T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
Linköping University
Department of Electrical Engineering
in collaboration with
Saab Aerosystems
Department of Aerodynamics and Flight mechanics
Investigation of a non-uniform
helicopter rotor downwash model
Master Thesis in Automatic Control
by
Berenike Hanson
LITH-ISY-EX--08/4188--SE
Linköping 2008
TEKNISKA HÖGSKOLAN SAAB AB
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
Ärende Subject Fördelning To
Investigation of a non-uniform helicopter rotor downwash model
TDAA, RL, PW, HJ, TDAA-EB, TDAA-EA, TDAF-OH, LiTH-Torkel Glad
ABSTRACT
This master thesis was carried out at the Department of Aerodynamics and Flight Mechanics at Saab Aerosystems, Linköping, Sweden. It makes up the author’s final work prior to graduation in the field Applied Physics and Electrical Engineering at the Department of Electrical Engineering at The Linköping Institute of Technology (LiTH), Linköping, Sweden.
The objective of the paper was to study a non-uniform helicopter rotor downwash model in forward flight for the unmanned helicopter Skeldar, which is under development at Saab. The main task was to compare the mentioned model with today’s uniform downwash model in order to find differences and similarities. This was done both from a modeling and a controlling perspective. To start with, an introduction is given which is followed by a helicopter theory chapter. The following three chapters deal with the theory of induced velocity, the helicopter model and the Linear Quadratic Regulator (LQR). Finally, the results are presented and discussed.
The downwash models were derived using Momentum Theory (MT) and Blade Element Theory (BET). These two theories were combined in order to find a connection between the induced velocity and the rotor thrust coefficient. The non-uniform downwash model that was studied is proposed by Pitt & Peters and describes a linear variation of the induced velocity in the longitudinal plane.
For the control, a LQ-regulator was chosen since it is easily implemented in MATLAB and it stabilizes the plant model by feedback and consequently creates a robust system. Before the controller could be implemented, the models had to be reduced and the states had to be divided into longitudinal and lateral ones.
The comparison between the open systems clearly shows that differences in the inflow models propagate to all states and consequently the helicopter behaves differently in all planes. Great discrepancies are apparent for the angular velocities p and q. For Pitt & Peters’ model those states are believed to be strongly affected by the system’s positive real pole, causing a rather unstable behavior. When the systems were closed by feedback, the differences were reduced dramatically. Pitt & Peters’ model resulted in greater overshoots than the uniform model, but the overall behavior of all states was rather similar for the two models.
It is concluded, that the adaption of Pitt & Peters’ inflow model does not make any substantial difference when a controller is implemented. The differences between the open systems, however, are reason enough to question Pitt & Peters’ model. In order to evaluate the non-uniform model properly, it has to be compared to suitable flight data which is still lacking up to this date.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. Acknowledgements
I would like to thank my examiner, Torkel Glad, who supported me throughout my work, my supervisor, Roger Larsson, who always knew how to use his whiteboard, my Skeldar specialist, Per Weinerfelt, who never hesitated to reach out a helping hand, my controlling specialist, Ola Härkegård, who gave me the feedback I needed, my friend and co-worker, Helena Johansson, to whom I could go and either cry or laugh, my LSA, Anders Paju, who was my guide through the world of UNIX (and through many other topics), my forerunner, Erik Backlund, whose master thesis has been my prototype and who narrowly examined my work, my mentor, Anna Freiholtz, who also proofread my work and gave me valuable advise along my way, my friend and colleague, Erika Amundsson, who encouraged my curiosity and showed me the world of Houston, all my other colleagues at TDA, who kept me going during breaks and last but not least, my boyfriend, Max Wilhelmsson, who always gave me a push when I was hovering.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. TABLE OF CONTENT 1 INTRODUCTION...7 1.1 BACKGROUND...7
1.2 OBJECTIVES AND LIMITATIONS...7
1.3 THESIS OUTLINE...7 2 HELICOPTER THEORY...9 2.1 CONTROLLING PRINCIPLES...9 2.1.1 Swash plate...9 2.1.2 Main rotor...10 2.1.3 Paddles...10 2.1.4 Tail rotor...11 2.2 AZIMUTH ANGLE Ψ...11
2.3 FLAPPING, FEATHERING AND LEAD/LAG MOTION...11
2.3.1 Flapping angle β...12
2.3.2 Phase lag...12
2.4 VELOCITY PROFILE...13
3 INDUCED VELOCITY THEORY...14
3.1 MOMENTUM THEORY (MT)...14
3.1.1 Momentum Theory in hover...14
3.1.2 Momentum Theory in forward flight...17
3.2 BLADE ELEMENT THEORY...19
3.2.1 Blade Element Analysis in hover and axial flight...19
3.2.2 Blade Element Analysis in forward flight...21
3.3 LINEAR INFLOW MODEL...22
4 MODELING THEORY...25
4.1 COORDINATE SYSTEMS...25
4.1.1 Coordinate transformation...25
4.1.2 Earth fixed coordinate system (FE)...25
4.1.3 Body fixed coordinate system (FB)...26
4.1.4 Hub and blade coordinate systems (Fh and Fb)...27
4.2 STATES AND INPUT SIGNALS...28
4.3 TRIMMING...29
4.4 EQUATIONS OF MOTION...30
4.4.1 Forces on the helicopter...30
4.4.2 Moments on the helicopter...32
4.4.3 Calculation of position, orientation and velocities...33
4.4.4 Calculation of paddle states...34
5 CONTROL THEORY...35
5.1 LINEARIZATION...35
5.2 LINEAR QUADRATIC REGULATOR (LQR)...36
5.3 SINGULAR PERTURBATION...37
5.4 BREAKDOWN INTO LONGITUDINAL AND LATERAL STATES...38
6 RESULTS...40
6.1 MODEL COMPARISON WITH SKELDAR TEST FLIGHT DATA...40
6.2 MODEL COMPARISON BETWEEN OPEN SYSTEMS...41
6.3 MODEL COMPARISON WITH CONTROL PERSPECTIVE...41
6.3.1 Linearization and state reduction...41
6.3.2 Control signals...43
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. 6.3.4 Simulation...44
6.4 ANALYSIS OF DIFFERENCES BETWEEN THE TWO MODELS...44
6.4.1 Railing of flight data...44
6.4.2 Open systems...44
6.4.3 Controlling behavior...46
7 DISCUSSION, CONCLUSION AND FUTURE WORK...48
REFERENCES ...50
APPENDIX A. TRANSFORMATIONS...51
APPENDIX B. OPEN SYSTEMS ...53
APPENDIX C. POLE-ZERO MAPS ...55
APPENDIX D. MODEL REDUCTION ...57
APPENDIX E. CONTROLLER...61
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. NOTATION dt dx
x& = Dot denotes differentiation with respect to time t F, x Bold style denotes vector or matrix
ABBREVIATIONS
2-D Two dimensional
AoA Angle of Attack
BET Blade Element Theory
CG Center of Gravity
DoF Degree(s) of Freedom
HP Hub Plane
LiTH Linköping Tekniska Högskola (The Linköping Institute of Technology)
LQR Linear Quadratic Regulator
MT Momentum Theory
RPM Revolutions Per Minute
TPP Tip Path Plane (The plane whose boundary is described by the blade tips)
UAV Unmanned Aerial Vehicle
NOMENCLATURE
Abbreviation Description Unit
A, B, C System matrices of helicopter -
A Rotor disk area m2
Ab Rotor blade area m2
CD Drag coefficient -
CL Lift coefficient -
α
l
C 2-D lift-curve-slope of the airfoil sections comprising the
rotor
-
CT Thrust coefficient -
c Local blade chord m
D Drag force N
e Flap hinge offset -
FE, FB, Fb, Fh Earth, body, hub and blade coordinate systems respectively -
β
I ,IP Inertia of one blade/one paddle kg·m2
kx, ky Weighting factors for inflow variations -
L Lift force N
m Mass of helicopter kg
mb Mass of one blade kg
Nb Number of blades of the rotor -
p Pressure Pa
1
Q Matrix that penalizes control errors -
2
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. E
r Position vector in Earth-fixed frame m
[
]
TB r q
p Scalar components of ωB rad·s
-1
R Rotor radius m
r Length ratio of blade -
S Control surface m2
T Rotor thrust N
U, U 2-D velocity vector and corresponding scalar velocity acting on blade element
m·s-1
UN ,UP Normal and parallel scalar components of U to the rotor disk m·s-1
[
]
T B w v u Scalar components of VB m·s -1V Velocity vector acting on blade m·s-1
B
V Velocity vector in body system m·s-1
Vc Climb velocity m·s-1
Vtip Scalar velocity acting on blade tip m·s-1
V∞ Free stream velocity m·s-1
vh Induced velocity in hover m·s-1
vi Induced velocity m·s-1
v∞ Induced velocity in far wake m·s-1
[
]
T E z y x Scalar components of rE M GREEK LETTERSAbbreviation Description Unit
α Angle of attack rad/deg
β, βcos, βsin, βcol,
βtlr
Blade flap angle, cosine/longitudinal flap angle, sine/lateral angle,
collective/coning angle, tail rotor flap angle rad/deg
θ, θcos, θsin, θcol,
θtlr
Blade pitch angle, cosine/lateral angle, sine/longitudinal angle, collective angle, tail rotor pitch angle
rad/deg
θsw,cos, θsw,sin,
θsw,col
Swash plate angles, cosine component of pitch angle, sine component of pitch angle, collective component of pitch angle
rad/deg
λ Induced inflow ratio -
µ, µx, µz Speed ratio, x-component of speed ratio, z-component of speed ratio -
ρ Density kg·m-3 σ Rotor solidity -
[
]
T E Ψ ΘΦ Euler angles, i.e. scalar components of ΩE rad/deg
φ Inflow angle rad/deg
χ Wake skew angle rad/deg
ψ Azimuth angle rad/deg
Ω Angular velocity of rotor rad·s-1
E
Ω Euler angle vector in Earth system rad/deg
B
ω Angular velocities of helicopter in body system rad·s-1
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. 1 INTRODUCTION
This master thesis is the final work prior to graduation in the field Applied physics and electrical engineering at the Department of Electrical Engineering at The Linköping Institute of Technology (LiTH), Sweden. It was carried out at the Department of Aerodynamics and Flight Mechanics at Saab Aerosystems, Linköping, Sweden. The examiner is Professor Torkel Glad at LiTH and MSc/Senior Aerodynamicist Roger Larsson is supervising the work at Saab.
1.1 Background
The helicopter Skeldar is one of the unmanned aerial vehicles (UAV) that are currently being developed at Saab. It can be used in the fields of reconnaissance, surveillance and intelligence where the risk of using a pilot in the aircraft is too high. In the work of making the helicopter fully autonomous, even in critical situations, it is necessary to be able to model it as accurately as possible in order to make it fully controllable. Among other things, this involves knowing the aerodynamic conditions around the helicopter in general and around the main rotor in specific. For this reason, two master theses have been carried out at the Department of Aerodynamics and Flight Mechanics. This thesis examines the aerodynamic surroundings of a helicopter rotor using theoretical relationships. The second master thesis covers the aerodynamic effects on the helicopter fuselage, using Computational Fluid Dynamics (CFD).
1.2 Objectives and limitations
The objective of this thesis is to investigate the induced inflow into the main rotor of the unmanned helicopter Skeldar. Today’s model uses an inflow distribution that is uniform over the rotor disk. By introducing an inflow model where the inflow varies linearly over the disk, the difference between the two models is studied. In addition, a controller that keeps a steady forward speed is implemented, with the aim to investigate differences between the two models from a controlling perspective.
This thesis only deals with the comparison of the uniform model and the linearly varying model in forward flight. Other flying conditions like climb/descent and turns are thereby excluded. As a consequence, the controller is only implemented for forward flight.
1.3 Thesis outline
In the following, the outline of this thesis is shortly presented.
Chapter 2 introduces general helicopter theory.
Chapter 3 describes the theory of induced velocity which is the main topic of this master thesis.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
Chapter 4 describes modeling theory and introduces the rigid body equations, as well as
rotor equations.
Chapter 5 describes control theory in general and the chosen controller in specific, as well as
necessary steps for the control of this particular helicopter model.
Chapter 6 presents the obtained results, as well as simulations and validations.
Chapter 7 discusses the results of the work and conclusions are made. Future work is
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. 2 HELICOPTER THEORY
On a helicopter, the lift force is produced by rotating “wings” or rotor blades that make up the
main rotor of the helicopter, Figure 1. The blades rotate with a rotational speed, Ω. A second rotor, the tail rotor, is used to counteract the torque of the main rotor and to control the yaw moment of the helicopter. The pitch and roll moments are controlled by changing the angle of attack (AoA) of the main rotor blades and thereby tilting the rotor disk. The rotor disk is the
imaginary surface swept by the rotor blades during their revolution [3]. This chapter
encompasses basic helicopter theory and starts out with controlling principles.
Tail rotor
Torque that wants to turn the fuselage around Force that counteracts the
torque of the fuselage
Main rotor Rotor disk
Rotor blades
Ω
Figure 1 - Overview of a helicopter.
2.1 Controlling principles
2.1.1 Swash plate
A helicopter is controlled through a so called swash plate that consists of two plates were the upper one is rotating and the lower one is non-rotating, Figure 2. A bearing lies between the two plates. The upper plate is connected to the rotor blades with pitch links. It can tilt in any direction and rotates at the same speed as the rotor. The non-rotating plate is attached to control actuators and can only tilt. By tilting the swash plate, the links move up or down, transferring their movement to the rotor blades.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. 2.1.2 Main rotor
On a helicopter, the engine turns the rotor which produces the lift force. This force depends on the shape, size and AoA of the rotor blades as well as the air speed and density. There are two different ways of controlling the lift, either by changing the rotor revolutions per minute (RPM) or by changing the collective pitch which simply means that the AoA of all rotor blades is changed simultaneously in order to either climb, descent or hover.
The rotor is also responsible for producing forces and moments that control the attitude and position of the helicopter. This is done by changing the AoA of the blades once per revolution which is called cyclic pitch. The rotor disk is then tilted in the desired direction with a phase lag of a quarter revolution or 90°. The tilting is achieved by a so called flapping motion of the blades which is described in chapter 2.3. The cyclic pitch is divided into longitudinal and lateral or sine and cosine cyclic pitch, which simply is a function of the azimuth angle ψ, which is described in chapter 2.2.
The tilting can be done both in the lateral and the longitudinal plane. The lateral cyclic pitch is used in order to tilt the disk left and right and thereby producing a side force and a rolling moment about the centre of gravity (CG) of the helicopter. It is achieved by applying a cosine cyclic pitch angle. The longitudinal cyclic pitch is achieved by applying a sine cyclic pitch angle and affects the disk in the fore and aft directions, thereby producing a longitudinal force and a pitching moment. The tilting of the rotor disk generates a moment about the CG since the thrust line of the rotor no longer passes through the CG, Figure 3. The moment tilts the fuselage until the thrust line again passes through the CG and thereby annulling the moment.
Figure 3 - Moments and forces during longitudinal cyclic pitch. βcos is the longitudinal flap angle and is
described in chapter 2.3.
2.1.3 Paddles
There are different ways of making a helicopter more stable against disturbances. Skeldar uses a so called Bell-Hiller mechanism which is a stabilizing bar with paddles at its ends. It is mounted at right angles to the rotor blades and rotates with them. If a rotor blade should flap due to a
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disturbance, the linkage to the stabilizing bar is such, that the paddle changes its AoA and thereby equalizes the flapping of the offending blade [3].
2.1.4 Tail rotor
The tail rotor is used to control the yawing moment. Its purpose is to counteract the torque produced by the main rotor that wants to turn the fuselage around, in the opposite direction of the rotor revolution. If no yawing is desired at all, the sideways force generated by the tail rotor multiplied by the lever arm to the rotational centre must equal the moment of the main rotor. The tail rotor thrust is controlled by changing the collective pitch of the tail rotor. The lateral force that arises from the tail rotor tends to move the helicopter sideways. This is counteracted by slightly tilting the rotor disk until the forces are balanced. [4, 7]
2.2 Azimuth angle ψ
The position of the rotating blades can be described by the azimuth angle ψ which is defined as
zero when the blade is above the tail, Figure 4. ψ increases counter-clockwise such that it is 180°
when the blade is pointing forward, in the same direction as the nose of the helicopter.
Figure 4 - Azimuth angle ψ.
2.3 Flapping, feathering and lead/lag motion
A rotor blade has three degrees of freedom (DoF) which implies that it can move around three axes. Flapping is the only movement that is implemented in the current model and therefore it is explained closer while the remaining DoF are only mentioned.
Flapping is the up and down motion of the blade tips, made possible by a flap hinge at the root of the blade, Figure 5. This motion is generated by the airloads acting on the blades, which vary as a function of the azimuth angle when the conditions around the disk vary by a cyclic pitch command. In this case, the pitch angle varies as a function of the azimuth angle and consequently the aerodynamic lift would vary in the same way if it wasn’t for the flapping motion. In order to reach an equilibrium condition and reduce the varying aerodynamic forces, the blades must be able to flap. In hover, the forces acting on the blades are constant as they rotate. The blades take up a steady flapping angle, called coning angle, such that there is no moment about the flap hinge caused by centrifugal and lift forces.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
The rotor blades are also free to move in the in-plane (lead/lag) and to vary their pitch angle by so called feathering as can be seen in Figure 5
Figure 5 - The degrees of freedom of a rotor blade. 2.3.1 Flapping angle β
The angle between a rotor blade and a plane normal to the rotational axis is called the flapping angle β. When β varies around the disk it is divided into the expression:
ψ
β
ψ
β
β
β
= 0 + coscos + sinsin (2.1)where
β
0 is the coning angle,β
coscosψ
is the fore-aft tilt andβ
sinsinψ
is the lateral tilt of thedisk, Figure 6. [10]
Figure 6 – When a longitudinal cyclic pitch command is given, the disk tilts in the fore-aft direction.
2.3.2 Phase lag
The cyclic flap motion lags the cyclic pitch motion by about 90° due to inertia of the blades. This means that an application of sine cyclic pitch results in a longitudinal tilt of the rotor disk, while an application of cosine cyclic pitch results in a lateral tilt.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
Figure 7 – There is a 90° phase lag between pitch angle and flap angle due to inertia.
2.4 Velocity profile
The flow velocity profile for hovering and forward flight is illustrated in Figure 8. As can be seen, the flow velocity profile is independent of the azimuth angle ψ in hovering flight. The velocity variation is axis symmetric and radially linear with the velocity being zero at the
rotational axis and the maximum velocity Vtip =ΩR at the blade tip. In forward flight, the
velocity profile looks differently since it varies with the azimuth angle. This is due to the fact
that the free stream velocity V∞ is added to the rotational velocity on the advancing side and
subtracted from it on the retreating side such that Vtip =ΩR+V∞sinψ . Thereby a reverse flow
region emerges on the retreating side, Figure 8.
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3 INDUCED VELOCITY THEORY
Just like the wings on an airplane, the blades of a helicopter trail tip vortices, which form a rotor wake. The difference is that the flow field in which a rotor operates is much more complex than that of an airplane. The reason for this is that the tip vortices of an airplane trail downstream of the aircraft while the vortices of a helicopter can remain close to the rotor for several revolutions. This phenomenon causes a strongly three-dimensional induced velocity field which in most
cases has a complicated nature, Figure 9. The induced velocity is denoted vi. This velocity field,
in turn, causes fluctuating airloads on the rotor blades which not only affect the rotor performance but also can cause high rotor vibrations and intrusive noise. In summary, these aerodynamic interactions can not be neglected in the modeling design of a helicopter and are therefore studied closer in this chapter.
Quiescent flow Quiescent flow Rotor wake TPP Tip vortices
Figure 9 - Velocity field caused by tip vortices along the rotor wake boundary.
3.1 Momentum Theory (MT)
In order to find an expression for the induced inflow, the basic performance of the rotor must be analyzed. This can be done using momentum theory. It encompasses three conservation laws of aerodynamics:
• Conservation of fluid mass
• Conservation of fluid momentum
• Conservation of energy
3.1.1 Momentum Theory in hover
Initially, the flight condition hover is studied since the rotor flow field is then azimuthally axis-symmetric, i.e. it is the same around a rotor revolution. The rotor gives rise to a slipstream or
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wake where the velocity increases further down in the wake as the diameter of the wake decreases. Consider the volume surrounding the rotor and its wake and let its surface have the
area S, Figure 10. dS is the unit normal area vector which points out of the control volume across
the surface S. vi is the induced velocity in the rotor plane and v∞is the velocity in the far wake.
Far above the rotor, in the reference plane 0, the flow is considered quiescent.
Figure 10 - Control volume surrounding the rotor and its wake.
It is assumed that the flow through the rotor is:
• quasi-steady (i.e. the flow properties at a point are constant with time)
• one-dimensional (i.e. the flow is uniform over the rotor disk)
• incompressible
• inviscid
The equation governing the conservation of fluid mass can be expressed as: 0
S
= ⋅
∫∫
ρV dS (3.1)where V is the local velocity and ρ is the density of the air. This equation, together with the
assumption of quasi-steadiness, state that the mass flow rate, m& , must be constant within the
control volume. If the area of the rotor disk is called A, as in Figure 10, the following is valid:
i i 2v Av A v A m& =ρ ∞ ∞ =ρ =ρ (3.2) i
v is here the induced velocity at the rotor disk.
The equation of conservation of fluid momentum is given by:
(
V S)
V S F=∫∫
+∫∫
⋅ S S d pd ρ (3.3)T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
For an unconstrained flow, which is the case here, the net pressure force on the fluid inside the control volume (i.e. the first term in equation (3.3)) is zero, which means that the net force on the
fluid, F, simply equals the net time rate of change of fluid momentum across the control surface,
S. According to Newton’s third law, there is an equal and opposite reaction to the force F which
in this case is the rotor thrust, T. This leads to the expression:
(
⋅)
−(
⋅)
= = −∫∫
∫∫
∞ V S V V S V F 0 d d T ρ ρ (3.4)In hovering flight, the second term is zero since the flow is quiescent far above the rotor disk. This results in the scalar equation:
(
)
∞ ∞ = ⋅ =∫∫
d mv Tρ
V SV & (3.5)Finally, the equation of conservation of energy states that “the work done on the fluid by the rotor manifests as a gain in kinetic energy of the fluid in the rotor slipstream per unit time” [7] as:
(
)
∫∫
⋅ = S d W 2 2 1 V S Vρ
(3.6)In this case, the work done on the rotor per unit time is given by:
(
)
∫∫
(
)
∫∫
⋅ − ⋅ = ∞ 0 2 2 2 1 2 1 V S V V S V d d v T iρ
ρ
(3.7)Analogical to the statement above, the second term is zero since the flow is quiescent far above the rotor disk. This results in the scalar equation:
(
)
2 2 2 1 2 1 ∞ ∞∫∫
⋅ = = ρ d mv v T i V S V & (3.8)Equations (3.5) and (3.8) now result in a simple relationship between the induced velocity vi in
the rotor plane and the velocity v∞ in the far wake:
∞ = v vi 2 1 or v∞ =2vi (3.9) This relationship can now be used in the expression for the rotor thrust, (3.5), as:
2 2 2 2vi ρAvi vi ρAvi m v m T = & ∞ = & = = (3.10)
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. A T v vh i
ρ
2 = ≡ (3.11)Finally, the relationship between the induced velocity and the thrust coefficient, CT, can be
determined. The thrust coefficient is a non-dimensional quantity that is given by:
2 2 T R A T C Ω = ρ (3.12)
Now equations (3.11) and (3.12) can be used to receive the following relationship:
2 2 1 T i i h C A T R R v = Ω = Ω = ≡ ρ λ λ (3.13)
This constitutes the relationship between the thrust coefficient and the induced inflow ratio under the assumption of an inflow uniformly distributed over the disk.
3.1.2 Momentum Theory in forward flight
In forward flight, the rotor wake trails backwards over the fuselage and is therefore no longer axis-symmetric as can be seen in Figure 11. In this case it is difficult to analyze the wake which among others depends on rotor thrust, blade flapping and blade collective and pitch angles.
Figure 11 - Rotor wake in forward flight.
There are different ways of modeling the inflow. The simplest way is modeling a uniform inflow, i.e. assuming that the induced velocity is constant over the rotor disk. A more explicated solution is the use of a linear inflow model. To start with, a uniform model is derived. The results are later used in the linear inflow model.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
(
) (
2)
2 sin cos V vi V U = ∞ α + ∞ α + (3.14)From the MT above it is known that:
(
)
i i 2 ρAU v v m 2 T = & = (3.15)(3.14) substituted into (3.15) yields:
(
) (
2)
2 sin cos i i V V v Av 2 T = ρ ∞ α + ∞ α + (3.16)In hovering flight (3.11) is valid:
A 2 T vh2 ρ = (3.17)
Using equations (3.16) and (3.17) gives the relationship:
(
) (
2)
2 sin cos i 2 h i v V V v v + + = ∞ ∞α
α
(3.18)Now, the following ratios are used, Figure 11:
R , R sin V , R cos V i = Ω Ω = Ω = ∞ ∞ i z x v λ α µ α µ (3.19)
Equations (3.19) substituted into (3.18) results in:
(
)
2 2 2 i i z x hλ
µ
µ
λ
λ
+ + = (3.20)From the momentum theory for hovering flight it is known that
2 CT h =
λ
which in equation(3.20) gives the induced inflow ratio as:
(
)
2 2 T 2 C i z x iλ
µ
µ
λ
+ + = (3.21)T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
3.2 Blade Element Theory
The blade element theory (BET) divides the rotor blade into sections, so called blade elements, which can be seen as 2-D airfoils that are independent of one another, Figure 12. The theory is used in order to estimate the distribution of blade aerodynamic loading over the rotor disk and thereby the overall rotor performance can be obtained. This means, that a second relationship between the rotor thrust and the induced inflow can be obtained.
Figure 12 - Top view of a blade element.
3.2.1 Blade Element Analysis in hover and axial flight
In the following, the relationship between the thrust coefficient and lift coefficient is derived. The local flow velocity, U, that acts on each element, consists of two components, one that is
normal to the rotor disk, UN, and arises due to climb/descent flow and induced flow, Figure 13.
The other component, UP, is parallel to the rotor and emerges due to the rotation of the blade.
The rotational speed varies with the radius of the blade and therefore also with individual blade
elements. The flow relative to the blade makes an angle
φ
, called the inflow angle, with the planeof rotation. This angle is mostly less than 10° and hence, the small angle approximation may be
used in calculations with
φ
. [7, 11]Parallel airflow UP Normal airflow UN α θ Total airflow U Reference plane dL dD dFz dFx x z y φ φ
Figure 13 - Flow velocities and aerodynamic forces around the cross section of a blade element.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. 2 2 N P U U U = + (3.22)
The induced inflow angle,
φ
, can be written as:{
}
P N P N U U ion approximat angle small U U ≈ ≈ = −1 tanφ
(3.23)The resultant incremental lift, dL, and drag, dD, per unit span can be expressed as:
dy cC U dD dy, cC U dL 2 l ρ 2 d 2 1 ρ 2 1 = = (3.24)
where Cl is the lift coefficient, Cd the drag coefficient and c is the local blade chord.
Continuing using Figure 13, the forces can be resolved as follows:
φ
φ
φ
φ
cos , cos sinsin dD dF dL dD
dL
dFx = + z = − (3.25)
Again, the small angle approximation can be used, such that sin
φ
=φ
and cosφ
=1.Additionally, the drag force is small compared with the lift force such that the approximation 0
sin
φ
=dD can be made. These simplifications reduce equations (3.25) to:
dL F dD, d dL
dFx =
φ
+ z = (3.26)The incremental thrust can be written as:
dL N dF N
dT = b z = b (3.27)
where Nb is the number of blades of the rotor. The increment in thrust coefficient can now be
obtained in the same way as above (equation (3.12)):
2 2 2 2 R A dL N R A dT dC b T Ω = Ω =
ρ
ρ
(3.28)Some further non-dimensional quantities are introduced in order to make the calculations independent of the rotor. This way, the rotor radius R and the angular velocity Ω are avoided. Lengths are divided by the radius, R, and velocities are divided by ΩR, such that:
r R y R y R U R y r = = Ω Ω = Ω = , (3.29)
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
Furthermore, the rotor solidity, σ, is introduced. It represents the ratio between rotor blade area
and rotor disk area such as:
R c N R cR N A A b 2 b b
π
π
σ
= = = (3.30)Equations (3.29) and (3.30) are now used in the equation for the rotor thrust which then yields:
( )
C r dr R y d R y C R c N R R dy cC U N R A dL N dC l l b l b b T 2 2 2 2 2 2 2 2 2 1 2 1 ρ 2 1σ
π
π
ρ
ρ
= = Ω = Ω = (3.31)In order to receive the total rotor thrust, the incremental thrust coefficient must be integrated along the blade, which for a rectangular blade yields:
∫
= 1 0 2 1 dr r C CTσ
l 2 (3.32)Cl in turn can be determined by using steady linearized aerodynamics:
(
φ
)
α
α α = − =C C θ Cl l l (3.33) where α lC is the two-dimensional-lift-curve-slope of the airfoil sections comprising the rotor.
Via a conversion of
φ
, the relationship between thrust and inflow ratio can be derived. Here,equation (3.23) is used: r r R v V Rr) v V y v V U U c i c i c i P N
λ
φ
= Ω + = Ω + = Ω + = = 1 ( (3.34)With the assumption that C is constant along the blade (i.e. an average value for the blade), the lα
thrust coefficient can finally be written as:
(
)
∫
− = 1 0 2 1 dr r r C CTσ
lαθ
2λ
(3.35)3.2.2 Blade Element Analysis in forward flight
Just as for hover and axial flight, the incremental lift and drag are given by:
dy cC ρU dD dy, cC ρU dL 2 l 2 d 2 1 2 1 = = (3.36)
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
By using equation (3.33), the incremental lift can be developed as:
(
)
dy cC(
U U U)
dy U U cC U dy cC U dL l P N P P N l P l P = − − = = 2 2 2 2 1 2 1 -2 1θ
ρ
θ
ρ
φ
θ
ρ
α α α (3.37)As before, the force perpendicular to the rotor disk is given by the incremental lift such as:
(
U U U)
dy cC dL dFz = = l P2 − N P 2 1ρ θ α (3.38)The force parallel to the disk is given by:
dy U C C U U U cC dD dL dF P l D N P N l x − = + = 2 2 2 1 α α θ ρ φ (3.39)
In forward flight, there are velocity components such that the expressions for the velocities from above are extended. The parallel component now includes a term that depends on the azimuth angle ψ such that:
( )
y,ψ y µ RsinψUP =Ω + xΩ (3.40)
The normal velocity component is extended with two parts that both are results of blade flapping:
( ) (
y,ψ µ λ)
R yβ( )
ψ µ Rβ( )
ψ cosψUN = z + i Ω + + xΩ
& (3.41)
The first term is the same as for hover and axial flight. The second term arises due to blade flapping velocity about a hinge while the third term is produced because of coning of the blades. The thrust coefficient is obtained analogically to the hovering case, such that:
(
)
(
)
(
)
∫
+ + Ω + + − + = 1 0 2 sin cos sin 2 1 dr r r r C CT l x z i x x α µ β ψ µ ψ β λ µ ψ µ θ σ & (3.42)3.3 Linear inflow model
Measurements made by Brotherhood & Stewart (1949) and Heyson & Katsoff (1957) indicated that the longitudinal inflow variation actually can be approximated as linear [7]. In the model suggested by Glauert, the lateral inflow variation is also considered to be linear. However, it is important to remember that these approximations are not true in many flight situations,
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
The following model is, according to [7], suggested by Glauert:
(
ψ ψ)
λ
λi = 0 1+kxrcos +kyrsin (3.43)
where the coefficient λ0 is the average induced velocity at the centre of the rotor as given by
equation (3.21), i.e.:
(
)
2 2 0 2 x z i T λ µ µ C λ + + = (3.44) xk and ky are weighting factors representing the deviation of the inflow from the momentum
theory value. Accordingly, for kx >1 an upwash-component is added on the leading edge of the
disk. For kx <1 the induced velocity field only constitutes a varying downwash field, Figure 14.
There are different expressions for the weighting factors. The expression suggested by Pitt & Peters (1981) is according to Leishman one of three models that best coincides with experimental data. Since it was the latest derived of the three models, it was used in this analysis. The weighting factors are given by:
0 , 2 tan 23 15 = = y x k k π χ (3.45)
which means that the lateral inflow is constant over the disk. The wake skew angleχ is given
by: + = − 0 1 tan
λ
µ
µ
χ
z x (3.46)See Figure 11 for the definition ofχ. [7, 10]
The difference between a uniform and a linear inflow model is apparent in Figure 14. The uniform inflow model represents an inflow that is constant over the entire rotor disk. The linear inflow model, on the other hand, describes the inflow as linearly varying over the disk with the
slope kx. Note that kx is not necessarily big enough to cause upwash components, as can be seen
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. kx λ0 TPP λ0 ψ = 180° TPP
b) Linear inflow model with upwash components a) Uniform inflow model
kx
λ0
TPP c) Linear inflow model
with-out upwash components
ψ = 180°
ψ = 180°
ψ = 0°
ψ = 0°
ψ = 0°
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. 4 MODELING THEORY
The model that has been used in the presented work is written in MATLAB and is made up of several different scripts that include a great deal of Maple-generated formulas. The original model was based on the uniform inflow model described above. In order to be able to compare the uniform inflow model with Pitt & Peters’ inflow model, the later sub-model had to be implemented in the existing helicopter model. Except for the inflow, the models are equal. The essence of the mathematical relationships is presented here in order to receive an overview of the helicopter model.
4.1 Coordinate systems
There are four types of coordinate systems used in the model for calculation of the helicopter
aerodynamics, Earth (FE), helicopter body (FB), helicopter hub (Fh) and helicopter blade (Fb). In
the following, the transformations between the coordinate systems as well as the systems themselves are described.
4.1.1 Coordinate transformation
Coordinate transformations are carried out between all of the four coordinate systems FE, FB, Fh
and Fb. Generally, coordinate transformations can be written as:
Q P Q Q P r A r r = ,0 + 2 (4.1)
where rP and rQ are coordinates in the P- and Q-systems, rQ,0 is the coordinate of the Q-origin
in the P-system and AQ2P is a rotation matrix from Q to P. The inverse rotation matrix is
expressed AP 2Q. The general transformation applied on the systems above results in:
b h b b h h B h h B B E B B E r A r r r Α r r r A r r 2 0 , 2 0 , 2 0 , + = + = + = (4.2)
4.1.2 Earth fixed coordinate system (FE)
For the formulation of the equations of motion an inertial coordinate system is needed. This means that the Earth is approximated as flat and stationary in inertial space. The x-axis points North, the y-axis points East and the z-axis points down, Figure 15. The position of the aircraft
relative the Earth-fixed frame (FE) is determined by:
[
]
TE E = x y z
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
The orientation of the helicopter relative the Earth-fixed frame is defined using Euler angles such that:
[
]
TE E = Φ Θ Ψ
Ω (4.4)
4.1.3 Body fixed coordinate system (FB)
The body fixed coordinate system originates at the centre of gravity of the helicopter. The right handed coordinate system has its x-axis through the nose of the aircraft, its y-axis pointing to the right (looking in the forward direction of the aircraft) and its z-axis pointing down through the floor of the helicopter as can be seen in Figure 15.
xB, u zB, w yB, v CG xE (North) yE (East) zE (Down) rE FB FE
Figure 15 - Relationship between Earth-fixed frame FE and body-fixed frame FB.
The angular velocities of the helicopter are given by:
[
]
T B r q p = B ω (4.5)where p, q, r are the rates of roll about the xB-axis, pitch about the yB-axis and yaw about the zB
-axis respectively. The Euler angle rates, which differ from the angular velocities, are given by the expression:
[
]
[
]
T B E B T E E Φ Θ Ψ T 2 p q rΩ& = & & & = (4.6)
where TB 2E is a transformation matrix from body system to Earth system given by (A.4) in
Appendix A. Thus, the Euler angles are obtained by integrating the Euler angle rates. The velocities of the helicopter are given by:
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
[
]
T B w v u = B V (4.7)where u, v, w are the velocities in the xB-, yB- and zB-directions respectively, Figure 15. In order
to receive the position of the helicopter in the Earth-fixed frame, the velocities in the body-fixed frame are transformed according to:
[
]
[
]
T B E B T E E x y z A 2 u v wr& = & & & = (4.8)
E B 2
A corresponds to the three rotations given by the Euler angles Φ,Θ,Ψ and is given by (A.3)
in Appendix A.
4.1.4 Hub and blade coordinate systems (Fh and Fb)
There are altogether four rotor coordinate systems, a hub coordinate system (index h) and a blade coordinate system (index b) for both the main rotor and the tail rotor. The hub system of the main rotor has its origin at the main rotor hub with its axes pointing in the same direction as the body axes, i.e. the x-axis pointing forward in the helicopter direction, the y-axis pointing to the right and the z-axis pointing down. The blade system follows the blade around the revolution. Its x-axis lies in the blade plane, pointing backwards, opposite to the rotational direction. The y-axis lies along the blade and points away from the hub while the z-axis is normal to the blade, pointing upwards. In Figure 16 the relationship between the hub and blade systems is illustrated.
The transformation from hub to blade system (denotedAh2b) is carried out in four steps. In the
first transformation, A0, all blade angles, i.e. the azimuth angle ψ, the pitch angle θ and the flap
angle β are zero, Figure 16 a). The following three transformations,Aψ, Aθ andAβ, are
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. θ yb xh zh yh xb zb yh xb zh xh zb yb β xh yh yb xb zb zh xh yh yb xb zb zh
a) Transformation A0 b) Transformation Aψ using
azimuth angle ψ c) Transformation Aθ using pitch angle θ d) Transformation Aβusing flap angle β ψ
Figure 16 – Transformation from hub to blade. Ah2b =AβAθAψA0.
The transformation matrices A0,Aψ,Aθ,Aβ andAh 2bare given by equation (A.6) and (A.7) in
Appendix A.
4.2 States and input signals
The model is made up of 24 states that describe the condition of the helicopter. The first twelve states are rotor states, describing the flap angles and their derivatives for the main rotor (mnr), tail rotor (tlr) and paddles (pad). The second indices, col, cos and sin stand for collective, cosine and sine respectively. The order is given by:
[
]
sin pad cos pad sin mnr cos mnr col tlr col mnr sin pad cos pad sin mnr cos mnr col tlr col mnr , , , , , , , , , , , ,β
β
β
β
β
β
β
β
β
β
β
β
∂ ∂ ∂ ∂ ∂ ∂ (4.9)As can be seen, the helicopter blades are only considered to have one DoF which is the flapping movement, although the blades in fact are able to both feather and move around a lead-lag-hinge. Those DoF are however neglected in the model.
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t.
The remaining states describe, in the given order, translational velocities, position, angular velocities and attitude of the helicopter such that the state vector is given by:
[
]
T E E E y z p q r x w v u Φ Θ Ψ (4.10)The input signals of the model are given by the following angles:
[
]
T tlr sw,col sw,sin sw,cos θ θ θ θ (4.11)The first three angles are swash plate angles that are controlled via three control actuators. They affect the main rotor and control the pitch, roll and vertical motion of the helicopter. The fourth angle is the blade angle of the tail rotor which controls the yaw motion. The swash plate angles are easily converted to blade angles which are of greater interest from an aerodynamic point of
view. The tail pitch angle θtlr needs no conversion since the tail rotor blades are directly
controlled via control actuators. The conversions are written as:
(
swcos)
sin bl sin sw cos bl , , , , factor factorθ
θ
θ
θ
− ∗ = ∗ = (4.12)The factor depends on linkages between swash plate and main rotor blades and its magnitude is not important in this analysis. More important is the switch between sine and cosine terms which will be seen in the analysis later on.
4.3 Trimming
If no settings are made, the model is trimmed for hover, i.e. x&E = y&E = z&E =0. If other flying
conditions are desired, the translational velocitiesx&E, y&E and z&E can be set and the model is
then trimmed thereafter. If for example the setting x&E =10m/s is made, the helicopter flies
straight forward, seen in the Earth-fixed frame. Since a forward velocity only is obtained by tilting the helicopter and thereby obtaining an AoA, the velocity in the body-fixed frame is
constituted by one component in the xB-direction (u) and one component in the zB-direction (w)
of the helicopter, Figure 17. The trimming sequence is performed on pitch and roll attitude as well as flapping angles. Flapping derivatives and angular velocities are set to zero.
u
E
x& w
Earth
T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. 4.4 Equations of motion
In the following, the calculation of the states is described. Initially, the motion of the rotor blades is calculated. This is done by evaluating the equation of the flap motion and thereby obtaining the various flap angles and flap angle rates. Additionally, forces and moments acting on the blades are obtained, which in turn are used for the calculation of the total forces and moments acting on the helicopter. Those forces and moments are then used in the Euler’s equations of motion which in turn evaluate the translational and angular velocities u, v, w, p, q and r. By
transformations the position, expressed inxE, yE and zE, as well as the Euler angles,
Ψ Θ
Φ, and are obtained.
4.4.1 Forces on the helicopter
The forces acting on the helicopter expressed in the body system are given by:
mg B E aero B tlr B mnr B B F , F , F , A 2 F = + + + (4.13)
The components on the right hand side are described in the following and illustrated in Figure 18. mg FB,tlr FB,mnr FB,aero CG
Figure 18 - Forces on the helicopter that are included in the model.
mnr B,
F is the resulting force acting on the main rotor. It is derived via the time-dependent
equations for the motion of the blades. Since this model neglects the lead-lag motion as well as the feathering motion, the only equation left describes the motion around the x-axis of the blade, i.e. the flapping motion. This motion depends on aerodynamic, centrifugal and gyroscopic forces and comes down to the following:
( )
(
)
( ) (
)
(
( )
( )
)
β β β I M e t q t p e e t e t = b,x + − Ω Ω − Ω + + + − Ω + − 1 sin cos 2 1 2 2 1 2 & & (4.14)The right hand side is given by the x-component of the moment around the hinge and the inertia of the blade according to:
(
)
(
y)
dyT h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. and: 2 2 ) 1 ( 3 1 − = m R e Iβ b (4.16)
where m is the mass of a blade and eR is the hinge offset as can be seen in Figure 19. b
Solving equation (4.14) gives an expression for β and thereby the flap acceleration,β&& , can be
obtained. β&& is then used in the following equation yielding the hinge force acting on one blade,
hinge b F : aero z b b hinge b m F F =
β
&&− , (4.17) hinge bF is located at the flapping hinge and lies in the z-direction of the blade, Figure 19. Fbaero,z is
the z-component of the aerodynamic force acting on the blade.
eR (1-e)R Hub shaft Flapping hinge ß Fbhinge Mbhinge Mh Fb,zaero zb xb yb
Figure 19 - Forces and moments acting on the flapping hinge and hub shaft.
The total force Fbaero can be calculated using the following relationship which is equivalent to
equation (3.25):
(
f dx)
dy[
(
)
]
dydS
fb b
aero
b =
∫∫
=∫
∫
=Aθ∫
Lsinφ+Dcosφ,0,Lcosφ−DsinφF (4.18)
Since Fbhinge acts normal to the blade, only the z-component of equation (4.18) is needed. Solving
the integral and substituting the resulting z-component into equation (4.17) yields an expression for Fbhinge which via two transformations gives the force FB,mnr. Since all calculations above are carried out for only one blade, the forces of all blades must be summed up such that:
[
]
∑
⋅ = Nb hinge T b h b B h B,mnr F 1 2 2 0 0 F A A (4.19)T h is d o c u m e n t a n d t h e i n fo rm a ti o n c o n ta in e d h e re in i s th e p ro p e rt y o f S a a b A B a n d m u s t n o t b e u s e d , d is c lo s e d o r a lt e re d w it h o u t S a a b A B p ri o r w ri tt e n c o n s e n t. , B tlr
F is obtained analogically to FB,mnr, only this time using tail rotor forces.
,
B aero
F is the aerodynamic force which arises due to the helicopter body. It is given as a function
of the dynamic pressure, the drag coefficient and its respective reference area such that:
ref D aero
B =q⋅C ⋅S
F , (4.20)
The last term of equation (4.13), AE2Bmg, is a transformation of the gravitational force from
Earth to body coordinates.
Summing up all forces above leads to the total force on the left hand side of equation (4.13) which can be written on component form such that:
[
BX BY BZ]
B = F F F
F (4.21)
This division is carried out in order to use the single components in Euler’s equations of motion, which is described further down.
4.4.2 Moments on the helicopter
The moments around the CG in body coordinates,rB,cg, are given by:
(
Bcg Bmnr)
Bmnr(
Bcg Btlr)
Btlr aero B tlr B mnr B B M , M , M , r , r , F , r , r , F , M = + + + − × + − × (4.22)The components of the right hand side of equation (4.22) are presented below and illustrated in
Figure 20. MB,aero is the aerodynamic moment which arises due to the helicopter body. In this
model, it is considered small enough to be neglected.
F
B,tlrF
B,mnrM
B,mnrM
B,tlrCG