A STUDY OF HELICOPTER STABILITY AND CONTROL INCLUDING …
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A STUDY OF HELICOPTER STABILITY AND
CONTROL INCLUDING BI/ADE DYNAMICS
Xin Zhao
H. C. Curtis!_, Jr.
Department of Mechanic i_l and Aerospace
Engineering
Princeton University
Princeton, NJ 08544-5263
Principal Investigator_ H. C. Curtiss, Jr.
Technical Report No. 1823T
NASA Ames Research Center Gra]_t No. NAG 2-244
Studies in Helicopter Aerodynamics and Control
NASA Technical Officer for th:Ls Grant: W. A. Decker
FINAL REP()RT
Covering the period July 1983-- September 1987
October 1988
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ABSTRACT
A linearized model oC rotorcraft dynamics has been
developed through the use of symbolic automatic equation
generating techniques. The dynamic model has been formulat-
ed in a unique way such that it can be used to analyze a
variety of rotor/body coupi_ing problems including a rotor
mounted on a flexible shaft with a number of modes as well
as free-flight stability and control characteristics.
Direct comparison of the tir_e response to longitudinal, lat-
eral and directional control inputs at various trim condi-
tions shows that the linear model yields good to very good
correlation with flight test. In particular it is shown that
a dynamic inflow model is essential to obtain good time
response correlation, especially for the hover trim condi-
tion. It also is shown that: the main rotor wake interaction
with the tail rotor and fixed tail surfaces is a significant
contributor to the respon_.e at translational flight trim
conditions. A relatively _imple model for the downwash and
sidewash at the tail surfaces based on flat vortex wake
theory is shown to produce cood agreement.
Then, the influence of rotor flap and lag dynamics on
automatic control systems feedback gain limitations is
investigated with the model. It is shown that the blade
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dynamics, especially lagging dynamics, can severely limit
the useable values of the feedback gain for simple feedback
control and that multivariable optimal control theory is a
powerful tool to design high gain augmentation control sys-
tem. The frequency-shaped optimal control design can offer
much better flight dynamic zharacteristics and a stable mar-
gin for the feedback systeln without need to model the lag-
ging dynamics.
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BIBLIOGRAPHY
The following papers were presented as part of this grant:-
Curtiss, H. C., Jro, Stability and :_ontrol Modelling, paper NOo 4],
Twelfth European Rotorcraft Forum, Garmisch-Partenkirchen,
Federal Republic of Germany, Sgptember 22-24, 1986.
Zhao, X., Curtiss, H. C., Jr., A Linearized Model of Helicopter
Dynamics Including Correlation with Flight Test, 2nd
International Conference on Rotorcraft Basic Research,
College Park, MD, February 198i_.
In addition two presentations consisting of figures only
were made:
Curtiss, H. C., Jr., McKillip, R. M_, Stability and Control
Modelling, Workshop on Dynamici{ and Aeroelastic Stability
Modeling of Rotor Systems, Atlanta, GA, December 4-5, 1985.
Curtiss, H. C., Jr., The Influence of Blade Degrees of Freedom
and Dynamic Inflow on Helicopter Stability and Control.
Second Technical Workshop on D[_,namics and Aeroelastic
Modeling of Rotorcraft Systems_ Florida Atlantic University,
November 18-20, 1987.
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I_JOR QUALiT K
C ONTE NT S
Abstract
Bibliography
List of Tables
List of Figures
Chapter I: Introduction and Background
I.I: Introduction
1.2: Approaches For Analysis of Coupled Rotor/
Fuselage System
1.3: Outline of Previous Work
Chapter 2: A Linear Dynamic Mathematical Model For
Coupled Rotor/Fuselage System
2.1: Background and Introduction
2.2: General Description of Model
2.3: Reference Frames
2.4: Rotor Blade Model
2.5: Inertial Analysis
2.6: Rotor Blade Aerodynamic Model
2.7: Dynamic Inflow
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2.8: Linearization
ORIQINAL PAGE iS
OF POOR QUALITY
Chapter 3: Influence of The Rotor Wake on The Tail
Rotor and Fixed Tail Surfaces
3.1: Introduction
3.2: Theory of Liftir_g Airscrews With a Flat
Vortex System
3.3: Influence of the Rotor Wake on the Tail
Surfaces
Chapter 4: Verification of the Model
4.1: Introduction
4.2: Hover
4.3: Forward Flight
4.4: Conclusions
Chapter 5: Influence of the Blade Dynamics on the
Feedback Control System Design
5.1: Introduction
5.2: Simple Feedback Control
5.2.1: Attitude Feedback at Hover
5.2.2: Attitude Feedback in Forward Flight5.2.3: Pitch Rate Feedback
5.2.4: Roll Rate _eedback
5.2.5: Summary
5.3: Multivariable Optimal Control
5.3.1: Standard Performance Index
5.3.2: Frequency-Shaped Performance Index
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5.3.3: Simplified Optimal Control5.3.4: Summary
Chapter 6: Conclusions and Recommendations
References
Appendix A: Derivation of 3ystem Equations of Motion
Appendix B: Pitt's Model of Dynamic Inflow
Appendix C: Flat Vortex Theory For Nonuniform Inflow
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ORIGINAL PAGE IS
OF POOR QUALITY LIST OF _iABLES
TABLE 5-1
TABLE 5-2
TABLE 5-3
TABLE 5-4
TABLE 5-5
TABLE 5-6
Table 5-7
Table 5-8
Table 5-9
Table 5-10
Table 5-11
Table 5-12
Table 5-13
Table 5-14
Table 5-15
Table 5-16
Poles, Zeros and Approxz_ate Transfer Functzon
of Lateral Helicopter Dynamics at Hover 71
Poles, Zeros and Approxzmate Transfer Function
of Longitudinal Helicopter Dynamics at Hover 76
Poles, Zeros and Approximate Transfer Function
of Lateral Helicopter Dynamics at 60KTS 79
Poles, Zeros and Approximate Transfer Functl on
of Lateral Helicopter Dynamics at 100KTS 8O
Poles, Zeros and Approximate Transfer Functzon
of Longitudinal Helicopter Dynamics at 60KTS 88
Poles, Zeros and Approximate Transfer Function
of Longitudinal Helicopter Dynamics at 100KTS 89
The Primary Feedback Gains and The Damping of Lag
Modes For Standard Optimal Feedback at Hover 105
The Primary Feedback Gains and The Damping of Lag
Modes For Standard Optimal Feedback at 60 KTS 106
The Primary Feedback Galns and The Damping of Lag
Modes For Standard Optimal Feedback at i00 KTS 107
The Primary Feedback Gains and The Damping of Lag Modes
For Frequency Shaped Optimal Feedback at Hover 112
The Primary Feedback Gains and The Damping of Lag Modes
For Frequency Shaped Optimal Feedback at 60 KTS 113
The Primary Feedback Gains and The Damping of Lag Modes
For Frequency Shaped Optimal Feedback at I00 KTS 114
The Poles Associated With The Short Period Flight
Dynamic Characteristics 9f The Helicopter Under
Standard Optimal Feedbac_
115
The Poles Associated Witi_1 The Short Period Flight
Dynamical Characteristic_ of The Helicopter Under
Frequency Shaped Optimal Feedback 116
The Poles Associated With The Short Period Flight
Dynamic Characteristics of The Helicopter Under
Simplified Standard Optimal Feedback 128
The Poles Associated With The Short Period Flight
Dynamical Characteristic_ of The Helicopter Under
Simplified Frequency Shal:)ed Optimal Feedback
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ORIGINAL PAGE iS
OF. POCR QUALITY
LIST OF FIGURES
Fig. 2-I The F System of Coordinates 19I
Fig. 2-2 The F System of Coordinates 19Z
Fig. 2-3 The H System of Coordinates 21
Fig. 2-4 The B System of Coordinates 21
Fig. 3-1 Lateral Distribution of Downwash 32
Fig. 3-2 Vertical Distribution of Sidewash 32
Fig. 3-3 Roll Moment Produced by Horizontal Tail as a
Function of Sideslip Angle 35
Fig. 3-4 Pitch Moment Produced by Horizontal Tail as a
Function of Sideslip Angle 36
Fig. 4-I Comparlson of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Right Cyclic Input, Hover) 39
Fig. 4-2 Comparlson of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Right Cyclic Input, Hover) 39
Fig. 4-3 Comparlson of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Right Cyclic Input, Hover) 39
Fig. 4-4 Comparlson of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 0.5-in Forward Cyclic Input, Hover) 41
Fig. 4-5 Comparlson of Calculated Responses and Flight-Test Data
(Roll Rate Response to 0.5-in Forward Cyclic Input, Hover) 41
Fig. 4-6 Comparlson of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 0.5-in Forward Cyclic Input, Hover) 41
Fig. 4-7 Comparlson of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Left Pedal Input, Hover) 43
Fig. 4-8 Comparlson of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Left Pedal Input, Hover) 43
Fig. 4-9 Comparlson of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Left Pedal Input, Hover) 43
Fig. 4-10 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 0.5-in Right Pedal Input, 60 KTS) 45
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Fig. 4-11 Comparlson of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 0.5-in Right Pedal Input, 60 KTS) 45
Fig. 4-12 Comparlson of Calculated Responses and Flight-Test Data
(Roll Rate Response to 0.5-in Right Pedal Input, 60 KTS) 45
Fig. 4-13 Comparlson of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Left Cyclic Input, 60 KTS) 47
Fig. 4-14 Comparlson of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Left Cyclic Input, 60 KTS) 47
Fig. 4-15 Comparlson of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Left Cyclic Input, 60 KTS) 47
Fig. 4-16 Comparlson of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Forward Cyclic Input, I00 KTS) 48
Fig. 4-17 Comparlson of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Forward Cyclic Input, i00 KTS) 48
Fig. 4-18 Comparlson of Calculate._ Responses and Flight-Test Data /
(Yaw Rate Response to 1-in Forward Cyclic Input, I00 KTS) 48
Fig. 4-19 Comparlson of Calculate<_ Responses and Flight-Test Data
(Pitch Rate Response to 1-in Right Pedal Input, I00 KTS) 50
Fig. 4-20 Comparlson of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Right Pedal Input, I00 KTS) 50
Fig. 4-21Comparlson of Calculate<1 Responses and Flight-Test Data
(Roll Rate Response to 1-in Right Pedal Input, i00 KTS) 50
Fig. 4-22 Comparlson of Calculate<_ Responses and Flight-Test Data
(Roll Rate Response to 1-in Lateral Cyclic Input, 140 KTS) 52
Fig. 4-23 Comparlson of Calculate<| Responses and Flight-Test Data
(Yaw Rate Response to 1-in Late::-al Cyclic Input, 140 KTS) 52
Fig. 4-24 Comparlson of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Laueral Cyclic Input, 140 KTS) 52
Fig. 4-25 Comparlson of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 0.5-in Pedal Input, 140 KTS) 53
Fig. 4-26 Comparlson of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 0.5.-in Pedal Input, 140 KTS) 53
Fig. 4-27 Comparlson of Calculated Responses and Flight-Test Data
(Roll Rate Response to 0.5-:Ln Pedal Input, 140 KTS) 53
Fig. 4-28 Comparlson of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Right Pedal Input, I00 KTS) 56
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POOR QUALITY
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O;-":C;G'.D!.:'-,LPt_,_E IS
.OE POOR QUALITY
Fig. 4-29 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Right Pedal Input, I00 KTS) 56
Fig. 4-30 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to l-in Eight Pedal Input, i00 KTS) 56
Fig. 4-31 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Left Pedal Input, i00 KTS) 57
Fig. 4-32 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Left Pedal Input, i00 KTS) 57
Fig. 4-33 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to l-in Left Pedal Input, I00 KTS) 57
Fig. 5-1 The Helicopter Rotor/Fuselage System Open Loop Roots
at Hover 67
Fig. 5-2The Helicopter Rotor/Fuselage System Open Loop Roots
at Hover (Detail) 68
Fig. 5-3
Fig. 5-4
Fig. 5-5
Fig. 5-6
Fig. 5-7
The Root Loci of The HeLicopter With Roll Attitude
Feedback to Lateral CycLic Input at Hover
Frequency Response of _'ZAis With Roll Attitude
Feedback to Lateral CyclLic Input, Hover
The Root Loci of The Helicopter With Pitch Attitude
Feedback to Longitudinal Cyclic Input at Hover
The Helicopter Rotor/Fu_elage System Open Loop Rootsat 60 KTS and I00 KTS
The Root Loci of The Helicopter With Roll Attitude
Feedback to Lateral Cyclic Input at 60 KTS
7O
73
74
78
82
Fig. 5-8
Fig. 5-9
The Root Loci of The Helicopter With Roll Attitude
Feedback to Lateral Cyclic Input at I00 KTS
Frequency Response of _zA1s With Roll Attitude
Feedback to Lateral Cyclic, 60 KTS
82
83
Fig. 5-10 Frequency Response of @/Ais With Roll Attitude
Feedback to Lateral Cyclic Input, I00 KTS
Fig. 5-11 The Root Loci of The Helicopter With Pitch Attitude
Feedback to Longitudinal Cyclic Input at 60 KTS
Fig. 5-12 The Root Loci of The Helicopter With Pitch Attitude
Feedback to Longitudinal Cyclic Input at I00 KTS
84
86
86
Fig. 5-13 The Root Loci of The Helicopter With Pitch Rate
Feedback to Longitudinal Cyclic Input at Hover 91
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Fig. 5-14 The Root Loci of The HeLicopter With Pitch Rate
Feedback to Longitudina I Cyclic Input at 60 KTS
Fig. 5-15 The Root Loci of The HeLicopter With Pitch Rate
Feedback to Longitudinal Cyclic Input at i00 KTS
Fig. 5-16 Effect of Pitch Rate Feedback on The Damping of
Advancing Lag Mode
Fig. 5-17 Effect of Roll Rate Fee_iback on The Damping of
Advancing Lag Mode
Fig. 5-18 Effect of Roll Rate Fee_Iback on The Damping of
Coning Lag Mode
Fig. 5-19 Effect of Mechanical Lag Damping on The Damping
of Advancing Lag Mode With Different Roll Rate
Feedback Gains
Fig. 5-20 Frequency Response of _/A1s With Roll Rate
Feedback to Lateral Cyclic Input, Hover
Fig 5-21 Frequency Response of _ "!A With Roll Rate• 18
Feedback to Lateral Cyclic Input, 60 KTS
Fig. 5-22 Frequency Response of ¢/AI, With Roll Rate
Feedback to Lateral Cyclic Input, I00 KTS
Fig. 5-23 Frequency Response of ¢/A1s With Standard OptimalFeedback at Hover
Fig. 5-24 Frequency Response of _,'AI_ With Standard OptimalFeedback at 60 KTS
Fig. 5-25 Frequency Response of _/A With Standard OptimalFeedback at I00 KTS ,s
Fig. 5-26 Frequency Response of _/A With Frequency ShapedOptimal Feedback at Hover I_
Fig. 5-27 Frequency Response of q>/A,, With Frequency Shaped
Optimal Feedback at 60 ]<TS
Fig. 5-28 Frequency Response of _IA1s With Frequency Shaped
Optimal Feedback at I00 KTS
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lO0
118
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OF. POOR QLJALiTY
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Chapter I
INTRODUCTICN AND BACKGROUND
I.I Introduction
Aeroelastic and aeromecnanical
response problems associated with
stability, control and
rotary-wing aircraft rep-
resent some of the most challenging problems in the area of
dynamic systems. Due to the complicated nature of the prob-
lem, stability and control analysis is usually treated sepa-
rately from aeroelastic and aeromechanical stability.
Aeroelastic analyses usually concentrate on the character of
the system eigenvalues and do not concern themselves with
system response characteristics. In many instances, stabil-
ity and control analyses are based on a quasi-static, rigid-
body stablity-and-control-derivative model in which the
blade dynamics are neglected and the rotor lag and flap
angles are determined from the instantaneous value of the
body angular and translational displacements, rates, and
accelerations.
Although use of the conventional quasi-static stability
derivative model is adequate for many applications associat-
ed with low-frequency and _teady-state flight behavior and
promotes physical insight, the true physical behavior of the
highly coupled rotor/fuselage dynamical system can only be
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OF POOR Q UALiYY
2
captured by developing a mathematical model based upon a
consistent formulation in _:hich the influence of the coupled
body blade motion is properly incorporated. Many years ago,
C.W.EIIis[I] found that the conventional quasi-static
stability-derivative mode] was not representative of the
higher frequency short-period dynamics, owing to the strong
influence of the unmodeled rotor
S.V.Cardinale[2] concluded that the
excite the body's natural modes,
modes. R.E.Donham and
oscillating rotor could
and showed that a body
attitude feedback system had an important influence on the
total system stability. Hansen[3] has noted the importance
of the flapping dynamics in parameter identification stud-
ies.
Along with the development of feedback control systems,
especially with an increasing emphasis on superaugmented,
high-gain flight control systems for military rotorcraft in
order to meet the requirements for demanding misson tasks
such as nap-of-the-Earth(N.gE) flight, blade dynamics are
increasingly important in the flight dynamic analysis of
helicopters. In the design and analysis of such high gain
control systems, it is essential that high-order dynamics of
the system components be adequately modeled. In theoretical
analyses, K.Miyajima[4] has found that the blade-flapping
regressing mode should be :,ncluded in a stability and con-
trol augmentation system de_ign, otherwise a very important
oscillatory mode with short period frequencies would not be
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included. H.C.Curtiss[5] has found that for helicopter con-
trol systems attitude fee_:,ack gain is limited primarily by
body-flap coupling, and _:'ate gain is limited by the lag
degrees of freedom. It i_ also shown that dynamic inflow
produces significant changes in the modes of motion and
response of the system. W.E.HalI[6] has shown that, for
tight control, neglecting the rotor dynamics in designing a
high gain feedback system results in unstable closed-loop
responses when the rotor flap dynamics are included. In
practice, the operators of variable-stability research heli-
copters have long been awame of severe limitations in feed-
back gain settings when attempting to increase the bandwidth
of flight control systems. These same limitations have also
been encountered in the helicopter industry, where achiev-
able stability augmentation system gains obtained from
flight tests have often been far below predicted values[7].
Even for the vibration analysis of helicopters it has been
concluded that the method of rotor induced vibration pre-
diction by applying the rot.mr forces and moments acting on a
rigid support to the flexible airframe can lead to large
errors of either over or under prediction of vibrations[8].
In addition, with the shift of emphasis in hingeless and
bearingless rotor design to soft-inplane configurations,
coupled rotor/fuselage mechanical instability becomes one of
the main concerns of designers and researchers. This is not
only because there is strong coupling between the rotor and
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fuselage and therefore the aeromechanical stability charac-
teristics are highly sensitive to aerodynamic and structural
feedback, but also because the influences of the aeroelastic
coupling, which can play a key role in alleviating aerome-
chanical instability, often are different on the coupled
rotor/fuselage system than on the isolated blade[9].
Hence, there is a wides}Dread need for analyses capable of
modeling coupled rotor/fuselage aeroelastic or aeromechani-
cal systems.
1.2 Approaches For Analy:_is of Coupled Rotor/Fuselage
System
For a subject as compl_x as coupled rotor/fuselage sys-
tems, an adequate understanding of physical phenomena can
not be attained unless a reasonably accurate analytical rep-
resentation of the system has been developed and verified.
Such a representation is necessary to provide a usable
design tool, to develop _n understanding of configuration
behavior through systematic parametric studies, and to
search for and evaluate the feasibility of particular
advanced configuration concepts. Because of the complexity
of the description of the coupled rotor/fuselage system, an
important element of the development of practical analytical
tools is to determine what is an acceptable level of approx-
imation for the various parts of the analysis. It is impor-
tant to avoid making a design tool impractically large for
efficient computation.
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Several mathematical approaches are available for heli-
copter analysts to perform a coupled rotor/fuselage analy-
sis. The most popular ones are mode displacement, force
integration, and matrix diEplacement methods.
The mode displacement method allows a completely coupled
rotor/fuselage system to be analyzed by replacing rotor
inertia couplings in the fuselage equations with stiffness
coupling; therefore the use of it enables a simplified
sequential solution of the coupled rotor/fuselage dynamic
equations. Most analysis methods result in inertial
coupling between the rotor and the fuselage in both sets of
equations. However, the mode displacement approach allows a
simpler stiffness type co_ipling of the rotor degrees of
freedom in the fuselage equations. This is possible since
modal coefficients are used to calculate hub shears and hub
moments, eliminating the acceleration terms in the fuselage/
pylon equations that are d1_e to the rotor degrees of free-
dom. The sequence of calculLation begins with three indepen-
dent computations for airframe and rotor aerodynamic forces,
and hub shears and moment:_. The computation of the hub
shears and moments is the process which actually uses'the
mode-displacement method. [?he aerodynamic forces acting=on
the airframe and the rotor are also calculated. These aero-
dynamic forces form the forc:ing function for the rigid-body
fuselage accelerations. After the rigid body fuselage accel-
erations are calculated, they are used in conjunction with
ORi_N_L _L:_: i_
OE PO0_ <_t:;:/_.;i'_
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the hub shears and moments to calculate the acceleration of
pylon coordinates at the fc,llowing step. Additional inertia
forces on the rotor which _,_ere not included in the calcula-
tion of the rotor modes are calculated from the fuselage and
pylon accelerations. These inertia forces are then added to
the rotor aerodynamic force calculated previously in order
to calculate the accelerations of the rotor modal coordi-
nates. Thus, acceleratior_s are calculated for all of the
degrees of freedom without having to solve a large set of
simultaneous algebraic equations.
The force integration method is used to compute hub
shears and moments by integrating dynamic and aerodynamic
forces along each rotor blade. The analysis treats the rotor
equations separately from those of the nonrotating system.
In the rotor equations, inertial coupling terms due to
pylon/fuselage motions are written explicitly and assumed to
be known at a particular time point. To solve the rotor
equations of motion at time t, the hub and pylon displace-
ment, velocity, and acceiieration vectors are obtained from
the solution of the equations of motion for the nonrotating
system at the previous tir_e point. A predictor-corrector
method is used for numerical integration of the rotor accel-
eration variables to obtain rotor velocity and displacement
components. Equations of the pylon/fuselage system are
derived with hub shears and hub moments appearing on the
right side of the equation_. The hub forces are calculated
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by integrating
tip to blade root for each blade
blades. The force integration is
at which the predictor-corzector
7
inertial arid aerodynamic loading from blade
and summing up for all the
performed to a time point
has converged the solution
of the rotor equations. C:iven the hub shears and moments,
the equations of motion for the pylon/fuselage are solved.
The results define the huh: motions which provide the fuse-
lage inertial coupling terms in the rotor equations at the
next time point.
These two approaches are widely used in the helicopter
industry to calculate the rotor loads and the response of
the coupled rotor/fuselage system. The disadvantage of these
approaches is due to the fact that time histories are
obtained by input integration so that quantitative stability
analysis is not applicable, and they are not convenient for
the systematic parametric studies as well. For most aeroe-
lastic and aeromechanical stability and control problems,
the matrix displacement method may be a good alternative.
The matrix displacement method uses a generalized
coupling procedure which allows analysis of structural com-
ponents in rotating and ncnrotating reference frames. The
method automates the dynamic couplings between the rotating
and nonrotating systems and takes advantage of the high
speed computer for the algebraic manipulation. Vector
transformations are used in the method to generate position
vectors for blade and fuselage points in fixed coordinates.
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Then by using the Lagrang_i.an approach,
inertial contributions of the equations
8
for example, the
of motion for the
coupled rotor/fuselage sys_em are obtained. The same trans-
formation also is used to generate air speeds and incidence
angles relative to local b[Lade sections, and through appli-
cation of strip theory, for example, to obtain the aerody-
namic generalized force co1_tributions for the system. The
disadvantage of the matrix displacement method is that some
of the dynamic coupling tel-ms carried in the component equa-
tion are cancelled if the equations are derived explicitly
for the coupled system. This consumes more computer time and
possibly degrades accuracy in the numerical solution.
Therefore, the matrix displacement method suggests a via-
ble engineering tool for solving coupled rotor/fuselage
problems. In this thesis, with the help of a symbolic com-
puter processor, the matrix displacement method is used to
obtain a coupled rotor-fu_elage helicopter system descrip-
tion.
1.3 Outline of Previous _iork
A number of powerful analyses which have been developed
by industry and the governn.ent are developed or verified for
only a particular technic_l problem that reflects the spe-
cific interest of the oricinating organization. A typical
example is shown in Ref.10. The coupled rotor/fuselage sys-
tem model is designed for the Black Hawk simulation. The
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9
model is a total system description and allows the simula-
tion of any flight condition which can be experienced by a
pilot. The mechanical and aerodynamic data used in the mod-
el are provided by wind tunnel tests for full angle of
attack range.
Several researchers constructed mathematical models for
the general coupled rotor/fuselage systems. W.Warmbrodt and
P.Friedmann [II] have derived the governing equations of
motion of a helicopter rotor coupled to a rigid body fuse-
lage, which can be used to study coupled rotor/fuselage
dynamics in forward flight. The final equations are pre-
sented in partial differential equations and the blade equa-
tions of motion are written in a rotating reference system
whereas the matching conditions between the rotor and fuse-
lage are written in a nonrotating reference frame.
W.Johnson[12] has developed a comprehensive analysis for
rotorcraft which is capable of modeling coupled rotor/
fuselage problems by an integrated Newtonian approach. A
modal representation is used to transform the partial dif-
ferential equations
which is equivalent
orthogonal modes of
to ordinary differential equations,
to _ Galerkin analysis based on the
free vibration for the rotating blade.
Its solution procedures for the transient, aeroelastic sta-
bility, and flight dynamics analyses begin from the harmonic
balanced trim solution. Then the flight dynamics analysis
calculates the rotor and airframe stability derivatives, and
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constructs linear
rigid body motions;
i0
differential equations for the aircraft
the _oles, zeros, and eigenvectors of
these equations define the aircraft flying qualities. The
transient analysis numerically integrates the rigid body
equations of motion for a prescribed control or gust input.
The aeroelastic stability analysis constructs a set of lin-
ear differential equations
rotor and aircraft; the
define the system stabilitF.
describing the motion of the
eigenvalues of these equations
Although their intentio-1 was to produce an analysis that
is applicable to a wide range of problems and a wide class
of vehicles, these nonlinear, periodic-coefficient, partial
or ordinary differential equations are too complex to get
physical insight for general understanding and the theoreti-
cal analysis. They are also not convenient for the system-
atic parametric studies. For analytical simplicity and an
use as basis of the desig::l of feedback control systems, a
linear description of the :_ystem is highly desirable, espe-
cially if it can be shown to agree with experiment.
Owing to the complexitlf of including blade dynamics in
forward flight, linearized models in the literature are lim-
ited to the hover case.
Hodges [13] has developed a system of linear, homogenous,
ordinary differential equations which is suitable for model-
ing the aeromechanical stability of both bearingless and
hingeless rotor in hover. The flexbeam equilibrium deflec-
ORIG/NAL PAGE IS
OF. POOR QUALITY
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II
tions are calculated through a nonlinear numerical iteration
process, and the flexbeam szructural loads for small pertur-
bation of the equilibrium are determined through numerical
perturbation of the equilibrium solution. By using the mul-
tiblade coordinate transformation, the terms with periodic
coefficient are removed; Therefore, the resulting constant-
coefficient equations can be solved as a conventional eigen-
value problem.
Another derivation of the air resonance problem in hover
of an N-bladed hingeless rotor helicopter has been developed
by Levin[14]. In his study the final equations of_dynamic
equilibrium are reduced to ordinary differential form by
using Galerkin's method wi_h a relatively small number of
rotating blade modes. Prow. sion for introducing active con-
trol of the rotor with the intent of eliminating the air
resonance instability is included in the formulation.
A third model[15] by Lytwyn and Miao is obtained by
means of the Lagrangian procedure. The virtual hinge repre-
sentation has been used fo:_ the first in-plane (lead-lag)
and the first vertical bend_ng modes of each of the blades.
The most important assumptions upon which these formula-
tions are based are: (I) the helicopter is in hover with low
disc loading (low inflow ratio), (2) the rigid fuselage has
only two translational degrees of freedom and two rotational
degrees of freedom; vertical translation and rotation about
the vertical axis (yawing) are eliminated, (3) the rotor
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consists of three or more hLngeless blades, (4)
can bend in two mutually perpendicular direction
the elastic axis.
12
each blade
normal to
Several kinds of model mentioned above have been used in
analyses of the aeroelastic and aeromechanical stability,
response, and control problem. Hodges has conducted a theo-
retical study of aeromechanical stability of bearingless
rotors in hover by comparin_ the hub-fixed motion, i.e. iso-
lated blade stability, with the case when coupled rotor/
fuselage motion is considered. His studies dealt mostly
with a soft-in-plane configlration using quasisteady aerody-
namics. Straub and Warmbrodt[16] have studied the use of
active blade control to increase helicopter rotor/fuselage
damping. The chosen feedback parameters include cyclic rotor
flap and lead-lag states, and the study focuses on ground
resonance. Curtiss[5] has studied the influence of rotor
dynamics and dynamic inflcw on the stability and control
characteristics of single rotor helicopter in hover, and
discussed the body attitu6e and rate feedback limitations
which arise due to rotor d_namics and dynamic inflow.
The restriction for obt_ining a linearized model for for-
ward flight is due partly to the complexity of the blade
motion of the helicopter _o that the algebra is increased
substantially. This has led to attempts to share the alge-
bra with computers through symbolic processors. Both general
and special purpose programs have been developed and are
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13
available. Another difficulty faced for extending the lin-
earized modeling to forw_rd flight is that the multiblade
transformation will not _emove
Therefore the resulting linear
will be time-varying. However,
all periodic coefficients.
dynamic system description
it has been found that the
constant coefficient approximation for the remaining period-
ic coefficients is satisfactory for low-frequency modes
under trimmed conditions[17]. Furthermore, all of the lin-
earized models for hover assume that yaw motion and vertical
motion of the helicopter are totally uncoupled; this is not
the case for forward flight. In addition, the tail rotor and
fixed tail surfaces, whicln operate in an extremely adverse
aerodynamic and dynamic environment, must be taken into
account. As a result, a celatively simple induced velocity
model at tail position due to the influence of the main
rotor wake is needed for a good overall system modelling.
With this background information established, the remain-
der of this thesis can be outlined. The first task undertak-
en will be to construct a linearized dynamic mathematical
model for coupled rotor/fuselage helicopter system for both
hover and forward flight by use of symbolic automatic equa-
tion generating techniques Also a relatively simple model
for the downwash and sidewash at the tail surfaces based on
flat vortex wake theory is employed to take the main rotor
wake interaction with the tail rotor and fixed tails into
account. The model will then be verified by comparing the
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14
response time histories :riot various prescribed control
inputs at various trim conditions with the flight test
results of Ref.18, which is obtained by a flight test pro-
gram solely for the purpose of validating mathematical mod-
els of the Black Hawk helicopter. In addition, a study will
be made of the influence of the blade flap and lag dynamics
on automatic control system feedback gain limitations at
hover and translational flight conditions.
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Chapter II
A LINEAR DYNAMICMATHZMATICALMODELFOR COUPLED
ROTOR/FUSELAGESYSTEM
2.1 Background and Introdaction
One distinction of the zoupled rotor/fuselage system is
the fact that the analysis must accommodate both rotating
and nonrotating coordinate systems. For the coupled rotor/
fuselage system, the equations for each blade, which are
usually written in a coordinate system rotating at a con-
stant velocity, are transformed to a nonrotating coordinate
system, to be combined with each other and with the fuselage
system. Consequently, it is difficult to obtain the system
description using a Newtonian approach because the required
blade acceleration terms are very complex. The Lagrangian
approach, which requires only velocity terms and position
terms, is much more convenient for overall system modeling.
Another distinction of the coupled rotor/fuselage system
is the increased number o_ degrees of freedom, which sub-
stantially increases the algebraic complexity in expressing
the inertia and aerodynamic loads. In order to generate
reasonably comprehensive aeroelastic equations of motion for
a helicopter rotor, seve]al axes of reference are usually
required in the analysis. Thus, a material point on a rotor
- 15-
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16
blade can most conveniently have its position coordinates
defined by means of successive axis transformations.
Although none of the transformations that change the co-
ordinates of a point from one axis system to another may be
particularly complicated, when equations of motion are
derived through the use of Lagrange's equations, the exer-
cise can prove quite arduous. The derivation of the equa-
tions involves a certain amount of differentiation, which,
combined with the succes:!_ive transformations, leads to an
enormous amount of work on paper for the analyst. Further-
more, the possibility of errors creeping into the analysis
is almost unavoidable.
Fortunately, there are serveral symbolic computer proces-
sors available for gener_l computer systems so that it is
possible to develop the system equations directly on the
computer. The program gc_nerates the steady-state and lin-
earized perturbation equEitions in symbolic form and then
codes them into FORTRANsubroutines. Subsequently the coef-
ficients for each equatior_ and for each mode are identified
through a numerical program. Through the use of symbolic
automatic equation generating techniques, the final system
equations are obtained in a systematic way. This also makes
it relatively easy to rigorously investigate the effect of
various ordering schemes cn the calculated motion dynamics.
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2.2
17
General Description of Model
The complete dynamic description of the multidimensional
system is formulated by means of the Lagrangian procedure.
The final set of linear second-order differential equations
is obtained by a perturbation analysis performed on the set
of original nonlinear equat:_ons.
The model has N degrees of freedom, each associated with
a generalized coordinate and a corresponding mode shape.
The model includes a number of rotor blades on one hub and a
fuselage. Each rotor blade undergoes flap bending and lag
bending; the torsional deflections are not included. Qua-
sisteady strip theory is _ised to obtain the aerodynamic
loads. Unsteady aerodynamic effects are introduced through
dynamic inflow modelling. Dynamic stall and reverse flow
effects are not included.
In this work, the model is of order 24 or 27 depending on
whether dynamic inflow is included for a better modelling of
unsteady aerodynamics. The fuselage has six degrees of
freedom: vertical, longitudinal, and'lateral translation,
pitch, roll, and yaw motions, each associated with two state
variables. The equations cf motion are formulated in such a
way that they can be extended to N degrees of freedom to
include the effects of flexibility between fuselage and the
hub without any change in the blade motion part of the mod-
el. Each blade has 2 deglees of freedom, flapping and lag-
ging, each corresponding 41o two state variables. When the
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18
blade motion is converted to the fixed frame through the use
of multiblade coordinates, six degrees of freedom result
(e.g. coning, lateral and longitudinal tilt of the rotor
plane for flap) each associated with two state variables.
Dynamic inflow adds three more state variables. Control
inputs are collective pitc_ of the blade, lateral and longi-
tudinal cyclic pitch of the blade, and collective pitch of
the tail rotor. Rotor RPM is assumed constant. Blade pitch
changes due to flapping, Lagging, and fuselage deformation
and motion by the rotor ]:hub geometry and elastic coupling
can be taken into account.
The equations of the _ystem are obtained by algebraic
manipulation performed with the symbolic system REDUCE on
the IBM computer at computer center at Princeton University,
and is checked with the syr_bolic system MACSYMA at the Labo-
ratory for Control and Aut¢:mation at Princeton.
2.3 Reference Frames
Because we use a Lagrargian approach, we have to begin
our systems of coordinates in an inertial frame, the E sys-
tem, the earth axis. The basic systems of coordinates are
the Fi and Fz systems which are shown in Fig. 2-1 and Fig.
2-2. The origins of these systems are placed at undisturbed
and disturbed hub centers respectively while Zf coincide
with the shaft direction, and Xf points toward the rear of
the helicopter. These are systems which do not rotate with
OR!_NAL P£,.GE _S
OF POOR QUALITY
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0e
O,R|_NAL pp.G_
OF pOOP, QUALITY
L-----JJ
V
Zfl
shaft
Fig. 2-1 The I-i
}Ze
System of Coordinates
L
19
Xfl
Ofl>
Zfl
Zf2
Z Z"
/_Yfl I " ' "Y Y
.'" _-<!'2 ',_ I Yf2
n
Xf2 X
Fig. 2-2 The F2
Syst_._m of Coordinates
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the hub. The F system moves due to the fuselagei
motion without perturbation, a Galilean frame. The FZ
2O
trim
sys-
tem moves relative to F due to small perturbations whichI
result from disturbed fuselage rotations and translations.
The equations are formalized so that the elastic deforma-
tions may also be included, depending on the definition of a
transformation expressing the hub motion relative to the
fuselage in terms of the generalized coordinates. In the
linearization of the system, this transformation is also
linearized under an assumption of small perturbations and is
assigned as a set of system input parameters to offer more
flexibility for the model. By selection of the set of the
input parameters, it is possible to investigate the dynamics
of a rotor on a flexible shaft or the free motion of a heli-
copter like that in this thesis, or some combinations of the
two.
The third system of coordinates is the H (hub) system,
which is rotating with the hub (See Fig. 2-3). The co-
ordinate axes Z h and Zfz coincide, while the H system
rotates about the Zfz axis with an constant angular veloci-
ty, relative to the F sys%em When the azimuthal angle of2
the H system relative to the F system is zero the two sys-2
tems coincide. The next system of co-ordinates is the blade
system B (See Fig. 2-4), which is fixed to the rigid blade
and is displaced from the H system by offset and rotates due
to lag and flap. See Appendix A for more detail.
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Yh
Yf2 21
Xh
0
-- Xf2
Fig. 2-3 The H System of Coordinates
Zh
!
Zh
_ Yh
_ xh
Fig. 2-4 The B System Of Coordinates
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22
2.4 Rotor Blade Model
The rotor blade is assumed to be a rigid beam with an
offset hinge for a fully articulated rotor. A proper combi-
nation of a hinge offset and springs about the hinge can be
used to represent a hingeless rotor. This model can incorpo-
rate the effects of blade and hub stiffness by two sets of
springs inboard and outboard of the hinge. Furthermore,
flap-lag-pitch-fuselage structural coupling can be easily
incorporated, Spring restrained hinges can be used to model
a bearingless rotor.
The coning angles both fgr flapping and lagging are con-
sidered as variables beca/se in forward flight there is
coupling between the coning and the first harmonic terms.
2.5 Inertial Analysis
The position of an element of a blade first is written in
the B frame. It is assume:_ that the blade can be modelled
as a slender rod with all of its mass located on [Xb, O,O].
Using a series of transformations, we can express the posi-
tion of the blade element in the inertia axis, the E system.
Then, it is straightforward to obtainthe local velocity and
the kinetic energy. The same approach applies to the fuse-
lage as well. After integrating along the blade, combining
the blade with the fuselage and taking required differentia-
tions, the contribution of the kinetic energy to the equa-
tions of motion is obtained The kinetic energy contribu-
tion of the tail rotor is neglected.
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23
For the potential energy, the terms due to the gravity of
the blades are neglected but those from the fuselage are
included. To model a hingeless or bearingless rotor,
other spring potential terms
elastic potential resultJ ng
flapping and lagging.
are added for including
from the deformations of
two
the
the
2.6 Rotor Blade Aerodynamic Model
First, we get expressions of the normal and tangential
velocities at a blade element in the B frame, then apply
strip theory to obtain the lift and drag at local blade sec-
tions. After integration along the blade and a transforma-
tion, we get expression_ of the aerodynamic forces and
moments of the blade in the hub axis H. The same approach
applies to the tail rotor, the vertical tail and the hori-
zontal tail as well, the only difference is that a little
algebra is used to treat _:he delta 3 feedback of the tail
rotor. Only the thrust oi: the tail rotor is considered in
this work.
The tail rotor and fixed tail surfaces can experience
aerodynamic interference effects from many sources. Only the
components of flow from the main rotor are included in this
model. However, the equations are formulated to allow easy
insertion of other components. The total velocity components
for the tail rotor and fixed tail surfaces are made up of
contributions from the basic body axes translational and
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24
angular velocities and rotor wash. Dynamic pressure loss is
introduced by factoring _he components of the free stream
flow. The actual total cynamic pressure is calculated from
the resultant velocity at tail rotor and fixed tail surfac-
es. This allows a more representative definition of dynamic
pressure at low speeds where the downwash velocities predom-
inates.
Then the total virtual work due to the aerodynamic forces
is expressed as a function of the generalized coordinates by
summarizing all virtual work done by each aerodynamic force
or moment, in which extre_ne care must be exercised_because
any inconsistency with the corresponding inertia term will
result in large errors at final dynamic equations after the
linearization. Taking re,fired differentiations, we get the
generalized forces for the equations of motion.
2.7 Dynamic Inflow
The rotor blade operate_!_ in an unsteady environment; con-
sequently unsteady aerod_,namics can have a significant
influence on the aeroelastic and aeromechanical stability
characteristics of helico_,ters. To describe the low fre-
quency unsteady aerodynamic behaviour of the rotor, there
are relatively simple unsteady aerodynamic models, known as
inflow models, which agree with experiment and can be con-
veniently incorporated in aeromechanical and aeroelastic
stability and control analyses of helicopters. These sim-
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ple models are
parameters which represent essentially
induced flow through the rotor disk.
The induced flow-field acting on
25
based upon the definition of certain inflow
the unsteady wake-
a helicopter rotor
affects both rotor equilibrium (trim loadings) and rotor
response (transient loading_). Hence, it is reasonable to
expect that the induced flc, w will also be affected by the
oscillations of the rotor. Following this assumption, the
inflow is written as a comh:ination of a steady inflow for
trim loadings and a dynamic perturbation for transient load-
ings. Then, the total induced velocity normal to the rotor
disk is expressed as
V = v _ v tl) 4 %'(t[i cosy ÷ V (t) sin_ (i)n no o c s
where V , V , and v are components of the dynamico c 5
inflow perturbation. Th_ cynamic inflow components can be
related to the perturbed thrust AF, the perturbed pitch and
roll moments AM , LM The equations are written in formy ×
[L] [M] {V'} ÷ {V> = [L] [D] {AF> (2)
where {V)_= [V , V , V ] and _ ZLlr}w [AT, 21M , AM ]o c s y x
The matrix [L] is the static coupling matrix between
induced velocity and aeroc[ynam_c loads, the matrix [M]
assumes the role of an inertia of the air mass, the product
of ILl[M] is a matrix of time constants, and the matrix [D]
is a dimension adjustor.
A number of such inflow _lodels are available in the lit-
erature. In this work, the steady inflow is obtained
OF. PO >R %' ,,.'__..,"<
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26
from momentum theory for hover and from classical vortex
theory for forward flight. It has a first harmonic distri-
bution as a function of wake skew angle. The dynamic model,
i.e. the [M], [L], and [D] matrices, come from Pitt's model
based on a rigorous solutior_ to actuator-disk theory. The
details of the model can be found in Ref.21.
There is no simple method available to include the
effects of the unsteady wake of the rotor on the tail rotor
and horizontal tail. Considering the dynamic inflow models
represent the global effects of the unsteady wake, the
effects of the unsteady wak_ on the horizontal tail and the
tail rotor are included by directly extending the dynamic
inflow components out of the rotor plane, which is done by
assuming that the dynamic inflow at tail rotor and tail sur-
faces are of two times of the value on the line Xfz=R ,
2.8 Linearization
The nonlinear equations of motion are of the form:
Q" = F( Q, Q', u, T ) (3)
Introducing multiblade coordinates, which transforms the
blade-fixed generalized cocrdinates to nonrotating hub-fixed
generalized coordinates, end omitting periodic higher har-
monic terms, a constant coefficient approximation to the
original periodic system i_!_ obtained:
Q" = F( Q, Q', u ) (4)
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27
The process of linearization consists of expressing the
time dependence of the generalized coordinates and inputs as
the sum of the steady-state value and the time-dependent
perturbation about the former.
Qi (t) = Q. @ _QjCt)Io (5)
U.Ct)_ = U._o * _Qi Ct) (6)
Equations (5) and (6) are substituted into the nonlinear
equations of motion, and terms containing squares of the
perturbation quantities a]-e neglected. The perturbation
quantities are set equal t(, zero to obtain the steady-state
values of the generalized (oordinates and the control inputs
in the trim condition.
F( Qo "Uo) = 0 >> Qo Uo (7)
The final form of the cynamic equations can be symboli-
cally written as
MCQo'Uo)_Q"+CCQo'Uo)_Q'+K(Qo'Uo)_Q=BCQo "Uo)_U(8)
This linear time-invaring system can
order form:
X' = AX + B U
be written in first
(9)
X and U are the state variables and control input vector:
X = [ #o" _I' _z' Co" C,, [:z"e, -@, -_, y, x, z
" . -G.#. £. £. vo.v=. v,,"
TU = [ Ai,, BI,, et]
The collective pitch angle of the main rotor is not
included in the control vector of the perturbation equations
because collective input wece not investigated.
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C1_apter III
INFLUENCE OF THE R(_TORWAKEON THE TAIL ROTOR
AND FIXED TAIL SURFACES
3.1 Introduction
It has been found fron wind tunnel tests that the rotor
wake influences the aerocynamics of the tail rotor in for-
ward flight [22] and that the effect of rotor wake on the
horizontal tail produces a significant contribution to yaw
pitch coupling [23], which arises because of the angle of
attack distribution across the span of the horizontal tail.
The angle of attack can vary by as much as I0 degrees from
one tip to the other.
In this work it has been found that the transient
response in forward flight, especially the pitch response,
is very sensitive to the _reatment of the influence of the
rotor wake on the vertical tail, the horizontal tail, and
the tail rotor. A simple "_heory based on a flat vortex wake
model has been employed to obtain estimates of vertical var-
iation of the sidewash at the tail rotor and vertical tail
and the horizontal variat_;on of the nonuniform downwash at
the horizontal tail. The mathematical details of the wake
model used in this work are discussed in Ref.24 and are also
given in Appendix C. Or_ly a brief description is given
here.
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29
Theory of Lifting Ai]'screws with a Flat Vortex System
is based upon the following
3.2
The flat vortex wake model
assumptions:
(I) The vortex wake folmed by free vortices leaving the
rotor blades moves downstleam without any downward motion.
This should be a good approximation for helicopters with
sufficiently high flight speeds. Experience and theoretical
considerations indicate that this assumption is reasonable
for a considerable range of flight speeds of helicop-
ters. J25]
(2) The intensity of the free vortices leaving the same
section of the blade at various azimuth angles is constant.
This means that an average value of the circulation at a
given radius r of the blade is used to replace the time var-
ying circulation value, which depends on the azimuth angle.
(3) The free vortices in the rotor wake form a continuous
surface of vorticity. This is due to the fact that for most
helicopters the cruise tip speed is at least 2.5 times high-
er than the velocity of flight, and the rotor has 3 or 4
blades so that the density of free vortices would be high
enough to be considered as _ continuous surface.
Under these three assumligtions, consider a free vortex
layer which springs from the blade at a given radius r with
a constant circulation. ,)ne can find, after some simple
mathematical derivation, the final vorticity surface is con-
sisted by a lateral vorticity surface within a circle of
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ORIGi[_AL FAGE IS
OF, POOR QUALITY
30
radius r and centred at hub, and a longitudinal vorticity
surface which coincides with the whole wake surface. The
circulation per unit len_rth of the lateral vortex layer is a
constant, which is propo1tional to the intensity of the free
vortex layer, and inversely proportional to the advance
ratio and rotor radius. The circulation per unit length of
the longitudinal vortex layer is a function of lateral posi-
tion y, which is also proportional to the intensity of the
free vortex layer, and inversely proportional to the advance
ratio and rotor radius. Then under the assumption that the
circulation distribution along the rotor radius, averaged
over the azimuth, is par._bolic, the distribution of vortex
strength can be determin_d and the induced velocity at any
point in space can be cal(;ulated by applying the Biot-Savart
law, and integrating ove_- the whole blade. This model pro-
duces lateral and vertica[i components of the induced veloci-
ty.
3.3Influence of the Rotor wake on the Tail Surfaces
To estimate the influence of the rotor wake on the tail
rotor and fixed tail surfaces, it is assumed that the down-
wash and sidewash distributions in the trim condition are
given by distributions calculated relative to the centerline
of the wake at X = R. Pertarbations in sideslip and angle of
attack cause the centerlin_ to move changing the correspond-
ing aerodynamic forces and the moments.
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The vertical
of the rotor disk due to
equivalent to the fore
classical vortex theory,
31
induced velocity contribution in the plane
the lateral vortices is physically
and aft variation produced by the
and the average induced velocity
due to the longitudinal vortices is equivalent to the uni-
form part from the classical vortex theory. These contribu-
tions are obtained by directly extending the classical vor-
tex results out the rotor plane to the required location.
The variable part of the vertical induced velocity contribu-
tion due to the lateral _ortices is symmetric with respect
to the vertical plane, as a result its first order variation
in the lateral direction will be zero. In addition, the
lateral vortices do not p::coduce a contribution to the later-
al component of the inducted velocity. Therefore, in this
work only the nonuniform contribution of the longitudinal
vortices is included as fc)llows.
The normalized nonuniform vertical induced velocity dis-
tribution along the later_l axis at the position of horizon-
tal tail in forward flig1_t with zero sideslip angle at an
advance ratio of 0.22 (10C KTS) for UH-60A is shown in Fig.
3-1. The antisymmetric pazt produces a steady rolling moment
and a pitch moment variation with the sideslip angle. The
symmetric part produces _ steady pitch moment and a roll
moment variation with the sideslip angle.
The shape of the distribution explains the phenomena
observed in the wind tunnel test of Ref.23: (1)the left hand
OR_I_5_ _'!,_.AGC ::_
OE POOR QUALITY
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4
16"
UP
CD
CD
C_
z_
ou_
_'-|.2D
ORW_WNAL PAGE IS
OE pOOR QUALITY
V = I OOICTS
Advance IKmtlo = 0.22
Sideslip AngZe = O.O
x
x
x x x x -- Flat Vortex Theory
Noment u_ Theory
x x
x
Span of HorlzontLt T_L1
x x
I"
-O. g5
Fig.
I I I t I I !
-O. 70 -O. _5 -O. 20 O. 05 O. 3D D.55 0.0OLRTERRL P05ITION YIF_
3-1 Lateral Distrioution of Downwash
!
I .05
32
{I_O
N=;
Z
ED
_0I,----
U..) _'"
I_. ,
¢r .
_,-
112
LLI
-D._D
W ,, I OOKTS
Advance RatLo = O. ZZ
Sldesl£p Angle • 0.0
-O. ]_D
xx
I¢
l=
;c
Tail I_otor Span
x x x x --InvJLs¢ld FluLd¥£scld Fluid ApproxLmtLon
I I i I I l 1
-D.2D -D.]O -D.DD {) In _.2D 0._0 D.N.D O. SD
NORMALIZED 5IDEWA5H
Fig. 3-2 Vertical Distribution of Sidewash
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33
panel encounters relative positive rotor induced angles and
the right hand panel enco1_nters negative values, (2)the yaw-
pitch coupling is general_Jy worse for a nose left slip than
for a nose right slip, (31_the right hand panel produces con-
siderably more coupling tlan the left hand panel when tested
separately.
The vertical variatior of the lateral component of the
induced velocity at the tail rotor in the zero sideslip case
for the same helicopter at the same flight condition at
advance ratio of 0.22 is shown in Fig. 3-2. As a conse-
quence the local angle of attack of the tail rotor and ver-
tical tail will vary with the angle of attack of the heli-
copter, producing roll and yaw moments. It can be seen that
there is a discontinuity 3n the wake, which is due to the
inviscid fluid assumption in the theory and is smoothed out
by taking the viscosity of the airflow into account.
Because the real distributLon is unknown and nonlinear, and
the assumption of tail rotor center located on the surface
of the wake is extremely poor for most flight conditions,
the corresponding derivati,,es are determined by correlation
with flight test. The d_rivatives used in this paper are
determined in one trim coI_dition at advance ratio of 0.14
(60 KTS). It has been found that the response prediction is
not sensitive to the value of these derivatives, however the
overall effect is important..
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34
Although the effects of nonuniform induced velocity are
strong nonlinear functions of the sideslip and angle of
attack, only the steady contribution and first order varia-
tion of these effects are included. For example, the overall
roll and pitch moment contributions by horizontal tail as
functions of the sideslip angle and their linear approxima-
tions are given in Figs. 3-3 and 3-4 for the helicopter at
the same flight condition as the induced velocities. As can
be seen, these nonlinear effects behave like linear effects
only in a small neighbor]_ood about the equilibrium point.
Therefore, the model is _iimited to the case of small sides-
lip motion.
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35
--[%
oo
c=
C:l
la.
I oG0_
z: T ,
_J• .,_/
T'
g&
I ,-i
ORIGINAL PAGE IS
POOR QUALITY
x • • • • • x • •
x x x _ Flat. Vortex Theory
Linear Approximat£on
• 1
• o I
V == I O0]CTS
Advance l_Itio m O. RR
I I i I
5.oo -2o.oo -',s.ooS_OESLIP'"°'°°-_.ooRNaC.E(0EG)°'c*°s.oo _o.oo'_ 2o.oo ,s.o=
Fig. 3-3 Roll Moment Produced by Horizontal Tail as a
Function of Sideslip Angle
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36
ORIG'!,N/_L p,C,GE ISOF POOR QUALITY
X_
w_
De_
m,
Im,m
wo
ZW
C:)=
I,--
0
0
V m I O01CTS
AdYance Ratio m 0.22
Oi
'-25. O0
• • • x
x x x x _ Flat Vortex Theory ,,
Linear Approx_,mation
S. DO I0. O0 X$. O0 20. O0 2_. O0
Fig. 3-4 Pitch Moment Produced by Horizontal Tail as a
Function of Sideslip Angle
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Chapter IV
VERIFICATION OF THE MODEL
4.1 Introduction
To correlate the model, the transient response of an
articulated rotor helicopter to small step-inputs of each
control at various trim co_iditions has been calculated and
compared with a large nonli_Lnear model currently used in a
simulator and with flight test data. The flight test
results are obtained by a flight test program solely for the
purpose of validating mathematical models of the helicopter.
The helicopter is a UH-60A Black Hawk, which has a fully
articulated rotor having fo_ir blades with lead-lag dampers.
The helicopter configurati:_n, structural and aerodynamic
properties are given in Ref.10. The trim conditions are
hover, 60 KTS level flight, I00 KTS level flight, and 140
KTS level flight. The time histories of the control inputs,
the test conditions, and the transient responses obtained
from flight test and the simulation are presented in Ref.18.
The trim values and the initial control settings used in
the simulation are directly obtained from flight test data.
After trimming to the test conditions, the time-histories of
the perturbed input, which are the differences between the
time-histories of test-aircraft control and its initial con-
- 37 -
OF POOR QUALITY
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38
trol positions, were used as direct input to the simulation.
In this way, the simulation begins each transient response
in the actual trim conditiDn; therefore, the trim errors
have a minimal influence Dn dynamic response comparisons.
All simulation performed in this work used a time step of
0.05 sec. The simulation oltput was recorded every 0.15 sec
to reduce the cost of plotting.
The calculated response time-histories are compared to
flight test data for small control inputs in Figs. 4-1 to
4-27. Correlations are discussed in terms of the fuselage
angular rate response since their quantities are of primary
interest in handling qualities. Calculated results from
models both with and without dynamic inflow are presented in
hover so as to illustrate the role of dynamic inflow in
response prediction. In forward flight, a third model, the
model including not only dynamic inflow but also the effect
of the rotor wake on the fixed tail surfaces and tail rotor,
is added.
4.2 Hover
Figs. 4-1, 4-2 and 4-3 present the roll, pitch and yaw
rate responses of UH-60A at hover to a 1-inch right cyclic
input, compared to flight test data of Ref.18. The results
obtained from this simulation including dynamic inflow
produce very good agreement with the flight test data, and
also represent an improvement over the nonlinear simulation
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ORW31NAL pAGE IS
OF pOOR QUAtITY 39
f,J
,_L./0
L_
_,-:,,.J
._I
er R
g6
%
a • •
L •
.00I i 3q iI.O0 2.00 . rO ¼t. OO 5. 2,0 6 G.O0
TIME 5EC
FLIGHT TEST OATR
a_88
WITHOUT DTNAMIC INFLOW
xwxx
WITH OTNAMIC INFLOW
I I gI.D O7.00 8.00
Fig. 4-I Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Right Cyclic Input, Hover)
._
.....
R
I i i I I LO_'O.O0 I • 00 2. CO ).OO 4. O0 5.30 5.00
TIME SEC
FLIGHT TE3T O_T_
688Q
WITHOUT OrNAM[C [NFLOW
xmxx
wITH OTNQMIC INFLOW
i I 94.007.00 8.00
Fig. 4-2 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Right Cyclic Input, Hover)
_DuJ
CD
L,J_
-r
6
'0. O0 l_.O0 . oo 3. O0 . O0 S'. oo 8'. O0
TIME SEC
FLIGHT TEST o_r_
A6•_
HI/flOUT DrNRM[C INFLOW
• • • v
WITH OFNAHfC INFLCN
i7.00 8_.00 0 _.00
Fig. _-3 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Right Cyclic Input, Hover)
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4O
model of Ref.18. For the on-axis response, the addition of
dynamic inflow gives a significant improvement in the agree-
ment between experiment and theory. The roll rate response
to the right cyclic input almost coincides with the flight
test data and reduces the error in the prediction of the
roll rate peak to zero fror, 25_ for the model without dynam-
ic inflow. The nonlinear simulation indicates about 40_
error in this important characteristic[18]. For the off-
axis response, the calculE_ted results are also quite close
to the flight test data, _tlthough dynamic inflow has little
influence on these respon_es. It is of interest _to note
that even though this is an articulated rotor helicopter,
dynamic inflow has a significant influence on the response.
Figs. 4-4, 4-5 and 4-6 present the pitch, roll and yaw
rate responses of UH-60A at hover to 0.5-inch forward cyclic
input. Although for the on-axis response, the model includ-
ing dynamic inflow gives a significant improvement, the
agreement in pitch rate response to the forward cyclic input
is not as good as in the lateral case. This discrepancy
implies that the effective Ditch damping is under-estimated,
tending to indicate that there is a significant additional
source of damping not accolnted for in the theory probably
due to the rotor wake horizontal tail interaction since the
good agreement for the lateral axis shows that the rotor
damping contribution is accLlrately estimated. For the off-
axis response, the calculated results are close to the
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r_J !
_=T
C:-I
OF pOOR QUALITY
1
A*a*_ • .a..
o=
5 + _ 5 __1 t b
'U.U0 I._L _ 7.._0 3.OD 4._D . : .UL]
T I_E 5EC
FLIGHT TESt D_TR
ASA_
W[THOUT OfN_IC INFLON
x_xx
WITH DTN_HIC [NFLON
I BI +7.0C .UO 9._0
41
Fig. 4-4 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 0.5-in Forward Cyclic Input, Hover)
e_ I ¼t'0. US _. OD .0Oq 21 + +.OD .0_3 5. r.3 5. O0
T I_E 5EC
FLIGHT TEST O_TA
HITHOUT OTNRHIC [NFLON
WITH D(NAH[C INFLON
, 8_.OO I7.U0 g. O0
Fig. 4-5 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 0.5-iz_ Forward Cyclic Input, Hover)
L_ D
iLU0
I/ .
ii_lilllliilllliJl
FLIGHT TE5T DRT_
WITHOUT OTN_MIC INFLOW
• x • x
WITH OTNRHIC INFLOW
i i I ' I R I 9 _I.U0 _. 00 3_.o0 i. 00 5. 30 6. O0 .i. 00 .00 .0O
TIME SEC
Fig. 4-6 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 0.5-in Forward Cyclic Input, Hover)
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42
dynamicflight test data, and as for the lateral input,
inflow does not give a significant change.
Figs. 4-7, 4-8 and 4-9 present the responses at hover to
1-inch left pedal input. The initial yaw acceleration is
under-estimated, and the _:oli rate response is close to the
flight test result. The pi:ch rate response is quite differ-
ent from the flight test result, again indicating a rotor-
tail interaction.
For each control input the dynamic inflow has little
effect on the yaw rate re_ponse because the dynamic inflow
is related only to downward inflow components. In addition,
because of pilot difficulty in maintaining trim of the
unaugmented aircraft, in n any cases flight test data drifts
from trim before the control input, causing differences
between the test data and the simulation responses, espe-
cially in the small amplitude off axis responses. This can
be clearly seen in Fig. 4-6, in which the control input
starts at 2.4 seconds; however at that time the yaw rate
response has drifted away a little more than 1 deg./sec,
which is almost equal to the difference between the flight
test and the simulation for the simulation period.
Generally speaking, the agreements obtained for the lat-
eral and directional respo:_ses by including dynamic inflow
are quite satisfactory. However the longitudinal responses
are not so good. The likeky source of this discrepancy is
the interaction of the roto:: wake with the large horizontal
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43
uJ_-
C
7?
_g
'0. OE i. U_ 2. 2D
TIME S[_
ORIGINAL P_.QE IS
OF, POOR QUALITY
a
• FLIGHT TEEr O_rq\
\' _ mm
W[THOUT DTNAMIC INFLOW
• • w •i WITH {]fNGHIC INFLC_
t 11
1 N , I _ b 813 L.DO 4. OD _._ _C ii II i $. (JO 7. _0 . OD 9 K.08
\
Fig. 4-7 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Left Pedal Input. Hover)
un_
cs5 FLIGHT TESt D;r_
.._ ,_ "I-T-_--. ..... _ _-_-...I_C_ J' .
r.r=,,, WITHOUT OfN_IV, IC INFLOW
'i • B • •
I WITH DfN_MIC INFLOW
_0 I. DD * _ I 4 I, L_O 6) I B I. DO I].OD 2.0D ).OO 5 DO .fro _.OO g. OO
TIME 5EC
Fig. 4-8 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Left Pedal Input, Hover)
U2
',-9 •
CD
tD
, ,._ 6_
6
• DO it. O0 OO ¼. DO -'3 OO . DO
_[HE 5EC
FLIGHT TEE[ DGTQ
WITHOUT DfNQMIC INFLOW
WITH 0TNRMIC INFLOW
I 8 I, OO?.OU 9.00
Fig. 4-9 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Left Pedal Input, Hover)
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44
tail, which is difficult to estimate due to lack of experi-
mental data. Even so, the longitudinal responses calculated
show better agreement than the nonlinear model of Ref.18.
4.3 Forward Flight
Figs. 4-10, 4-11 and 4-3.2 present the pitch, yaw and roll
rate responses of the UH-60A at 60 KTS level flight to
0.5-inch right pedal input. The traces of Fig. 4-11 show
that the yaw rate responseE, the on-axis response, obtained
by the models without dyn_mic inflow and with only dynamic
inflow predict a larger pe_,k yaw rate and a higher damping.
Including the effect of the rotor wake on the tail surfaces
improves the correlation producing excellent agreement. In
Fig. 4-10, the predicted pitch rate responses with and with-
out dynamic inflow depart from the flight test data to the
same degree as the nonlinear simulation. Considerable
improvement in the agreement is obtained by including the
effect of the rotor wake. The improvement arises primarily
from the addition of yaw pitch coupling due to nonuniform
downwash at the horizontal tail. The roll rate response is
shown in Fig. 4-12. The main rotor wake has a significant
effect on the response; hcwever this simulation predicts a
significantly larger roll coupling. The initial roll accel-
eration due
the theory,
response.
to application of rudder is over-estimated by
resulting in a larger amplitude roll rate
In this case, the model including the effect of
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45
L, i
u_
I
OmSi'_t _AGE ';:;
OF pOOil QUALITY
c:
C_OL I.DL £.D0
! l III I l I I ; " _ •
lA*" t •
o
_ [: ? S[5 .LL __.r. 3
FLIGHT TE[T D_,;;
a t • •
I.; THC'r'7 [!)N;- _'( !r:
i-i TH _?r_,_l:![ !hrLC;
i-;TH TH r !NFLL'EP,_[ n_
HAIN ROTC _ l,iCqi'[
7. DC L_ . DU {a. fq.
Tit,',-. c_-
Fig. 4-I0 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 0.5-Ln Right Pedal Input, 60 KTS)
, S " J_
u-il _ _ kFTl'" r t,lr..r 1, rr,rl n!,
i:![h [tN_ ri, It,r! ';c-
N! [H I,HF INFLUFNi _" Or
_I_,DO I ['[ Z.OC 3.00 _.OC 5.DC .50 7.0D .[JC
Fig. 4-11 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 0.5-in Right Pedal Input, 60 KTS)
uc
L_ e-U ,,:5
W
a_ E ,
JCw_
D
l
/ \ , :
6n,
i _i i , ,'0. DO | . DlJ .U0 .1. [,.IC _. 00 b] UD 6. DO
TIME 5FE
FcIC.HT TE._T [_ll_
I',tITHOIIT DTNC-_I!," INrl hi!
WITH OTN_'I r IN_I nil
WITH THE [NFt. tlUN [ ric
H_lr,, HOTO_ t,i,qr'F.
I [J. UU Ii. UU '_. llll
Fig. 4-12 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 0.5-1n Right Pedal Input, 60 KTS)
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46
the rotor wake indicates excellent performance by predicting
the yaw-pitch coupling. Dynamic inflow has less effect at
this translational flight condition than in hover.
Figs. 4-13, 4-14 and 4-15 show the roll, pitch and yaw
rate responses to 1-inch left cyclic input. The roll rate
response, the on-axis response in this case, shows very good
agreement for the initial roll rate with some drift away
with time. Note that for the pitch response, the model with-
out dynamic inflow does noel give the right direction for the
response. Adding the dynamic inflow reverses the sign of
the response and includinq the influence of the rotor wake
gives a response very close to the flight test. For the yaw
rate response, although all three are close to the flight
test data, the model including the influence of the rotor
wake shows no improvement over others. However, the shape
of these traces suggests that trim drift may be present in
the flight test as mentioned earlier. Therefore, generally
speaking, the correlation between theory and experiment
including the influence of the rotor wake still is much bet-
ter in this case; it not only gives a correct pitch rate
response by taking the yaw pitch coupling into account, but
also gives a improvement in roll rate correlation.
Figs. 4-16, 4-17 and 4-18 show the angular rate responses
at I00 KTS level flight to 1-inch forward cyclic input. Once
again the pitch rate responses, the on-axis response,
obtained by the models with and without dynamic inflow drift
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47
ORIGINAL PAGE IS
OF POOR QUALITY
I FLJGHT TEST [dq[_L_ ,
L£c£i--6-_m = " | W = II =| m m W N I _ i_[ THOI'T [1N-' ' 1,_,
rr- 2 ! _\ WITH, [_} NE'" ] r ' t4_ _ ¢_,)
I 6=:"r"_6_[" WIIH THE INrLUEN'F_ Pr
c: i _<,, ROIO =, iu'-_'L
,o.o_ ,'.:,:.- -T-.o_ _.o: ,Y_L----2t-_i: _to_ _'.T::-----_:o: _4_T ] t,I£ BEE
Fig. 4-13 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Left Cyclic Input, 60 KTS)
IbJtN c
LD
uJ
i.-
_dL3
,-._,.u_,J-,-,_ ......... ,_:__.
FLIGHT TEST DAT&
HITHOLiT DTNC-r_!C !N';[_r_ d
ii = ii •
kl [H DTNCIrI!E INFL_._
WITH THE INFLUENCE P;
Kq]N ROTOR _qKL
].o_l I ql • I 51 I I at_.07 _t.OC .O0 5.00 .DD ?.OO .DD _D7 ] P'_[ S [ C -- 1
Fig. &-14Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Left Cyclic Input, 60 KTS)
I
oh, 'I
L_A_
_e2:
oo i.oo _.oo _.oo ,_ oo _.oo _.ooIIHF SEE
FLI{:HT TE2T Dql_
WITHOUT DINAHI{ NF LOI:
WITH DINI=P'IIC INFLOk
WITH THE INFLUENCE OF
MAIN ROTOFI W_hE
}. OO DO a Ol_
Fig. &-15 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Left Cyclic Input, 60 KTS)
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4,1
C-
7T
OREWNAL PAGE _
OF POOR QUAUTY
FLIGHT TEST DqTH
• I
c_/ W_TI_C_T DTN_H;C INFLOw,
m_
K_T_ D_N_':IC INFLOL
=c _J_T_ Th[ INFLUENZ[ OF
_" MGIN RC]TO_ N_qKE
48
Fig. 4-16 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Forward Cyclic Input, I00 KTS)
_cl
_-- I::) r c¢cI:CC _J¢¢
_=_...-_-_.....,...........,_,_ ,--,.....'o_.
m
_T
FLIGHT TEST D_T_
KITHOL, T D[I_'_<],] IN;'I_PI'
WITB DTN;_tIC IN_Ov
c KITH THE INrLUEN_IF OF
H_IN ROTOF KGKE
I _ I I =t Eli DO 4 I. O0 I_lIo. OC IIDD _,DD _.DC ¼,OC _, EiC 7.OD ,.DO
7]_E SE_
Fig. 4-17 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Forward Cyclic Input, I00 KTS)
un 'I
r-,
L_C! CI_ @ I!: ¢ ¢ ¢ i_ ¢ i_ ¢ m
I_ ,_ i I i . = i III iI_ } I I I II IIIlll
%'.°° }.Dn 2. DD _.DO 4. OO
lIME SEC
FLIGHI TEST DQT_
_IT_C_uT Olh,,_HIC INFLOF
WITH I]_ N_I", _[ "1 NFL ON
WITH THE INFLUENCE OFMQIN ROTOR WRKE
5. DO 6. O0 "/.DO 8. DO @. O0
Fig. 4-18 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Fo::ward Cyclic Input, I00 KTS)
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away from the flight test data.
inflow is quite small at this
improved correlation of pitch rate
49
The effect of dynamic
airspeed. Considerably
is obtained by including
the effect of the rotor wake. It has a similar shape to the
flight test data, which has two peaks instead of one. For
the yaw rate response, the model including the influence of
the rotor wake presents a better shape but still an over-
shoot, which is believed due to the lack of the dynamic
inflow modelling for the tail rotor. The overshoot of yaw
rate response, through the yaw roll coupling by the horizon-
tal tail caused by the nonuniform downwash, results in a
small wrong positive roll rate response. Even so, it is
still reasonable to say thaZ the model including the effect
of the rotor wake is better because it can predict the sec-
ond peak in the primary response.
Figs. 4-19, 4-20 and 4-21 present responses of UH-60A at
I00 KTS level flight to 1-inch right pedal input. The
traces of Fig. 4-20 show that for the yaw rate response, the
on-axis response, the models without the effect of the main
rotor wake show poor agreement after the first peak. After
3.5 seconds, both of the responses drift away from the
flight test data, there is an error in dominant frequency.
Including the influence of the rotor wake gives a much bet-
ter agreement with flight test data for the 6 second test
period. In Fig. 4-19, once again the pitch rate response
which shows best agreement to the flight test data comes
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L_
c
OE pooP,QUAUTY
/ c
1
]"7_ _ET.
t; ' Tk,$.. r [j' I_:_'_i, J i'17 L _,_
K_T_ Dlt,,-_'i - INCLO_t
c c E
KET_ l _-: IN__U_N_'[ 0;
i gl i;. [,[ . CL ._. Ot"
50
Fig. 4-19 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to l-..in Right Pedal Input, I00 KTS)
t/
E
'i" [. _ 1. O;, Z. OL _, DC Ii, DC 5i.5_ E, (_7.
,mit!l o
II t / i
t% ,('-_ ...."i_" l,]Ik" 01N-"jC IN:L_,;
c c _
_TF TH-[ I_=-LLIE_,. "-'- OF
7. OC . OO !t I. OO
Fig. _-20 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Right Pedal Input, I00 KTS)
U_
U_
U=6_
d_Dc_
61 s_,
i
i@@ml
FLIGHT TEST DI_TIq
'T"
_n. OO
WITHOUT DTNAHIC INFLOw
WITH DTNRMIC INFLON
i@ • D @
WIT_ THE INFLUENCE OF
MAIN ROTOA WAKE
li. OO i ' I2. DO DO _t DO St O_ 5[ DO i 8'. DD @'• _. DO O0 "l]_-" 5£[
Fig. 4-21 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Right Pedal Input, I00 KTS)
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51
from the model including the effect of the rotor wake on the
tails and tail rotor. The same improvement also can be seen
at Fig. 4-21 for the roll rate response, but again as in the
60 KTS case the initial roll acceleration due to pedal input
is over-estimated by about a factor of two. A similar dis-
crepancy appears in the nonlinear simulation model of
Ref.18.
Figs. 4-22, 4-23 and 4-24 show the responses of UH-60A at
140 KTS level flight to 1-inch lateral cyclic input. In this
case, the responses obtained by all three models show very
good agreement with the flight test data, the model_includ-
ing the influence of the rotor wake produces only a little
improvement in long term trends.
Figs. 4-25, 4-26 and 4-27 give the responses of UH-60A at
140 KTS level flight to 0.5-inch doublet pedal input. The
traces of Fig. 4-25 sho%' that the yaw rate responses, the
on-axis response, obtained by all models are very close to
the flight test. In this case, the improvement including the
effect of the rotor wake is significant for both yaw and
pitch rate. The roll rate responses are shown in Fig. 4-27,
and again the roll acceleration due
estimated by this model.
It should be pointed out that in these six
flight cases the variations of the sideslip angle
to pedal input is over-
forward
are all
within 15 degrees, and the reason for only one longitudinal
input case being chosen is that the longitudinal inputs usu-
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ORIGINAL PAGE IS
OF POOR QUALITY
FLIGHT TEST D_TA
52
Fig. 4-22 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Lateral Cyclic Input, 140 KTS)
FLIGHT TEST D_TP
HITHOUT DTN,q_'_IF_. INFLO_-J
NITH DTNAH[E INFLOw;
_ED
al
g WITH THE INFLUENCE Of
MAIN ROTOR WAKE
i. , 3' ' ' ' 5'.00 ' 1. I.OC 1 O0 2._C .r.._. 4.00 5.OC 7.00 8 O0 g. ODlIME SEC
Fig. 4-23 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Lateral Cyclic Input, 140 KTS)L'
U3
CE_
I
_0{00 ] [DO _ _ 5_• DO 3. DO ¼.DO .DO 5. O0
lIME SEC
FLIGHT TEST DAT_q
i" I THOUT DTN_FI [ I NF L [')H
WITH DTN_HIC INFLOH
m D #
HITH THE INFLUENCE OF
MAIN ROTOR NqK[
I
7.00 _I. OD _l.[.i0
Fig. 4-24 Comparison of CalculatedResponses and Flight-Test Data
(Pitch Rate Response to 1-in Lateral Cyclic Input, 140 KTS)
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ORIGINAL F-AGE ISOF POOR QUALITY
IL-'_i
,I--
L--
LJ'cu
_r
I;!
U.U,?
/
I 5* 5 °I I. OD _. OC' 3 I. Eli3 4 I. UD . UC . D_
I ] r',E 5EC
FLTLHT TESI DA;G
NITH.qdT L'IN_*'![ ]NFLOK
WITHD;'NZ_'rC"l_ LO,:
WITH THE It'FLUEN'..E OFwl_lt; _['TO p NqFE
'7I. U0 B). (J(J 9 _.OC
53
Fig. 4-25 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 0.5-in Pedal Input, 140 KTS)
u_,u2
12u_, __-
L_
CC_
L,F--
EL:
==c5
%_.oo
........... ,,_,.. . _._ .I_,_,'"
FLIGHT TEST DRT_
N]THOdT DFN_F',[C INFLOb4
WITH DTN_qH[c INFLOW
Iif. Do 2.0C 31. OO
T;_E SEC
I_]TH THE INPLUEI42E OF
_,_g!N F_OTC'P NRr.[
. DO S. O0 6. OD 7. DO B. DO DO
Fig. 4-26 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 0.5-in Pedal Input, 140 KTS)
• • mmm_mm_ Ill=
_J_m_m
FLIGHT TEST DrlT_q
WITHOUT DTNAHIE INFLOW
• I,, • x =
• W[TH DTNRMIC INFLOWmm
WITH THE INFLUENCE OFMPlIN ROTOR HPlKE
D. O0 I. O0 .OO 11!DO l&. OD 5. DO 6. DO "/.DO B. O0 g. DO
TIME SEC
Fig. 4-27 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 0 5-in Pedal Input, 140 KTS)
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54
ally result in a significant variation of rotor speed, which
is not included in this model.
Generally speaking, the calculated results are close or
very close to the flight test data for all three models used
in the correlation study. The on-axis directional responses
to the pedal input show excellent agreement with flight test
data for all three models, although taking the effect of the
rotor wake on the tail rotor and fixed tails into account is
still beneficial. The longitudinal off-axis response to
pedal input is strongly influenced by the effect of the main
rotor wake on the tail rotor and fixed tails. The models
without the effect of the rotor wake give the similar dis-
crepancies as the nonlinear model used in Ref.18. The model
including the effect of the rotor wake gives significant
improvement and shows excellent agreement with flight test
data, especially at low speed. In general the roll acceler-
ation due to pedal input is over-estimated at all airspeeds.
The agreement is reasonable only in hover but in this case
the yaw acceleration is under-estimated. The reason for this
discrepancy is not clear. A possible source of error could
be the estimation of the inertial characteristics of the
vehicle. The on-axis lateral response and off-axis direc-
tional response to lateral cyclic input show very good
agreement with the flight test data for all of three models.
The model including the effect of rotor wake shows a little
improvement in long term trends. The off-axis longitudinal
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55
response to lateral cyclic input is also strongly influenced
by the main rotor wake, the model including this effect
works quite well, offering noticeable improvement. As for
the longitudinal input, if there is not much variation in
the main rotor speed, the models show good agreement with
flight test data as well.
The pitch-yaw coupling mentioned in Ref.23 is estimated
by the flat vortex mode3 of the rotor wake at moderate
speeds and at high speeds. The estimate of yaw-roll
coupling is not so obvious because the roll rate responses
due to pedal input are not: satisfactory. In contrast to the
hover cases, the additic;n of dynamic inflow has a small
effect in both moderate speed and high speed flight.
Finally, to illustrate the nonlinear nature of the
coupled rotor/fuselage system and the nonlinear nature of
the influence of the rotor wake on the tail rotor and fixed
tails, transient responses of the helicopter for two moder-
ate 1-inch pedal inputs at I00 KTS are calculated. The var-
iations of sideslip angle in two cases are all 20 degrees.
The yaw rate, pitch rate, and roll rate responses are pre-
sented in Figs. 4-28, 4-29, and 4-30 for the right pedal
input and in Figs. 4-31, 4-32 and 4-33 for the left pedal
input. For both cases, the sideslip angle reached 15 deg.
at the fourth second[18]. Before that time the model includ-
ing the influence of the rotor wake gives very good respon-
ses for all three rates. After that, the yaw-pitch coupling
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r.
L21.6C
rr =
-r
2-
7'
/
t.j_li, i..i j J j 41i_ /
ORIGINAL PAGE IS
OF POOR QUALITY.
iI/'
/_f=ee ¢
q< •ooN w _
x
FLIGHT TL._T D_TP
• • • m
W[THS_T OrNt_{ F" [NF,.C,.
WITH DTN_'iC INFLOW
I • I •
WITH THE INFLU[4Z[ OF
F.q M_IN ROTO_ wnY,E
r_ I q I i I
'0. DO I. O0 2. DO _* O_ 4. O,: 5.OO 6.0_ ?.0F; C0 .0c
"TIME SEE
56
Fig. 4-28 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Right Pedal Input, 100 KTS)
_ • • /
° i" FLIGHT TEST DQTAa
- ..--L,: • .,-.\•,,, . • / WITHOUT DTN_MIC INFLOW
% * I • '_ m / •
• WITH OTN_MIC INFLOw
7
II••
_! _._;._,'o•" WITHT,E ,NFLU_,<_OFH_IN ROTOR W_KE
O0 ].O0 200 3 O0 4._0 5.OO S.00 7._ 8._ I._
lIME SEE
Fig. 4-29 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to 1-in Right Pedal Input, I00 KTS)
I•
_ ..-li • •
tl • • aa •
J • I • •
°;i '
:lr. I
I I i11._ 21. {J_ ]i. {I_ 4.Ol gO 5.OC
lIME 5FC
FLIGHT TEST DQTQ
_6_m
NITH_U T _T_&F'IC [NFLC_
WITH DTNQ_IC INFLON
WITH THE INFLUENCE 0FMAIN ROTOE WAKE
i 8'.O_ '?.0L' g. O0
Fig. 4-30 Comparison of CalculatedResponses and Flight-Test Data
(Roll Rate Response to l-in Right Pedal Input, I00 KTS)
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=_ ORIGINAL PAGE IS_" OF POOR QJALITY
AI_,
=
_=i __'/°
=r
r I .I ,,se_" • o _ •
WITH THE INFLUENCE OF
6, MQIN ROTOR WAKE
-._ , 21 , q J _ *. J'0. _C I. r,'c . 0(] 3.00 4. DO 5. OO 5.80 7. O_ 8 DO 9. O0TIHE 5EC
FLIGHi" TESt OAT_
WITHOUT DTNAH[C INFLOW
WITH0;N_"I'C'INFL0"
57
Fig. 4-31 Comparison of Calculated Responses and Flight-Test Data
(Yaw Rate Response to 1-in Left Pedal Input, I00 KTS)
s /° \ ° "'°/° 1 °me°
! ,° . \.,;_ • .__ t"---_"............ '', "" • \• %,
"" "2 _DO Ii.O(D .O0
I IME 5EC
FLIGHT TEST D_qTA
WITHOUT DTNAHIC INFLOw
WITH CTN_HIC INFLCW
o ° m O
WITH THE INFLUENCE OF
-• MAIN ROTOR WaKE
I. OO _&*.O0 5'. C)O 5 _.DO "/.I DO 8I.DO 9.1DO
Fig. 4-32 Comparison of Calculated Responses and Flight-Test Data
(Pitch Rate Response to l-i_i Left Pedal Input, I00 KTS)
• x
j x_ o.. o>-,,?.._./M•° •
ev
; I , 1'O.D0 i'.[..'(_ 2I.EL0 3.DO q.L_O 5'.DO 5.DO
[ I HE 5EC
FL{GHT TEST DQTQ
W[THOUT DINQMIC INFLOW
WITH 01"N_MIC [NFLON
I ° I•
WITH THE INFLUE4CE OFMQIN ROTOR WaKE
..LC
Fig. 4-33 Comparison of Calculated Responses and Flight-Test Data
(Roll Rate Response to 1-in Left Pedal Input, i00 KTS)
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not valid when
15 degrees.
On the other
responses
58
and the yaw-roll coupling are overestimated as expected.
For the left pedal input, the overestimation mainly happenes
in pitch rate response, and for the right pedal, the roll
rate has the largest overshoot in all three over responses.
Consequently it is concluded that the linearized approxima-
tion to the nonuniform down wash and sidewash at tails is
the sideslip angle variation is larger than
hand, it should be noticed that all three
obtained by the models without considering the
influence of the rotor wake have the same accuracy and simi-
lar shape or trend with the nonlinear dynamic model used in
Ref.18. Therefore, it seems that even under moderate control
inputs, the simulation deficiencies still mainly are results
of insufficient modelling of the rotor/tail interaction, and
have little to do with the small perturbation assumption
under which the system is linearized and the approximation
by replacing the periodic term with its time average.
4.4 Conclusions
From the correlation results given in the last section,
it is clear that the linearized model of helicopter dynamics
developed in this work is a good description for helicopter
free-flight dynamic characteristics in both hover and trans-
lational flight trim conditions. The flight test data con-
firmed the analytic prediczions with excellent accuracy for
small inputs.
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v q
v
59
The comparison of the transient responses with flight
test data shows that in hover the effects of dynamic inflow
are significant and can be correctly taken into account by
momentum theory, and that the inclusion of dynamic inflow is
not important as expected in forward flight. The rough
agreement of the transient responses between the models
without considering rotor tail interaction in this work and
the nonlinear simulation _odel used in Ref.18 for small and
moderate control suggested that for the flight dynamical
analysis the coupled rotoz/fuselage system still can be con-
sidered as linear time invariant in a wide range of flight
conditions. The significant improvement obtained by the mod-
el including the influence of the rotor/tail interaction
suggested that for forward flight the sidewash variation at
tail rotor and vertical tail and the nonuniform downwash at
horizontal tail are more important for flight dynamic analy-
sis than the inertia nonlinear coupling, the mechanical non-
linearities associated with moderate elastic deflections,
the servo dynamics, the effects of sweep, the compressibili-
ty, and the nonlinear lag damping, all of them are included
in the nonlinear simulator model. Therefore, the inclusion
of the static influences of the rotor wake on the tail rotor
and fixed tail surfaces are very important and may be the
most important factor for forward flight dynamical analyses
after the basic configuration modelling. It also has been
shown from the comparison that the simple linear flat vortex
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60
theory employed here is a good description of the phenomenon
and its linearized approximation can be used in a wide range
of flight speed for small control inputs.
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Chapter V
INFLUENCEOF THE BLAZIEDYNAMICSON THE FEEDBACK
CONTROLSYSTEMDESIGN
5.1 Introduction
In the design of high-gain control systems for the heli-
copter, it is essential to consider the influences of the
blade dynamics. Although it has been recognized for quite
some time that the flapping dynamics of an articulated rotor
system imposes limitations in the design of automatic con-
trol systems for rotorcraft,
analytical research has been
impact on the
limited number
into account.
design of automatic control
and a significant amount of
performed to investigate their
systems, only a
of studies take the lag degrees of freedom
Furthermore, all investigations to date are
based on incomplete system modelling under assumptions that
yaw motion and vertical motion of the helicopter are uncou-
pled, the fuselage center of gravity is on the shaft, and
the effects of the tail rotor are not included. In Ref.26,
R.T.N.Chen and W.S.Hindson investigated the limitations in
control gain encountered when flapping dynamics are included
and presented experimental verification of these trends. In
Ref.5, H.C.Curtiss investigated the high frequency charac-
teristics of the transfer functions describing the response
- 61 -
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modelling, for
hover.
Due to the
62
of helicopters associated with the rotor degrees of freedom
including the lag degrees of freedom and examined the impact
of those on the design of automatic control systems in hov-
er. The results showed that if the simple roll attitude or
roll rate feedback is employed on the helicopter model with
rotor dynamics neglected, uhere is no gain limitation. For
a model including flapping dynamics, there will be limita-
tions for roll rate feedback due to the effect of the feed-
back on the regressing flap mode, and for the roll attitude
feedback due to the effect of the feedback on the advancing
flap mode. When both flapping and lagging dynamics are
included in the model, the maximum allowable gain of the
roll rate feedback is much smaller, and the corresponding
unstable mode is advancing lag instead[5]. This study
extends these results by an analysis on a complete system
model described in Chapter 3 which includes all the low fre-
quency degrees of freedom, the effects of center of gravity
location, the effects of the tail rotor and fixed tail sur-
faces, and the unsteady aerodynamics through dynamic inflow
forward flight trim conditions as well as
multivariable nature of the helicopter sys-
tem, linear optimal regulator theory has also been used to
design stability augmentation systems for helicopters.
Although successful flight control systems[27,28] have been
designed by optimal control procedures based on conventional
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63
quasi-static stability derivative models. These studies are
limited to helicopters that have relatively high fuselage
inertia and small hinge offset. Therefore the rotor-body
coupling is small. And these designs do not reflect the more
demanding bandwidth requirement for very agile rotorcraft.
Attempts to design a model-following flight control system
for a hingeless rotor helicopter to achieve moderately high
bandwidths have worked well[ in ground-based simulation, but
have been less successf_l in flight[29,30]. Since the
ground-based simulation, which is based on a stability
derivative model, and flight results do not agree, it must
be assumed that better models of such rotor-system dynamics
are required. Several investigators have shown that for the
application of the linear cptimal regulator theory to high-
gain, full-authority controller, the inclusion of the flap-
ping dynamics is essential. In particular, Miyajima has
found that the blade regressing flap mode should be included
in the stability and control augmentation system design[4].
Hall has shown that if an optimal control system devised
based on the quasi-static flapping assumption is applied to
a model with flap dynamics included, instabilities
result[6]. This study extends previous studies by examining
the closed-loop responses of the model including both flap
and lag when the controller design is based on a model which
includes only flapping dynamics, and the closed-loop con-
troller is designed by standard and frequency-shaped per-
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formance indexes.
for eliminating
64
The latter is shown to be very effective
the destabilizing effect of unmodelled
dynamics. It has been found that in addition to rotor dyna-
mics, the sensor dynamics, the actuator dynamics, and the
transport delay associated with the digital implementation
also can severely limit the usable values of feedback
gain[26]. One investigation[31] has found that the simula-
tion of the feedback controller design based on the semi-.
empirical stability deriw!_tive model shows an instability
due to interaction between actuator dynamics and sensor
dynamics. However, these open-loop modes have higher damping
and frequency than the rotor lag
seems more reasonable to include
basic model before examining the
the lag
dynamics. Therefore, it
dynamics in the
destabilizing effects of
the actuator and sensor dynamics. For the requirement of
better modelling, some semi-empirical models are obtained by
numerically adjusting time constants, damping factors, and
natural frequencies in an assumed model structure untill the
frequency response of the model matched flight test data
[32]. This approach may be useful in the design of feedback
control system for a specific aircraft, but it can not pro-
vide the physical insight to the helicopter designer for im-
proving the basic configuration design for next step in the
development. Furthermore, a series of simplified control-
lers are developed through successive reduction in the num-
ber of feedback loops while using the feedback gain factors
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65
obtained for the optimal control, which makes it possible to
reduce significantly the number of feedback loops required
by optimal control design without any noticable effect on
the overall system dynamical.
All the active control simulations in this study were
performed on a UH-60A Black Hawk helicopter. All the
results use the complete model which includes the dynamic
inflow at hover and both ¢[ynamic inflow and the influences
of rotor wake on the tail rotor and empennage at forward
flight unless noted.
5.2 Simple Feedback Control
Although a variety of output variables are possible
sources of closed loop feedback information for control
actuation, the rotation attitude and rotation rate variables
have been considered to be highly effective for stabilizing
helicopters by some investigations [33,34], and are most
frequently used in practice. Therefore, both have been cho-
sen as feedback variables.
In this section, the influences of rotor dynamics and
dynamic inflow have been studied by examining the root loci
and the closed loop frequency response
the system with simple state feedback.
eigenvector analysis has been used to
insight.
characteristics of
The eigenvalue and
promote physical
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66
5.2.1 Attitude feedback at hover
Fig. 5-1 shows the eigenvalues of the helicopter at hov-
er, their numerical values are listed in Table 5-1. It can
been seen that the regressing flap mode, roots 17 and 18,
has moved away from the position directly below the coning
flap, which suggests that it is strongly coupled with the
fuselage modes. Fig. 5-2 shows the eigenvalues associated
with free-flight stability-and-control-characteristics in
detail. According to eigenvector analysis, the left complex
conjugate pair, roots 17 and 18, represents the mode coupled
by regressing flap and body roll motion. The two right com-
plex conjugate pairs near each other, roots 22, 23 and 24,
25, represent modes having coupled pitch and longitudinal
velocity, and coupled ro_l and lateral velocity, the two
so-called longitudinal and lateral phugoid modes. Four zero
roots, roots I, 2, 3, and _!7, are associated with vertical,
lateral, longitudinal and yaw position each. The root 27 is
not exactly zero due to a lack of complete cancellation of
terms, and should be a zero if the equations are derived
explicitly for the coupled system. The smallest negative
real root near zero, root 26, is associated with yaw damp-
ing, and the one next to it, root 21, is associated with the
vertical damping; the left two roots, roots 19 and 20, rep-
resent the modes coupling body pitch and regressing flap.
All of them are coupled t_gether through the canted tail
rotor.
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! !
0 2_
::Or-
(_ j.7.
!
m_
ol_
o0_d
0
0
o
00
't/
00
1-55.00
I
_t
_5
m
_.
g-
IMnGE-25.00
PRRT OF-10.00
I
E IGENVRLUE55.00 20. O0! ----I
35. OO 50. O0 65. O0
X
X
X
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cc
6,
CDCD
CD
C9UD_
bJ_-
>
L_-_
_4-
_ ,=
CC_
LIj r'_
l'l- r_l.
I
e-,
u_I-6.00
Fig. 5-2
ORIGINAL PAC-,EIS
OF POOR QU;_,LITY
X X X X
I I i l t I-5. DO -U,.DO -3. CO -2.DO -I .DO
REFIL PRRT OF E IGENVRLUES (HOVER)O.OO
The Helicopter Rotor/Fuselage System Open Loop Roots
at Hover (Detail)
68
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69
The root loci of the helicopter with roll attitude feed-
back with in a range from zero to 1.5 Deg/Deg are shown in
Fig. 5-3. The modes shown in the figure are those locations
that are significantly affected by the feedback. Those
include the body roll/regre_sing flap, roll/lateral velocity
and yaw damping. The longitudinal modes, lag modes and con-
ing and advancing flap modes are hardly affected by the
feedback.
Roll attitude feedback stabilizes the roll/lateral veloc-
ity mode very effectively at low gain, and makes it over-
damped at gain K = 0.12. This will improve the helicopter
lateral dynamic character;Lstics because the roll/lateral
velocity mode dominates the low frequency lateral dynamics.
However, the body roll/_l_egressing flap mode is destabi-
lized and finally becomes unstable at feedback gain a little
higher than critical gain }[ = 1.0 Deg/Deg, which has been
predicted to be the theoretical feedback limitation by a
simple model used in Ref.5. This is physically reasonable
because the model here incllud extra damping contributed by
the tail rotor. The roll a%titude feedback destabilizes the
yaw damping mode as well, implying there is a sizeable
coupling between the late:cal and directional dynamics at
hover.
The pole-zero locations of open-loop roll angle to lat-
eral cyclic transfer function at hover were also calculated.
The results are shown in Table 5-1. As can be seen, the
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( | _ A I ! ( ! 1 _ (
0 C
_Nc2 ,,_
._ .,_
oUloW '
d
tlJ
u_o
oo
_ °.
n-'-'
klJCD
(3 o_rd
D121
OO
(_-7. UO
Body
Ro11/Lateral
I-5.UO
I-5. UO
REAL
Fig. 5-3
Roll/Regres_ir_q
V_I oci ty
_
Fla_
Stable Unstable
_X
Y___w
O. UO
! I I I-q. DU -3. DO -2. UD -I. DO
PFIRT OF E IGENVFILUE5
The Root Loci of The Helicopter With Roll
Feedback to Lateral Cyclic Input at Hover
I1 .DO
Attitude
I2.O
0
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POLES ZEROS
71
1
2
3
4
5
6
7
8
9
I0
II
12
13
14
15
16
17
18
19
2O
21
22
23
24
25
26
27
0 0000E+00
0 0000E+00
0 0000E+00
-0 9427E+01
-0 9427E÷01
-0 1973E+01
-0 1973E+01
-0 8447E+01
-0 8447E+01
-0 2375E+02
-0.2375E+02
-0.1275E+01
-0.1275E÷01
-0.1960E+02
-0.2028E+01
-0.2028E+01
-0 3258E+01
-0 3258E+01
-0 4947E+01
-0 I065E+01
-0 5168E+00
-0 8410E-02
-0 8410E-02
0 5203E-01
0 5203E-01
-0 1380E÷00
0 5839E-06
°
0.
O.
0.
--0.
0.
--0.
O.
-0
0
-0
0
-0
0
0
-0
0
-0
0
0
0
0
-0
0
-0
0
0
O000E+00
0000E'00
O000E-00 -0
5229E_02 -0
5229E-02 0
3974E+02 0
3974E+02 -0
2545E_02 -0
2545E+02 -0
2171E+01 -0
2171E+01 -0
1791E_-02 0
1791E_-02 0
0000E_-O0 -0
7586E_01 -0
7586E_'01 -0
4257E+01 -0
4257E_01 -0
0000E+00 -0
O000E _00 0.
O000E +00 O.
4062E +00 -0.
4062E+00 -0.
2813E+00 0.
2813E+00 -0.
O000E+00 -0.
O000E+O0 O.
3843E+01 0
3843E+01 -0
2413E+01 0
2413E+01 -0
8447E+01 0
8447E+01 -0
2617E+02 0
1960E+02 0
2258E+02 0
1474E+01 0
1474E+01 -0
2011E+01 0
2011E+OI -0.
I149E+02 0.
4905E+01 0.
I026E+01 0
5178E+00 0
6182E-01 0
6182E-01 -0
2112E+00 0
3537E-02 0
1961E-05 0
5199E-I0 0
1342E-12 0
1893E-II 0
I179E÷03
I179E+03
5588E+02
5588E+02
2545E+02
2545E+02
O000E÷00
0000E+00
O000E+00
1739E+02
1739E+02
7574E+01
7574E+01
0000E+O0
0000E÷O0
0000E+00
0000E+00
2928E+00
2928E+00
O000E÷00
O000E÷O0
O000E+00
O000E+00
O000E+O0
0000E+O0
APPROXIMATE TRANSFER FUNCTION :
Roll(s)
AIs(s)
2.057(S+0.O03537)('S+0.2212)
(S+0.138)(S+O.30841+j0.4062)(S+O.OO841-jO.4062)
(S+ii.49)
(S+3.258+j4.257)(S+3.258-j4.257)
TABLE 5-1 Poles, Zeros and Approximate Transfer Function
of Lateral Helicopter Dynamics at Hover
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longitudinal low frequency poles have very close
hence will be canceled ir_ the overall transfer
72
zeros and
function.
The low-frequency modes remaining after the cancellation
will be the body roll/recTressing flap, the roll/lateral
velocity and the yaw dampirtg. In Table 5-1, a reduced-order
approximate transfer function consisting of the remaining
low-frequency modes is also presented. From the frequency
response of the helicopter roll attitude to the lateral
cyclic input with roll attitude feedback shown in Fig. 5-4,
it can be clearly seen thele are significant improvements by
both reducing the resonanc6 ratio of the system and increas-
ing the bandwidth in the _!,ode amplitude characteristics and
by reducing the phase shi_t in the phase characteristics.
Due to the presence of several nonminimum phase poles and
zeros, the standard interr.retation of gain margin and phase
margin is not valid here, and there is a phase lead at low
frequencies. In addition, the root loci using the approxi-
mate transfer function with same feedback will coincide with
the root loci shown in Fig. 5-3. This implies that the roll
response is primarily determined by the lateral modes of
fuselage and the mode of regressing flap and is coupled with
the mode of directional damping. The cross coupling from
longitudinal dynamics to the roll attitude response is one
order smaller.
The longitudinal root 2oci of the helicopter with pitch
attitude feedback in the same gain range is shown in Fig.
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73
613.
6
tm=,
TW
<.n
PJ
?-
,4i-m
_ A 1,2 x x x x
__ K=O.O
,, ,, ,, K=0.2
,, . , K=O. q
• _ _ _. ! i i 1 i',.111 , I
FREOUENC'r RAD.'SEC (ROLL ATT]IUDE FEEDBACK)
,,I
Fig. 5-4 Frequency Response of _/A With Roll Attitude
Feedback to Lateral Cyclic Input, Hover
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o
._1C1S
.
0Do
CIZ "-"C_C
UJo(.D o
CJ-"r,0 _u
0D
I
00
t-7.U Dt I I
-6.00
Fig. 5-5
Pit c h/Lonqi tudi nal
F_it.ci,JR_gro_zog FI_p
Velocity ?,
_-x-x
/
\
._l. a b I o
l t I-q. 00 -3. UU -2. DU -I . UD 0. UO
PRFIr OF EIGLNVRLUE5
The Root Loci of The Helicopter With Pitch Attitude
Feedback to Longitudinal Cyclic Input at Hover
Unstable
O0
OzO>:PE-
C_
r-tg
II .00
I2.DO
-,,.I
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75
5-5. The modes shown in the figure are those whose loca-
tions in the complex plane are significantly affected by the
feedback. They include the body pitch/regressing flap, the
pitch/longitudinal velocity and the yaw damping modes. In
addition, the damping ratio of the roll/lateral velocity
mode is also significantly changed by the pitch attitude
feedback, from -0.18 at K=0.0 Deg/Deg to 0.0132 at K=I.5
Deg/Deg, although this variation of the damping is too small
to be shown in the root loci map. The pitch attitude feed-
back stabilizes the longitudinal oscillatory mode at gain
K<0.15 Deg/Deg; destabilizes it thereafter; makes it unsta-
ble about K=I.4 Deg/Deg; and increases the oscillatory fre-
quency quite rapidly. The flapping velocity component in
the corresponding eigenvector is increased rapidly with the
feedback, suggesting that the feedback limitation physically
results from the coupling with the flapping dynamics. The
pitch attitude feedback de:reases the damping of the right
body pitch/regressing flap mode and increases the damping of
the left one.
The pole-zero locations of open-loop pitch angle to
longitudinal cyclic transfer function at hover are presented
in Table 5-2. For this case, only the vertical damping and
regressing lag have a very close zero, hence will be can-
celed in the overall transfer function. Therefore the sim-
plified transfer function, presented in the table, is much
more involved. This implies that the pitch response of the
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O_i:_N_,L PAOE JS
OF POOR QUALW_'76
POLES
1
2
3
4
5
6
7
8
9
I0
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
0
0
0
-0
-0
-0
-0
-0
-0
--0.
--0.
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
-0
0
0
-0
0
0000E+00
0000E+00
0000E+00
9427E+01
9427E÷01
1973E+01
1973E+01
8447E+01
8447E+01
2375E+02
2375E+02
1275E+01
1275E+01
1960E+02
2028E+01
2028E+01
3258E÷01
3258E+01
4947E+01
I065E+01
5168E+00
8410E-02
8410E-02
5203E-01
5203E-01
1380E+00
5839E-06
0.0000E+00
0.0000E+00
0.000OE+00
0.5229E+02
-0.5229E+02
0.3974E+02
-0.3974E+02
0 2545E+02
-0 2545E+02
0 2171E+01
-0 2171E+01
0 1791E+02
-0 1791E+02
0 O000E+O0
0 7586E+01
-0 7586E+01
0 4257E+01
-0 4257E+01
0 O000E+O0
0 O000E+O0
0 O0001!:+O0
0 4062E+00
-0 4062E+00
0.2813E+00
-0.2813E÷00
O.O000E+O0
O.O000E+O0
APPROXIMATE TRANSFER FUNCTION :
ZEROS
-0.9378E+00
-0.9378E+00
0.1431E+01
0.1431E+01
-0.8453E+01
-0.8453E+01
-0.2609E+02
-0.2254E+02
-0.1960E+02
0 1485E+01
0 1485E+01
-0 I148E+02
-0 2025E+01
-0 2025E+01
-0 I189E+01
-0 I189E+01
-0 3728E+01
-0 5275E+00
-0 2315E+00
-0 3231E-01
-0.2872E-02
0.9957E-06
-0.4581E-I0
0.2050E-II
-0.2433E-II
0.1239E+03
-0.1239E+03
0.5591E+02
-0.5591E+02
0.2543E+02
-0.2543E+02
0.0000E+00
O.O000E+O0
0.O000E+O0
0.1694E+02
-0 1694E÷02
0 0000E÷O0
0 7593E+01
-0 7593E+01
0 4844E+01
-0 4844E+01
0 O000E÷00
0 O000E÷O0
0 O000E÷O0
0 0000E÷O0
0 0000E+00
0 0000E+O0
0 O000E+O0
0 O000E+O0
0 O000E÷O0
Pitch(s)
BIs(s)
-0.2954(S÷0.002872)
(S-O.052C'3+j0.2813)(S-O.05203-jO.2813)
(S+0.03231)(S+0.2315)(S+3.728)
(S+0.00841-jO.4062)(S+0.00841+j0.4062)(S+0.138)(S+I.065)
(S+I.189-j4.644)(S+I.189+j4.844)
(S+3.258-j4.257)(S+3.258+j4.257)(S+4.947)
TABLE 5-2 Poles, Zeros and Approximate Transfer Function
of Longitudin_Jl Helicopter Dynamics at Hover
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helicopter, which is
dynamics, is strongly
tional dynamics.
77
a key figure for the longitudinal
coupled with the lateral and direc-
As same as roll attitude feedback control, the yaw damp-
ing mode is destabilized s:_gnificantly by the pitch attitude
feedback.
5.2.2 Attitude feedback :_n forward flight
Fig. 5-6 shows the eigenvalues associated with free
flight stability and control characteristics of the helicop-
ter for forward flight at 60 KTS and I00 KTS; their numeri-
cal values are listed in Tables 5-3 and 5-4. According to
eigenvector analysis, the left complex conjugate pair, roots
17 and 18, represent the mode coupled by regressing flap and
body roll. The complex conjugate pair in the middle, roots
22 and 23, as well as the left real mode, root 19, repre-
sent a short period mode which involves primarily pitch
angle and angle of attack and is strongly coupled with the
regressing flap. The right complex conjugate pair, roots 20
and 21, represent the dutch roll mode which involves prima-
rily the yaw degree of freedom with a number of small trans-
lation velocities; another complex conjugate pair very near
zero, roots 24 and 25, is the spiral coupled with an unreal-
istic yaw mode due to the remainded error mentioned in the
introduction about the disadvantage of the matrix displace-
ment method. This mode is the one that is somewhat inaccu-
rate due to a lack of complete cancellation of terms, and
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GOl'--.
s_oo_ dooq uodo
5d "0 ll__; "i1- 5?" l- rill "?-
I I I I
v
W X '4
X
¥X
"4
v
X
S_H OOl pue SIM 09 _emo%sAs ebel_sn3/=o%o H a_%dooTleH eqL
S_N-1UAN-g.GIt _-ID I YUd -It-Jilt5L " {7- t15 " I-'" 57 " h" 111] " _,"
I I I I
9-S "_T3
ll.l --_
ZO
____.oO
v
X
_l "h-
x)
,,)
AS "0-I
LL_
0
!
C_O
i
C.--_
D1
-OQ
-.
__-Dc2
nl
QC_
I"11
nC2
Q gl3
UI
' ) I t i t ) il l I b
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ORIG/NAL
,OF 'POORPAGE ISQUALITY
79
i
2
3
4
5
6
7
8
9
i0
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
POLES
0.0000E÷00
0.0000E+O0
0.0000E+00
-0.9359E+01
-0 9359E+01
-0 2012E+01
-0 2012E+01
-0 8313E+01
-0 8313E+01
-0 2428E+02
-0 2428E+02
-0 I177E+01
-0.I177E+01
-0 1949E+02
-0 2092E+01
-0 2092E+01
-0 4444E+01
-0 4444E+01
-0 3830E+01
-0 2137E+00
-0 2137E÷00
-0 9324E+00
-0 9324E+00
0 1490E-01
0 1490E-01
0 2209E+00
-0 3006E+00
0.O000E÷00
O.0000E+00
0.0000E+00
0.5207E+02
-0.5207E+02
0.3954E+02
-0.3954E+02
0.2503E+02
-0.2503E+02
0.7011E+01
-0.7011E+01
0.1774E+02
-0 1774E+02
0 0000_i+00
0 7671_i+01
-0 7671_]+0i
0 3883[+01
-0 3883_+01
0 0000[+00
0 1254[i+0!
-0 1254[i+01
0.109311+01
-0.109311+01
0.5367E-01
-0.5367E-01
O.O000E+O0
O.O000E+O0
APPROXIMATE TRANSFER FUNCTION
ZEROS
-0.5844E+02
-0 5844E+02
0 7133E+01
0 7133E+01
-0 5050E+02
-0 8648E+01
-0 8648E+01
0 1648E+01
0 1648E+01
-0 2276E+02
-0.1571E+02
-0 1571E+02
-0 1919E+01
-0 1919E+01
-0 3892E+01
-0 8986E+00
-0 8986E+00
-0.2731E+00
-0.2731E+00
-0.2916E+00
0.2168E+00
0.2771E-03
0.2678E-09
0.1802E-12
-0.2330E-12
0.9794E+02
-0.9794E+02
0.5395E+02
-0.5395E+02
0.O000E+00
0.2530E+02
-0.2530E+02
0.1876E+02
-0.1876E+02
0.O000E+00
0.7746E+01
-0.7746E+01
0 7638E+01
-0 7638E+01
0 O000E÷O0
0 I125E÷01
-0 I125E+01
0 III8E+OI
-0 III8E+01
0 O000E+O0
0 0000E+O0
0 0000E+00
O.O000E+O0
O.O000E÷O0
O.O000E+00
Roll(s)
AIs(s)
1.052
(S-0.0149+j0.O5367)(Si0.0149-jO.05367)
(S-O. 0002771)
(S+4.44-j3.88)(S+4.44-j3.88)
TABLE 5-3 Poles, Zeros and Approximate Transfer Function
of Lateral Helicopter Dynamics at 60KTS
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OF POORQUALITY 8O
POLES ZEROS
123456789I0ii12131415161718192021222324252627
O.O000E+O00 O000E+O00 O000E+O0
-0 902SE+01-0 9025E+01-0 2083E+01-0 2083E+01-0 3312E+02-0 3312E+02-0 7759E+OI-0 7759E+01-0 2418E+02-0 I167E+0!-0 I167E+01-0 2269E+01-0 2269E+01-0 5138E+01-0 5138E+01-0 4890E+01-0 3618E+00-0 3618E+00-0 I146E+01-0 I146E+010 I137E-010 I137E-010 1528E+00
-0 9152E-01
O.O000E+O00 O000E+O00 O000E+O00 5194E+02
-0 5194E+020 3897E+02
-0 3897E+020 1683E+02
-0 1683E+020 2517E+02
-0 2517E+02O.O000E+O00.1738E+02
-0 1738E+020 7914E+01
-0 7914E+010 4588E+01
-0 4588E+010 O000E+O00 1440E+01
-0 1440E+010 1503E+01
-0 IBO3E+OI0 4161E-01
-0 4161E-010 O000E+O00 O000E+O0
APPROXIMATETRANSFERFUNCTION :
-0.2156E+03-0.2047E+02-0.2047E+02-0.2612E+01-0.2612E+01-0.8965E+01-0.8965E+01-0.2221E+02-0.2221E+02-0 2660E+02
0 I175E+010 I175E+01
-0 2016E+01-0 2016E+01-0 4889E+01-0 I146E+01-0 I146E+01-0 3910E+00-0 3910E+000 1449E+00
-0 8595E-010 II16E-020 8511E-II
-0 1040E-IO-0 8388E-12
O.0000E+O00.7327E+02
-0.7327E+020.5333E+02
-0.5333E+020.2499E+02
-0.2499E+020 1608E+02
-0 1608E+020 0000E+000 1937E+02
-0 1937E+020 7886E+01
-0 7886E+010 0000E+O00 1526E+01
-0 1526E+010 1312E+01
-0 1312E+010 0000E+00O.0000E+000.0000E+O00.0000E+00O.O000E+000.O000E+O0
Roll(s)
AIs(s)
TABLE 5-4
0.7014
(S-0.OII37+j0.04161)(S-O.01137-j0.04161)
(S-0.001116)
(S+5.138-j4.588)(S+5.138-j4.588)
Poles, Zeros and Approximate Transfer Function
of Lateral Helicopter Dynamics at IOOKTS
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81
should be a zero and a real root if the equations are
derived explicitly for the coupled system. Three zero roots,
roots I, 2, and 3, are associated with vertical, lateral,
longitudinal position each; the two real roots at right rep-
resent the phugoid mode coupled with lateral damping due to
the canted tail rotor.
The root loci of the helicopter dynamics with roll atti-
tude feedback for a gain range from zero to 1.5 Deg/Deg for
60 KTS and I00 KTS forward flight are shown in Fig. 5-7 and
Fig. 5-8. The modes shown in the figures are those signifi-
cantly affected by the feedback, including the body roll/
regressing flap, dutch roll and spiral. It is shown that
the roll attitude feedback stabilizes the spiral mode very
effectively but has very little effect on the dutch roll.
The stabilization of the spiral mode improves the low fre-
quency characteristics of the helicopter's lateral dynamics.
As can be clearly seen from the frequency responses present-
ed in Figs. 5-9 and 5-10, the feedback reduces the resonance
ratio of the system, increases the bandwidth in the Bode
amplitude characteristics and reduces the phase shift in the
phase characteristics. It should be noticed that the spiral
mode is unstable without the feedback. The low frequency
response peak shown in the Bode plot is only a measurement
about how close to the imaginary axis of the pole, instead
of the classical magnitude of steady-state response for a
sinusoidal input. This means that the improvement obtained
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ORKMNAL PAGE IS
OF POOR QUALITY
82
ua_-
(I=>
z_U_.l_(_9
b_l
LI-
{D o(D
rr--CIQ...
LJKDL.9C }CI r_J,,;- ,
'-]2.00 -ID.SO
JStable
X
Spiral
Dut ch Rol I
Body Roll/Regressing Flap
Uns t abl e
t I i I I I I-g. U0 -7.50 -5. UCJ -_&. 50 -3. O0 -) . 50 0.00 ] . 50
REAL PART OF EIGENVFLUE5
Fig. 5-7 The Root Loci of The Helicopter With Roll Attitude
Feedback to Latera! Cyclic Input at 60 KTS
c
uJ_
JE_
£_9 ..
uJ
Lu
ED o
__o.
E_O-
L_D(_90
/
Ur_table
x
Spir al
Dutch Roll
Body Roll/Regressing Flap \I I I _ l I I I
-12.OO -10.50 -9.00 -7.50 -5,DO -4.50 -3.00 -1.50 0.00 1.50
REAL PART OF EIGENVALUE5
Fig. 5-8 The Root Loci of The Helicopter With Roll Attitude
Feedback to Lateral Cyclic Input at I00 KTS
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83
ORi_NAL PAGE IS
OF POOR QUALITY
6
03tm
r'm_
'-6-Z
£..0
(It3
:c=
i "
P_6
,,,,. K=O, LI
, , . ,I . . .I
FREOUENc'r RRD/SFC [ROLL GTT]'i'L'3E FEEDBRCK)
6
_"6O=,"
Uj 1
(J3
CE_"to
_6
?-
?
"'..8" _
x• x
• x
,, , K=O. 2
,,x,, ff=O.q
_ _ _ _ _h'_' '_ '__ _h'o _ _ _ _ _'o,' _ _ _ i_h'o*_FREOUENCT RRDISEC (ROLL RTT]TUDE FEEDBRCK}
Fig. 5-9 Frequency Response of _/A With Roll AttitudeImFeedback to Lateral Cyclic, 60 KTS
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84
CD
De;
Z
(_9
CEc)
?-
OIlIIGINAL PAGE tS! _ POOR QLJAL[T'i
'
• , , K=0.2
x x , K=O.
r_
- _910+I
FREOUENC1 RADISEC _RC_LL ATTITUDE FEEDBALK;
I I
IAtAAI* Am|Ill il|i|j||x xwX x x
. 8 _ A A x _ X _ Xx x
I •
X
D
u.jT _ K=O.O . ,u_
C£D A.:]E;D
_"_ . , I_=0.2
. ,, K=0.q
FREOUENCr RFID/SEC (ROLL ATIIT JDE FEEDBF-.CK;_
Fig. 5-10 Frequency Response c:f _/AI, With Roll Attitude
Feedback to Lateral Cyclic Input, I00 KTS
.... I I
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85
larger than it seems to beby the roll attitude feedb_ck is
in the frequency responses_
The same as at hover, the roll damping/regressing flap
mode is destabilized and finally becomes unstable at feed-
back gain a little higher than K=I.0 Deg/Deg.
The pole-zero location_ of open-loop roll angle to lat-
eral cyclic transfer function at 60KTS and 100KTS are pre-
sented in Tables 5-3 and 5-4. The same as at hover, most
nonlateral low frequency poles have a very close zero,
therefore will be canceled in the overall transfer function.
Furthermore, even the dutc_ roll mode, which traditionly is
strongly coupled with the lateral dynamics, has a close zero
pair, this explains why the roll attitude feedback hardly
affects it's position. T_erefore it is suggested that for
this helicopter, the low frequency roll response to the lat-
eral cyclic input at forw6rd flight is only determined by
the roll damping/regressirg flap and spiral modes. The
reduced order approximate transfer functions are also pre-
sented in Tables 5-3 and 5-4. Also, the root loci of the
simplified transfer function with same feedback will have
same shapes with those shcwn in Figs. 5-7 and 5-8. This
shows that low frequency l_teral dynamics of the helicopter
at forward flight is well separated from the longitudinal
and directional dynamics but coupled with the flapping
dynamics.
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C_
(_9 ,-
W
U_
[DcDCD
Q_
Wo
crg5"- ,
7.00
ORIGINAL PAGE ,_S
OF POOR QUALF,'Y
Short Period/Regressing Flap
4 X
-5.00
Stable Unstable
I I I I I I I
-5.00 -q. DO -3 D -2.00 -].00 0.00 1.00 2.00
REAL PART OF EIGEN:_LUES
86
Fig. 5-11 The Root Loci of The Helicopter With Pitch Attitude
Feedback to Longitudinal Cyclic Input at 60 KTS
C_
[-9 ,.
UJ
U_
cD
cc--
C[
O_
tJo
Short Period/Regres__,ing Flap
Stable
// _
Unst able
+ x _---X _-X
Phugoi d
¢_ i ! I I I I t iI-7.00 -E. Do -5.0C} -L&. Do -3. (0 -2.00 -I. 00 0. DO I. O0 2. OD
RERL PQRT OF EIGEN/QLUE5
Fig. 5-12 The Root Loci of The Helicopter With Pitch Attitude
Feedback to Longitudinal Cyclic Input at i00 KTS
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87
The longitudinal root loci of the helicopter with pitch
attitude feedback in the same gain range is shown in Fig.
5-11 and Fig. 5-12. The modes which are significantly
affected by the feedback are those traditional longitudinal
modes, phugoid and short period.
real roots, one divergent; and
coupled with the regressin_ flap.
The phugoid mode is two
the short period mode is
The less damped spiral
mode which has coupled with longitudinal velocity is also
affected although it's numerical variation is too small to
be shown in the figures. The pitch attitude feedback stabi-
lizes the phugoid mode as well as the real root of short
period mode but destabilize_ the complex pair. The spiral
mode is also destabilized _t low gain range and is stabi-
lized after the gain K=0.4 [or 60KTS and K=O.II for 100KTS.
The final feedback limitatLon gain due to the destabilized
short period mode is increa3ed with forward flight velocity
because the stable effect of the horizontal tail on the
longitudinal dynamics is increased with flight velocity.
The pole-zero location_ of open-loop pitch angle to
longitudinal cyclic transf_r function at 60KTS and 100KTS
are presented in Tables 5-5 and 5-6. There is only one
pole-zero close pair, which is associate with regressing lag
mode, in the overall transfer function. Hence the simpli-
fied transfer function pre:_ented in the table is much more
involved. As at hover, it is implied that the pitch
response of the helicopter which is a key figure for the
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88
1 02 03 04 -05 -06 -07 -08 -09 -0.I0 -0.II -012 -013 -014 -015 -016 -017 -018 -019 -020 -021 -022 -023 -024 O.
25 O.
26 O.
27 -0.
POLES
O000E+O0 0.0000_+00
O000E+O0 0.0000_+00
O000E+O0 O.O000i_+O0
9359E+01 0.5207]_+02
9359E+01 -0.5207]_+02
2012E+01 0.3954]_+02
2012E+01 -0.39541!_+02
8313E+01 0.2503]i_+02
8313E+01 -0 2503}C+02
2428E+02 0 7011]i_+01
2428E+02 -0 701115+01
I177E+01 0 1774_+02
I177E+01 -0 1774!_+02
1949E+02 0 0000}]÷00
2092E+01 0 76711_+01
2092E+01 -0 76711_+01
4444E+01 0 3883}_+01
4444E+0i -0 38831]+01
3830E+01 0 0000}il+00
2137E+00 0 12541!_+01
2137E+00 -0 1254}<+01
9324E+00 0.I0931_i+01
9324E+00 -0.I093}_Z+01
1490E-01 0.5367_i-01
1490E-01 -0.5367}i-01
2209E+00 0.0000_i+00
3006E+00 0.0000_+00
ZEROS
-0.2293E+03
0.I055E+03
-0.1500E+OI
-0.1500E+OI
-0.5593E+02
-0.5681E+01
-0.5681E+01
0.5003E+00
0 5003E+00
-0 1596E+02
-0 1596E+02
-0 2118E+01
-0 2118E+01
-0 6366E+01
-0 6366E+01
-0 1676E+01
-0 1676E+01
-0 3050E+00
-0 3050E+00
-0 6778E+00
0 1721E-I0
-0.2397E-02
-0.2378E-01
0.5138E-II
0.1640E-13
0.0000E+00
0.0000E+00
0.5544E+02
-0.5544E+02
0.0000E+O0
0.2826E+02
-0.2826E+02
0 1814E+02
-0 1814E+02
0 2315E+01
-0 2315E+01
0 7736E+01
-0 7736E+01
0 I082E+01
-0 I082E+01
0 4423E+01
-0 4423E+01
0 III8E+OI
-0 III8E+OI
0 O000E+O0
0 0000E+O0
O.0000E+O0
O.O000E+O0
O.O000E+00
0.0000E+00
APPROXIMATE TRANSFER FUNCTION :
Pitch(s) 0.I16(S-0.01721)(S+0.002397)(S+0.6778)
Bls(s) (S-0.0149+jO.C5367)(S-0.OI49-jO.05367)(S-0.2209)
(S+0.305+jI.II8)(S+0.305-jI.IIS)(S+I.676-j4.423)
(S+0.9324-jl.093)(S+0.9324+jl.093)(S+0.2137+jl.254)
(S+I.676+j4.423)(S+6.366+jl.082)(S+6.366-jI.082)
(S+0.2137-jl.254)(S+4.444-j3.883)(S+4.444+j3.883)(S+3.83)
TABLE 5-5 Poles, Zeros and Approximate Transfer Function
of Longitudinal Helicopter Dynamics at 60KTS
OF PO_ Q_AL_f
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OF POOR QUALITY89
1 0
2 0
3 0
4 -0
5 -0
6 -0
7 -0
8 -0
9 -0
I0 -0
II -0
12 -0
13 -0
14 -0
15 -0
16 -0
17 -0
18 -0
19 -0
20 -0
21 -0
22 -0
23 -0
24 0
25 O
26 0
27 -0
POLES
O000E+O0 0
O000E+O0 0
O000E+O0 0
9025E+01 0
9025E+01 -0
2083E+01 0
2083E+01 -0
3312E+02 0
3312E+02 -0
7759E+01 0
7759E+01 -0
2418E+02 0
I167E+01 0
I167E+01 -0
2269E+01 0
2269E+01 -0
5138E+01 0
5138E+01 -0
4890E+01 0
3618E+00 0
3618E+00 -0
I146E+01 0
I146E+01 -0
I137E-01 0
I137E-0! -0
1528E+00 0
9152E-01 0
0000E _00
O000E _00
O000E _00 -0
5194E _02 -0
5194E _02 -0
3897E _02 -0
3897E _02 0
1683E _02 -0
1683E _02 -0
2517E _02 -O
2517E _02 -0
O000E _00 -0
1738E _02 -0
1738E _02 -0
7914E _01 -0
7914E _01 -0
4588E _O1 -0
4588E _01 -0
0000E _00 -0
1440E _01 -0
1440E _01 -0
1503E _01 -0
1503E_01 -0
4161E-01 -0
4161E "01 -0
0000E _00 0
0000E_00 0
ZEROS
2013E+03
8070E+02
4372E÷01
4372E+01
3553E+02
2014E+O1
2014E+01
1983E+02
1983E+02
5780E+00
5780E+00
2321E+01
2321E+01
9728E+01
8057E+01
1563E+01
1563E+01
3449E+00
3449E+00
9088E+00
1261E-01
I152E-09
I158E-02
2535E-II
4516E-12
O.O000E+O0
O.O000E+O0
0.5287E+02
-0.5287E÷02
O.0000E+00
0.3259E+02
-0.3259E+02
0.6674E+01
-0.6674E+01
0 1764E+02
-0 1764E+02
0 8041E+01
-0 8041E+01
0 O000E+O0
0 O000E+O0
0 4835E+01
-0 4835E+01
0 1338E+01
-0 1338E+01
0 O000E÷O0
0 0000E+O0
0 O000E+O0
0 O000E+O0
0 0000E+OO
O O000E+O0
APPROXIMATE TRANSFER FUNCFION :
Pitch(s) 0.3924(S+3.001261)(S+0.01261)
Bls(s) (S-O.OII37+jO.O4161)(S-O.OII37-jO.04161)
(S+0.9088)(S+0.34%9-jl.338)(S+O.3449+jl.338)
(S+0.09152)(S+0.3618+jl.44)(S+O.3618-jl.44)(S+l.146+jl.503)
(S+I.563-j4.835)(S+I.563+j4.835)(S+8.057)(S+9.728)
(S+I.146+jl.503)(S+4.89)(S+5.138-j4.588)(S+5.138+j4.588)
TABLE 5-6 Poles, Zeros and Approximate Transfer Function
of Longitudinal Helicopter Dynamics at IOOKTS
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90
longitudinal dynamics, is strongly coupled with the lateral
and directional dynamics.
5.2.3 Pitch Rate feedbac]:
The longitudinal root ]oci of the helicopter with pitch
rate feedback in a gain ral_ge from zero to 1.5 Deg/(Deg/Sec)
are shown in Figs. 5-13, 5-14 and 5-15 for the hover, 60KTS
and 100KTS forward flight _espectively.
At hover the pitch rate feedback increases the damping
ratio of the long perioc_ mode very effectively by both
increasing the damping and reducing the frequency, and sta-
bilized the unstable mode _t gain K=0.5 Deg/(Deg/Sec). In
contrast to the attitude f_edback, the pitch rate feedback
moves the two coupled body pitch/regressing flap real roots
closing to each other ant becoming a complex pair at the
gain K=0.18 and finally ccupling with the lateral coupled
body roll/regressing flap rode, making it more stable. This
means that at low gain rarge, the feedback stabilized the
dominant less stable one cf the two body pitch/regressing
flap modes. Furthermore, the feedback slightly stabilizes
the yaw damping mode as well, although the effect is too
small to be shown in the root loci. All of these make the
pitch rate feedback more keneficial than the corresponding
pitch attitude feedback.
For forward flight, t_e pitch rate feedback offers the
same beneficial improvements. The feedback not only increas-
es the damping of the unstable long period mode but also
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(-,,j91
t
0
I-
_L
t-O
X
_Lff
t I r !
O0 "L C_ "?" OP_ "_. _ "_- •O0 -r_.-
,,,m
:.m
I
12_'"_-
:,m
,m
7
tm."m
b.J
_2r1"-,
.=>"qz_tu.J
_(_.9
t'-l-C,_
."-, .j
_u.Jr'r"
:m
,..4I
I>,z::Oo_
.p•r..I 4._
•,,-,I 1_
l--4
4...t ._
0 OO
•_ (,.)
® ,-.-_
.P
0 l_
,._ 0O_0
0
O_0 00_ m
,n
!
m
g
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ORIGfNA" ,=.,_-=
Of PO0_ QUALiTy92
.J
u_l
127--
Q-
W_
b_
B
I
-7.0_ -C.U_
/
Short PeriodX
_x
Stable
Phugoid
Unstable
-S. Lr_ -q LIO -_. +d:2 -c] . Om - I , L'Z2 O. 03 . (20 go
REqL FRF_[ OF EIGENVAL,JES
Fig. 5-14 The Root Loci of The Helicopter With Pitch Rate
Feedback to Longitulinal Cyclic Input at 60 KTS
_5
J
Z_W_
uJ
u_
_D(:3
Q_
W_
O_
a:_d_-,
Short Period
Phugoi d
Stable Unstable
I-7.OD -5.UU -5.UO -4.00 - UO -2.00 -I,00 0._0 l.O0 2.UO
BEqL PABT OF EIGENVALLES
Fig. 5-15 The Root Loci of The Helicopter With Pitch Rate
Feedback to Longitud:_nal Cyclic Input at I00 KTS
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93
increases the damping of _he oscillatory short period mode,
both of them are dominant modes in the long period response
and the short period re_ponse respectively, although it
decreases the damping of t_he stable long period mode and the
real root of short period mode.
The feedback limitation for pitch rate feedback comes
from the destabilized laci[ motion. The advancing lag mode
becomes unstable at feedb_!_ck gain about K=I.5 Deg/(Deg/Sec)
for hover. The effect on the damping of the advanced lag
with the pitch rate feec_ack gain for hover and forward
flight are shown in Fig. 5-16. As can be seen, the limita-
tion is relaxed significa1_tly at forward flight.
5.2.4 Roll Rate feedbach
So far all of the feedback gain limitations encounted by
the rotor/fuselage coupli_Lg are quite high compared to those
conventionally used in tle rotorcraft. However, the gain
limitations in roll rate Jeedback are far lower. At hover
the advancing lag mode becomes unstable.with a feedback gain
0.23 Deg/(Deg/Sec), in iorward flight the coning lag mode
becomes unstable at about the same feedback gain at I00 KTS.
Figs. 5-17 and 5-18 present the effect of the roll rate
feedback on the dampings of the advancing lag mode and the
coning lag mode. As can be seen, although the destabilized
mode changes from advancing lag at hover to the coning lag
at high speed, the limitation in feedback gain which will
destabilize the rotor/fu_elage system does not change sig-
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OF POOP, QuALm, t' 94
gL..) _ "
UJU_
,'-7 _,C['L
(_3Z
O:
"77
g
?
UN5 fABL.
/
HOVER
V=6OKT_
-U. UU O. BD D. 9: ] . 20 ! . SO I BO 2 10
PITCH RATE FEEDBACK GRIN [3EO/IBES/SEC)
Fig. 5-16 Effect of Pitch Rate Feedback on The Damping ofAdvancing Lag Mode
gu_L_
(-9
Z
17
rnc_'
UNSi'QBLE
/STQBLE
"7
HOVER
v=BoKr s
I I I I I I I I
-0.00 O, IU 0.20 0.30 O.q n D.50 0.50 0.'/0
ROLL RqTfi FEfiDBI:qLK GAIN OF 5/{DEC,/SEC)
Fig. 5-17 Effect of Roll Rate Feedback on The Damping ofAdvancing Lag Mode
O i. @O
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nificantly. Therefore, the roll rate feedback has
used with caution in the _hole flight speed range.
The precise value of the limiting gain is of course sen-
sitive to the estimation cf the mechanical lag damper char-
acteristics. The effect of the estimated mechanical damping
on the advancing lag damping is shown in Fig. 5-19 for roll
rate feedback near the stalDility boundary of the helicopter
at hover. The increase of the mechanical damping will
result in a increase in the allowable rate gain before
instability is encounted. Therefore, for the nonlinear dam-
per whose estimated dampi_g increases with the oscillatory
velocity, the slightly un!3table mode only means a moderate
oscillation limit cycle.
Unfortunately the roll rate feedback is very beneficial
for the helicopter lateral, dynamics. At hover the roll/
lateral velocity mode and body roll/regressing flap mode are
stabilized by the feedback this can not be done simultane-
ously by the attitude feedback. In forward flight, the body
roll/regressing flap mode, which is unstable for the high
gain roll attitude feedbac]:_, and dutch roll mode, which can
not be stabilized by the 2oll attitude feedback, are both
stabilized by the feedback. The spiral mode, which can be
effectively stabilized by the roll attitude feedback, is
affected very little by the roll rate
quency response of the hel_copter roll
lateral cyclic input with roll rate
95
to be
feedback. The fre-
rate at hover to the
feedback is shown in
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96
guT_O _o_ct_a__:i_l TTo_I _ua=_TC I tl_,T_4_poH S_q SuTOu_ApV _o
6uTctm_(3 _q,T, UO 5uTdweCi 6_"I IeOTueqoel_l _o q.oa_ 6I-S "6.r.;l'
(hOIX) 1310/OUUI/fS87 1..-II ON[_tl4UO OU7 'IU3INUN]]N130"2 011" I 0g" I 0h" I 02" I 01_" [ 011 "0 09"0
I i i I i i i I dh io o2 0_
"0=)I , "37_idI_
\\ I_apoH Beq SuTuoo
08"0 0Z "0 0g'0
i I I
(3]_10]0) I0]] NIUO _3U8033_ 31UU 770U05"0 oI_o [_ "o o2"o Ol o ooo-
I I i I I I
_7_UISNO
,u,rlvnOaood :io
01 "0-
o_
o;D
-O
zE3
7
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(.2
(.f)_
-9 . ORI_NAL PAGE !S
OF POOR QUALITY
o
lm
_ u8 _
I " ,_ _=0.0
• , , t I
i , , _ g iiit'O_ 2 s_J 2 3 l
FREOUENCT RADISEC IROLL RATE FEEDBACK)
. , I
' _ , i,,_L'o.2q
97
6
7
C3c:,
W_
r-_5
Ua"7,"
:!_o-
K=O.O
•., I_=0.I
,, , K=O. 2
FREOUENCT RADISEC (ROLL RATE FEEDBACK)
• • t
3 a
o
Fig. 5-20 Frequency Response of @/Ai, With Roll Rate
Feedback to Lateral Cyclic Input, Hover
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98
Fig. 5-20. As can be seen, the roll rate feedback increases
the bandwidth of the rate command system by both moving the
low frequency response peak, which associated with lateral
phugoid mode, to lower frequency and increasing magnitude of
the rate response at high frequencies. This will signifi-
cantly improve the ablit:y of the helicopter for the
manoeuvre requirement. The corresponding phase characteris-
tics also have the same extension. The frequency response
of the helicopter roll rate for forward flight to the later-
al cyclic input with roll rate feedback is shown in Figs.
5-21 and 5-22. Although the peak to be smoothed is small
itself, the roll rate feedback makes the improvement in the
high frequency range, which is the most important for the
manoeuvre capability.
5.2.5 Summary
For the simple feedback control,
severely limit the useable values of
especially for the roll rate feedback.
tude gain limitations arise primarily
the blade dynamics can
the feedback gains,
The fuselage atti-
from the stability
limits associated with the coupled body-flap modes, the
fuselage rate gain limitations arise primarily from the sta-
bility limits associated with the lag modes. It should be
noted that rate feedback always stabilizes those fuselage/
flap modes which produce the limitations in the attitude
feedback. The proper combination with rate feedback will
hence increase the attitude feedback limitation, which is
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S_ 09 "_nduI OTTOXD i_ze_q o_ _o_qpas_
|
:°I_!!_ _ _
(_3U9033_ 31UU 770W) ]_S/QUW ION3003W_
_.O=W _ .
L ! "0=)4 • • •
x x,._,.._ O-O=W --
oo
c_7-C_:D
t_
j_
oc7c_
c_<D
C_
)4 U8C]3_3 _IU_ 770UI ]_S/OUU IDN3CIO3Y_
A.L_..I',+:+++P>_OOd ..!0
I I
_2
Z
0-.-I
CO
B_C _.
ool
66
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I00
UA
UJCD
___ CD °
Z
I_9
=Ec3o
,-;-
7
ORIGINAL i,:iAGE iS
OF POOR QUALITY
FREOUENCr RRD/SEC (ROLL RATf FEEDBACK1
q
" _ a x x
ID • • x
.__ K=O.O • •
IE3__. ,,.. K=O.t
ILl _
:Z:'_ _ x ,, K=O. 2
(:Dtry.
I
I # I _ I I I 11 _ I _1 t I t{ill t 1
FREQUENCY RAD/SEC (ROLL RATE FEEOBACK) i/
I
Fig. 5-22 Frequency Response of _/Ai, With Roll Rate
Feedback to Lateral Cyclic Input, I00 KTS
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I01
already quite high. Therefore the limitation on the atti-
tude feedback should be not a real problem for the automatic
control design. However, the effect of attitude feedback on
the lag modes is very small, although it is stabilizing.
Therefore the limitations on the rate feedback can not be
relaxed by addition of attitude feedback. Consequently it
can be concluded that rec_ucing the destabilizing effect of
the feedback control on the lag dynamics will be required to
raise the feedback gain limitations.
For the low frequency longitudinal dynamics, the
improvement obtained by the pitch attitude and/or pitch rate
feedback is limited because of the coupling with the lateral
dynamics. It seems that good lateral dynamics, especially a
stable spiral mode, is essential to achieve satisfactory
longitudinal dynamics. _he gain limitations due to the
blade dynamics are not critical for their relative high val-
ues. For the lateral dynamics, in contrast, the simple roll
attitude or roll rate feedback offers a significant improve-
ment to the lateral dynamics. The proper combination of both
can give perfect lateral dynamical characteristics.
5.3 Multivariable Optimal Control
Active control considered in this section is based on the
deterministic linear optimal regulator problem[35,36]. The
purpose of the present study is to show the effect of the
lag dynamics on the overal_ system controller design. For
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102
the sake of clarity, it s assumed that all of the states,
including the dynamic inf ow, are available for measurement.
Optimal control theor _, is applied to the linear, con-
stant coefficient differ,_ntial written in first
order form
Eq.(8)
X' = A X + I_ U
Y = C X
The objective is now to find controls U,
cyclic control inputs to the swashplate and
control input to the tail rotor, which will
quadratic cost function.
_O ,T-j = _ Q Y _ t_T;: U dt (II)
where the weightlng m;_tlices Q and R are assumed to be
symmetric and positive definite. The solution is the deter-
ministic optimal controli el- with linear feedback of all
state variables.
U = K X (12)
where
(9)
(I0)
that is the
the collective
minimize the
K = - _-IBTS _ (13)
and the matrix S is t]_e constant, symmetric, positive
definite solution of the algebraic Riccati equation.
SA • ATs - SBR-IBTS - CTQC _O (14)
The closed loop dynamic:s equation is then defined as
X' := (A+BK) X (15)
For the multivariable optimal control, the choice of a
performance index, rather than feedback gains, to obtain the
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103
desired response is the _:entral feature of the method. The
solution of the optimal regulator problem is well defined,
if the equations for a complex multi-input, multi-output
plant is in hand. The p:incipal difficulty lies not in the
solution but in the choi(e of a suitable performance index.
The solution is optimal in the sense that the chosen per-
formance index is minimi=ed, but different optimal solutions
can be obtained by alter:ng the Q and R matrices. The per-
formance index may be interpreted as a quantitative measure
of the system performance. The R matrix penalizes the con-
trol input required. T_e Q matrix penalizes the error in
maintaining a desired tr_jectory.
A system model which includes only flapping dynamics is
obtained from the systen model developed in Chapter 3 by
simply letting the pertLrbation variables associated with
lag degrees of freedom be zero. This kind model has been
used for controller design of helicopters by many previous
investigators. The feedback controller then was designed by
using the MacFarlane-Potter concept of eigenvector decompo-
sition instead of integrating matrix Riccati equations.
5.3.1 Standard Performance Index
The quadratic perfornance index used here is of the form
FJ : q(YTI Y) + rfUTI U) dt (16)_O
In the present study the output scaling matrix C is cho-
sen so that the output vector y only corresponds the three
fuselage rotation attituces, i.e. the system velocities, the
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104
translational displacements, and the blade dynamic variables
are not included in the coEt function. The weighting matrix-
es Q and R are assumed to be diagonal, a production of a
number and a unit matrix, qI and rI. Some investigaters
consider that q/r=l gives a good choice in terms of balanc-
ing control effort, system stablity, and system
response[37]. For the design of tighter controllers, which
tend to hold fuselage pitch, roll and yaw angles to smaller
deviations, weighting factor q/r on the fuselage rotation
angles is increased from ] to 5, and then to 25. These
tighter controllers are then evaluated on the complete sys-
tem model including the lag dynamics.
The main feedback gains obtained by applying the linear
optimal regulator theory on the model which does not include
the lag degrees of freedom, and the dampings of the advanc-
ing lag and coning lag modes obtained by applying the same
feedback on the complete mcdel including the lag degrees of
freedom are presented in Tables 5-7, 5-8, and 5-9 for the
cases of hover, 60KTS and 100KTS level flight respectively.
The resulting eigenvalues obtained from the complete sys-
tem model by applying the feedback law obtained from the
model without the lag degrees
ing the weighting factor q/r
the lag degrees of freedom.
lag mode and the coning lag
obtained for the simple roll rate feedback.
of freedom show that increas-
results in an instability in
The dampings of the advancing
mode vary with the same trend
The correspond-
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105
The Primary Feedback Gains:
q/r 1 5 25
Als
Roll Attitude
BlslllmO
Pitch Attitude
Tot
--lllll_lllmll
Yaw Attitude
Als
Roll Rate
Bls
lll_l,
Pitch Rate
Tot
ll--llllllllll
Yaw Rate
0 65
0 966
0 98c_
0 1225
0 44£ 7
0 69£ 5
I. 736 4. 143
2.087 4.529
2. 198 4. 869
0.285 0.5635
0.7718 1.3
1.076 1.633
The Damping of Lag Modes:
Advancing Lag -0.26_ 5 I. 143 2. 784
Coning Lag -1.35_ -1.039 -0.2897
Table 5-7 The Primary F,_edback Gains and The Damping of
Lag Modes For Standa:_-d Optimal Feedback at Hover
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106
The Primary Feedback Gains:
q/r 1 5 25
Als
Roll Attitude
Bls
Pitch Attitude
Tot
Yaw Attitude
Als
Roll Rate
Bls
Pitch Rate
Tot
Yaw Rate
0.8_) 1.792
0.8_,7 1.972
0.7_75 1.905
0. I[i 0. 290
0.3(.!96 0. 744
0.4(137 0.6848
4 195
4 439
4 434
0 5685
1 316
1 079
The Damping of Lag Modes:
Advancing Lag -0.9_ 48 -0. 0147 I. 197
Coning Lag -0.9 (.01 -0.5625 0.1967
Table 5-8 The Primary I eedback Gains and The Damping of
Lag Modes For Stand_;rd Optimal Feedback at 60KTS
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107
The Primary Feedback Gains:
q/r 1 5 25
Als
Roll Attitude
Bls
Pitch Attitude
Tot--_--------------------_
Yaw Attitude
Als
Roll Rate
Bls
Pitch Rate
Tot
Yaw Rate
O. 6_ 16 I. 775 4. 189
0.7678 1.82 4.194
0.6661 1.799 4.333
O. 1248 0.2797 0.5506
0. 3418 0. 6623 I. 217
0.3678 0.6597 1.067
The Damping of Lag Modes:
Advancing Lag -1.35.1_ -0.6556 0.3157
Coning Lag -0.86w_7 -0.3597 0.4177
Table 5-9 The Primary F,_edback Gains and The Damping of
Lag Modes For Standa_-d Optimal Feedback at IOOKTS
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108
ing feedback boundary is also the same as the boundary for
the simple roll rate feedbazk. In fact, when the weighting
ratio q/r increases, the primary feedback gains increase
together by about the same factor. The attitude feedback
gains increase beyond the limiting value, which produce
instability for the simple attitude feedback, at q/r=5 with-
out resulting the flappin_ instability because the rate
feedback gains increase as well, which stabilizes the
coupled fuselage/regressing flap mode as shown in the last
section. However, when
limiting value, there is
from other feedback loops.
the roll rate gain is beyond the
no significant stabilizing effect
Hence the instablity that occurs
in these cases has the sane trend as roll rate feedback
alone studied in the last section.
It is worthwhile to mention that at hover the limiting
q/r ratio for the instability due to unmodeled lag dynamics
is less than 5. This number is much smaller than a similar
limiting boundary due to unmodeled flapping dynamics given
in Ref.6. This suggests that as far as stability is con-
cerned, the inclusion of the lagging dynamics in the system
modelling for the controller design has more practical sig-
nificance than the
although the latter
control and response.
inclusion of the flapping dynamics,
may be more important in terms such as
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109
5.3.2 Frequency-Shaped Performance Index
The poles associated with the suppressed lag degrees of
freedom have low open-loop damping and are relative high
frequency. They lie vet} close to the imaginary axis in the
s-plane. Therefore any nisplaced control energy (spillover)
will push them quickly irto instability. Readjusting Q and R
to prevent this (the only means available for the standard
performance index design) can cause a drastic loss of
closed-loop damping in t_e design mode poles, in some cases
to the point where almcst no closed-loop improvement in
damping is possible. The problem arises from the fact that
penalty matrices Q and R penalize the states and controls by
the same amount at all frequencies.
One way to avoid constant penalties is the use of
frequency-shaped cost fu_ctionals, an extention of standard
linear optimal regulator design[38]. In this method, the
performance index to be minimized is assumed to be a func-
tion of frequency as follows:
J = rYTCj_)Qcj_)YCj_) + UTCj_)RCj_)UCj_) de (17)_-_
Note that here the weighting matrices Q and R are func-
tions of frequency, rather than constant matrices. The
detailed discussion of this method is given in Refs.39 and
40. The physical concept of the frequency-shaped cost func-
tionals is that the performance index is defined such that
the low frequency error in maintaining a desired trajectory
and the high frequency inputs are more heavily penalized
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such that
frequency
ii0
the feedback energy is mainly placed on the low
fuselage dynamics. The cost function can be
defined in three ways: (I) frequency-shaped response penal-
ty with constant control p_nalty, (2) frequency-shaped con-
trol penalty with constant response penalty, and (3)
frequency-shaped both resi3onse and control penalties. It
should be noticed that the frequency-shaped response penalty
leads to increasing degrees of freedom of the system to be
augmented, and the frequenzy-shaped control penalty leads to
feed forward of the derivatives of the control inputs. The
numerical study shows that introducing new degrees of free-
dom of the system, whose 9rder is quite high already, not
only results in difficultLes for system analysis but also
requires extremely high feedback gains which are physically
unrealistic. Therefore, only a frequency-shaped control
penalty is used in this stldy.
The performance index _hen is defined as:2
_, _ • b 2
j = [ q yTy + r f ) U_U d_ (18)b 2J
The frequency-shaped c)ntrol penalty used here is equiv-
alent to inclusion of a slnaping filter in the forward path
of a standard optimal co:_trol problem. The corresponding
shaping filter has a transfer function of the form:
U = b/(jw+b) Uc (19)
The low pass characteristics of the filter will reduce
the high frequency component in the feedback so as to penal-
ize the high frequency con:rol. This is physically conven-
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III
ient for the implementatioll because actuator dynamics can be
thought of as having the form of shaping filters. Since the
frequencies of the coning and advancing lag modes are about
38 and 16 rad/sec, the cor_er frequency of the low pass fil-
ter used in the study is chosen to be I0 rad/sec.
The ratio q/r here can not be directly compared with the
ratio in the eq.(16) because the frequency-shaped control
penalty has changed the spectrum distribution of r in fre-
quency domain. Thus the 1_atio q/r is decided independently
here and the chosen ratios for the frequency-shaped cost
functions are I0, I00, I000, and I0000.
The main feedback gain_!_ obtained by applying the linear
optimal regulator theory with frequency-shaped cost control
penalty on the model which not includes the lag degrees of
freedom and the dampings oi! the advancing lag and coning lag
modes obtained by applying the same feedback on the complete
model which includes the lag degrees of freedom are present-
ed in Tables 5-10, 5-11, and 5-12 for the cases of hover,
60KTS and 100KTS level flicTht respectively.
The resulting eigenvalues obtained from the complete sys-
tem model show the same trend for an instablity in lag modes
as standard optimal control. However corresponding feedback
gains for the instablity are much higher than the standard
cost function. The freq_ency shaped-optimal feedback has
introduced new poles into the overall system, the Butter-
worth configuration of the system has been changed so that
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112
The Primary Feedback Gains:
q/r I0 i00 i000 I0000
q
!
Als
Roll Attitude
Bls
Pitch Attitude
Tot
Yaw Attitude
Als
Roll Rate
Bls
Pitch Rate
Tot
Yaw Rate
0.94 5.3 22.2 78.0
3.07 9.44 29.0 88.77
3.13 9.84 30.9 96.37
0.2 1.16 4.22 12.1
1.71 4.28 10.37 24.37
2.53 5.64 12.4 26.86
The Damping of Lag Modes:
Advancing Lag -2.00 -2. Ol -I. 63 0.6
Coning Lag -1.40 -1.78 -2.36 -I.II
Table 5-10 The Primary Feedback Gains and The Damping of
Lag Modes For Frequenc) Shaped Optimal Feedback at Hover
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113
The Primary Feedback Gains:
q/r I0 I00 I000 I0000
I
w
Als
Roll Attitude
Bls
Pitch Attitude
Tot
Yaw Attitude
Als
Roll Rate
Bls
Pitch Rate
Tot
Yaw Rate
1.00 5.6
2.58 8.56
1.96 7.76
0.27 1.27
1.44 3.94
1.55 3.85
22.8 78.3
27.7 87.59
26.8 86.98
4.3 12.16
10.09 24.75
8.88 19.6
The Damping of Lag Modes:
Advancing Lag -2.04 -2.09 -2.03 -1.07
Coning Lag -1.26 -1.48 -1.56 -0.44
Table 5-11 The Primary Feedback Gains and The Damping of
Lag Modes For Frequency Shaped Optimal Feedback at 60KTS
ORIGINAL PAGE rS
OE POOR QUALITY
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ORIGINAl. PAGE IS
OF POOF_ QUALITY114
The Primary Feedback Gains:
q/r I0 I00 I000 I0000
!
Als
Roll Attitude
Bls
Pitch Attitude
Tot
Yaw Attitude
Als
Roll Rate
Bls
Pitch Rate
Tot
Yaw Rate
0.83
2.23
i. 49
0.25
1.20
1.38
5 24
7 68
6 92
1 21
3 45
3 66
22.27 77.88
25.63 82.85
25.57 85.43
4.14 11.83
9.22 23.54
8.70 19.5
The Damping of Lag Modes:
Advancing Lag -2.10 -2.13 -2.08 -1.07
Coning Lag -1.23 -1.37 -i. I0 0.046
Table 5-12 The Primary Feedback Gains and The Damping of
Lag Modes For Frequency Shaped Optimal Feedback at IOOKTS
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Jt
q/r
Hover
60 KTS
I00 KTS
q/r
Hover
60 KTS
I00 KTS
Table 5-13
115
] 5
Damping Frequency Damping Frequency
-0.99 1
-1.29 1
-2.93 4
-2.22 7
-5.80 0
-6.68 0
243
971
227
5O4
000
000
-1.46 1.862
-1.76 2.737
-3.63 5.495
-2.33 7.414
-7.46 0.000
-9.30 0.000
-I. 03
-1.50
-3.15
-2.34
-4.94
-9.33
1.807
2.025
4.255
7.335
0.000
0.000
-1.40 2.454
-2.16 2. 897
-4.24 6. 123
-2.23 6.920
-5.78 0.000
-16.2 2.228
-I
-I
-4
-2
-6
-9
08 1.829
62 2.443
09 4.708
52 7.420
12 0.000
61 0.000
-1.56 2.568
-2.26 3.228
-5.15 6. 729
-2.30 7.039
-7.19 0.000
-15.6 0.000
.'_:5
Damping Frequency
-2.10 2.808
-2.45 3.716
-5.29 6.687
-2.44 7.261
-8.82 0.000
-11.5 0.000
-I .96 3
-3.12 4
-2.11 6
-6.19 7
-6.72 0
-17.5 5
370
264
768
457
000
903
-2.20 3.700
-3.31 4.531
-2.33 6.729
-6.82 8.212
-8.69 0.000
-23.2 0.000
The Poles Asscciated With The Short Period Flight
Dynamic Characteristics of The Helicopter Under
Standard OptiH1al Feedback
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i
J
-- I
v I
q/r
Hover
60 KTS
I00 KTS
I0 i00
Damping Frequency
-0.79
-0.89
-1.85
-4.45
-4.43
-2.07
1 545
1 891
1 745
2 338
5 137
7 672
116
Damping Frequency
-I
-I
-2
-5
-6
-2
-1.80 1
-0.81 1
-1.29 2
-3.78 2
-5.27 5
-2.24 7
14 2.208
34 2.638
49 3.224
29 3. 702
92 6.951
15 7.768
130 -2.12
844 -I.i0
496 -2.14
I01 -4.72
312 -7.32
747 -2.48
-1.98 0
-0.89 1
-1.32 2
-4.25 1
-5.56 6
-2.47 7
687 -1.20
954 -2.95
654 -2.02
944 -5.13
063 -2.76
972 -7.61
q/r I000 I0000
Hover
60 KTS
2.791
2.387
3.683
3.032
7.800
7.773
Table 5-14
I00 KTS
2.593
3.075
3.516
3.255
7.969
8.690
Damping Frequency Damping Frequency
-1.61 3.101 -2.26 4
-1.96 3.625 -2.47 5
-3.40 4.539 -4.51 5
-6.68 5.618 -9.48 8
-2.41 7.962 -3.35 8
-12.4 11.19 -15.4 0
297
087
268
727
358
000
-1.57 3.192 -2.40 4.398
-2.62 4.166 -4.27 5.313
-2.79 5.226 -1.84 5.757
-6.29 4.302 -I0.0 6.375
-3.21 7.921 -5.00 9.365
-10.9 11.63 -18.2 11.49
-1.67 3.571 -2.99 4.888
-2.51 4.563 -1.57 5.491
-3.79 5.204 -5.14 5.745
-6.31 5.047 -5.42 9.550
-3.64 8.092 -8.72 8.234
-11.5 13.06 -17.9 17.57
The Poles
Dynamical
Frequency
Associated With The Short Period Flight
Characteristics of The Helicopter Under
Shaped Optimal Feedback
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117
damping ratio can not be used as a measurement of the system
augmentation. The dampings and frequencies of the short
period modes associated closely with the fuselage dynamics
are presented in Table 5-13 and Table 5-14 for both standard
and frequency-shaped cost functions. If the lowest damping
of these modes is used as a measurement for the system's
augmentation, the case q/r=1000 with the frequency-shaped
cost function will be more stabilized by the feedback than
the case q/r=5 with the standard cost function, and the case
q/r=10000 with the frequency-shaped cost function will be
more stabilized by the feedback than the case q/r=25 with
the standard cost function. The feedback limitations for
the frequency-shaped optimal control due to the unmodelled
lag degrees of freedom therefore are not only numerically
much larger but also offering much stronger system augmenta-
tion in stability and control characteristics. In addition,
the feedback gains required by the q/r=10000 case are far
higher than those that are physically practical. This sug-
gests that by applying the frequency-shaped cost function on
the helicopter automatic control system design, the unstable
effect due to the unmodelied lag degrees of freedom can be
removed.
To illustrate the advantages of the optimal feedback con-
trol, the frequency responses of helicopter roll attitude to
lateral cyclic input are shown in Figs. 5-23, 5-24, and 5-25
for the standard optimal feedback control and in Figs. 5-26,
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A117vn_) _ood _0
_=, ,,_ ....
(_3U80333 7bHildO ObUONUISI+Otl,e_ 9 1
,,:::_, _ _ i _l_ _23S/0_W 12N3i303W_
l=WIO " " "
x• _ 0 "O=U/O
x x
_i x x v"
.X_xxxxxx_x_xxx:_x_xxxxxxxxxxxxxxxXX_..
• • v
c=
IC3
oz
o_-tl
==00
811
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==
==,4
e36"
laJ_
o _,
Z
C_
CIZ(:
_co.
o
6
ORiL_,%P.L PA_E IS
OF POOR QUALITY
_t
• ,, O/R=!.O
,xx 011_=5.0
FREQUENCY RAD/SEC ISTAND-qRD DPTJMAL FEEDBPlCK)
3.19
o
,4-
==i
c_
w=
'2
T (_2
• x• • x
-- 011_:0.0• ,, O/FI:I.O
_,_ OIFI:5.0
• xx
FREOUENF. T RADISEC ISIRNDPlRD OPIIMAL FEEDBRCK)
Fig. 5-24 Frequency Response of _/As, With Standard OptimalFeedback at 60 KTS
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120
OF_ pOOR QUf_L|TM
oD6"
C)
uJcD c
Z7Z
._.
o
•., O/R-- I
3 _LC _
FF_ECJUENC.T F_RDISEC I5TRND_,_D OPT]MRL FEEDBqCK)
, _ _'_,__' '_l 0 ''2
[D
CD
U')
_CD_CD
"-6_ID
--ST"t_D
n-"
"'c_r-_c,
1',4,
(leo"r-_
,?
[:D
6e-,
_ 01_=0.0
•. • 0/R: !
x x x 0/F_=5/
x
FFIEOUENC'r I_I_D/SEC (5"[SND_IF_D OPTI_I_L FEEDB_CKI
Fig. 5-25 Frequency Response of @/A With Standard OptimalFeedback at I00 KTS
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v
#6-
{D
=Jl
r_
CD
UjC
Z
tD
CZ_
OF POOR QUALITY
........................ii....."iiiiiiiiIt
__ Q/R:IL_
... O/R=IO0
b
,4
i 2
,,, C/_=1000
7 0 910.*[
FREQUENCY RqD/SEC {FRE-SHqPE OPT]MPlL FEEDBRCK)
121
2 3
DC3-
t ¸
uJ_
hl t"
CE_"rc_
,?,.
oo
=T,
Jl _x x_
• • x
__ " Q/R--]O " "z
x Jt
.,. O/R=IO0 _ A _
\ J x x
... O/F_ lO00 _,
...... , : : , _tlc_ ' , ,, ...... ,FREOLJEN[Y RiqD/SFC (FRE-SH_PE OPTIMRL FEEDBF_CK)
Fig. 5-26 Frequency Response of _/Ass With Frequency ShapedOptimal Feedback at Hover
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6
6
_D
Wo
DE
Z
fro
_r-c
OF.pOOR QUALI_f
__ O/R=IO
iCr2
_. 018=100
x #x OIR--1000
' ' _ _."_'_Ic_ ' ' .'_k'_&l _ : ..... 'FREQUET4ET RI:qDI5EC IFRE-SHRPE OPTII'4RL FEEDBCICK)
122
i
6M
00
L_°o
m'm_¢,'_.
I.LI I
0"3
C_Z_
,?,-
6
... Oll_= 1 O0 _.i= _
,, _,, O/l_=lO00 "x_
FREOUENCT I:IIqFJISEC (FAE-SHrlPE OPTIMAL FEEDBACK)
Fig. 5-27 Frequency Response of #b/a_s
Optimal Feedback at 60 KTSWith Frequency Shaped
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w
(:3(D
4.
ID
6'
6'I',!
6
:T,
ID
I.,UcD
:Dc_
,.__,Z
(-9
er_
o!
OR_INA{_ P!_,_ZiS
.OF.POOR QUALITY
123
OIR=lO
... Q/R=tDO
,,,,x Q/B--.IDO0
FREOUENCY RRO/SEE (FRE-SHQPE OPIIMRL FEEDBF_CK)
, . , ,|
IDID
6{D
irr_
?-
IDID
6c=
'T
68&X x .
QIB=|
&imm
..l O/B=lOO0 " : "_.
FBEQUENET RRDISEC (FRE-SHIqP[ OPI'IMRL FEEDBRCK)
Fig. 5-28 Frequency Response of 4_'As. With Frequency Shaped
Optimal Feedback at I00 KTS
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5-27, and 5-28 for
control.
124
the frequency-shaped optimal feedback
By comparing with the corresponding frequency responses
obtained by the simple attitude feedback, it can be seen
that the main improvements obtained by the standard optimal
feedback control are limited in phase characteristics. The
Bode amplitude characteristics by the standard optimal feed-
back is only improved a little by smoothing the peak at high
frequency end in the bandwidth obtained by the strong atti-
tude feedback. The phase characteristics, in contrast, have
a significant improvement by reducing the phase shift at
high frequency range 5-15 rad/sec. At these high frequen-
cies, the simple attitude feedback cannot offer any reduc-
tion. As for the very low frequencies, the standard optimal
feedback gives the same trend in both amplitude and phase
characteristics as the simple attitude feedback.
The frequency-shaped optimal feedback control, however,
has improved the frequency responses in both amplitude and
phase characteristics. The frequency responses obtained by
the frequency-shaped optimal feedback have perfect low fre-
quency characteristics. The amplitude characteristics at
low frequencies up to 2 rad/sec is a straight line for any
q/r ratio. For phase characteristics, the frequency-shaped
optimal control has totally removed the phase lead resulting
from the nonminimum phase characteristics; this is especial-
ly obvious in the hover case. In addition, at a quite wide
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range of frequency,
significant reduced
125
the phase shift is very small and is
along with the increasing of the q/r
ratio. For the high frequency part, in spite of the fact
that the frequency-shaped optimal control is designed to
reduce the high frequency augmentation, the obtained
improvements seem still better for the high gain simple
attitude feedback. The amplitude characteristics obtained
by the frequency-shaped optimal feedback are not only better
than the simple attitude feedback by removing the high fre-
quency peak but also better than the standard optimal feed-
back in term of the maximum achievable bandwidth. Take hov-
er case as a example, the q/r=5 standard optimal feedback
case has almost the same bandwidth with the q/r=1000
frequency-shaped optimal feedback case. However for standard
optimal feedback, the advancing lag mode has became unstable
far below the q/r=5, in contrast, for the frequency-shaped
optimal feedback, the system will be stable until q/r=10000.
For the high frequency phase characteristics, the reductions
of phase shift obtained by the frequency-shaped optimal
feedback are smaller than those obtained by the standard
optimal feedback but still offer improvements which can not
be obtained by the simple attitude feedback because for the
simple attitude feedback, there is no phase shift reduction
at frequencies higher than 6 rad/sec. Therefore, it is
clearly suggested that the improvements obtained by the
frequency-shaped optimal feedback are much more practical.
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v
126
5.3.3 Simplified Optimal Control
The design of linear controllers in this section results
in a feedback structure which requires the measurement and
feedback of all state variables except translational posi-
tions. The obvious impracticality of this requirement has
led to several research efforts directed toward the synthe-
sis of simplified controllers which are more easily imple-
mented than would be the optimal control. As a result, a
series of simplified controllers are developed through suc-
cessive reduction in the number of feedback loops while
using the feedback gain factors obtained for the optimal
control. The sequence of the loop reduction is determined by
both the difficulties for measurement and the importance of
the feedback requirement of the loop;the latter is naturally
measured by the amplitude of the corresponding gain for the
optimal control. In addition, particular emphasis is placed
on eliminating the feedback of rotor degrees of freedom. The
gain constants for these reduced state feedback controller
are chosen as the values obtained for the corresponding
states in the optimal controller.
The first loop reduction is eliminating the feedback of
dynamic inflow because these state variables can not be
measured, and removing the feedback of translational fuse-
lage velocities because the corresponding gains for this
group state variables are far smaller than others, conse-
quently they are considered the least important for the
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127
feedback control. The number of remaining feedback loops
for this simplified controller, called controller A, is 12.
For the controller B, flapping velocities of the blade are
also eliminated from the feedback. This is due to both the
difficulties in measurement and the importance considera-
tion. The number of remained feedback loops for this sim-
plified controller is 9. The controller C is formed by
eliminating flapping attitudes from the system feedback
loops. The required gains for these attitude feedback loops
are in the same order as those for fuselage attitudes, but
the measurements are more difficult and more expensive than
the corresponding fuselage attitudes because the resolution
of the measurements from rotating to nonrotating axes is
required. The number of remained feedback loops for this
simplified controller C is 6. The final loop reduction is
eliminating the feedback of three fuselage angular veloci-
ties simply because for standard optimal control, removing
these feedback loops result in
the dampings of the lag modes,
resulting stablity limitations
The number of remained feedback
controller D is 3.
a favourable increasing of
which are the very modes
for the feedback control.
loops for this simplified
The poles associated with the short period flight dynami-
cal characteristics obtained by various reduced loop con-
trollers are shown in the Table 5-15 for the simple standard
optimal feedback control with q/r=l and in the Table 5-16
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128
Number offeedback loops
Hover
12
Damping Frequency
-I 01-i 30-2 99-2 22-I 35-0 28
1.2391.9794.5787. 50217.1339.42
Damping
-I.01-1.30-2.96-2.22-12.7-0.33
Frequency
1.2391.9634.5637.50117.2040.01
60 KTS -1.06 1.833-1.54 2.056-3.07 4.302-2.35 7.333-0.99 17.16-0.94 39.13
-I .06-1.56-3.06-2.35-0.94-0.84
I. 8282. 0494.2127.32817.2139.54
I00 KTS -1.09 1.847-1.64 2.553-4.01 4.683-2.51 7.423-0.87 16.88-1.36 38.73
-I .09-1.66-3.94-2.52-0.83-1.33
1.8452.5414.5787.41916.9238.99
Number offeedback loops
Hover
Damping Frequency
-1.02 1.239-1.89 2.871-3.01 7.061-2.23 7.460-1.60 16.79-0.07 39.11
Damping
-I. 02-0.03-0.66-2.02-1.53-2.03
Frequency
1.5892.5215.8257.64317.9839.51
60 KTS -1.20 1.834-1.75 2.674-1.96 7.005-3.52 7. 601-1.15 16.96-0.91 38.93
-I 79-0 32-5 88-2 12-I 37-2 09
2.0422.5835.4537.88117.7539.43
I00 KTS -1.14 1.917 -0.23 2.048-2.00 3.168 -0.44 2.826-1.97 7.371 -0.63 5.980-4.27 9.031 -2.28 8.249-1.04 16.52 -1.39 17.36-1.33 38.60 -2.13 38.89
Table 5-15 The Poles Associated With The Short Period FlightDynamic Characteristics of The Helicopter UnderSimplified Standard Optimal Feedback
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Number offeedback loops
Hover
60 KTS
I00 KTS
12
Damping Frequency Damping
129
Frequency
-1.61 3.127 -1.64 3.133-1.90 3.628 -2.00 3.532-3.44 5.186 -3.13 5.177-8.10 3.895 -7.33 4.355-2.43 7.960 -2.44 7.956-10.3 0.000 -ii.0 0.000
-1.43 3.194 -1.49 3.192-2.74 4.104 -2.92 3.971-2.33 5.336 -2.17 5.272-6.59 3.712 -9.98 5.907-3.23 7.932 -3.27 7.943-11.6 10.55 -16.2 8.544
-1.49 3.540 -1.55 3.537-2.59 4.872 -2.45 5.204-3.30 5.058 -3.29 4.458-8.32 3.304 -8.14 4.703-3.60 8.088 -3.67 8.151-10.9 13.26 -12.7 13.53
Number offeedback loops
Hover
60 KTS
I00 KTS
Table 5-16
3
Damping Frequency Damping Frequency
-1.62 3.398 0.326 2.814-0.74 5.392 0.770 3.561-3.13 0.000 -6.87 1.022-13.6 9.951 0.695 6.575-2.33 7.999 -1.92 7.661-11.5 0.000 -11.8 0.000
-1.22 4.297 0.453 3.103-1.19 5.453 0.545 3.705-1.55 7.849 0.619 6.324-3.04 0.000 -1.87 8.001-3.15 9.618 -11.5 2.551-19.5 7.610 -24.9 7.190
-1.28 4.307 0.405 3.154-0.97 5.978 0.538 3.891-1.92 8.438 0.623 6.665-3.18 0.000 -11.6 0.000-2.25 10.99 -1.95 8.359-24.2 11.13 -15.6 11.96
The Poles Associated With The Short Period FlightDynamical Characteristics of The Helicopter UnderSimplified Frequency Shaped Optimal Feedback
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for the
q/r=1000.
130
frequency-shaped optimal feedback control with
Both of them can offer stabilized system dynamics
at all of three chosen flight conditions if all of state
variables are available for measurement. Since the standard
optimal control with high q/r ratio usually results in lag
dynamical instability, the poles associated with the lag
degrees of freedom are also given in the Table 5-15.
The elimination of the feedback of the dynamic inflow,
the translational velocities and the flapping velocities
together has little effect on the system dynamics. Compared
with the baseline optimal controller, the variation of the
short period fuselage dampings and frequencies is quite
small, less than 5% for the standard optimal control and I0%
for the frequency-shaped optimal control. These results
suggest that at least half of the feedback loops theoreti-
cally required by the optimal control method, dynamic
inflow, fuselage translational velocities and blade flapping
velocities, are not necessary for practical implementation.
The 18 loop feedback control system studied here can be
replaced with 9 loop implementation without any significant
impact on the system dynamics. The state variables involved
in these 9 unnecessary loops are difficult to measure and to
reconstruct. Therefore the difficulties in implementing the
optimal control methodology is greatly reduced by simply
eliminating these feedback loops. In the remaining feedback
loops, the only blade state variables left are the flapping
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131
angles. Although without feedback these state variables, the
overall system is still stable, the dynamic characteristics
are almost the same as the original system without any aug-
mentation of the damping ratio. It seems, therefore, that
the flapping angle feedback has to be part of the optimal
control implementation. Fortunately these state variables
need not be measured because it has been found they can be
estimated sufficiently accurately from fuselage state meas-
urements[6].
5.3.4 Summary
Results in this section show that the multivariable opti-
mal control theory is a powerful tool to design high gain
augmentation control systems. The frequency-shaped optimal
control design can offer much better flight dynamic charac-
teristics than either the simple feedback control or the
standard optimal feedback control. The feedback gains com-
puted from the optimal control theories can be used to
develop reduced state feedback systems. The feedback loops
required can be significantly reduced to the half of the
original optimal control designs without any noticable
effect on the overall system dynamics.
Results in this section also show that the lagging dynam-
ics has a more significant impact on the automatic control-
ler design than the flapping dynamics. If a standard design
method is used, the lag degrees of freedom must be included
in the system modelling. Otherwise a high gain control sys-
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132
tem design can lead to unstable close-loop responses due to
the unmodelled lag dynamics. Using a frequency-shaped con-
trol penalty in the system performance index is an effective
way to obtain a stable margin for the feedback system design
without need to model the lagging dynamics.
This margin is essential for any real implementation of a
control system because the actual structure has an infinite
number of modes, and any finite description of the actual
system, though very high order, still has modeling errors.
These are errors which cannot be modeled generally due to
limited knowledge of the structural behavior at high fre-
quencies. Thus if a system controller is important to the
stability and performance, a very robust control system is
required. This is especially true for the helicopter sys-
tems because the difficulties for modeling the high harmonic
blade dynamics and aerodynamics.
shaped cost penalty seems to be
the helicopter controller design.
Therefore the frequency-
a very good methodology for
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Chapter VI
CONCLUSIONSAND RECOMMENDATIONS
This thesis has had four fundamental objectives:
(I) By applying the matrix displacement method and with
the help of a symbolic computer processor, to develop a lin-
ear description of helicopter system including blade dynam-
ics.
(2) To take the rotor/empennage interaction into account
without destruction of the linearity of the system model.
(3) To investigate the effects of blade dynamics on the
automatic control system design with the model developed.
(4) By using the modern optimal control technology, to
find a control methodology capable of removing the limita-
tions due to the unmodelled high frequency blade lag dynam-
ics on the flight control system design of the helicopter.
As indicated in the correlation results with flight test
data shown in Chapter 4, the first and second objectives of
the study have been largely achieved. The excellent corre-
lations for all kinds of small control inputs
for lateral and directional control inputs at
ward flight speeds are evidence of the success
earized model
the effects
at hover and
various for-
of the lin-
and the simple but effective description of
of the main rotor wake on the tail rotor and
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134
fixed tails. It is the first linear model of the helicopter
including blade dynamics for forward flight, that makes the
analysis of flight stability and control by the convenient
eigenvalue and eigenvector analysis and the feedback control
design by modern linear control theory possible. The vali-
dation of the model reveals many new ideas.
(I) A linearized model of the helicopter is quite satis-
factory for predicting the stability and control character-
istics. The proper linear model produces a good representa-
tion of helicopter control responses for forward flight as
well as hover.
(2) For forward flight the sidewash variation at tail
rotor and vertical tail and the nonuniform downwash at hori-
zontal tail are more important for flight dynamic analysis
than the most inertia, mechanical, and aerodynamic nonli-
nearities. Therefore, better understanding of the influence
of rotor wake on the tail surfaces and tail rotor is one of
the most important factors needed to improve the representa-
tion of helicopter motions. Consequently the proper simple
method to treat the influence will be a key breakthrough for
development of helicopter simulation.
(3) The simple flat wake model employed in this paper
although crude appears to be a good approximation when
sideslip angle remains relatively small.
(4) The influence of the dynamic inflow is most signifi-
cant in hover and somewhat less significant in translational
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135
flight. Its effect can be estimated by methods given in the
literature.
The disadvantages of the model mainly result from the
assumption of constant rotating speed of the main rotor,
which have not been shown to be the case for longitudinal
control inputs at forward flight. It seems that the inclu-
sion of the engine and drive train will be the logical next
step for better modeling. Changes of rotor speed are a
result of an imbalance of the main rotor torque required and
the engine torque available. Therefore the rotor speed
degree of freedom must involve the engine dynamics and fuel
control system. As for enlarging the range in which the
model is validated for lateral and directional control, the
first thing to do should be improving the modeling of the
influences of the main rotor wake on the tail rotor and
fixed tail surfaces.
assumption there is
inputs.
The third
achieved by
Chapter 5.
methodology
obtained by
It seems that for small perturbation
still room left for increasing control
and fourth objectives also have been largely
the classical and optimal control studies of
The results obtained by simple feedback control
have very good agreements with the results
previous works for hover flight condition and
have physically consistent results for translational flight
conditions. The results obtained by the optimal control
methodology has successfully introduced a new feedback con-
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136
trol concept originally developed for space structure stabi-
lization into the helicopter control system design. The
following conclusions may be drawn.
(I) The control feedback gain limitations due to unmo-
deled lagging dynamics are much closer to those currently
used in the helicopter industry than those due to unmodeled
flapping dynamics. Therefore much attention has to be given
to the lag degrees of freedom.
(2) Most of the feedback gain limitations are quite
high except the roll rate, which has the same order as those
currently used in the helicopter industry.
(3) The application of frequency-shaped optimal control
methodology gives us a practical robust control design meth-
od for high gain tighter controller design without need to
worry about the effects of spillover by the unmodeled high
frequency modes.
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REFERENCES
[I] C.W.EIIis, "Effects of Rotor Dynamics on Helicopter
Automatic Control System Requirements" Aeronautical Engi-
neering Review, July 1953
[2] R.E.Donham,S.V.Car_inale and I.B.Sachs, "Airborne and
Ground Resonance Characteristics of a Soft In-Plane Rigid-
Rotor System" J. Am. Helicopter Soc. 14 (1969)
[3] R.S.Hansen, "Towards a Better Understanding of Heli-
copter Stability Derivatives" J. Am. Helicopter Soc. 29
(1984)
[4] K.Miyajima, "Analytical Design of a High Performance
Stability and Control Augmentation System for a Hingeless
Rotor Helicopter" J. Am. Helicopter Soc. 24 (1979)
[5] H.C.Curtiss Jr., "Stability and Control Modelling"
Paper No. 41 in Twelfth European Rotorcraft Forum Sept. 1986
[6] W.E.Hall Jr. and A.E.Bryson Jr., "Inclusion of Rotor
Dynamics in Controller Design for Helicopters" J. Aircraft
Vol.10. No.4 April 1974
[7] S.J.Briczinski and D.E.Cooper "Flight Investigation
of Rotor/Vehicle State Feedback" NASA CR-132546
[8] K.H.Hohenemser and Sheng-Kuang Yin, "The Role of
Rotor Impedance in the Vibration Analysis of Rotorcraft"
Vertica Vol 3. No. 3/4 1979
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138
[9],R.A.Ormiston, "Aeromechanical Stability of Soft
Inplane Hingeless Rotor Helicopters" Third European Rotor-
craft and Powered Lift Aircraft Forum, Paper No 25 (1977).
[I0] J.J.Howlett "UH-60A black Hawk Engineering Simula-
tion Program: Volume I - Mathematical Model" NASA CR-166309
1981
[Ii] W.Warmbrodt and P.Friedmann, "Formulation of Coupled
Rotor /Fuselage Equations of Motion." Vertica 3, 254-271
(1979).
[12] W.Johnson, "A Comprehensive Analytical Model of
Rotorcraft Aerodynamics and Dynamics." NASA TM 81182
(1980).
[13] D.H.Hodges, "Aeromechanical Stability of Helicopter
With a Bearingless Rotor--Part I: equations of motion" NASA
TM-78459 (1978).
[14] J.Levin, "Formulation of Helicopter Air Resonance
Problem in Hover With Active Controls" M.Sc. Thesis, Univ.of
California,Los Angeles (1991).
[15] R.T.Lytwyn,W.Miao and W.Woitsh, "'Airborne and Ground
Resonance of Hingeless Rotors." J. Axn. Helicopter Soc. 16
(1971)
[16] F.K.Straub and W.Warmbrodt, "The Use of Active Con-
trols to Augment Rotor/Fuselage Stability" J. Am. Helicop-
ter Soc. 30 (1985)
[17] Gopal H. Gaonkar and David A. Peters "Flap-Lag Sta-
bility With Dynamic Inflow by the method of Multiblade Coor-
dinates" J. Aircraft 17 ppil2-119 (1980)
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139
[18] Mark. G.Ballin "Validation of a Real-Time Engineering
Simulation of the UH-60A Helicopter" NASA TM-88360 1987
[19] B.Etkin, "Dynamics of Atmospheric Flight" John Wiley
& Sons,lnc. New York 1972
[16] M.P.Gibbons and G.T.S.Done, "Automatic Generation of
Helicopter Rotor Aeroelastic Equations of Motion" Vertica
Vol 8. No 3.1984
[17] G.H.Gaonkar and D.A.Peters, "Effectiveness of Cur-
rent Dynamic-Inflow Models in Hover and Forward Flight" J.
Am. Helicopter Soc. Vol 31 No 2 (1986)
[18] Wayne Weisner and Gary Kohler, "Tail Rotor Perform-
ance in Presence of Main Rotor , Ground, and Winds" J. Am.
Helicopter Soc. 19 (1974)
[19] Dean E. Cooper, "YUH-60A Stability and Control" J.
Am. Helicopter Soc. 23 (1978)
[20] V.K.Baskin, "Theory of The Lifting Airscrew" NASA TT
F-823 1976
[21] Vil'dgrube, L. S. "Theory of Lifting Airscrew With
a Flat Vortex System" All-Union Meeting on Theoretical and
Applied Mechanics. Proceedings of Lectures USSR, 1960
[26] R.T.N.Chen and W.S.Hindson "Analytical and Flight
Investigation of the Influence of Rotor and Other High-order
Dynamics on Helicopter Flight Control System Bandwidth"
Paper presented at International Conference on Rotorcraft
Basic Research, Research Triangle Park, NC, February 1985
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140
[27] David R. Downing and Wayne H. Bryant "Flight Test of
a Digital Controller Used in a Helicopter Autoland System"
Automatica, Vol. 23, No. 3, May 1987
[28] R. F. Stengel, J. R. Broussard and P. W. Berry
"Digital Controllers for VTOL aircraft" Proceedings of the
1976 IEEE Conference on Decision and Control, Dec. 1976
[29] K.B.Hilbert and G.Bouwer "The design of a Model-
Following Control System For Helicopters" AIAA Paper
84-1941, 1984
[30] D.L.Key and R.H.Hoh "New Handling Qualities Require-
ment and How They can be Met" Proceedings of the Annual
Forum of American Helicopter Society, 43rd, 2, 1987
[31] W.L.Garrard and B.S.Liebst "Design of a Multivaria-
ble Flight Control Systen_ for Handling Qualities Enhance-
ment" Proceedings of the Annual Forum of American Helicopter
Society, 43rd, 2, 1987
[32] Dale,Enns "Multivariable Flight Control for an
Attack Helicopter" Proceedings of the 1986 Automatic Control
Conference, Williamsburg VA. June, 1986
[33] M.I.Young, D.J.Bailey, and M.S.Hirschbein, "Open and
Closed Loop Stability of Hingeless Rotor Helicopter Air and
Ground Resonance" Paper No.20. NASA SP-352, pp 205-218.
[34] W.Johnson, Helicopter Theory. Princeton University
Press, Princeton, New Jersey (1980)
[35] A.E.Bryson, and Y.C.Ho "Applied Optimal Control"
Hemisphere Publishing, Washington D.C. 1975
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[36] T.Kailath "Linear Systems" Prentice-Hall,
Cliffs, N.J. 1980
141
Englewood
"Optimal Control of Helicopter Aerome-
C.D.V.Palmer,
and Cost of
Guidance and Control
[37] F.K.Straub,
chanical Stability" Vertica Vol ii. No 3.1987
[38] B.D.O.Anderson and J.B.Moore "Linear Optimal Con-
trol" Prentice Hall, Englewood Cliffs, N.J. 1971
[39] N.K.Gupta, "Frequency-Shaped Cost Functionals:
Extensions of Linear Quadratic Gaussian Design Methods" AIAA
J. Guidance and Control, Nov./Dec. 1980
[40] Lt.D.B.Ridgely, Siva S.Banda, and
"Reduced Order Control Design, Benefits
Frequency-shaped LQG Methodology" AIAA
Conference, Gatlinburg, TN, Aug. 1983
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142
APPENDIX A
Derivation of System Equations of Motion
1) The Transformations between frames
The notations for transformation matrices are
I O O ]IxC#) = O cos_ -sln_
O sin_ cos_[°0 ]ty_ e) = 1 0
sine 0 cose
cos%# --sln_ O ]• zC_) 1 sin_ cosw O
O O I[o0, 0'--sin_ 0 cos/3
_os[-sin[ 0 ] _COSPk --sJ.n_k Ol•=c:_. isis: cos_o ,=c_ - _,i._k c_ k oLO 0 I LO O I
The fig. R-1 shows the relationship between the F frame1
and the inertial frame:
z e z fl Vt sinC oOC J)
Fig. R-_. shows the relationship between the F frame and
the F z frame. Beside the translational perturbations, which
carry the frame origin from the trim hub center to the pertur-
bated hub center, the perturbated rotations have following
sequence:
C13 A rotation -F about Zft , carrying axes to OX'Y'Z'.
CR) A rotation e about Y', carrying axes to OX"P'Z".
C3) A rotation -# about X", carrying axes to the F frame.2
Therefore, the transformation relationship for a v.ctor
is
{X)fl " (AXhu b) + _zf-vO_),(e)_xC-_#) (X)fz CR)
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14:3
where (AXhu b) is the translational perturbation at hub.
Fig. 2-3 shows the relationship between the F z frame and
the H frame, it is easy to obtain
(X}f z = _z(@ k) {X)_ C3)
Fig. 2-4 shows the relationship between the H frame and
the B frame. Beside the hinge offset displacement, the rota-
tions have following sequence:
¥'Z'.CI) A rotation -[ about Z h, carrying axes to Oh'X h" h h
Ca) A rotation -f9 about Yh'" carrying axes to the B frame.
The tranformation relationship for a vector is
e I C4)(X) h = 0 + @z(-[) _(-_) (X>b
0
2) Kinetic Energy
The position of a fuselage element is given by
<Xm} = {Xcg) _ _z(-_)_)Ke)_x(-@) {Xm}f CS)o o
(Xm)f contains the body axis coordinates of the point,
(Xcg} and (Xm) are the locations of the center of gravlty
mnd the point in the inertial axis system'-
The position of an element of a blade in the B frame is
given as:
T
(Xm) b = [ r, O, 0 ]
Then the position of an element of the blade in the
CO)
inertial axis system is given by:
<_r=-[-.... ....... .'_,
OF POOR QUALjTf
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144
O
I]}]+ 0
0
+ {Xh} C73
(Xh} is the location of the rotor hub center in the
inertial axis system
The kinetic energy can be written as:
I dx dz
If = dYmT = -- lC--) z + C------) z + C--
2 +b dt dt dt
.)z] dm C8)
3) Potential Energy
The potential energy is given by
v = N g CZcg) + c E K_ O_ + E K_ _" 1,2• k k k
CO)
4) Generalized Force:
Using the position vector given in Eq. C6), it is straight
forward to get the velocity of an element of a blade in the in-
ertial axis (Wbfr)) . The expression for the velocity components
in the B frame is
(WbCr)) b = _y{8) _zC[) _zC-¥, k) '_xC4,3 _yC-e) _zC_,) .CWbCr))
CI0)
(Vbfr)) is the velocity components of the rotor ele_nt
relative to the inertial axis in the inertial frame. (%qbCr)}b
is the velocity components in the B frame.
Then we get the normal and tangential airfoil velocities
of the element of the blade in the B frame from the relationship:
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Vt
Vt I = {YbCr)>bVp
145
CI13
From quasisteady strip theory we I_ve:
P
dD k m -- c b Vt:Cr) Cd °
2
dr (1R)
dLk
P
a Vt:(r) (Ok(r) -= -- C t c b
2
VPk(r) + VnkCr)
Vtk(r)
) dr C13)
where 8kCr) is the local pitch angle of the blade:
OCt) k 1 80 -- 8' r - _s cosp k - Bzs sln_u k(14)
where VnkCr) is the total local downwash of the blade:
VnkCr) = Vnokfr) + Vndk(r,t)
= Ynot • Vv, r cos_ k ( Steady In_flow )
V Ct) • V Ct) r cosw k- .+ V (t)o ¢ C
C Dynamic Inflow )
r sln_ k(15)
The aerodynamic moments and forces of the kth blade
are obtained by:
Fn k m dFnkfr) -
JO
-edLk(r)
JO
Ftk - r) = r) * (
JO JO
VPkCr) 4" Vnk(r)
VtkC r)
,) dL k
Mfk = r) MLk m
JO JO
(18)
Where Mr and Mt are the blade flap and lag aerod)n_amic
moments and Fn and Fi are the blade normal and £nplane aero-
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245
dynamical shear forces. The expressions for these forces and
moments in the H frame are:
_Fp 1 [o][ _F_ h . _y(fl) _zC_) -F_
Fn h k Fn k
_Mth -Mr k
C 173
Then the virtual work terms due to the aerod)_nmmtc forces
acting on the rotor blades can be obtained by
6Wr " _ [ M6k 6/_k _ ML k 6[ k "_ ( Mthk 4 Fi.hke)( 6_ )k
-I-( ]Mt'hk ,IFFnhke )( -6@ sinw k -- 68 cos@' k )
4 MXhk ( 6@ COS_U k - 68 sin_u k )
FPhk ( - 6Xhe c°s_ k -
+ Fi.hk ( + 6xhe slny_ k --
Following the same procedure, the virtual
6Yhe $1n_ k ) + Fnhk 6Zhe
6yhe COS_k) ] (18)
work term
due to the tall rotor, the fixed tall surfaces, and the fuselage
can be obtained.
53 The Lag Damper Modelling
The lag damper is modeled bM a dissipation functions
D - _ C_. C d[ )z / p.
k d¢
Clg)
63 The equations of motion
Then, the final system equations are
d 8C T-V) _ T-V)
,.) -- +
dt dC_.
dt
8D 6W
dQ_ 6(_
8<-------)dt
(P-O)
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147
7) The Multlblade Coordinates
The multlblade coordinates are defined as follows
/_k = _o - /_s COS_k -- _z sinPk (21)
[k I Co -- _, COS_k -- _Z slnPk (22)
Then, one obtains the multiblade coordinates An term of
blade flapping and lagging angle.
Collective flapping and lagging (coning)"
I I
N k N k
CE5)
First order cyclic flapping and lagging (tiltlng)!
kN
I
_z = _ _ -_ksinWkN k
C263
1 I
[i " -- _ -_kCOS_Uk CZ = -- _ --_ksln_k
N k N k
C29'3
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ORIGINAL PAGE !_
POOR QUALITYAPPENDIX B 148
Pitt's Model of Dgn_amic Inflow
The static coupling matrix, the air mass Inertial matrix
and the dimensional adjustor mtrix are glwen below:
[M] =
128R
0
7R.
-I 6R0
0
45_
0
0
45n
[L] =
Vso l
Vso
2 84 41 +slnaa
15n| 1-si no_ -4sl naa
04 41*sinacx I +sinaa
0 0
[D] =
0
_R z
10
0
0
0p_R 4
1
_R 4
CVnot-Vsi noO C 2Vnot-Vsl noO ._V2c os a
_/VZcosZa + CVnot-VstnoO z
a_ = Tan -s I Vnot-Vsln_vcosaI
0
0
-4
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_;i_ i,'...._ *'*_......
OF PO0_ _L_AL_V
APPENDI X C
149
The Flat Vortex Theory For Nonuniform Inflow
I) Flat vortex model
Under ass_mptios mentioned in Chapter 3, it will be more
convenient to use alr-traJectory reference frame for the deri-
vation of the flat vortex theory. Thlsreference frame has an
origin fixed to the hub center of the hellcopter, and the OX
axis is directed along the velocity vector but backward. The OZ
axis is directed up. As a result, the vortex layer will me-- in
the plane z=O in this frame. Therefore, all position components
in this appendix are in this alr-traJectory frame.
Circulation of a free vortex layer of width Ar which
springs from one blade is
dUCr)
AFCr) = Ar C13
dr
The equation defining the shape of a single free vortex
is given under the following form
x = _ R (_o-p) + r cos_ y = r sinp (2)
where W is the azimuth angle at which the free vortex0
left the blade.
Let us single out an elemental vortex layer associated
with two azimuth positions being different by an angle AI#o-
Circulation per unit length in this vorticity layer w111 be
AF(r) AP°
A_ = C 33
2_ As
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ORiGiNAL P:/:,;?: i_;
OF POOR Q:JALLTY
where As is the distance between the cyclolds.
r cos%uAs - _ siny tgy - (43
From (2) we have:
Ax i _ R A_O
Then:
O
i
As M R sin_
R + r sln_
Ar'C r)
C53
and A_ = CB)
En p R slny
Let us replace the free cycloidal vortex layer wlth two
systems of vortex layers; one system of free longitudinal
vortices and one system of free lateral wortlces, Circulation
per unit length of the lateral vortex layer Is:
A_y i A@ sinz =
and the longitudinal one:
AVC r)
2n _ R
L_C r) R + r sln_
2n _u R r cos_
C73
C83
150
Aq[,'x = A_ cos:y =
Free lateral rectilinear vortices will exist only within
a circle of radius r. Outside of this
votices will disappear as a result of
the incremental circulation, and only
circle, the lateral
geometric smxmatton of
longitudinal ones will
remain with doubled value of the circulation per unlt length.
R) Nonuniform induced velocity at tall surfaces
As mentioned in the Chapter 3, the nonuniform induced
velocity contribution of the longitudinal vortices is the only
one being co_idered.
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OF POOR Q ;,! -_;'f151
It is assumed that in the trim condition the weloclty
field experienced by the tall rotor and tall surfaces is that
on the wake. The induced velocity field of the longitudinal
vortices can be determined by replacing the actual system
of longitudinal vortices by rectilinear vortices extending
from X " -- CO to X " .4,
Then by applying the Biot-Savart Law, the wertlcal com-
ponent of the induced velocity at a point (xT, yT, O) on the wake
by a free vortex layer of width Ar is given as
AVz(r) =_F(r) _r @ R * r sin_
an H nR J-r (y - yT) r cos_
dy (Ig)
Let: y = r sinw ; dy = r cosw dw , Then we have:
AVzC r) =
-AVC r)
2n /._ nRz/z _ R 4 r slnw/z r sinv, - Yv
dy
-I -C-- *"r
YT) 1 j_n/_.z d_r n sin_ - yT/r
(10)
Since:
I
Then we obtain:
AVzCr) I
_M'C r)
(-1
2n /_R
AVCr)
En /_ R
0 -r-_ YT--" r
yv) r
2 + rZ
(11)
/_ R "b Y T
)
/ yt*._ r z
YT ( --r
-r -< YT --"r (lm-)
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OF. pOOR QUALITY
A_J-Cr) /a R + YT
2n /_ R /yTZ._ r z
C-1 + .) y_ > r
152
The vertical induced velocity due to all free vortlces
springing from the blade wall be
Vzfy) = _ AVzCr)
dVC r)
dr
dr
- /J R - y I_ dVCr) dr
f 2n /_ R dr _tyZ _ r z
JoI /_ R + y _/ dFCr) dr C13)
_.., R Jo dr _ yz _ r z
The distribution of circulation along the rotor radlus
is assumed to be parabolic:
ZFCr) = a r C R - r ) C14)
dFCr)
dr
= a r C _.R - 3r ) C153
Then we have:
/a R 4. y 3n y
- a C 2R- 3 y CR>y>O)
2n /_R 4
VzC y3 - C 103
/J R • y 3n y
- a C 2R -I. -3 y CO>y>-IO
2n _ R 4
The parameter a is determined by setting llft I weight|
T = p _rCr) VTCr) dr
JO
_0 ZI p a r C R - r ) 0 r dr = W C173
a I = C_-O_) -- CT
RC18)
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OF POOR QUALITY
153
The normalized nonuniform downwash Is defined as:
VzC y) Y Y 31_yCz = - - C_u • -----3----C2 _ )
a R z R R 4 R( -----3
VzC y) I VzC ),)= __ ClO)
20(CTC'_R_)/C2/a) I 0 2 Vnot
2 Vnot is the uniform downwash at tail surfaces from
the momentum theory. The lateral distribution of nondlmen-
sional downwash of eq.(10) for p=O. 22 is shown in F_g. 3-1.
Following the same procedure, we obtain:
AV_ r) = AVCr) I ; r zT ( p R • r sln_ ) dy2n p R _ r ( z z + yz) r cos_
T
AFCr) 1 Fn/z z T ( /_ R 4 r slnp ) dp2. _u R n a./z z z 4 rZstnZ_
T
im
Al"C r) 1
r 2 4 z zT
(ao)
Then we have:
Vy(z) =
a r ( 2R - 3r )
r2 + z z
dr
a R = /1 z 4z- C + C-----)2n R R
z R 4 _'R z 4 z =4 3(---) z In ( ) )
R zCz>O) C21 )
The normalized nonuniform sldewash is defined as:
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154
Cy =Vy(z)
a RzC
P,_ /a
= _ ( ._j z+ C--)R
z
• 3 C-----) z In CR
R + /RZ_ z z
z
Vy(z)
) )
- Cz> O) C _2)10 2 Vnot
For Z<O, we have VyC-z) = - VyCz).
The longitudinal distribution of the sidewash gi_n by
eq. CR2) for H-O. 22 is shown in Fig. 3-2.
For [z [ < O. 2 R, we use:
a Rz 4y> O) C23)
VyCz) _ _+ C 1 ) Cz <2_ R
3) Effects on Tail Surfaces
When there is sideslip, the position of a spanwlse
section of the horizontal tail relative to the center llne
of the wake at the wake layer will be:
y - C Yh + Thx /?f ) CR4)
Then we have the roll moment contribution of the horl-
zontal tail:
Mxh -
= C
mhP C t VzC
Vh Hc a Yh_Thx{_f ) YhdYh
2 J -mh
p Vh H aC
960n H R
• C 135n
) C 45n R /_ CRh) 4- 160 /_ RZCRh) ll
ThxCRh)4 - 320 R Thx C Rh) 3 ) _]
OF POOR QU,-,L_
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DP.
DF' -'
r_r4AL F'_u_.-.,_ '_
OE POOR QUALITY 155
4 gOn _u R C Rh) z T z /9z 4. gOn C Rh3 z T s _shx hx
- 15n R T 4 f14 _ gn T _hx h× fl_ )C 2_3
The terms for forming the llnearlzed approximation are
given belows
|
Mxh I_h =O =
p V h Hc C to a Rh s
lgE n
COn Rh - 32 R )
C 26)
p V h H c C t_ a ThxR.ha
gl¢
I gan _ R
C27n Rh - 64 R )
Similarly, the pitch moment contribc_clon_
Myh -p C L f rh
- --V h Hc _ Thx
2 J -rh
Vz(Yh+Thxflf) dy h
= C
pVhH aTc hx
) C 3a R CRh) s- gn CRh) 4
J gan _ R
+ C g6_ ThxRh R z - 36_ _u R ThxCRh) z) /3
+ Cg@ R T_xRh - 54- T_xCRh)Z) flz
- 12n /_ R T s /_B -- gr_ T 4 /94 )hx hx
C 273
and the required terms
J p V h H c Thx C t a R CRh) s
cx
Myh 0 (fir = l gen /._ R
) C3_. R - On Rh )
C_'8)
@Myh [ p V h H c T z C thx _
m
8/9f _f=O J 15n
aRRh
C 8R -3n RI_
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OF POOR QUALITY
156
For the tail rotor thrust contribution, we begin from
the local angle of attack of the tail rotor:
Vy(z) a R z 4z> O) (_)
oT = -_ -+ C1 - ------3 Cz <
V 2_ V RT T
Thrust contributions of the tail rotor due to 1_ontsn_L-
form sidewash at zero sideslip angle and zero angle of
attack can be given as:
8T
= -- Acx (z) dsCz) = 0 C30)TT T
O_T
When there is an angle of attack variation, the tail :
rotor thrust will vary as:
AT = KT _10TT
8_T
TCz) I ds II1 1
C31)
K is a parameter decided by the correlation wlth flight
test.
The vertical position of a point of the tall rotor rela-
tlve to the wake is
z = z + _T C323T Tx
Th.o: IAZ = TTx A_ ds I = dx I _z C33)
From experimental data[21], the effect of viscosity will
be significant at z < 0.1R. Therefore, let AoT=O for lzl < 0.1R.
The thrust variation of the tall rotor due to the angle of attack
variation can be obtained by integrating Eq. C313 with the relation
Eq. C33). The same procedure also applies to the vertical tail.