96685625 realtime flightsim helicopter

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DEVELOPMENT OF A R EAL-TIME FLIGHT SIMULATOR

FOR AN EXPERIMENTAL MODEL HELICOPTER

Diploma Thesis

Cand. aer. Christian Munzinger

Atlanta, December 1998

Georgia Institute of Technology

School of Aerospace Engineering

advised by:

Dr. Anthony J. Calise

Dr. J. V. R. Prasad

IFR

University of Stuttgart


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SUMMARY

This work first describes the development of a real-time flight simulator for an R-50

experimental model helicopter. A mathematical model of the helicopter is developed to

represent the dynamics of the real system. This simulation model is used to investigate

and analyze the helicopter dynamics in a hovering flight condition. The importance of a

control rotor used for stability augmentation of the helicopter is emphasized and

investigated in more detail. Combining the model with further flight software and

hardware, a flight simulator is obtained that is capable of real-time flight simulation. This

simulator will be used in the future for detailed studies on new modern control algorithms

used for helicopter flight control.

Experimental flight tests with the real helicopter are performed and analyzed and

allow the identification of a simplified linear model valid close to the hover flightcondition. Results are shown and compared to the linearized model obtained from the

simulation. The system identification employs a frequency response and a step response

method that result in an approximate model for the helicopter dynamics.

Linear models from simulation and flight tests are then used in a recently developed

Neural Network Adaptive Nonlinear Flight Control System and applied to the real-time

simulator and the real helicopter. The results of both applications are then briefly

presented.


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iii

ACKNOWLEDGEMENTS

This work was conducted by the School of Aerospace Engineering at the Georgia

Institute of Technology in Atlanta.

I want to thank Dr. Anthony J. Calise of the School of Aerospace Engineering in Atlanta

and Dr. Klaus H. Well of the Institute of Flight Mechanics and Control in Stuttgart for

their support. They both made it possible for me to study at Georgia Tech and enabled me

to realize this work.

I also want to thank Dr. Anthony J. Calise and Dr. J. V. R. Prasad who guided and

assisted me throughout my work and studies and gave me all the support I needed duringthis time.

Special thanks to Dr. Eric J. Corban of Guided Systems Technologies, Inc., who went

with me through numerous hardware and software problems related to the flight test

program.

My respect and thanks to the pilot, Mr. Jeong Hur, and the whole flight team, who

suffered only minor heart-attacks during some critical flight maneuvers that would have

brought my work to a sudden end.

Finally I want to thank all my colleagues and friends who supported me during this one

year at Georgia Tech.

Christian Munzinger Atlanta, Georgia

December 1998


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Contents

iv

CONTENTS

Summary .............................................................................................................................ii

Acknowledgements.............................................................................................................iii

Contents ............................................................................................................................iv

Nomenclature ......................................................................................................................vi

List of Figures......................................................................................................................x

List of Tables......................................................................................................................xii

Chapter 1 Introduction.....................................................................................................13

1.1 Simulation of Flight.......................................................... ........................................ 131.2 Experimental Flight ................................................. ................................................. 15

Chapter 2 Helicopter Flight Dynamics............................................................................17

2.1 General Equations of Unsteady Motion.. ........................................................ ............ 17

2.2 The Small-Disturbance Theory.............. ........................................................ ............ 21

Chapter 3 Helicopter Theory ..........................................................................................23

3.1 Main Rotor Reference Frames and Notations.... ...... ..... ...... ..... ...... ...... ..... ...... ..... ..... ... 23

3.2 Hover and Vertical Flight..................................................................... ..................... 26

3.3 Forward Flight ........................................................ ................................................. 283.3.1 Rotor Theory in Forward Flight....................................................................................................283.3.2 Influences of Rotor Effects and Rotor–Helicopter Interference ..................................................29

Chapter 4 Helicopter Stability and Control ...................................................................35

4.1 Helicopter Control............................................................................... ..................... 35

4.2 Helicopter Stability......................................... ....................................................... ... 374.2.1 Hover ...............................................................................................................................................384.2.2 Forward Flight................................................................................................................................42

4.2.3 Stability Augmentation with a Control Rotor ...............................................................................46

Chapter 5 Mathematical Modeling .................................................................................48

5.1 General Helicopter Model...................... ....................................................... ............ 48

5.2 Rigid Body Model....................... ........................................................ ..................... 49

5.3 Main Rotor Model....................... ........................................................ ..................... 49

5.4 Control Rotor Model ............................................... ................................................. 585.5 Model of Fuselage, Wing and Tail ........................................................ ..................... 63

5.7 Simulation Results for the Linarized Model...... ...... ..... ...... ..... ...... ...... ..... ...... ..... ..... ... 65

Chapter 6 Real-Time Simulation: Hardware and Software .........................................73

6.1 Simulation Elements.. ....................................................... ........................................ 73

6.2 Flight System Elements .................................................... ........................................ 76

6.3 Hardware-In-The-Loop-Simulation ...................................................... ..................... 77


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Contents

v

Chapter 7 Validation of the Helicopter Simulation Model ...........................................79

7.1 Flight Test Data and Requirements ....................................................... ..................... 79

7.2 System Identification Procedures ................................................. .............................. 817.2.1 Static Trim Values...........................................................................................................................817.2.2 Frequency Response Analysis........................................................................................................847.2.3 Step Response Analysis...................................................................................................................95

Chapter 8 Modern Adaptive Nonlinear Flight Control in

Simulation and Real Flight ..........................................................................101

8.1 Flight Control System........................... ........................................................ .......... 101

8.2 Simulation and Experimental Results................................. ...................................... 105

References........................................................................................................................108

Appendix A – R-50 Helicopter Data..............................................................................110

Appendix B – Equations of Unsteady Motion of the Rigid Body................................112

Appendix C – System and Control Matrices ................................................................113

Appendix D – Results of Linear System Analysis for Hover.......................................115

Appendix E – R-50 Helicopter System Components....................................................118

Appendix F – Simulated and Experimental Bode Plots...............................................120


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Nomenclature

vi

NOMENCLATURE

( L, M, N ) Components of moment about the CG, in body frame ft lb

( p, q, r ) Angular helicopter body rates, in body frame rad/sec

( u, v, w ) Velocity components relative to air expressed in body frame ft/sec( u, v, w )E Velocity components relative to the Earth fixed frame ft/sec

( X, Y, Z ) Components of force acting along the (x, y, z) B axes lb

( x, y, z )a Helicopter aerodynamic coordinate frame

( x, y, z )B Helicopter body coordinate frame

( x, y, z )E Earth fixed coordinate frame

( φ, θ, ψ ) Euler angles rada Two-dimensional constant lift curve slope 1/rad

a0 Coning angle rad

a1s First harmonic coefficient of longitudinal blade flapping rad

with respect to shaft (positive for tilt back)

a p, aq, ar Uncoupled stability derivatives with respect to 1/sec

uncoupled body angular rates

jâ Estimated system parameters

A Rotor disk area ft2

A1, A2 First and second harmonic coefficient of lateral rad

blade feathering

A1,SP Lateral swashplate tilt relative to HP rad

(positive for tilt right)

AR Blade aspect ratio b1 First harmonic coefficients of lateral blade flapping rad

with respect to feathering plane

B Number of blades

B1, B2 First and second harmonic coefficient of longitudinal rad

blade feathering

B1,SP Longitudinal swashplate tilt relative to HP rad

(positive for tilt forward)

b1s First harmonic coefficient of lateral blade flapping rad

with respect to shaft (positive for tilt right)

b p, bq, br Uncoupled control derivatives with respect to inputsresulting in uncoupled body angular rates

c Mean blade chord length ft

cD0 Mean profile drag coefficient

cm Mean profile moment coefficient of control rotor

dhub Horizontal hub distance from helicopter CG ft

eMR Flap hinge offset of main rotor blade ft


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Nomenclature

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E Vector of output errors

E0 Mean harmonic coefficient of blade lag motion rad

E1, E2 First and second harmonic cos-coefficients of rad

blade lag motion

F Matrix of state derivatives

F1, F2 First and second harmonic sin-coefficients of rad

blade lag motion

f wake Wake-function for low/high speed effects ondv

db s1 ,du

da s1

G Matrix of control derivatives

hhub Vertical hub distance from helicopter CG ft

is initial shaft tilt (positive back) rad

I b Moment of inertia of blade about flapping hinge slug ft2

Ixy Product of helicopter inertia ∫ dm xy slug ft2 Ixz Product of helicopter inertia ∫ dm xz slug ft

2

Iyz Product of helicopter inertia ∫ dm yz slug ft2 k MR Coefficient defining main rotor blade pitch due to swashplate tilt

k β Coefficient defining main rotor blade pitch due control rotor tilt

K 1 Cross-coupling coefficient due to delta-three-angle

K 2 Cross-coupling coefficient due to hinge offset

K c Total cross-coupling coefficient

l b Length of aerodynamic blade section of control rotor ft

lCR

Length of control rotor bar ft

β& M Non-dimensional aerodynamic moment due to blade flapping velocity

Mgust Control rotor moment due to wind velocity ft lb

MT Torque ft lb

Mµ Non-dimensional aerodynamic moment derivative with

respect to rotor advance ratio

R Blade radius ft

T Thrust lb

Td Time delay sec

ν̂ Total airspeed ft/sec

V Velocity vector relative to the atmosphere ft/secw blade Average velocity of main rotor blade relative to air ft/sec

wr Velocity of rotor disk relative to air ft/sec

x State vector of helicopter rigid body motion


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Nomenclature

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Important Derivatives

du

da s1 Derivative of longitudinal TPP tilt with respect to u-velocity rad/(ft/sec)

dvdb s1 Derivative of lateral TPP tilt with respect to v-velocity rad/(ft/sec)

1dA

dL Roll moment due to lateral cyclic pitch change ft lb/rad

sdb

dL

1

Roll moment due to lateral TPP tilt ft lb/rad

sda

dM

1

Pitch moment due to longitudinal TPP tilt ft lb/rad

1

dB

dM Pitch moment due to longitudinal cyclic pitch change ft lb/rad

Greek Symbols

α̂ Parameter vector in system identification

β Blade flapping angle radβc First harmonic coefficient of longitudinal blade flapping rad

of control rotor with respect to shaft

βc,CR Control rotor longitudinal TPP tilt rad

βs First harmonic coefficient of lateral blade flapping radof control rotor with respect to shaft

βs,CR Control rotor lateral TPP tilt radδ3 Delta-three-angle radδcoll,MR Collective main rotor input radδcoll,TR Collective tail rotor input radδlat Lateral cyclic input radδlong Longitudinal cyclic input radδu Input vector to helicopter rigid body motion radγ Lock numberϕd Phase shift due to time delay radλ Directional parameter (-1=clockwise, 1=counterclockwise) νi induced velocity ft/secθ Blade pitch angle radθcoll Blade pitch due to pilot collective input radθtwist Blade twist radθ0 Collective main rotor blade pitch rad


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Nomenclature

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θ0,CR Constant initial control rotor blade pitch radρ Density of air slug/ft3 τ Time constantω Frequency rad/sec, Hz

ω in Flap rate coefficient for in-axis-motion

ωoff Flap rate coefficient for off-axis-motionΩ Rotor rotational speed rad/secΩf Coefficient defining change in natural main rotor frequency

due to hinge offset

ξ Limited extension parameter of control rotorψ b Rotor blade azimuth rad

Abbreviations

CG Center of gravity

coll Collective pitch

DOF Degree of freedom

HP Hub plane

lat Lateral

long Longitudinal

MR Main rotor

NN Neural network

rpm Rotor rotational speed

TR Tail rotor

TPP Tip path plane


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List of Figures

x

LIST OF FIGURES

Figure 2.1: Body axes of the helicopter and notations.............................................................................18

Figure 2.2: Wind axes of helicopter in forward flight .............................................................................. 19

Figure 2.3: Block diagram, vehicle with plane of symmetry, body axes, flat-earth approximation,

no wind [4] ..................................................................................................................................20

Figure 3.1: Rotor disk and notations...........................................................................................................24

Figure 3.2: Hub plane, tip-path plane, body axes and notations ........................................................... 24

Figure 3.3: Fundamental blade motion....................................................................................................... 25

Figure 3.4: Rotor blade velocity in forward flight .................................................................................... 28

Figure 3.5: Offset of rotor blade flap hinge ...............................................................................................31

Figure 3.6: Cross-coupling due to the delta-three-angle ........................................................................31

Figure 3.7: Rotor-Fuselage interference in (a) hover and (b) forward flight .......................................33

Figure 3.8: Mechanical linkages of the control rotor for the R-50 experimental helicopter ............. 34

Figure 4.1: Longitudinal hover poles dependent on normalized flap frequency ν =ωn / Ω ................ 40

Figure 4.2: Typical hover poles for decoupled longitudinal and lateral motion .................................41

Figure 4.3: Typical hover poles for coupled longitudinal and lateral motion ..................................... 41

Figure 4.4: Influence of forward speed and horizontal tail on longitudinal poles .............................. 44

Figure 4.5: Influence of forward speed on lateral poles .......................................................................... 45

Figure 4.6: R-50 control rotor providing rate f eedback [29] ................................................................. 47

Figure 5.1: Control Rotor of the R-50 Helicopter (view from top).......................................................... 60

Figure 5.2: Poles of coupled longitudinal and lateral motion, no control rotor.................................. 68

Figure 5.3: Poles of coupled longitudinal and lateral motion, with control rotor............................... 70

Figure 5.4: Hover poles of longitudinal motion, with and without control rotor.................................71

Figure 5.5: Hover poles of lateral motion, with and without control rotor........................................... 72

Figure 6.1: Elements of Simulation Software............................................................................................. 74

Figure 6.2: Display of the R-50 real-time simulator on PC-screen ........................................................ 75

Figure 6.3: Joint GST/Ga Tech real-time hardware-in-the-loop simulation facility [28] .................. 78

Figure 7.1: Trim table for R-50 (simulation), rearward to forward flight ............................................. 83

Figure 7.2: Block diagram of approximated linear helicopter dynamics for hover.............................85

Figure 7.3: Frequency response in the pitch channel, ω= 0.75Hz, quδ̂ =0.035 rad...........................87

Figure 7.4: Experimental Bode plot for the pitch channel ....................................................................... 88

Figure 7.5: Experimental Bode plot for the roll channel ......................................................................... 89

Figure 7.6: Experimental Bode plot for the yaw channel ......................................................................... 90

Figure 7.7: Experimental and simulated frequency response, 0.31sec time delay, pitch dynamics .. 94


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List of Figures

xi

Figure 7.8: Response of identif ied model (7.17) compared with measured data ..................................98

Figure 7.9: Response of identified model (7.18) compared with measured data..................................100

Figure 8.1: Neural Network Augmented Model Inversion Architecture................................................. 102

Figure 8.2: Multilayered network with one hidden layer......................................................................... 104

Figure 8.3: Simulated system response of R-50, doublet inputs in long. cyclic .................................... 105

Figure 8.4: Pitch response of R-50 in flight test with NN controller in pitch channel ......................... 106

Figure E.1: R-50 fully equipped during flight test .................................................................................... 118

Figure E.2: R-50 fully equipped on transport cart; GST ground control station in background ...... 118

Figure E.3: R-50 avionics box with on-board PC and sensor packet .....................................................119

Figure E.4: R-50 horizontal tail for improved handling characteristics in forward flight ................. 119

Figure F.1: Experimental and simulated frequency response, 0.31sec time delay, roll dynamics ....120

Figure F.2: Experimental and simulated frequency response, 0.31sec time delay, yaw dynamics ....121


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List of Tables

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LIST OF TABLES

Table 4.1: Single rotor helicopter coupling sources [15]........................................................................36

Table 5.1: Trim values in simulated hover for R-50, with and without control rotor .......................... 66

Table 5.2: Analytically obtained system matrix in hover, no control rotor ...........................................67

Table 5.3: Analytically obtained control matrix in hover, no control rotor .......................................... 67

Table 5.4: Eigenvalues, damping and frequencies of hover modes, no control rotor ......................... 68

Table 5.5: Analytically obtained system matrix in hover, with control rotor ........................................ 69

Table 5.6: Analytically obtained control matrix in hover, with control rotor .......................................69

Table 5.7: Eigenvalues, damping and frequencies of hover modes, with control rotor .....................70

Table 7.1: Measured and simulated trim values for R-50 in hover and forward f light ....................... 82

Table 7.2: Identified parameters and system delay in hover using experimental Bode plots ............92

Table 7.3: Identified parameters for decoupled lateral motion, step response ....................................97

Table 7.4: Identified coupled model parameters for system including only angular rates.................99


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Chapter 1 Introduction

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Chapter 1

INTRODUCTION

During the early phase of design and development of an aircraft, it needs to be tested

and performance limits need to be validated. For manned flight vehicles it is very

common to use flight simulators before real flight tests, not only to reduce time and costs

during the testing phase, but also to aid in the avoidance of possible loss of pilot and

aircraft in case of a failure. The use of real-flight simulators is therefore mostly related to

manned flight. In this work a simulation for a remotely controlled aircraft of much

smaller size needs to be developed to test a modern controller. The approach and the

mathematical model that will be used are similar to that of a manned helicopter; in fact

the main part of the simulation is taken from a simulation of a full size helicopter. Several

aspects arise from the very different size and therefore result in different dynamic

behavior which need to be taken into account. A brief description of desired capability

and capacity of real flight simulators in general will follow, and the still necessary results

of experimental flight for validation purpose and to build up confidence in handling the

real flight system will be pointed out.

1.1 Simulation of FlightRotorcraft and its unique capabilities of vertical take-off and landing, hover, vertical

and forward flight are playing an important roll in commercial and military applications.

Superior hover and low speed performance and agility are coupled with good flight

characteristics even in fast forward flight. The rotor of a helicopter generates the

predominant aerodynamic forces in all flight conditions and is source of forces and

moments on the aircraft that control position, attitude and velocity. An increase of

complexity with respect to rotor and blade dynamics and the still not sufficiently

explained aerodynamic effects of rotor aerodynamic or rotor-body interferencecomplicate the development of detailed mathematical models of rotors and helicopters in

general.

The use of mathematical models for simulation is therefore limited to some degree of

accuracy and depends on the final objective of the simulation. In general, examining

structural dynamics of single blades or blade sections needs a more accurate model than a

simulation of flight mechanics and aircraft performance. The effort and expense put into


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modeling and simulation are justified by the helicopter’s unique capabilities, and the

objective of simulation influences the level of math model fidelity directly related to the

effort and expense required for a given task.

High performance computers made an increase of accuracy of simulation possible,

but the complexity of simulation still rules the capability of operating in real-time. The

development of high-fidelity real-time simulators for research and design, concept

validation and training is strongly dependent on available time and money. Once

developed and validated, the simulator can be very efficiently used to support flight tests

to evaluate handling qualities during the development and design phase of new or

modified helicopter configurations or helicopter components. More advanced

applications, as in unconventional configurations like tilt wings and tilt rotors, unmanned

flight of full size aircraft, simulating emergency situations, validating and testing new

control systems or simple pilot training, underline the usefulness of such a high-fidelityreal-time simulator. Nevertheless, a highly accurate model of one component of a

complex system does not necessarily mean that the simulation of the whole system

behaves like the real physical system. This is especially true for a helicopter, since

components like rotors, fuselage, wings, horizontal or vertical tail, engine and actuators

interact with each other and influence the system response to external and internal

disturbances.

For application in helicopter controls, where the main objective is to control the

dynamic behavior of the helicopter over some desired flight envelope, it is necessary to

find a representative model that shows the same dynamic characteristics as the real

aircraft. On the one hand a detailed model of the main rotor is desired since the dynamics

are governed mainly by the main rotor, but on the other hand a too detailed description

increases the complexity of the simulation and limits the capability of real-time

simulation. Furthermore, most of the existing simulations make a very detailed

knowledge of the simulated system necessary. This knowledge covers exact physical data

of the aircraft geometry, airfoils of blades and wings and aerodynamic data that is gained

in wind tunnel tests. For competitive reasons this data is handled by companies with care

and is therefore generally not available.If a basic math model can be developed that depends only on basic data sources, then

the inflexibility of very sophisticated models that are now in common use can be

overcome. Additional individual vehicle components can be added and the existing

model can be easily extended or refined. Such a so-called "Minimum-Complexity

Helicopter Simulation Math Model" has been developed by NASA [1]. A modification of

this simulation model will be used and applied to the remotely controlled experimental


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model helicopter R-50 (see Appendix E), built by YAMAHA [29]. Aerodynamic data

from wind tunnel testing does not exist and the physical data of system components is

known only approximately. The capability of the minimum-complexity math model to be

flexible and easily extendible to other helicopter configurations is not only desired, it is

of great importance for this application. A small-scale model helicopter shows a different

aerodynamic behavior, since it is designed for a very different flight envelope than full-

scale helicopters. Further undesirable effects of excessive model complexity are

computational system delays, a great number of system parameters that need to be

determined for each aircraft, inability to easily observe relationships between modeling

parameters and model response (very important for handling quality simulation) and the

inflexibility in temporarily removing undesired dynamics for debugging. The most

important benefit of a minimum-complexity math model is the potential for a more clear

understanding of the cause and the resulting effect. This aspect can become veryimportant if the dynamic system itself is additionally complicated by a modern control

system that does not allow a fast and easy engineering understanding of the control and

response features.

In summary, the main attribute of a simulator as an effective tool for controller design

is the ability to produce desired results for a specific application and to operate over the

full flight envelope (forward, rearward and sideward flight, hover, transition from hover

to forward flight, vertical climb) with representative handling qualities. Through a man-

in-the-loop simulation it also becomes a very powerful tool to identify critical man-

machine or controller-machine interface issues and allow pilot training within a

reasonable amount of time, costs and risk until confidence in flying with a new system or

flight controller is gained.

Even if satisfying results can be achieved with a high-fidelity real-time simulator, the

results will not be sufficient unless they are confirmed in real flight. In the following, the

objective and the importance of flight tests to validate results will be briefly described.

1.2 Experimental Flight

To increase the fidelity of the mathematical model and to decrease effort required to

create and validate the model, a systematic system identification approach is used to

identify parameters and data that tunes the simulator to fit the highly complex and

nonlinear characteristics of helicopter flight. This approach will be described in Chapters

5 and 7. The data used for the identification process is produced during flight tests. The

aircraft motion is measured, and a model that reflects the physical behavior of the system


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needs to be found and investigated. The complexity of helicopters, and for the application

described in this work, the reduced size and low payload capability further reduce the

possibility of adding numerous sensors and hardware to measure velocities, rates or

angles in the rotating as well as in the non-rotating frame.

The on-board system of the experimental helicopter consists of a 200 MHz Pentium

based flight control processor and an integrated avionics system. Available on-board

sensors include a 3-axis gyro and accelerometer package, differential GPS with 2 cm

accuracy, 3-axis magnetometer and an 8 channel ultrasonic ranging system. A wireless

digital data link provides a communication link with a mission control ground station.

Representative flight tests are mainly basic maneuvers about hovering, level forward

flight and step inputs to the control stick from carefully trimmed flight conditions. Once

this basic validation is done, more complex maneuvers could be flown, measured and

compared with the simulation results. It is also important to understand operating rulesand human interaction with the flight system. The pilot information given in reports about

handling qualities are not measurable with sensors and are also dependent on personal

experience and pilot skills. However, to evaluate the overall accuracy of the simulation

and its real-time capability, this aspect needs to be investigated. It is obvious that the

visual channel strongly influences the pilot’s response to a directly observed change in

aircraft motion. This is only one weak point of some real-time simulations, since the hard

and software might not provide the pilot with the 3-D picture he is used to during flight.

Flight tests will give us further information how these facts have to be evaluated and how

they influence the simulation results. Engineering and piloted validation is therefore

necessary, and the quantitative and qualitative handling quality flight tests are described

in more detail in Chapter 7.

In the following work flight tests will mainly be used to create the basis on that a

fine-tuned model can be built on. This is a first important step for the procedure that will

follow to evaluate and investigate advanced control algorithms on model helicopters. The

Uninhabited Flight Research Facility (UFRF) located in the School of Aerospace

Engineering at Georgia Tech was initiated in June 1997 and is dedicated to flight testing

for this purpose. It presently contains two Yamaha model helicopters of the Type R-50.Testing capability and performance of the simulator is the main objective of this work.

After validation, this simulator will be used to test a neural network (NN) based, adaptive

flight controller, recently developed at Georgia Tech. More details on theory and design

of the developed control algorithm are given in [25, 26, 27]; Chapter 8 only gives a brief

overview of important aspects and should provide the reader with the basic understanding

of Feedback Linearization and Adaptive Neural Networks in flight controls.


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Chapter2 Helicopter Flight Dynamics

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Chapter 2

HELICOPTER FLIGHT DYNAMICS

Like most other flight vehicles, the helicopter body is connected to several elastic

bodies such as rotor, engine and control surfaces. The physical nature of this system is

very complex in shape and motion, and simple mathematical modeling seems not to be

very precise. Nonlinear aerodynamic forces and gravity act on the vehicle, and flexible

structures increase complexity and make a realistic analysis difficult. Several

assumptions can be made to reduce this complexity to formulate and solve relevant

problems. This chapter describes assumptions necessary for a satisfactory modeling of

the helicopter motion and introduces the fundamental motion of the flight vehicle in

general. Some features for the helicopter case are emphasized and explained with respect

to stability analysis and system identification as needed.

2.1 General Equations of Unsteady Motion

Derived from first principles, equations can be found that describe the aircraft as a

rigid body with six degrees of freedom (DOF’s), free to move in the atmosphere.

Aerodynamic forces and moments and gravity are incorporated directly in those

equations. In [5] the derivation of the general motion of a mass particle is given, then the

dynamic and kinematic equations for an arbitrary deformable vehicle in flight are

derived. Treating the earth as flat and stationary in inertial space simplifies the model

significantly. For most problems of airplane flight this is acceptable, and for an

experimental helicopter flying at low speed at very low altitude this approximation is

valid. The derived equations contain only a few further assumptions:

• The aircraft can be treated as a rigid body with any number of rigid spinning rotors.

• There is a plane of symmetry, so that Ixy = Iyz = 0.• The axes of spinning rotors are fixed in the direction relative to the body axes andhave constant angular speed relative to the body axes.

The third assumption seems to be rather crude for the helicopter case since helicopter

motion is mainly controlled by tilting the main rotor relative to the body axes and

therefore creates additional moments and forces. This assumption is justified by assuming


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only changes in rotor tilt of a few degrees relative to the body axes. This is in general true

and no modification needs to be made at this point.

For deriving the equations of motion, it is convenient to choose body axes since all

of the inertias remain constant in the body frame. The x-axis is fixed to a longitudinal

reference line in the aircraft; the y-axis is oriented to the right and the z-axis downward.

A common assumption is the exact symmetry of the aircraft with respect to the xz-plane.

The used body reference frame and its notations for forces, moments and angular rates

about its axis are shown in Figure 2.1.

xB, X, L, p

yB ,Y, M, q

zB, Z, N, r

Figure 2.1: Body axes of the helicopter and notations

Another reference frame that is very useful in formulating the equations of motion is

the earth-fixed frame. With the assumption of a flat and stationary earth, this frame

becomes an inertial system in which Newton's laws are valid. The origin is arbitrary, the

x-axis is horizontally pointing in any convenient direction, the z-axis is pointing

vertically downward. The y-axis is perpendicular to both. For all frames of reference a

right-handed coordinate frame is assumed. The center of gravity (CG) of the aircraft is

equal to the mass center, and its location is given by its Cartesian coordinates relative to

the earth-fixed frame.

To describe the aerodynamic forces the wind frame becomes important, since all of

these forces depend on the velocity relative to the surrounding air mass. This wind frame

is as well fixed to the aircraft, but the x-axis is now oriented along the velocity vector V

of the vehicle relative to the atmosphere. The z-axis lies again in the plane of symmetry;


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the y-axis is perpendicular to both. The origin is again located at the CG of the aircraft.

The wind frame and notations for a helicopter in forward flight are shown in Figure 2.2.

xB

yB ,ya

zB

xa

za

V

Figure 2.2: Wind axes of helicopter in forward flight

In the case of no velocity relative to the atmosphere, which occurs for a helicopter in

hover with no additional wind, the wind frame is not defined. That makes it necessary to

change back to the body axis if the helicopter motion from or to hover condition needs to

be computed. This again increases the complexity of the computations, but high performance computers used in flight simulation allow an easy transformation to any

desired frame without remarkable effort. Important transformation matrices can be found

in [4, 5]. The resulting general equations of unsteady motion are also taken from [4] and

listed in Appendix B. This set of equations includes kinematic and dynamic equations

and is presented with respect to body axes. For the case of no wind the equations can be

viewed in a block diagram shown in Figure 2.3. The mathematical system consists of 12

independent equations and the same number of dependent variables.

These variables are:

Position of the Center of Gracity (CG): xE, yE, zE

Attitude angles (Euler angles): φ, θ, ψVelocities relative to the earth fixed frame: u

E, v

E, w

E

Angular velocities: p, q, r


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Force Equations p , q , r

u , v , w

Control forces

θ , φ u

w

v

Moment Equations

u , v , w

p , q , r

Control Moments

p

q

r

Kinematics 1 p , q , r

φ

θ ψ

θ , φ

Kinematics 2

u , v , w x E

θ, φ ,ψ y E z E

Figure 2.3: Block diagram, vehicle with plane of symmetry, body axes, flat-earth approximation,

no wind [4]

Each block represents a set of equations with inputs and outputs. The generated

outputs on the right-hand side are the inputs to the left-hand side. Control forces and

moments are dependent on the control inputs. For a helicopter, these inputs are collective

and two cyclic (lateral and longitudinal) stick inputs to the main rotor, pedal inputs

controlling the tail rotor and the throttle controlling the power. In Chapter 5 the effects on

the main rotor tilt and body reactions due to control inputs are examined in more detail.

For stability and control analysis these equations of motion are frequently linearized

about a specific flight condition. From some steady flight condition it is assumed that the

aircraft motion consists of only small deviations from this reference condition. Since the

system of linear equations can also be used for identification purposes of the developed

simulation math model for several flight conditions, relevant aspects of the small-

disturbance theory will be formulated in the following. With respect to the control aspectsit should be pointed out that various flight control structures require at least an

approximate linear model for a vehicle, valid for one or even various flight conditions.

This allows vehicle dynamics to be easily inverted and used, for example, in methods

based on an inverting control scheme [20].


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21

2.2 The Small-Disturbance Theory

The nonlinear equations of motion listed in Appendix B are still accurate and

therefore very useful for engineering purposes when linearized about some unaccelerated

steady-state flight condition. Aerodynamic effects can be assumed to be linear functions

of disturbances, and the values of linear and angular velocity perturbations are usually

small for many cases [4]. Limitations for this method are disturbances or flight

maneuvers that result in large changes of angles and rates and cause large nonlinearities,

as for example in flight at high angle-of-attack. The detailed derivation of the basic

equations is given in [4].

The linearization process assumes small disturbances, so only first-order terms are

kept, and squares and products are assumed to be negligible. For a steady-state flight

condition all disturbances are set equal to zero. Linear relations to eliminate reference

forces and moments acting on the vehicle in this trimmed flight condition are obtained.Then the classic assumption of linear aerodynamic theory allows us to express

aerodynamic forces in terms of stability or, more generally, in aerodynamic derivatives.

In determining these derivatives more effort is necessary, and engineers use analytical

and experimental means to find reasonable and accurate results. At this point the

equations of motion for airplanes and helicopters need to be treated separately. An

assumption of decoupled equations of longitudinal and lateral airplane motion is in

general not valid for the helicopter. Derivatives of lateral forces and moments (Y, L, N)

with respect to longitudinal motion variables (u, w, q) are no longer zero, and some

derivatives with respect to rate changes of variables, often negligible in the airplane case,

need to be considered for a helicopter.

The system and control matrices F and G for hover listed in Appendix A show all of

the important gravitational terms that can be obtained analytically and partial derivatives

arising from aerodynamic forces and moments necessary to describe the linear set of

equations for a helicopter. The linear, first-order set of differential equations is then of the

form

uG x F x δ⋅+⋅=& , (2.1)

where x represents the perturbation of state variables uB, wB, θB, qB, vB, pB, φB and r B from a steady-state reference flight condition, the trim state. The control vector δucontains deviations from the trim control inputs δlong, δcoll,MR , δlat and δcoll,TR . The trimstates as part of the elements inside the matrix are noted with the subscript 0. It is


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convenient to use only the variable names in this matrix form instead of adding another

subscript to denote perturbations. This linear representation is only valid for the initial

angular velocities (pB, qB and r B) equal to zero. The system matrix F includes derivatives

due to small perturbations of system states; the control matrix G represents the

derivatives due to small perturbations of control inputs. It can be seen that the throttle is

not considered to be a control input. For a wide range of flight conditions, the rotational

speed of the rotor does not change, and a variation of throttle is made only to adjust

power keeping some desired rotational rotor speed constant.

One way to obtain the force and moment derivatives is to sequentially perturb the

states and control inputs, positively and negatively from trim values by some small

amount ∆. Then the forces and moments due to both perturbed conditions are computed,and the derivatives can be obtained by the following equation.

( ) ( )X

X

u

X u u X u u

uu = ≅

+ − −⋅

∂∂

0 0

2

∆ ∆∆

(2.2)

The force X and the state u in this equation represent all the forces, moments, states and

control inputs in the equations of motion.

This approach is used in the later described simulation routine to compute the linear

system matrices for any desired trimmed flight condition. Linear system analysis is very

useful and convenient to examine eigenvalues or eigenvectors, system responses to step

inputs, frequency response and other stability characteristics of a dynamic system. The

system matrices for hover are analyzed in more detail in Chapter 5.

As this chapter has summarized the dynamics of the helicopter treated as a rigid body,

the following chapter introduces the main features of helicopter theory. The main rotor is

essentially responsible for thrust, control forces and moments and is therefore the main

subject of the following investigation. Rotor dynamics and aerodynamics influence the

previously mentioned rigid body dynamics, result in cross-coupling of longitudinal and

lateral motion, and affect the stability of the dynamic system. Chapters 3 and 4

summarize the most important aspects of helicopter theory with respect to the main rotor

and its dynamics.


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Chapter 3 Helicopter Theory

23

Chapter 3

HELICOPTER THEORY

As previously mentioned, helicopter dynamics and aerodynamics are mainly affected

by the main rotor. Therefore this chapter will first introduce new main rotor reference

frames and notations that are useful to describe the main rotor in detail. Then it will cover

important aspects of hover and vertical flight and discuss forward flight. A section on

helicopter stability and control follows. Finally the influence of an additional control or

servo rotor to improve stability and handling qualities is described, and the importance of

this dynamic subsystem is pointed out. In general simplifying assumptions are introduced

to allow a faster analysis of the complex main rotor system, and some of the introduced

aspects will be neglected in some applications.

3.1 Main Rotor Reference Frames and Notations

Basically there are three different rotor reference frames used in this main rotor

analysis. The first reference frame is the rotor disk as shown in Figure 3.1. Basic

variables necessary to define and derive the basic equations of main rotor and blade

motion are described. The orientation of rotor rotation in the equations derived is

counterclockwise. Since the main rotor for the experimental helicopter, as for most

remotely controlled helicopters, rotates clockwise, this major difference can be

compensated with a simple sign change in some equations. This additional parameter

describing the direction of rotation will be pointed out when necessary. Further sign

changes in force and moment components due to torque and tail rotor will also be

necessary, as explained in the mathematical modeling of main and tail rotor in Chapter 5.

It is important to mention that the azimuth angle ψ of the blade is defined as zero in thedownstream direction. Azimuth and rotational speed Ω are for now defined positive

counterclockwise.The two additional rotor reference frames are shown in Figure 3.2. The hub plane

(HP) axes are defined with respect to the main rotor hub that remains fixed relative to the

rigid body of the helicopter. The tip-path plane (TPP) axes are defined with respect to the

motion described by the main rotor blade tips. A first simplifying assumption is that the

thrust vector is always perpendicular to the TPP, which is true in hover and vertical flight

and still very accurate in forward flight. The tilt of the TPP with respect to the HP can be


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24

defined by the two angles, a1s and b1s, referring to longitudinal and lateral tilt of the TPP.

For some helicopters the shaft and therefore the HP is designed with a forward tilt by a

small angle, is, with respect to the helicopter body axes frame. This shaft tilt is not shown

in Figure 3.2, but it will be included in the final force and moment equations for the

helicopter math model.

ψ = 0°

ψ = 270°

ψ = 90°

Forward

Velocity

V

ψ = 180°ψ

Rotor

Disk

Blade

Figure 3.1: Rotor disk and notations

xB

yB

zB

xHP

zHP yHP, yTPP

xTPP

a1s

T

zTPPzTPP

xHP, xTPP

b1s

zHP yHP

yTPP

T

TTR

b body axes

TPP tip path plane

HP hub plane

Figure 3.2: Hub plane, tip-path plane, body axes and notations


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The motion of the single main rotor blades governs the final tilt of reference planes.

The basic blade motion as treated in this analysis is essentially rigid body rotation about

the root attached to the hub. The degrees of freedom are the angles β, ζ, θ shown inFigure 3.3. The angle of rotation β about an axis in the disk plane, perpendicular to the

blade axis is called flap angle. The lag angle ζ is defined by the rotation about an axisnormal to the disk plane, parallel to the rotor shaft, and the pitch angle θ is the angle ofrotation about an axis in the disk plane parallel to the blade spar. For a detailed main rotor

analysis more complex motion than this fundamental blade motion needs to be

considered. In this work it is assumed that only the basic flap and pitch motion contribute

to major force and moment calculations and describe the most important rotor

characteristic influencing stability and control of the rigid helicopter.

ζ

Rotor

Shaft

Ω

β

θ

Blade

Figure 3.3: Fundamental blade motion

The steady-state blade motion is periodic around the azimuth. Using a Fourier series

expansion, the flap, lag and pitch motion can be written as

...2sin2cossincos

...2sin2cossincos

...2sin2cossincos

22110

22110

22110

+⋅−⋅−⋅−⋅−=

+⋅+⋅+⋅+⋅+=−⋅−⋅−⋅−⋅−=

ψ ψ ψ ψ θθ

ψ ψ ψ ψ ζ

ψ ψ ψ ψ β

B A B A

F E F E E

babaa

(3.1)

Since the mean and first harmonics (subscript 0 and 1) are most important to rotor

performance and control, all higher order terms will be neglected. The accuracy of the


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model remains high whereas the analysis is simplified. The most important angles used in

Equations 3.1 are the coning angle a0, pitch and roll angles of the TPP, a1 and b1,

collective pitch, θ0, and cyclic pitch, A1 and B1, commanded by the pilot. The notation ismainly taken from [1, 6].

It is convenient to compute main rotor forces and moments in the TPP and then

transform them into the HP or directly into the body axes frame. The components can

then be easily added to aerodynamic and gravitational forces and moments acting on the

rigid body. The nature of main rotor forces and moments is briefly described in the

following for different flight conditions, hover, vertical flight and forward flight.

3.2 Hover and Vertical Flight

Hover and vertical flight implies axial symmetry of the rotor and can therefore be

treated as a special case. Analysis is greatly simplified compared to forward flightdescribed later, and the necessary equations can be written in a nearly closed form.

Momentum theory and lift line theory will be used to determine inflow velocities, thrust

and power for main and tail rotor.

Momentum theory treats the rotor as an actuator disk with zero thickness and circular

surface, able to support a pressure difference and thus accelerate air through the disk.

This resulting airflow is called induced inflow. An approximation of uniform inflow over

the rotor disk is valid in hover and vertical flight and also represents a rough estimation

for forward flight. A more accurate modeling, as described by vortex wake theory and

dynamic wake theory [6, 8], will increase computational cost and is therefore not always

useful for most real-time simulations. For modeling forward flight, a triangular induced

velocity field can be used to increase accuracy. To compute thrust and induced velocity

in general, momentum theory is applied to a specific inflow model. In any case the

iteration of thrust and inflow velocity converges quickly and gives a good estimate of

induced velocity for most flight conditions. More detailed information on rotor wake

theory can be found in [6, 8, 13, 14].

Uniform induced velocity yields minimum induced power loss of an ideal rotor for

given thrust. Other power losses due to non-uniform inflow, non-optimal rotor design,

blade drag and swirl in the wake need to be considered if accuracy is to be increased. For

a finite number of blades additional tip losses reduce effective thrust, affect inflow

characteristics and increase power losses of the rotor. Engine and transmission losses also

affect power calculations, but for most applications a detailed engine model is not

necessary. The constant rotor speed assumption through the entire flight envelope


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27

simplifies this influence on power calculations. Especially in hover, interference between

main rotor, fuselage and tail rotor cause additional power losses and need to be

considered. For forward flight this effect becomes less important due to increasing other

aerodynamic forces. In the case of vertical flight an additional climb or descent velocity

of the helicopter needs to be added, and total power must be further changed due to

necessary climb power.

To compute forces and moments acting on the single blades due to velocity relative to

the surrounding air, blade element theory is used. Linear lifting-line theory is applied to a

rotating wing, the rotor blade, and it is assumed that the aerodynamic forces are produced

by the two-dimensional airfoil of each blade section. The induced angle of attack at a

blade section influences again induced velocity, and therefore again wake-body

interference and further related aerodynamic aspects. Blade element theory is capable of

dealing with the detailed flow and blade loading and allows a very accurate description ofrotor aerodynamics, rotor performance and flight characteristics. To compute the blade

angle of attack a first estimation of the induced velocity at the rotor disk is needed and

can be provided by the momentum theory. The mathematical modeling of the previously

mentioned aspects of rotor aerodynamics is described in Chapter 5. Furthermore,

parameters might be used to adjust constraints or aerodynamic limits like stall speed or

maximum side velocity. If the mathematical modeling of some phenomena can not be

easily done, the existing model needs to be fine-tuned for this very specific experimental

helicopter.

Helicopter stability and control is based on the equations of motion for force and

moment equilibrium on the entire aircraft. In hover, all of the forces and moments are due

to gravity, main rotor, tail rotor or rotor-body interference. Thus the main rotor primarily

governs the stability characteristics of the helicopter for this flight condition. More

detailed remarks about stability and control for hover and forward flight are made in

Chapter 4 . The following section introduces some very important aerodynamic features

of the main rotor in forward flight. The resulting forces and moments strongly influence

flight characteristics of the helicopter and are therefore very important for a desired

accurate modeling and the overall simulation of flight.


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3.3 Forward Flight

Rotational motion of the rotor and translational motion of the helicopter are combined

and become a source of additional complexity of rotor theory in forward flight.

Axisymmetry as assumed in hover or vertical flight is no longer valid, because the

aerodynamic environment varies periodically with rotation of rotor blades. The velocity

of the blade relative to the air now governs blade motion. The resulting blade motion and

its influence on forces and moments are the subject of this chapter.

3.3.1 Rotor Theory in Forward Flight

Necessary background of rotor theory in forward flight is provided in this section and

important aspects are explained. For more detailed derivations see [6, 8, 9].

In Figure 3.4 the rotor blade velocity is shown dependent on the azimuth. The

velocity of the advancing blade relative to the air is higher than that of the retreating

blade due to forward velocity and rotation of the blade. Because of this periodic motion,

with the fundamental frequency equal to the rotor speed Ω, the blade aerodynamics aswell as the blade dynamics are first considered.

V

Ω

Advancing

Side

Retreating

Side

Ω r + V

x

yReverse

Flow

Figure 3.4: Rotor blade velocity in forward flight


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Momentum theory can again be used to obtain the power due to induced velocity. At

high forward speed this power loss is small compared to other components due to high

velocity, and the assumption of uniform inflow over the entire disk is again valid. But for

transition from hover to fast forward flight the inflow model must be treated more

carefully. This approach is given in [8]. In [1] a mathematical formulation is presented

for the entire range from hover to fast forward flight assuming uniform inflow over the

rotor disk and a triangular velocity field for the rotor wake acting on the fuselage (see

also Chapter 5).

Lifting-line theory for single blade sections integrated over the blade length is used to

derive the force and moment equations for forward flight. Several assumptions can be

made to simplify the analysis and to obtain the equations of blade motion. The following

chapter introduces and explains the influences of blade motion on rigid body dynamics

qualitatively. This provides also the background for the equations used to describe thoseinfluences if desired and necessary for the analysis in Chapter 5.

3.3.2 Influences of Rotor Effects and Rotor–Helicopter Interference

Influences on power losses, thrust and blade motion depend on blade and rotor

geometry, aerodynamic phenomena and dynamic characteristics of the rotating system

and the rigid body motion. The most important effects are briefly mentioned in the

following sections.

Nonuniform Inflow

Using a linear variation over the disk instead of a uniform inflow can extend the

computation of induced velocity. Additional coefficients can be found dependent on the

forward speed of the helicopter that define a linear distribution of inflow at the rotor.

Typical coefficients result in an inflow model with a small induced velocity at the leading

edge of the rotor disk and about twice the mean value at the trailing edge. The mean

value can still be obtained with the assumption of uniform inflow. This kind of analysis

can be implemented in existing models, and improvements are expected in computation

of mean and first harmonic quantities influencing rotor performance and blade flapping.If higher harmonics are subject of the analysis, the much more complicated and nonlinear

inflow models become very important. Inflow variation mainly affects the rotor cyclic

flapping and cyclic pitch trim; longitudinal flapping due to inflow variation is small.


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Tip Loss and Root Cutout

Finite number of blades instead of a solid rotor disk result in an additional

performance loss. At blade tips the lift decreases to zero, thrust will therefore be reduced

and induced power loss will increase. This tip loss can be accounted for by introducing a

tip loss factor such that there is drag but no lift starting at the radial position defined by

this factor. Major influences can be found on the thrust magnitude. Near the blade root

the airfoil of the blade is different compared to the rest of the blade. Due to reverse flow

at the blade root in forward flight (Figure 3.4), the blade profile is cut out near the blade

root in order to decrease drag. In general, the influences on thrust and flapping moment

are small and can be neglected as long as performance and control aspects are concerned.

Effects of Natural Frequency on Flap Motion

Assuming that the flap hinge of a blade is located at the center of rotation without aspring producing any additional moment, the natural frequency of the flapping motion is

equal to the frequency of the rotational speed. The resulting moment at the hinge is zero

since the blade is free to move about this hinge. As soon as a spring, some hinge offset or

both (Figure 3.5) is introduced, the resulting moment on the rotor hub is no longer equal

to zero. The additional force of the spring and the offset of the hinge must be considered

in the derivation of the flapping equation of motion. The offset results in a moment arm

for centrifugal, inertial and aerodynamic forces and the additional hub moment must be

added to the moments acting on the rigid body. The natural frequency of the flap motion

for a blade with hinge offset and spring becomes larger than the rotational natural

frequency. The primary effect of the hinge offset and the spring on the flap response is a

coupling of longitudinal and lateral control due to this change of natural frequency. Some

rotor systems instead are designed without a flap hinge and are called hingeless rotors.

They can be treated similarly as described in this section, and [6, 8] deal with further

details of these kind of rotors. At this point it should be pointed out that an additional

control or servo rotor described later, strongly influences the helicopter stability and

performance. This kind of rotor system can approximately be treated like a teetering

rotor. In most applications, for a teetering rotor the coning angle, a0, of the blades inEquation 3.1 can be disregarded.


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31

Rotor

Shaft

Ω

B l a de

β

Flap Hinge

Flap Hinge Offset

e

Figure 3.5: Offset of rotor blade flap hinge

A further coupling of blade pitch and flap motion arises from the geometry used to

control blade pitch. For a pitch bearing outboard of the flap hinge, the blade experiences

a pitch change due to flapping displacement of the blade if the pitch link is not in line

with the flapping hinge (see Figure 3.6).

δ3

Blade

ΩHub

Flap Hinge

Pitch Horn

Figure 3.6: Cross-coupling due to the delta-three-angle

The δ3-angle between the virtual hinge axis and the real flap hinge axis defines thenature and magnitude of this coupling effect. It also introduces an aerodynamic spring,

and the effective natural frequency of the flap motion is again increased and influences

the flapping response.


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Flap Motion due to Pitch and Roll Velocities

This effect becomes important to flying qualities since additional damping is added

due to the effect of roll or pitch velocities on the rotor tilt. Gusts change the attitude of

the helicopter and therefore the tilt of the rotor shaft. The change in main rotor tilt lags

behind by an amount proportional to pitch or roll rate and rotor moment of inertia due to

this additional damping term. The TPP wants to follow the shaft tilt due to a pitch or roll

rate of the body as long as there is a hinge that provides a moment from the shaft to the

rotor blades. Asymmetric flapping velocities over the azimuth with respect to the shaft

results in different aerodynamic forces and moments over the azimuth, and the TPP

follows the shaft. Furthermore, looking at pitch or roll rates it can be seen that the blade

angle-of-attack also changes. To maintain equilibrium, the blade compensates for this

with off-axis flapping. A pure pitch rate for example results in a change of lateral

flapping. It is obvious that this aspect introduces an additional cross-coupling term and becomes important for evaluating flying and handling qualities.

Dihedral Effect

As in the airplane case, this effect is desirable since it helps the pilot to fly the

aircraft. For wind producing sideslip angles, the aircraft tends to roll away from the

approaching wind. For helicopters, the source for this positive dihedral effect is the blade

flapping. Since for zero side slip angle the blades over tail and nose experience no

velocity due to pure forward flight, a non-zero side slip angle results in additional

velocities for those blade positions over the azimuth. As a result of wind coming directly

from the right, for example, the blade over the nose will become the retreating blade, and

the blade over the tail the advancing blade (for a rotor spinning counterclockwise). The

rotor plane will flap down on the left. Therefore thrust is tilted to the left, and the

helicopter rolls away from the wind. Even so the advancing and retreating blades are

different for a rotor spinning clockwise, the dihedral effect results in a tilt of the rotor

plane to the same side. This dihedral effect is mainly responsible for the phugoid-like

response of the helicopter in forward flight.

Rotor-Body Interference

Due to induced velocity of the main rotor, the additional airflow over the surface of

the helicopter body causes drag counteracting the thrust created by the main rotor. For

some helicopter configurations with additional wings or other aerodynamic surfaces, this

effect becomes even more important. As Figure 3.7 illustrates, this aerodynamic aspect is


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different for hover and forward flight [7]. Assuming that the shape of the rotor wake can

be described in such a simple manner, the influence of this wake in hover contributes

mainly to the horizontal force component. In forward flight this wake is diverted due to

the airspeed and also acts on the tail rotor, horizontal tail and vertical tail. The transition

from hover to forward flight is an intermediate condition partially affecting all

components and causing great difficulty. The exact shape of this rotor wake is

furthermore not known and its influence on body, wings or horizontal tail can only be

approximated if a simple model is desired.

(a) (b)

Figure 3.7: Rotor-Fuselage interference in (a) hover and (b) forward flight

Main Rotor Control

Among different types of rotors, different types of rotor control schemes were

developed. The conventional way is through cyclic pitch changes of the individual

blades. A pilot stick input is transformed by means of links and actuators into a tilt of the

swashplate. The swashplate tilt occurs in the non-rotating and in the rotating frame and is

then transferred to the individual blades through further mechanical links. This can be

achieved directly over linkages and joints, or, as in the case of a so-called servo or control

rotor, via an additional smaller rotor mounted on top of the main rotor. In some literaturethis smaller rotor is referred to as a fly bar, servo or Hiller rotor. The mechanical linkages

of the R-50 hub from swashplate to control and main rotor are shown in Figure 3.8. The

TPP of the control rotor then governs the lateral and longitudinal inputs of the main rotor.


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Figure 3.8: Mechanical linkages of the control rotor for the R-50 experimental helicopter

One advantage is, that rotor systems of this kind prevent a feedback of rotor forces into

the control system of the non-rotating frame. Forces and therefore the loads on

swashplate actuators are minimized. Another even more important effect is the additional

stability produced by the control rotor if arranged properly. Additional damping is added

to the rotating system, which, as seen in Chapter 4, provides pitch and roll rate feedback

as well as translational velocity feedback to the main rotor. The dynamic effects of the

control rotor on stability are described in general. In Section 5.7 the influences of the

control rotor on the rigid body dynamics of the helicopter are evaluated and investigated

for the linearized equations of motion for the R-50 helicopter.

This chapter referred to individual and coupled effects of aerodynamic, dynamic and

kinematic features and their influence on blade motion or body dynamics. The overall

stability of a helicopter is discussed in the next chapter. Since, for simulation, the

response of the physical system is much more important than individual blade motion, the

background for stability and control analysis important for helicopter dynamics are

provided.


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Chapter 4 Helicopter Stability and Control

35

Chapter 4

HELICOPTER STABILITY AND CONTROL

This work refers to a single rotor two bladed helicopter with a tail rotor. Therefore

any further explanations and analysis are related to only this kind of helicopter. To keep

the model more general, non-rotating surfaces generating additional lift through wings

and a horizontal tail are not excluded from the analysis. For the investigation of the

simulated model helicopter additional wings are not present. To provide better flight

characteristics in forward flight a simple horizontal tail was added. In the mathematical

model the general case of a single main rotor and tail rotor helicopter is treated.

This chapter will first describe how control is accomplished for this kind of helicopter

and summarizes how the control inputs can influence off-axis motion. Stability aspects of

the decoupled and coupled rigid body dynamics are then presented for hover and forward

flight. A final section on stability augmentation with a control rotor follows.

4.1 Helicopter Control

Direct control of the helicopter is obtained mainly by controlling moments. Except

the vertical force component of the main rotor thrust, which can be controlled directly, all

main rotor control inputs result in a tilt of the main rotor and produce a moment about the

aircraft CG. Commanded are therefore changes in pitch and roll angles, resulting in

lateral and longitudinal forces and finally in the desired translational helicopter motion.

Controlling particular moments also makes some compensating control inputs in other

axes necessary, since most inputs are coupled with off-axis motion as mentioned in the

previous chapter.

The pilot's controls consist of a collective stick to control the vertical force, a cyclic

stick to control longitudinal and lateral moments, pedals to control the yaw moment via

the tail rotor thrust and the throttle to adjust rotor speed. Collective stick is used to trimthe thrust of the main rotor for some desired forward flight condition and for height

control in hover. This input changes the collective pitch of all blades equally, so that only

the magnitude of thrust, not the orientation of the thrust vector, is influenced; similarly

for the pedal input, which provides torque balance and directional control from the tail

rotor. Only the collective blade pitch of the tail rotor is changed to control tail rotor thrust

and the resulting yaw moment due to the moment arm relative to the center of gravity. To


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vary thrust and forward speed, and to maintain the desired constant rotor speed, the

required rotor power changes and throttle must be adjusted. Since a speed governor on

the engine can manage this, the assumption of constant rotor speed over the entire flight

envelope in regards to helicopter performance, is valid. The engine dynamics are

assumed to be fast compared to the rigid-body dynamics and are therefore neglected. An

increase in throttle and engine power results in higher available power and does not

directly influence the helicopter stability. Cyclic pilot stick displacements are connected

to the blade pitch, such that the rotor tilts to the desired direction. In small manned

helicopters this can be done directly by mechanical linkages, for bigger helicopters

electro-hydraulic actuators can be used to convert rotor control inputs. In the case of the

remotely controlled model helicopter, purely electric actuators are mounted on the

aircraft. For full-sized helicopters it is important for the pilot to have a proper feedback of

control forces due to pitch moments of the blades to the pilot's stick to improve handlingqualities. A mechanical linkage automatically provides this feedback. If actuators are

used, an artificial-feel-unit can simulate these forces in the pilot's stick.

ResponseInput

Pitch Roll Yaw Climb orDescent

LongitudinalStick

Pure (Prime) 1. Lateral flappingdue to longitudinalstick

2. Lateral flappingdue to load factor

Negligible Desired forvertical flight

path control

in forwardflight

Lateral Stick 1. Longitudinal flapping

due to lateral stick2. Longitudinal flappingdue to roll rate

Pure (Prime) 1. Undesired in

hover, caused bydirectionalstability

2. Desired for turncoordination andheading control

in forward flight

Descent with

bank angle atfixed power

Pedals(Rudder)

Negligible 1. Roll due to tailrotor thrust

2. Roll due to side slip

Pure (Prime) Undesireddue to powerchanges in

hover

Collective 1. Transient longitudinalflapping with loadfactor

2. Steady longitudinal

flapping due to climband descent in forwardflight caused by rotorflapping

3. Pitch due to change inhorizontal tail lift

1. Transient lateralflapping with loadfactor

2. Steady lateral

flapping due toclimb and descent

3. Side slip induced by power changecauses roll due todihedral effect

Power changevaries requirementfor tail rotor thrust

Pure (Prime)

Table 4.1: Single rotor helicopter coupling sources [15]


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To find the controls required in a trimmed flight condition, all forces and moments on

the helicopter must be zero. An iterative routine is necessary that varies the pilot inputs

until the six force and moment components are simultaneously zero. Predicting the

control inputs for a trim condition is difficult due to the complexity of the rotor dynamics.

For validation purposes it is therefore necessary to compare simulated results with flight

experiments. Since for some identification studies linear models about steady-state flight

conditions are computed, it is very important to verify that the system response about

these steady-state trim conditions correspond with the real system. The basic behavior of

rotor control from hover to forward flight is briefly described in the following.

Summarizing sources of inertial and aerodynamic coupling of longitudinal and lateral

helicopter motion, Table 4.1 describes various sources of cross-coupling.

In forward flight, a longitudinal cyclic input creates a lateral moment on the rotor disk

necessary to cancel the changes of blade-angle-of-attack due to flapping and tocompensate for the higher velocity of the advancing blade. Due to hinge offset and hinge

spring there is also a cross-coupling effect and the phase shift of input and TPP tilt is less

than 90°. A lateral tilt of the TPP and a resulting lateral moment for longitudinal cyclicinput proportional to the natural frequency of the rotating system develop and partially

compensate for the effect of the higher velocity on the advancing blade. To maintain

forward flight a longitudinal cyclic input (cyclic forward stick) is required to change the

TPP tilt as speed increases. A lateral cyclic input (cyclic left/right stick, dependent on

direction of rotor rotation) is required to compensate for the lateral TPP tilt due to lateral

flapping.

4.2 Helicopter Stability

In terms of dynamic stability and response to control inputs, the rigid body degrees of

freedom are mainly involved in the flight dynamic analysis. Separate longitudinal and

lateral motion can usually be assumed to simplify analysis and to observe the most

important stability characteristics. A further simplification is to use only low frequency

dynamics of the main rotor. No additional degrees of freedom are therefore added to the

system. In fact, the low frequency model for the main rotor response is a very good

approximation even for a more complex analysis. Later it will be shown how the dynamic

characteristics of the rigid body are influenced by an additional control rotor. In contrast

to the main rotor dynamics, these rotor dynamics are then coupled with the rigid body

dynamics. Also the coupling of longitudinal and lateral rigid body motion can be

considerable and important for handling qualities. The use of a fully coupled simulation


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model for the experimental helicopter characteristics will allow a more realistic

representation of the dynamics.

In general, a helicopter shows different characteristics in hover and forward flight.

The most important stability characteristics will now be investigated for decoupled

longitudinal and lateral dynamics to show the basic features of helicopter motion. If

principal aircraft axes are assumed, inertial cross-coupling of yaw and roll can be

neglected. Furthermore yaw motion is assumed to be fully decoupled from the other

degrees of freedom.

The following analysis is a summary of the basic helicopter motion for the flight

conditions hover and forward flight and explains the typical helicopter behavior. More

details can be found in [6, 8].

4.2.1 Hover

Vertical force equilibrium is given by the equation of motion for the helicopter

vertical velocity E z & . Collective pitch control is directly related to main rotor thrust, and

for now it is assumed that there is no pitch-flap-coupling. The resulting first-order

differential equation describing vertical dynamics has only a single pole. The time

constant increases with rotor speed, blade loading and gross weight. This root is in

general small, justifying the low frequency rotor response with respect to vertical motion.

Rotor speed will always be assumed to remain constant to simplify analysis. A variable

rotor speed would add another degree of freedom and modeling height control would

become more difficult.

For the yaw motion, only moments due to main rotor torque and tail rotor thrust will

be considered. Perturbations due to side velocity will be included. The low frequency

response of the tail rotor thrust leads to a first-order differential equation for the yaw rate.

The time constant is approximately the same as the time constant of vertical motion.

Since sideward velocity changes with a change in tail rotor thrust, the lateral translation

and yaw motion in general are coupled. This coupling is small compared to other

coupling effects. A change in lateral cyclic causes a sideward velocity and requires small

pedal input to maintain heading. For constant rotor speed, a change in thrust will alsovary the main rotor torque, and therefore couple vertical and yaw control. To maintain the

heading while thrust is changed, a coordinated pedal input is necessary. The tail rotor is

operating in adverse aerodynamic environment due to the main rotor wake, fuselage and

vertical tail. Modeling all these aspects is very complex and can often be simplified by an

approximation only. However, these effects will become important in forward flight since


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yaw damping and directional control are greatly influenced. Because of only low yaw

damping in hover, a helicopter is very sensitive to tail rotor thrust changes. Most real

helicopters therefore require at least yaw rate feedback to show handling qualities that

allow a reasonable control over the helicopter.

Longitudinal dynamics consist of the pitch motion, longitudinal velocity and vertical

velocity. Corresponding longitudinal inputs are longitudinal cyclic stick, longitudinal

gust velocity and collective pitch. The characteristic equation has three solutions

representing the open loop poles of the longitudinal dynamics. One is a stable root on the

real axis, the other two are a mildly unstable complex pair. This instability is a result of

the coupling between pitch moment due to longitudinal velocity and the longitudinal

component of the gravitational force due to pitch. The s

96685625 realtime flightsim helicopter

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