wing and tail surfaces
pozitivna ukrivljenost
Cm0 < 0
simetriþQL�SURILOCm0 = 0
negativna ukrivljenost
Cm0 > 0
Krilo s pozitivno ukrivljenostjo pri CZ=0
Krilo s pozitivno ukrivljenostjo pri CZ=0
a)
b)
Višinski rep s CZ>0
Višinski rep s CZ<0
MS
MS
Wing contributionZk
αk
M0k
XklSAT
aerodinami
� � �
center SAT
srednja aerodinami
� ��
tetiva krila – SAT
Principles of Flight – Modular ATPL(A) Course
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αk
Zk
XkM0kMS
V
Aerodinami
�� �
center SAT
Srednjaaerodinami
�� �
tetiva krilahnk lSAT
hlSAT
lSAT
zlSAT
sinαk ≅ αk , cosαk ≅ 1
( )( )....
....
cacammk
caZcammk
hhaCC
hhCCC
−α+=−+=
Fuselage contribution
Vsinα
a)
b)
Principles of Flight – Modular ATPL(A) Course
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Tail contribution
V
αkt
MS
V
V’ ε
zh
αh
Zh
Xh
Mach
ih
αkt-ε
Srednja aerodinami
� tetiva višinskega repa
Aerodinami
� �� � �
srednje aerodinami
�
tetive višinskega repa
xh
Srednja aerodinami
�
tetiva krila (SAT)
( )hkhhhhhmh iiaVaVC +ε−−αη−=αη−=
Pitch moment of complete airplane
( ) ( ) DmFmhkhhcaamfusmm CCiiaVhhaCCC ...c. +++ε−−αη−−α++=
Balance or equilibrium: Cm = 0
Static stability: 0<∂∂
z
m
CC
or 0<α∂
∂ mC
Neutral point: N0 = hn
( )nm hha
C −=α∂
∂
Principles of Flight – Modular ATPL(A) Course
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Cm
α
Cm0
0
masno središþHzadaj
masnosredišþH�VSUHGDM
h = hn
h > hn
h < hn
Cm = Cm0 + a(h-hn)α
9SOLY�OHJH�PDVQHJD�VUHGLãþD�QD�JUDGLHQW�NROLþQLND�PRPHQWDPitch control
Višinskistabilizator
Višinskokrmilo
Šarnirna oskrmila lb
lhk TrimerŠarnirna ostrimerja
yh
A
A
Šarnirna oskrmila
Šarnirna ostrimerja
lb lhk
lh
b)
a)
Principles of Flight – Modular ATPL(A) Course
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višinski stabilizator
δh
a)
b)
c)
0
Cm
za
�� �� �
αuravnote
�� � ��
δh = 0
δh > 0kon
� � �
αuravnote
�� � ��
α
CZ
δh > 0
δh = 0za
�� �� � �� � � � �� �� �
to
� �� � �� �
kon
� � �
RT∆CZ
0 α
višinsko krmilo
Vpliv odklona višinskega krmila na Cm in CZ: a) pozitiven odklon krmila,b) diagram Cm - α, c) diagram CZ - α
Principles of Flight – Modular ATPL(A) Course
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višinskistabilizator
višinskokrmilo
šarnirna oskrmila V
V
α
δh
b)
a)
Porazdelitev normalne sile na višinskem repu pri:a) spremembi vpadnega kota α ob δh = 0; b) odklonu krmila δh ob α = 0
αh
δleb
V
Floating elevator
Principles of Flight – Modular ATPL(A) Course
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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. limitThe permissible aft C.G. limit is determined by the stability considerations. It is based onthe location of the stick-free neutral point h’n when manual controls are employed, and onthe stick-fixed neutral point hn if the elevator control is irreversible. Conservative practice isto keep the aft limit a small distance forward of the computed relevant neutral point due tothe effects of wing flaps, the propulsive system, aeroelastic deformation and to provide safehandling characteristic.
Principles of Flight – Modular ATPL(A) Course
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The forward C.G. limitAs 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 istherefore based on the control considerations and may be determined by one of thefollowing 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 and4. the elevator angle required to land shall not exceed maximum up elevator.
Airplane directional static stability
Sideslip
0>β∂
∂ nC
V
N
y
x β
Principles of Flight – Modular ATPL(A) Course
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Airplane lateral static stability
G
Vzgon
φ
L
y
MS
z γ
Sile na letalo v nagibu
Ravnina tetive krilay
Vx
Vy
Vz
Vn
xz
Komponente
hitrosti letala
β γ
Vpliv diedra oz. V-loma krila na vpadni kot krila
Principles of Flight – Modular ATPL(A) Course
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Vβ
Nizkokrilnik
Visokokrilnik
9SOLY�WUXSD�QD�XþLQHN�GLHGUD���Clβ
V V
Vn Vn
Λ
β
V
9SOLY�SXãþLFH�NULOD�QD�XþLQHN�GLHGUD
V
MS
zv
aerodinami
! "# $ ! % $&
smernega repa
Vpliv smernega repa na Clβ
Principles of Flight – Modular ATPL(A) Course
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Rigid Airplane Dynamic Stability
Equations of motion for rigid airplane (6 DOF)
• for inertial reference frame
dtvd
mF cv
v
=dthd
Gv
v
=
• for airplane-fixed reference frame
ωx
z
y
iv
kv
jv
P
ivdtid
P
vvv
v
×ω==
ω
cc vmtv
mF vvv
v×ω+
δδ= h
dthd
Gvv
vv
×ω+=
Symmetrical airplane assumption
• longitudinal dynamic stability (pitch)
• lateral-directional dynamic stability (roll-yaw)
Principles of Flight – Modular ATPL(A) Course
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Small disturbance theory
vv
vv
FFu
uF
uuF
F δ∆
δ∆∂
∂+δ∆
δ∆∂
∂++
∂∂+
∂∂=∆ &&
&&&
&L&
&0000
Stability derivatives
KKK
KKK
1
1
1
1
1
1
000
000
∂∂=
∂∂=
∂∂=
∂∂=
∂∂=
∂∂=
rN
IN
wM
IM
pL
IL
wZ
mZ
vY
mY
uX
mX
zr
yw
xp
wyu
Linearised system of equations:
• eigenvalues, eigenvectors
Aperiodic motion
• first order linear differential equation
Oscillatory motion
• second order linear differentialequation
0=++ xmk
xmd
x &&&
02 200 =ω+δω+ xxx &&&
• PIOTime
Am
plitu
de
Time
Am
plit
ud
e
Principles of Flight – Modular ATPL(A) Course
39
Airplane Longitudinal Dynamic Stability – 2 oscillatory modesPhugoid mode
u1 ≈ 0.85 θ1
α1 ≈ 0.02 θ1
(ni viden)
θ1
Re
Im
ω
Vector diagram of phugoid mode
x’
x
x
x' – u0t
a)
b)
Phugoid motion path in (a) fixed reference frame (b) moving reference frame
Principles of Flight – Modular ATPL(A) Course
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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 fallingcorresponding to an exchange between the kinetic and the potential energy
• long period (T ≈ 2min) and lightly damped mode (Nhalf = 2)
Short-period mode
Reu2
(ni viden)θ2
α2
ω2
Im
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)
Principles of Flight – Modular ATPL(A) Course
41
Short-period motion path
• Root locus plot
Puš
' ()* +, -* ./ 01 * 23 2* 1 ( + , 4* 5*
korenov kratkoperiodi
'6 * / 7 4 ( +* 3 2 (
pomikanju masnega središ
' , 4* 8 , 4 , 6 ,9 , .Root locus plot of short-period motion
Puš
: ;<= >? @= AB
smer pomika legekorenov fugoidneoblike pripomikanju masnegasrediš
: ? C= D ? C ? E?F ? A
Tretja oscilatorna oblika
Root locus plot of phugoid motion
Principles of Flight – Modular ATPL(A) Course
42
Phugoid motion summary:• the motion is approximately one of constant total energy, the raising and falling
corresponding to an exchange between the kinetic and the potential energy
• change of angle of attack is negligible – velocity of airplane is approximately tangent tothe path
• long period and lightly damped mode
• moving CG back lowers static stability and consequently reduces frequency of thephugoid mode
• increase in equivalent airspeed reduces frequency of the phugoid mode
• at higher altitude the damping of the phugoid mode is reduced
Short-period motion summary:• the motion is approximately pure oscillatory pitch motion of the airplane
• negligible speed variation, short period and highly damped motion
• as for the phugoid mode, shifting the CG back lowers static stability (aerodynamicstiffness) and therefore reduces frequency of the short-period motion
• damping and frequency of the short-period mode are proportional to the equivalentairspeed
• with increasing altitude the damping of the short-period mode is reduced
• motion should be considerably damped in order to prevent PIO
Principles of Flight – Modular ATPL(A) Course
43
Airplane Lateral–Directional Dynamic Stability
– 2 aperiodic modes and oscillatory mode
Roll mode• very heavily damped, almost pure single DOF rolling motion
• damping is reduced with decrease in airspeed and increase in altitude
• CG position has no effect on roll motion
• it is very important to determine the roll response characteristic of the airplane
Time
p
Variation of roll rate p with time for pure rolling motion
Spiral motion• usually weakly damped motion in bank and yaw, with negligible sideslip
– approximately a correctly banked turn of increasing radius; the airplane flies along aslightly curved path and approaches initial heading
• often this mode is unstable; the path of motion of the airplane is then a tightening spiral– approximately a correctly banked turn of decreasing radius (graveyard spiral)
• due to large time to double/half the amplitude, there is no quantitative standard of spiralstability; however, time to double the amplitude should exceed 20 sec
Principles of Flight – Modular ATPL(A) Course
44
• effect of fin and dihedral
• increase in airspeed (decrease of AOA) increases stability of the spiral mode
• CG position does not effects the damping of the mode
• spiral divergence vs. directional divergence
ψ
divergentna
spiralna oblika
asimptota
y’
x’
Dutch Roll oscillation• Dutch Roll motion consists of a relatively short period oscillations, which may be either
damped or divergent, involving rolling yawing and sidesliping motions
• roll/yaw ratio is important characteristic of Dutch Roll because it affects the pilot‘sassessment of the handling qualities
• snaking – the motion consists mainly of yawing
Principles of Flight – Modular ATPL(A) Course
45
Re
ψ
φ
β
Im
ω
Vector diagram of Dutch Roll mode
Sketch of Dutch Roll motion
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Principles of Flight – Modular ATPL(A) Course
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• Increase in equivalent airspeed increases frequency of Dutch Roll motion
• At higher altitudes damping of the Dutch Roll motion reduces considerably (yaw damper)
Effects on Dutch Roll motion
• Increase in dihedral stability
G
slightly increase frequency
G
decrease damping
G
increase roll/yaw ratio
• Increase in weathercock stability
G
increase frequency
G
increase damping
G
decrease roll/yaw ratio
Principles of Flight – Modular ATPL(A) Course
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Control balance
• aerodynamic balance of controls
• mass balance of controls
Modification of directional stability characteristicsdorsal (zgoraj) fin – increase in fin stall angleventral (spodaj) fin – increase in fin area effects stability in stall characteristics
Inertial coupling