t. c. corke „design of aircraft” aircraft design 1 d.p. raymer...
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Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Horizontal tail – static stability
References
T. C. Corke „Design of Aircraft”
D.P. Raymer „Aircraft Design, a Conceptual Approach”
J. Roskam „Airplane Design”
D. Stinton „The Design of the Aeroplane”
Why we need a horizontal tail?
• to satisfy trim (equilibrium)
conditions
• to satisfy stability• to satisfy stability
Equilibrium problem
• Classical configuration
• Tailless configuration
Equilibrium problem
• Classical
configuration
• Canard
configuration
Equilibrium problem
0)( =−−+=∑ SCHZHSCZYSC xxPxPMM
Equilibrium and stability problem
∂C
0)(2
1
2
1
2
1 222 =−−+=∑ SCHZHHHSCZmbHaSC xxCVSxCSVCCSVM ρρρ
02
12
22 =
−−+=∑
a
SCHZH
HH
a
SCZmbHaSC
C
xxC
V
V
S
S
C
xCCCSVM ρ
)( 0αα −= aCZ where: α∂
∂= ZC
a
KHHZH aaaC δδα 321 ++=
0110111 )1()()( εααε
εααε
αεαα aaaaa H −∂∂
−=−∂∂
−=−=
constconst KH == δδ ,
Equilibrium and stability problem
)1(1
αε
α
α∂∂
−=
∂∂∂∂
=∂∂
a
a
C
C
C
C
Z
ZH
Z
ZH
0)1(1
2
2
=−
∂∂
−−+∂∂
=∂∂ SCHHHSCmbHmSC
C
xx
a
a
V
V
S
S
C
x
C
C
C
C
αε
0)1(2
=∂
−−+∂
=∂ aaZZ CaVSCCC α
0)1()1(1 1
2
2
1
2
2
=∂∂
−−
∂∂
−++∂∂
=∂∂
αε
αε
a
a
V
V
C
x
S
S
a
a
V
V
S
S
C
x
C
C
C
C H
a
HHHH
a
SC
Z
mbH
Z
mSC
Equilibrium and stability problem
)1(1
)1(
1
2
2
1
2
2
αε
αε
∂∂
−+
∂∂
−+∂∂
−==
a
a
V
V
S
S
a
a
V
V
C
x
S
S
C
C
C
XX
HH
H
a
HH
Z
mbH
a
��
Equilibrium and stability problem
Assumption: free stick
0321 =++ KHH bbb δδαThe equilibrium equation for hinge moments:
thus:
3
21
b
bb HHK
δαδ
−−=
or:
2
31
b
bb KHH
δαδ
−−=
Equilibrium and stability problem
if incidence angle is sufficient, the angle of
trimming/balancing tab is equal to:
0=+ bb δα
0=Kδthen we can write:
021 =+ HH bb δαthen the elevator deflection is equal to:
2
1
b
b HH
αδ
−=
Equilibrium and stability problem
The pitching moments equation is the same:
using:
0)( =−−+=∑ SCHZHSCZYSC xxPxPMM
and:
HHZH aaC δα 21 +=
HHH
HZH aab
aba
b
baaC αα
αα '
1
12
211
2
121 )1( =−=−
+=
Equilibrium and stability problem
The neutral point position in free stick case:
)1(1
)1(
'
1
2
'
1
2
2
εαε
∂−+
∂∂
−+∂∂
−==
aVS
a
a
V
V
C
x
S
S
C
C
C
XX
HH
H
a
HH
Z
mbH
a
��
)1(21
121
'
1ba
baaa −=
)1(1 1
2 α∂−+
aVSHHa
where:
How to improve the stability?
• to increase a1 and decrease a2
– comment: it is not realistic, because decreasing of
a2 , because the problem with trimming could
occuroccur
• to increase b2 and decrease b1
– comment: usually increasing of b2
increases b1
How looks stability in canard
configuration
Equilibrium and stability problem
0)( =−+−=∑ SCHZHSCZYSC xxPxPMM
0)(2
1
2
1
2
1 222 =−+−=∑ SCHZHHSCZpmbHapSC xxCVSxCSVCCSVM ρρρ
122 −∑ xxSxVV
02
12
2
2
2
2 =
−+−=∑
a
SCHZH
H
a
SCZ
p
mbH
p
aSCC
xxC
S
S
C
xC
V
VC
V
VCSVM ρ
αααε
1)1( aCaC ZHZ =∂∂
−=
Equilibrium and stability problem
2 − ∂∂ xxaSxCVC
)1(
1
αε
α
α
∂∂
−=
∂∂∂∂
=∂∂
a
a
C
C
C
C
Z
ZH
Z
ZH
0
)1(
1
2
2
=−
∂∂
−+
−
∂∂
=∂∂
a
SCHH
a
SC
Z
mbHp
Z
mSC
C
xx
a
a
S
S
C
x
C
C
V
V
C
C
αε
)1()1(
1
2
2
1
2
2
αε
αε
∂∂
−+
∂∂
=
∂∂
−+
a
a
C
x
S
S
C
C
V
V
a
a
S
S
V
V
C
x
a
HH
Z
mbHpHp
a
�
Equilibrium and stability problem
)1(
)1(
1
2
2
1
2
2
αε
αε
∂∂
−+
∂∂
−+
∂∂
==
a
a
S
S
V
V
a
a
C
x
S
S
C
C
V
V
C
XX
Hp
a
HH
Z
mbHp
a
��
Important remark
• canard configuration is more stable in
free stick case
• it is opposite to classical configuration
Downwash angle
Downwash angle behind wing ε (in the symmetry plane).
Airfoil USA 45; taper ratio 2, Cz = 1,35; λ = 6
Downwash angle behind wing
Downwash behind wing ε in distance from aerodynamic center (0,25c) equal to 1,3 (left) and 3,4 MAC (right)
Downwash behind the canard and decreasing of the
lift coefficient of main wing
Lift distribution in canard configuration (SAAB Viggen)
Stinton
Stinton
Controllability neutral point
• definition:
0=∂Cm
0=∂
∂
n
C�m
Controllability neutral point
V
qCCCSVxxCSVCCSVM a�
mqa�MZ�maM
22
,
2
2
1)(
2
1
2
1ρρρ +−+=∑
V
qCCCSVxxPCCSVM a�
mqa�MZ�maM
2
,
2
2
1)(
2
1ρρ +−+=∑
Controllability neutral point – cont.
2 −
derivative Cmq depends mainly on horizontal tail:
12
2
, aV
V
S
S
C
xxC
A
HH
a
�H�
Hmq
−−=
Controllability neutral point – cont.
the forces equation on “z” direction is:
zy PRmmgnQ =+= 2ωthus:
g
R1n
2
yω=−
Controllability neutral point – cont.
as we know:
RV yω=we obtain:we obtain:
V
ngq y
)1( −==ω
Controllability neutral point – cont.
2
2
,
2 )1(
2
1)(
2
1
V
CngCCSVxxnQCCSV a�
mqa�M�ma
−+−+ ρρ
after differentiation with respect to n
02
1)(
2 =+− �
mqa�M CgSCxxQ ρ
the position of neutral point of controllability M is equal to:
Q
CgSC
C
xx�
mqa
a
�M
2
ρ−=
−
Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1
Attitude of an aircraft
Ψ
Y0
X0
YawX1
First rotation:
Yaw
(heading)Y0
Y1
−=
0
0
0
0
1
1
100
0cossin
0sincos
k
j
i
k
j
i
ψψψψψψψψ
ψψψψψψψψ
Transformation matrix ->
Second
rotation:
pitch
X2
X1
Pitch
Transformation matrix ->
θ
Z1 Z0
−
=
0
1
1
1
1
2
cos0sin
010
sin0cos
k
j
i
k
j
i
θθθθθθθθ
θθθθθθθθ
Third rotation:
roll
φ
Y2
Y1
Roll roll
Transformation matrix ->
Z2
Z1
−
=
1
1
2
2
2
2
cossin0
sincos0
001
k
j
i
k
j
i
φφφφφφφφ
φφφφφφφφ
Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1
Dynamic Dynamic Dynamic Dynamic stabilitystabilitystabilitystability
Modes of motion – Short Period oscillations
Modes of motion – Short Period oscillations Modes of motion – Short Period oscillations
Simplified mathematical model of the Short Period
Assumption:
0,0 VUuu === & 0,0 VUuu === &
Equation:
+
+=
−
−
q
w
MVSM
ZmVZ
q
w
JMS
SZm
qxw
qw
ywx
xw
0
0
&
&
&
&
Modes of motion – Phugoid
simplified model
assumption: 0,0,0 VUqw eee ===== ααααθθθθ&&
−=
−− θθθθθθθθ
u
Z
mgXu
ZmV
m
u
u
q 00
0
0&
&
Modes of motion – Phugoid
Modes of motion – RollModes of motion – Roll
Roll – simplified model
assumptions: 0===== rqwvu assumptions: 0===== rqwvu
equation: pLpJ px =&
Modes of motion – SpiralModes of motion – Dutch roll
Modes of motion – Dutch roll Modes of motion – Dutch roll
Dutch roll – simplified model
assumptions:
0,,,0 Vvrpp ββββββββψψψψψψψψφφφφφφφφ ======= &&&
equation: )( 0VS%v%rJ xrvz ++=&
Lateral stability diagram – TS-11 Iskra
Nv
0.20
0.40
0.60 Lv
-0.10 0.00 0.10 0.20 0.30
Nv
-0.60
-0.40
-0.20
0.00
Boundaries of stability
duch roll
spiral
What say the regulations?
CS 23.181 Dynamic stability
(a) Any short period oscillation not including combined lateral-
directional oscillations occurring between the stalling speed and the
maximum allowable speed appropriate to the configuration
of the aeroplane must be heavily damped with the primary controls –
(1) Free; and (2) In a fixed position, except when compliance with
���� SAS
���� SAS
(2) In a fixed position, except when compliance with
CS 23.672 is shown.
(b) Any combined lateral–directional oscillations (“Dutch roll”)
occurring between the stalling speed and the maximum allowable speed
appropriate to the configuration of the aeroplane must be damped to 10
amplitude in 7 cycles with the primary controls –
(1) Free; and (2) In a fixed position, except when compliance with
CS 3.672 is shown.
What say the regulations?
(c) Any long-period oscillation of the flight path (phugoid) must not be
so unstable as to cause an unacceptable increase in pilot workload or
otherwise endanger the aeroplane. When, in the conditions of CS
23.175, the longitudinal control force required to maintain speeds
differing from the trimmed speed by at least plus or minus 15% is
suddenly released, the response of the aeroplane must not exhibit any
CS 23.175 – static stability conditions
suddenly released, the response of the aeroplane must not exhibit any
dangerous characteristics nor be excessive in relation to the magnitude
of the control force released.
Other regulations:
Norm:
MIL-F-8785C
the most coherent document about
flying qualities of an aircraftflying qualities of an aircraft
New (confident) version:
MIL-STD-1797A
Aircraft class
Class Definition
I Small, light airplanes (m ≤≤≤≤ 5000kg) such as:
Light utility, Primary trainer, Light observation
II Medium (5000÷÷÷÷30000 kg) weight, low-to-medium ma-
neuverability airplanes such as:
Heavy utility/search and rescue, Light or medium trans-
port/cargo/tanker, Early warning/electronic counter-
measures/airborne command, control, or communica-measures/airborne command, control, or communica-
tions relay, Antisubmarine, Assault transport, Recon-
naissance, Tactical bomber, Heavy attack, Trainer for
Class II
III Large, heavy, low-to-medium maneuverability airplanes
such as: Heavy transport/cargo/tanker, Heavy bomber,
Patrol/early warning/electronic countermea-
sures/airborne command control, or communications re-
lay, Trainer for Class III
IV High-maneuverability airplanes such as: Fighter (inter-
ceptor), Attack, Tactical reconnaissance, Observation,
Trainer for Class IV
Flight phases
Phase definition Typical flights
A. Those nonterminal Flight Phases
that require rapid maneuvering, preci-
sion tracking, or precise flight-path
control
a. Air-to-air combat (CO)
b. Ground attack (GA)
c. Weapon delivery/launch (WD)
d. Aerial recovery (AR)
e. Reconnaissance (RC)
f. In-flight refueling (receiver) (RR)
g. Terrain following (TF)
h. Antisubmarine search (AS)
i. Close formation flying (FF).
B. Those nonterminal Flight Phases
that are normally accomplished using
a. Climb (CL)
b. Cruise (CR)
c. Loiter (LO) that are normally accomplished using
gradual maneuvers and without preci-
sion tracking, although accurate
flight-path control may be required.
c. Loiter (LO)
d. In-flight refueling (tanker) (RT)
e. Descent (D)
f. Emergency descent (ED)
g. Emergency deceleration (DE)
h. Aerial delivery (AD).
C. Terminal Flight Phases are normal-
ly accomplished using gradual ma-
neuvers and usually require accurate
flight-path control
a. Takeoff (TO)
b. Catapult takeoff (CT)
c. Approach (PA)
d. Wave-off/go-around (WO)
e. Landing (L)
Flight quality levels
Level The ability to complete the operational missions
1 Flying qualities clearly adequate for the mission
Flight Phase
2 Flying qualities adequate to accomplish the mission
Flight Phase, but some
increase in pilot workload or degradation in mission
effectiveness, or both, exists
3 Flying qualities such that the airplane can be con-3 Flying qualities such that the airplane can be con-
trolled safely, but pilot workload
is excessive or mission effectiveness is inadequate, or
both. Category A Flight
Phases can be terminated safely, and Category B and
C Flight Phases can be
completed.
Cooper-Harper rating scale
Cooper-Harper scale vs. MIL flight levels
Relation between Cooper-Harper scale and Flight levels
Pilot rating Level Definition
1 - 3½ 1 Clearly adequate for the mission
flight phase
3½ - 6½ 2 Adequate to accomplish mission
flight phase flight phase
Increase in pilot workload, or loss of
effectiveness of mission, or both
6½ - 9 3 Aircraft can be controlled
Pilot workload excessive – mission
effectiveness impaired
Category A flight phases can be
terminated safely.
Aircraft Design 1Aircraft Design 1Aircraft Design 1Aircraft Design 1
Introduction to controlIntroduction to control
Definition of the control surfaces
deflection and sign conventionControls – sign convention
The general rule:
• positive stick deflection causes positive
aircraft reaction (moment)
• positive stick deflection causes negative • positive stick deflection causes negative
control surface deflection
• negative control surface deflection causes
positive moment
Controls – sign convention
General rule:
force and
stick control surfacestick control surface
deflection deflection moment
(+) � (-) � (+)
Controls – sign convention
which moment are positive:
• roll - X axis forward – right wing downward
• pitch – Y axis on the right wing – nose upward
• yaw - Z axis downward – turn right• yaw - Z axis downward – turn right
Controls – sign convention
• roll (+) � stick right (+)
� right aileron up, left aileron down (-)
� positive rolling moment
• pitch (+) � stick back (+) • pitch (+) � stick back (+)
� TE of elevator up (-)
� positive pitching moment (nose up)
• yaw (+) � right pedal forward (+)
� rudder right (-)
� positive yawing moment (on right)