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See [3] pag.93-141
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ED'HILQLWLRQVDFURQ\PVDQGDEEUHYLDWLRQV
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MR Main rotor STA Station line
TR Tail Rotor VF Vertical Fin WL Water line CW Clockwise
CCW Counter clockwise
6\PEROV
'HVFULSWLRQ
8QLW
BF TR maximum blockage effect ad
C
TTThrust coefficient of TR ad
K
3Tan (tail rotor δ
3angle) ad
K
λCorrection coefficient for induced velocity ad
K
BETR blockage effect coefficient ad
L,M,N Moments along x, y, z axis respectively N*m
Q TR torque N*m
R TR radius m
T TR thrust N
X,Y,Z Forces along x, y, z axis respectively N
a Blade lift-curve slope 1/rad
c Blade chord m
p,q,r Angular velocities rad/sec
u,v,w Linear velocities m/sec
β
TRTail rotor sideslip angle rad
Λ
βBlade stiffness to the flapping motion,
( )
2 2
1 K I
β β β
Λ = + ⋅Ω
ad
Ω Tail rotor rotational velocity rad/sec
0
,
1c,
1sβ β β Rotor blade coning, longitudinal and lateral flapping angles
rad
δ TR blade mean drag coefficient ad
δ
3Pitch-flap coupling angle rad
γ Blade Lock number, ρ⋅a⋅c⋅R
4/I
βad
ϕ Tail rotor sideslip angle rad
λ
0Tail rotor uniform inflow component ad
T
*
θ
0Tail rotor collective pitch angle after delta 3 correction
rad
T
*
θ
1sTail rotor cyclic pitch angle applied through δ
3angle
rad
0T
θ Tail rotor collective pitch angle rad
tw
θ Tail rotor blade twist rad
ρ Air density Kg/m^3
σ TR solidity, n⋅c/π⋅R ad
µ Rotor advance ratio,
X2+
Y2( Ω ⋅
5) ad
µ
zTotal normalized tail rotor inflow velocity ad
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AC
Aircraft
TIP
TIP of blade rotor
b
Body axis
w
Wind local axes
TR
Tail rotor
H
Hub axes
EE2YHUYLHZ
This model has the scope to calculate the forces and moments acting on a Tail Rotor in a classical helicopter configuration (TR mounted on or near the Vertical Fin surface).
It takes account the Main Rotor wake influence and the interaction between the TR and the Vertical Fin; the rotor dynamic is calculated with the same theory used for MR but using some simplifying assumption ([13],[14]).
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Purpose of this model is to calculate the tail rotor force and moment contributions to the total force acting on a classical helicopter configuration (TR mounted on or near the Vertical Fin surface).
It takes account the Main Rotor wake influence and the interaction between the TR and the Vertical Fin.
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The following reference systems are used:
• Body axes system
All forces and moments are expressed relative to the body axes system for use in the six-degree-of-freedom rigid body equations of motion, later on. This axis system has its origin at the tail rotor hub with the X axis aligned with the longitudinal axis of the helicopter toward the nose and the Z axis lying on the plane of symmetry, directed down.
• Hub-Wind axes
The hub-wind axis system is used in the calculation of rotor forces and moments.
The origin of the system is the rotor hub and the T(thrust) axis is aligned with the tail rotor shaft. The H (horizontal) axis is aligned with the component of relative
Wwind normal to the shaft axis and the Y (side force) axis completes the right-
Whanded orthogonal set. This axis system is shown in figure B-1 along with the components of relative wind.
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The hub-body system coincides with the hub-wind system when the sideslip angle β
Wis zero. Thus, the T axis is aligned with the shaft axis and the force H lies in
Rthe x -z plane (see fig.B-1).
B B• Aircraft Reference
The aircraft reference axes are used to locate all force and moment generating components. The aircraft reference axes are parallel to the body axes. The axis origin is typically located ahead and below the aircraft at some arbitrary point within the plane of symmetry. Stations are measured positive aft along the longitudinal axis. Buttlines are lateral distances, positive to the pilot’s right.
Waterlines are measured vertically, positive upward.
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To calculate the forces and moments acting on TR the first step is to calculate the flow
field around it. The contribution of the MR wake is calculated using the module
INTERACT [20], then the total linear velocity in body axes is obtained with the following
formula:
( ) ( )
( )
−
⋅
−
⋅
⋅
−
=
⋅
−
−
⋅
− +
=
=
67$
67$
T Y
. Z Z
S :/
:/
U 67$
67$
Y Y
X X
0
λ
λ
These velocities are transformed from body to TR hub axes to calculate the a-dimensional velocities, µ, as follow:
⋅
−
= ⋅ +
=
rotation CCW
has MR the if 1
rotation CW
has MR the if 1
2 2
Y Y
Y Y
Y Z X
µ µ
After to have transformed in Hub-axes also the angular velocities, in the Hub-wind axis system:
W
W
TR H TR H TR
TR H TR H TR
p p cos r sin
q r cos p sin
= ⋅ β + ⋅ β
= ⋅ β − ⋅ β
The TR control angles (the first step in hub-wind axes, then in the hub axes) are obtained with the next formulas:
( )
T
T
3
0 0 z
*
0 2
3
K 4
8 3
1 K 1 8
β
β
γ
θ + ⋅ Λ ⋅ ⋅ λ − µ θ = − ⋅ ⋅ γ + µ Λ
( )
( )
w
0 3 0 z 3
1s
2 2 2
3 3
1 1
8 K 2 K
3 2 9
1 2 K K 1 2
9
β β
β
µ ⋅θ ⋅ ⋅ + ⋅ Λ γ + ⋅µ⋅ λ − µ ⋅ + ⋅ Λ γ
β = + ⋅ ⋅ Λ γ ⋅µ + ⋅ + ⋅µ
( ) ( )
w w
2
1c 0 0 z 3 1s
8 2 K 1 2
β = − ⋅µ⋅θ − ⋅µ⋅ λ − µ − 3 ⋅ + ⋅µ ⋅β
T T T
*
1s 1s
k
3 0θ = θ + ⋅β
Then the control angles are transformed in Hub axes:
w w
w w
1c 1c 1s
1s 1c 1s
cos( ) sin( ) sin( ) cos( ) β = β ⋅ ϕ −β ⋅ ϕ
β =β ⋅ ϕ +β ⋅ ϕ
The tail rotor was modelled as a teetering rotor without cyclic pitch. For this case the forces in the hub-wind system may be obtained from the expressions derived for the main rotor by setting the lateral and longitudinal cyclic pitch terms equal to zero. Further, since the tail rotor flapping frequency is much higher than that of the main rotor system, the tip- path plane dynamics may be neglected. Thus, for tail rotor applications, the firsts and second derivatives of the blade flapping non rotating coordinates are set to zero in the force equations. The result is a set of basic quasi-static force expressions similar to these in classical work.
The thrust coefficient and the inflow are obtained iterating the following equations
( )
( )
( )
T
w
*
0 2 z 0 * 2
TT 0 1st t
2 2
0 TT z 0
1 3 p
C a 1 1
2 3 2 2 2 4
C 2
= ⋅ ⋅σ⋅ θ ⋅ + ⋅µ + µ − λ + ⋅θ + θ ⋅ + µ + ⋅ µ µ
Ω
λ = ⋅ µ + µ − λ
After the C
TTand λ
0calculation, it is evaluated the blockage effect on the thrust due to the VF presence near to the TR:
<
<
⋅
− +
=
>
=
⋅
⋅ Ω
⋅
⋅
⋅
=
X X X
X
%)
%)
X
%)
X X .
.
&
5 7
0 )
1 (
0 for
1 where
2
π
4ρ
Where
%)is a coefficient calculated as in [5].
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To calculate the other component it is necessary to define the mean blade drag coefficient as:
2 2
0
! &
⋅ + = δ δ δ
and using the formulas illustrated:
( ) ( )
( ) ( )
( ) ( ) ( )
( )
w T
T T T
T T T T T
T
T T T T T T
w T
*
t 0
H 0 z 0 1c z 0 1c
0
3 *
0 0 1c 1s z 0 1c
1s 2
* * 2
1c z 0 0 0 0 0 1c
* t 0
z 0
1 2
C a
2 2a 2 3 4
k 2
4 3 3
4 6 4
1 3
6 8 2 16 3 k
δ⋅µ θ θ
= ⋅ ⋅σ⋅ − ⋅ µ − λ ⋅µ − ⋅β − ⋅ µ − λ ⋅µ −β
− ⋅ ⋅ θ − θ ⋅β + β ⋅ µ − λ − µ⋅β
β µ
+ ⋅β ⋅ µ − λ − θ − θ ⋅ + θ − θ + β ⋅ θ
− θ + + µ − λ ⋅ + ⋅ ⋅ ( )
( ) ( )
T T
T T T T
TR
3 1s 1c
TR
* TR
3 1c 1s 0 0
TR
p
q k 1
16 6
⋅β + β ⋅µ ⋅
Ω
µ
− ⋅β + β ⋅ + θ − θ ⋅ Ω ⋅
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
T T T T T T T
w
T T T
T T T T T T
T T T
* * * 2
Y 0 0 3 0 0 0 0 1s
t * 2
0 0 1s
2 *
3 1c z 0 1c 1s 1s 0 0
*
1s z 0 0 0
1 1 2
C a k 3 2
2 4 3
2 1
4
k 2 1
3 3
4
= ⋅ ⋅σ⋅ − ⋅ θ − ⋅ θ − θ ⋅ ⋅ θ − θ ⋅µ − ⋅β ⋅ µ +
θ
− ⋅ θ − θ ⋅µ − β ⋅ + µ
+ ⋅ β ⋅ µ − λ + µ ⋅β + ⋅β ⋅ µ⋅β − µ + ⋅ θ − θ
+ ⋅β ⋅ µ − λ + θ − θ ⋅ ( ) ( )
( ) ( ) ( )
( )
( ) ( )
T T T T T
T T T T T T T
w
T T T T T
*
1c 0 0 z 0 1c 1s
* 2 * TR
0 0 1c 3 1c 1s 0 0
TR
* * t
0 3 0 0 3 1s z 0 1c
1 6
6 4
p
k 5 1
16 6
1 1 7
k k
6 8 16 2 16
µ
β ⋅ − ⋅ θ − θ ⋅ µ − λ − β ⋅β ⋅
µ
− θ − θ ⋅β ⋅µ + ⋅β + ⋅β ⋅ − θ − θ ⋅ Ω
θ
µ
+ θ − ⋅ θ − θ ⋅ + + ⋅β ⋅ + µ − λ ⋅ + ⋅µ ⋅β
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
w
T T T T
T T T T T
T T T T T T T T T
2 * t 3 2
Q 0 0 z 0 z 0 1c 1s
0
* 3 2
1c 0 0 1s z 0 z 0 z 0 1c
2 2 2
* 2 2 2 * 2 2
0 0 1c 1s 0 0 1s 1c 1s
k
1 1
C a 1 4.7
2 4 a 3 4 8
k 1
6 4 2 2
1 3
4 4 3 8
δ θ
= ⋅ ⋅σ⋅ ⋅ ⋅ + ⋅µ − θ ⋅ µ − λ ⋅ − ⋅ µ − λ − ⋅β ⋅β ⋅µ
µ µ
+β ⋅ ⋅ θ − θ −β ⋅µ⋅ µ − λ ⋅ − µ − λ ⋅ − ⋅ µ − λ ⋅β
µ µ µ
− θ − θ + ⋅ ⋅β + β + ⋅ θ − θ ⋅β − β + β ⋅
− ( )
( )
T T T T T T T
* 3 TR 3 * TR
0 1s 1c 1c 1s 0 0
TR TR
2
2 2
TR TR
k 1 p k 1 q
6 8 4 8 4 3
p q
8
µ µ
⋅θ + ⋅β − ⋅β ⋅ − ⋅β + ⋅β − ⋅ θ − θ ⋅
Ω Ω
− + ⋅ Ω
the X, Y force and torque coefficients are calculated.
Before to write some output control at the end of the model, the forces and moments are calculated in hub axes and converted to aircraft axis system.
H TR TR
H TR
H TR TR
X Y sin( ) H cos( )
Y T
Z Y cos( ) H sin( )
= − ⋅ ϕ − ⋅ ϕ
=
= ⋅ ϕ − ⋅ ϕ
H
H TR
H
L 0
M Q
N 0
=
= −
=
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The content of this chapter is taken from [12]. The following section will give an overview on how the vertical fin and horizontal tailplane is modelled.
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$5
Aspect ratio ad
L"
Aerodynamic setting angle deg
L#
Geometric setting angle deg
89:
Velocity component along X, Y, Z axes m/sec
&$3
Centre of aerodynamic pressure
&$
Drag coefficient
&$&%
Induced drag coefficient ad
&$('
Profile drag coefficient ad
&$&)
Drag constant at α=0° ad
&$&*+)
Drag constant at α=90° ad
&$&,
2nd order profile drag coefficient at α=0° ad
&-
Lift coefficient ad
&-/.0" 1
Maximum lift coefficient ad
73
Tailplane
9)
Vertical fin
D
Aerodynamic incidende deg
E
Sideslip angle deg
/
Sweepback angle deg
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2
Body axes system
%
Induced velocity
354
Main rotor
6
Local surface axes system
7897
Total velocity
:<;
Tailplane
:<4
Tail rotor
=&>
Vertical fin
?
Local wind axes system
Table Legends for the vertical fin and horizontal tailplane models.
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This chapter will explain the models used for the Vertical Fin and the Horizontal Tailplane in the EUROPA code. The following requirements are used to define the models for vertical fin and tailplane:
• simply models, using bidimensional characteristics of profile adopted
• full angle models (±180°)
• stall effect
• sweepback effect
• main rotor and tail rotor wake interference
• blockage effect for vertical fin
• variable incidence with collective control input
• use of multiple surfaces to take account different effects at different locations (i.e. the horizontal surface could be considered as two different to introduce asymmetrical effect of wake interaction on it).
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The following reference systems are used:
•
• Local surface axes, defined in the airfoil section containing the surface aerodynamic pressure centre, origin in it, X axis along the line of lift=0 (leading to trailing edge), Y normal to section and spanwise oriented, Z normal to X in the section, to complete the system.
• Local wind axes, defined in the section passing on surface aerodynamic pressure centre and the origin in it, X axis along section wind velocity, Z axis orthogonal in the section to X axis, Y normal to section and spanwise oriented.
The last two system are showed in Figure A and Figure B.
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chord
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YI
α
Iα
Iα
JLKM(N O+P
O
N9QR+ST
P O
M(UV
V9W R QS
VI VXYZ
β
[L\Λ
[C\X@
Z@ V]]
W@ W^CY
V_`_
CAP Xa
Za
Xb Zc
Xa Yc
Yd
Xc U@ CAP
V]]
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The aerodynamic angles are defined as follow:
• the aerodynamic incidence angle (α) is the angle between the reference line and the parallel component of velocity at the section;
• the sideslip angle (β) is the angle between the normal component of velocity to section and the normal itself.
Then, for the VF (see Figure A) the incidence angle is obtained with α = +
Leα
f, where i
ais the setting incidence and
α
g g hig
9 9
8
j
= +
tan
−1is the aerodynamic angle evaluated in body axes.
The sideslip angle is
( ) ( )
β = +
+ + + +
tan
−12 2 2
: :
8 9 9 : :
k lnm
k k oCm k l(m
p
p p
.
For the TP (see Figure B) the incidence angle is obtained with α = +
Lqα
r, where i
ais the setting incidence and
α
s s t(us
: :
8
v
= +
tan
−1is the aerodynamic angle evaluated in body axes.
The sideslip angle is
( )
β = + + +
tan
−12 2 2
9
8 9 : :
w
w w w xny z
.
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To build the models for vertical fin and tailplane, it was used the following assumption (ref.17 and 18):
1. The lift and the drag are applied at the chord quarter of the airfoil section containing the aerodynamic pressure centre.
2. The lift curve slope before stall is calculated by simple lifting line theory (18), introducing corrections for sideslip and sweepback (19).
3. It is request C
Lmaxinput, defined as maximum C
Lobtained at the stall angle. If this value is reached at an angle of attack (measured from the reference line of C
L=0, see Figure A and Figure B) higher than π/4, C
Lmaxis assumed to occur at π/4.
4. Post stall variation is obtained from a linear decrease of 20% as α is increased by 20%.
From this point it is assumed a 2nd order variation with α, to reach a maximum of 0.8*
C
Lmaxat π/4, then a value of 0 at α=π/2.
5. The lift in the rearward flights C
L(π/2<α≤3π/2) =0.8*C
L(-π/2≤α≤π/2).
6. Profile drag before stall varies with a 2nd order law.
7. Profile drag after stall varies with another 2nd order law respect α and reaches the maximum value C
D90at α=±π/2.
8. Induced drag varies as the square of the calculated lift (18).
9. Aerodynamic moments, acting on the tail surfaces, are not considered because their contribution is negligible for the scope of EUROPA code
Previous assumption give the following equation (for the range 0 ≤α≤π /2):
{|}~~
α α ≤
&= ⋅
Dα
α α α < ≤
&=
& max− ⋅ −
D( α α
L)
α α ≥
max( 2 )
0.8 ( 2 )
& &
π α α
α π α
= ⋅ ⋅ − ⋅
⋅ −
last formula is the result of the next system:
U
W
W
V
X
XL
X
iC i i
Z
α
α α
Ln¡£¢+¤
¢ ¡9¥¦§ ¨
¤ ¢
n©ª ª9« ¦ ¥§
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( )
0 1 2 2¬& D D D
α α α = + ⋅ + ⋅
( ) 0.8
max ®
&
α = ⋅
&( ) 2 0
¯&
π =
( ) 4 0
°&
∂ π
∂α =
The profile drag is given by:
0 ≤ ≤ α α
± ² 0 2³ ³ ³
&
=
&+
& α⋅ α
±
α α π < ≤ ( ) ( ) ( )
290 0 901 4 1 1 4
´µ µ µ µ ¶·¸¹¹ µ¶L·¸º¹¹ ¶·¸¹¹
& & & & α &
α π α π
π α
α α π π
− − ⋅
= − ⋅ + ⋅ − ⋅
⋅ − −
Last formula is obtained from the solution of the following system:
2
0 1 2
»
¼& E E E
α α = + ⋅ + ⋅
( ) 2
90Ã
Ä Ä
&
π =
&2 0
Å
Æ&
∂ π
∂α =
The 1st order coefficient, a
s, is defined as
( )
D
$5
Ç
= ⋅
+ 2 ⋅ +
1 2
π cos
2β Λ ,
which take account of sideslip and surface sweepback (evaluated at 1/4 of chord). If the C
Lmax, with the a
sdefined before, is reached at α>π/2 it is imposed an α
stall=π/2 and C
lmax=a
s⋅π/2. The induced drag is evaluated, for any α, with the formula
& &
$5
È É
Ê
=
⋅ ⋅
2
0 8 . π .
Then the total drag is
C
D= C
Dp+C
Di.
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The content of the following chapter is taken from [14]. The aerodynamic model of the fuselage must fulfil two requirements . The first is to provide an estimate of the forces and moments at small angles of attack and sideslip that will be encountered at substantial forward speeds. This provides a representation of the import effects of fuselage aerodynamics on performance and stability at these speeds. The second requirement is to provide a continuous variation of forces and moments over the entire range of angle of attack and sideslip (0° to ±180°) that can be encountered in approach to hovering flight or in hover. Continuity is required to avoid sudden unrealistic linear or angular accelerations in response to a small change in attitude. Accuracy of the model at extreme attitudes is considered to be of secondary importance, since the fuselage forces at these speeds are very small compared to the rotor forces.
A technique has been developed to provide a continuous model by fitting typical variations of the forces and moments through data points obtained at specific widely separated angles of attack and sideslip in a wind tunnel. However, even this sparse level of test data for the fuselage is typically unavailable and an alternative technique must be employed.
The model employed herein relies on separate representations for angles of attack and
sideslip in the range from -15° to 15° and from ±25° to ±180°. Continuity is provided by a
linear interpolation for forces and moments in the angle range not covered. The forces and
moments for the lower angle range are obtained from test data or from estimates based on
data from similar fuselages. The data for the high angle of attack and sideslip range are
based on the estimated magnitude and location of the drag force vector when the fuselage
is in a 90° cross flow, and on an approximation to its observed variations with attitude
from wind-tunnel tests of bodies of revolution.
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The content of the following chapter is taken from [14]. The rpm governor model provides for an rpm degree of freedom with simple engine and governor dynamics. When the rpm governor option is used, the main rotor and tail rotor speeds are changed based on current torque requirements and engine power available. Trim initial conditions establish baseline values of rotor speed and engine torque. Flight variations from that trim condition result in changing torque requirements which cause rotor speed variations which feed through the governor control laws which are converted to engine torque available.
The rpm degree of freedom may be described by the equation:
Ë Ì
4
−
4= Ω
-where:
Í4
= engine torque
Î4
= required rotor torque
-
= rotor rotational inertia
Ω = rotor speed rate of change
For low flapping hinge offsets, the rotational inertia,
-, may be approximated from the blade flapping inertia,
,β:
-
=
1,βwhere
1is the number of blades.
The main rotor torque required is a complicated function of many variables including blade pitch settings, airspeed, inflow velocity, flapping angles, and rotor speed. The torque equation provides the necessary calculation. The rotor speed governing of this model uses linear control theory based on an operating point. Thus, the rotor speed, Ω , takes the from:
Ω = Ω + ∆Ω
0where Ω
0is the trim rotor speed and ∆Ω is the rotor-speed variation.
A detailed representation of engine torques requires a complicated nonlinear function of many variables including operating power setting, ambient pressure and temperature, and fuel flow, ω
Ï. For this simplified model, rpm governing acts on fuel flow to control engine torque. A gas turbine produces power which must be converted to torque based on the current rotor speed.
550
Ð4 +3
=
Ω
It is important to note that the current rotor speed is used, not the operating point speed, Ω
0. Thus, the power-to-torque conversion factor is always changing.
The engine power response to fuel flow is simply modelled as a first order lag:
1
Ñ Ò
Ñ
+3 .
V
ω
∆ = τ ∆
+
The time constant, τ
Ó, and gain,
.Ô, are selected based on engine characteristics and operating point.
Note that a throttle on/off switch has been added to allow turning the power on or off. A
constant for the throttle and a small power change time constant for normal engine operation.
The governor control law is given by:
2
1 3
Õ
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