A STUDY OF GUST RESPONSE
FOR A ROTOR-PROPELLER
IN CRUISING FLIGHT
by
MASAHIRO YASUE
B.S. UNIVERSITY OF TOKYO, JAPAN
May 1969
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF SCIENCE
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1974
Signature of Author
Department of Aeronautics
and Astronautics, May 1974
Certified by
Thesis Supervisor
Accepted by
Chairman, Departmental
Graduate Committee
ARCH VEs
ss. aiS.rc
DEC 23 1974
8
RIES
ISRA
A'
-
A STUDY OF GUST RESPONSE
FOR A ROTOR-PROPELLER
IN CRUISING FLIGHT
by
Masahiro Yasue
Submitted to the Department of Aeronautics and
Astronautics on May, 1974 in partial fulfillment
of the requirements for the degree of Master of
Science.
ABSTRACT
Equations of motion for a rotor-propeller aircraft in cruising flight have been developed and implemented in a computer program. The formulation is based on Galerkin's method using coupled
mode shapes for the blade and wing. This procedure is applied to
the analysis of two types of rotors, gimballed rotor and hingeless.
The results are evaluated by means of eigenvalue analysis of the
stability of the system and frequency response analysis of the gust
and control response. In general, the results show that:
(a) The choice of mode shape (rigid-body mode with
spring restraint at the root, or elastic coupled
mode) to construct the equations of motion affects
the damping of the system significantly. The dependency of the damping on the mode shape is estimated for the first beam bending mode. The frequencies of the system have little dependence on the
mode-shape type.
(b) The results of the frequency response are quite
similar to those of Johnson, in spite of the use
of different models. However, the amplitude of the
response is slightly different.
(c) Addition of higher mode degrees-of-freedom has little
influence on the stability of the rotor and wing.
ACKNOWLEDGMENTS
The author would like to express his deep appreciation
and gratitude to his thesis supervisors, Professor Norman D. Ham
and Professor Pin Tong, who provided their invaluable advice and
guidance throughout this study.
The author is deeply indebted to Professor J. Dugundji
and to Dr. W. Johnson who contributed various useful suggestions.
Professor E.A. Witmer's advice and assistance at various
times, and especially in the thesis-writing and review phase of
the work is acknowledged gratefully. Also, Mrs. G. Hubbard and
Miss Marion West are especially thanked for their excellent
typing of this thesis.
The author wishes to express his sincere thanks to his
colleagues, T. Shimizu and A. Tada for their invaluable help
and suggestions.
Finally, the author would like to thank his wife, Kazuko
and daughter Sonoko for their invaluable help and encouragement.
SUMMARY
This study has been devoted to the development and evaluation of a theoretical model of the proprotor on a cantilevered
wing, operating in normal cruising flight. This theory expresses
the wing and blade motion in coupled form, and can include any
number of mode shapes required to describe the motion accurately.
It has been applied to the investigation of the dynamic characteristics of the Bell and of the Boeing design. The Bell rotors are
gimballed and the Boeing rotors are hingeless. The analysis includes the frequency response to gusts and cyclic pitch, and an
eigenvalue analysis of the dynamic system.
Based on the theoretical results included in this study, the
following conclusions may be stated:
(a) The choice of mode shape (rigid-body mode or elastic-coupled
mode) affects the damping significantly. The dependency of
the damping on the mode shape can be estimated for the first
beam bending mode. The blade inplane deflection opposing
the rotor direction of rotation, accompanied by the forward
out-of-plane deflection, increases the damping, comparing it
with the rigid-body calculation.
The inplane deflection proceeding in the rotor direction of rotation decreases the
damping. The mode shape has little influence on the frequencies
of the system.
(b) The results of the frequency response are quite similar to those
of Johnson, in spite of the difference in the mode shapes. The
amplitude of the response is slightly different, since structural damping was not included in the present calculation, and
the mode shapes used were different.
(c) The analysis of the eighteen degree-of-freedom system showed
that the higher-frequency degrees of freedom have small influence on the basic degrees of freedom.
iii
Stresses or bending moments of the wing or blade can be
predicted from the motions of the wing and blade obtained from
this analysis. In addition, this analysis may be applied to the
development of an automatic control device to alleviate the gust
response of the vehicle.
CONTENTS
Section
1
2
3
4
Page
INTRODUCTION
1
1.1
General
1
1.2
Brief Survey of Past Work
3
1.3
Objectives of the Present Study
5
THE EQUATIONS OF MOTION
7
2.1
Model and Coordinate System for the Analysis
7
2.2
Basic Formulation for Powered Flight
9
2.3
Supplement for Autorotational Flight
15
AERODYNAMIC FORCES
16
3.1
Rotor Aerodynamic Forces
16
3.2
Wing Aerodynamic Forces
22
MODAL ANALYSIS
26
4.1
The Variational Functional
26
4.2
Free Vibration of the Blade and Wing
28
4.3
Variational Functional Described in
Modal Form
33
4.4
Rotor Non-Rotating Coordinate System
36
4.5
Aerodynamic Forces Described in
Modal Form
40
4.6
Final Equations of Motion
48
4.7
Equations for Gimballed Rotor
67
CONTENTS CONCLUDED
Section
5
6
Page
NATURAL FREQUENCIES AND MODE SHAPES OF THE
ROTOR AND WING
71
5.1
Rigid Rotor
71
5.2
Gimballed Rotor
72
5.3
Wing
74
PROPROTOR DYNAMIC CHARACTERISTICS
77
6.1
Introduction
77
6.2
Eigenvalues and Eigenvectors
Frequency Response
79
82
6.3
7
CONCLUSIONS AND COMMENTS
85
7.1
Conclusions
85
7.2
Suggestions for Future Research
86
REFERFNCES
87
TABLE
89
FIGURES
92
APPENDICES
A.
B.
DETAIL DERIVATION OF EQUATIONS OF MOTION
STRESS ANALYSIS OF THE ROTOR BLADE AND THE WING
213
217
LIST OF ILLUSTRATIONS
Figure
1
Page
Configuration of Analytical Model, Proprotor
on a Cantilever Wing Operating in the Airplane Mode
92
Proprotor Geometry and Motion with Gust Velocity
Definition
93
3
Blade Geometry
94
4
Wing Geometry
95
5
Velocities at the Blade Element and Resulting
Aerodynamic Forces when Looking Outboard
96
Wind Velocities in the Disc Plane of the
Rotor, Looking Forward
97
2
6
7
Velocities at the Wing Element and Resulting
8
9
10
11
12
13
.
14
Aerodynamic Forces when Looking Outboard
98
Structural Characteristics of Two Proprotor
Blades
99
Blade Natural Frequencies, Nonrotating, for
Boeing Rotor
103
Blade Rotating Natural Frequencies for
Boeing Rotor
104
Blade Natural Frequencies for Boeing Rotor at
Constant Rotational Speed (A = 386 RPM)
106
Mode Shapes for Boeing Rotor at A = 0.7
and A = 386 RPM
107
Blade Natural Frequencies, Nonrotating,
for Bell Rotor
111
Blade Rotating Natural Frequencies for
Bell Rotor
113
vii
LIST OF ILLUSTRATIONS CONTINUED
Figure
15
P.age
Blade Natural Frequencies for Bell Rotor at
Constant Rotational Speed (Q = 458 RPM)
117
Collective Mode Shapes for Bell Rotor at
X = 0.7 and A =458 RPM
119
Cyclic Mode Shapes for Bell Rotor at
X = 0.7 and Q = 458 RPM
123
Structural Characteristics of the Wing (the
Same Wing is Used by Both Bell and Boeing)
127
19
Mode Shapes for Boeing Wing
129
20
Mode Shapes for Bell Wing
135
21
Influence of Addition of Higher Degrees of
Freedom on Eigenvalues Compared with Basic
Nine Degrees of Freedom
141
22
Eigenvalues for 18 Degrees of Freedom
143
23
Influence of Blade Mode Shape Types on
tem Eigenvalues and Damping Ratios for
Mode Shapes, Compared with Those Based
Mode Shapes and Uncoupled Elastic Mode
24
Eigenvalues and Eigenvectors for Nine Degrees
of Freedom
149
Frequency Response of Bell Rotor to Vertical
Gust uG Input at Frequency w
155
16
17
18
25
26
Proprotor SysCoupled Elastic
on Rigid-Body
Shapes
145
Frequency Response of Bell Rotor to Lateral
Gust vG input at frequency w
27
164
Frequency Response of Bell Rotor to Longitudinal
Gust wG Input at Frequency w
viii
173
LIST OF ILLUSTRATIONS CONCLUDED
Page
Figure
28
29
30
31
32
Frequency Response of Boeing Rotor to Vertical
Gust uG Input at Frequency w
182
Frequency Response of Boeing Rotor to Lateral
Gust VG Input at Frequency w
191
Frequency Response of Boeing Rotor to Longitudinal
Gust wG Input at Frequency w
200
Frequency Response of Bell Rotor to Longitudinal
Cyclic Pitch els Input at Frequency w
209
Frequency Response of Boeing Rotor to Longitudinal
Cyclic Pitch
els
Input at Frequency w
211
LIST OF SYMBOLS
aw
Blade sectional lift-curve slope
Time function of the wing motion corresponding to
the jth mode
Wing sectional lift-curve slope
CDo
Blade-sectional drag coefficient
CDow
Wing-sectional drag coefficient
Cmo
Cma
Wing-sectional pitching moment coefficient
dD
Sectional aerodynamic drag
dL
Sectional aerodynamic lift
dH-.
Wing-sectional pitching moment at the quarter
chord
(EI)B
Blade bending stiffness perpendicular to the
chordline
(EI)C
Blade bending stiffness along the chordline
(EIw)B
Wing bending stiffness perpendicular to the
chordline
(EI )C
Wing bending stiffness along the chordline
e
Exciting force defined in Eq. 4.53
e
Distance between the elastic axis and the
aerodynamic center of the wing
a
a.
Fx,
f.
Wing-sectional pitching moment curve slope
Fz
Wing resultant forces per unit length in the
x and z directions
Aerodynamic loading of the wing due to the
rotor aerodynamic forces, corresponding to
the ith wing mode shape
(GJw)
Wing torsional rigidity
g.
Aerodynamic loading of the wing due to the wing
aerodynamics, corresponding to the ith wing
mode shape
h
Mast height
Nondimensional mast height: h/R
I
Unit matrix
IB
Blade mass moment of inertia about the virtual
flapping hinge
I
Pylon pitching mass moment of inertia
p
Pylon rolling mass moment of inertia
I
r
I
Pylon yawing mass moment of inertia
Iw
Wing mass moment of inertia about the elastic
axis per unit length
K .
Pitch-flap coupling coefficients corresponding
to the ith blade mode shape defined in Eq. 4.26
L
Wing length (semispan)
MB
M4P
Mass of one blade
Pylon mass
M
y
Wing resultant pitching moment about the elastic
axis per unit length
m
Spanwise mass of the blade per unit length
m,
Spanwise mass of the wing per unit length
N
Number of blades
n
Blade index
Pz
Resultant force per unit length in the z direction
on the blade
Pg
Resultant force per unit length in the circumferential direction of the rotating blade
Pe.,
P
.
Defined in Eq. 4.17
pw
Wing torsion (positive nose up)
p
Time function corresponding to the wing torsion
mode
Qj
jth blade collective motion degree of freedom
Qjc
jth blade cosine cyclic motion degree of freedom
Qjs
jth blade sine cyclic motion degree of freedom
qnj
Time function corresponding to the jth mode of the
nth blade
q,
Time function corresponding to the wing vertical
bendina
q2
Time function corresponding to the wing chordwise
bending
R
Blade radius
r
Blade running-spanwise coordinate
r,xPylon
vertical displacement (positive upward)
rz
Pylon longitudinal displacement (positive
forward)
Sa
Static mass moment of the wing segment about
the elastic axis
s
Distance between the elastic axis and local
center of gravity of the wing
xii
LIST OF SYMBOLS CONTINUED
T
Matrix defined in Eq. 4.11
T
Centrifugal force of the blade
t
Time
U
Blade sectional resultant velocity
UpPBlade
sectional inplane velocity
Blade sectional out-of-plane velocity
UT
UwWing
sectional resultant velocity
UxB
Blade inplane velocity component in the
x B direction
UyB
Blade inplane velocity component in the
uG
yB direction
Vertical gu:t
el
ty
uwVertical
bending deflection of the
wing (positive upward)
V
Aircraft forward velocity
V..
Defined in Eq. 4.16
V.
J
0
V
J
Blade jth inplane bending mode shape
VScjth
J
cyclic mode shape of the inplane
bending of the blade
v
Rotor induced flow
vG
Lateral gust velocity
vn
jth collective mode shape of the inplane
bending of the blade
-
Inplane deflection of the nth blade
(positive clockwise)
xiii
LIST OF SYMBOLS CONTINUED
WB
B
Pylon vertical translation
Pylon rolling angle
WC
Pylon longitudinal translation
WN
Pylon yawing angle
W
Defined in Eq.
W.
Blade jth out-of-plane bending
mode shape
W0
3
jth collective mode shape of the
out-of-plane bending of the blade
W.
)
jth cyclic mode shape of the out-ofplane bending of the blade
WWB
Work done on the wing by the blade
aerodynamic forces
Www
Work done on the wing by the wing
aerodynamic forces
wG
Longitudinal gust velocity
wn
Out-of-plane deflection of the nth
blade (positive forward)
ww
Chordwise bending of the wing (positive
forward)
x
Blade nondimensional running spanwise
coordinate r/R
x,y,z
Coordinates fixed to the wing root (see Fig. 2)
xB'yB
Coordinates fixed to the 'rotor (see Fig. 6)
4.16
xiv
LIST OF SYMBOLS CONTINUED
G
Blade sectional angle of attack
a0Blade
sectional angle of attack in the
equilibrium state
%w
Wing sectional angle of attack
B
y
Blade flapping degree of freedom
paCB R4
Lock number -B
y
Wing jth vertical bending mode shape
A*n
Phase angle between blades
6
Variation
03
Rotor blade pitch-flap coupling
C
Blade lagging degree of freedom
Ci
Wing jth chordwise bending mode shape
eAT
Built-in angle of twist of the blade
eB
Angle of twist of the blade
6c
Blade pitch angle due to pitch control
6D
Blade pitch angle defined according
to the performance analysis
0 PF
Blade pitch angle due to pitch-flap coupling
0w
Angle of twist of the wing
90
Blade collective pitch control
e1c
Blade cyclic pitch control
LIST OF SYMBOLS CONTINUED
ols
Blade cyclic pitch control
o .75
Blade pitch angle at 75% span
K
Integral operator defined in Eq. 4.37
A
Wing jth natural frequency
Inflow ratio V/(QR)
17
Absolute value of the inflow ratio
.
J
lXI
jth natural frequency of the rotating blade
VP
Pylon pitching motion (positive nose up)
v RBlade
rigid-body mode rotations in autorotation case (positive for clockwise rotation, when looking forward)
Vr
Pylon rolling motion (positive for
clockwise)
VyPylon
yawing motion (positive for
counterclockwise)
ir
Variational functional
p
Air density
Tr
Defined in Eq. 3.15
0
Pylon pitching angle
Blade sectional inflow angle
Wing jth torsional mode shape
$w
Wing sectional angle formed by the
horizontal and vertical velocity components (see Fig. 7)
xvi
LIST OF SYMBOLS CONCLUDED
Azimuth position of the nth blade
Azimuth position of the nth blade excluding
the pylon roll motion
Rotational speed of the rotor
Direction of rotation Q/|101
Value is evaluated in the equilibrium state
o
('),
(00)
Time derivative
a/3t, a2/t2
00
0
Nondimensional time derivatives,
()n
nth blade
Superscript
Collective mode of the rotor
o
Cyclic mode of the rotor
c
Subscript
o
c
s
i,j,k
Collective mode of the rotor
Cyclic mode of the rotor
Cyclic mode of the rotor
Indices
xvii
3322
'
a
SECTION 1
INTRODUCTION
1.1
General
It has been recognized that the tilting proprotor aircraft,
one of the composite aircraft family, is a very promising concept
that combines into one aircraft the hover efficiency of the helicopter and the high-speed efficiency of the fixed-wing aircraft
(Refs. 1-3).
The typical tilting proprotor aircraft is a twin-engine aircraft with tilting rotors mounted on each wing tip. Its configuration consists of a fuselage, a high swept-forward wing, and an empennage. The empennage has a vertical stabilizer and rudder, and
a horizontal stabilizer and elevator. The large diameter rotors
are three bladed. hingeless or gimbal-type rotors which are mounted
on the rotor shaft. The rotor shaft is connected through the gearbox to each engine in the pylon attached at the wing tip. The conversion system provides the rotation of the rotor pylon from the
vertical position to the horizontal position and return, in order
to obtain the helicopter mode or airplane mode corresponding to the
desired flight regime.
When the aircraft takes off or lands, the rotor pylon is rotated to the vertical position to achieve vertical takeoff or landing similar to the helicopter. The flight controls apply pitch
changes to the rotor to provide the longitudinal and directional
control corresponding to helicpoter rotor cyclic pitch, while the
collective pitch controls vertical flight and roll motion.
In high-speed flight, the rotor pylon is rotated to a horizontal position similar to that of the conventional propeller type
aircraft. The thrust is produced by the rotor, and the lift by the
wing. The flight controls are provided by the conventional aircraft
control surfaces such as the elevator, rudder and aileron.
In addition to the above modes, various conversion modes
can be obtained. At that time the rotor pylon tilts to some position between the vertical and horizontal, where it can be safely
locked. This makes STOL-type operations possible.
The tilting proprotor is exposed to a severe aerodynamic
environment including gusts, the wake of preceding blades, and
harmonic airloading like a helicopter. But its dynamic and aeroelastic characteristics are in many ways unique; for example, the
large flexible blades with a large amount of twist experience significant coupled out-of-plane (flapping) and inplane (lagging)
motion.
As described later in Subsection 1.2, several years of experimental and theoretical analyses have been conducted to establish a fundamental understanding of the dynamic and aeroelastic
behavior. However, it is necessary to understand the aeroelastic
response of this aircraft to atmospheric turbulence more adequately
and to predict it more accurately, since during the preliminary
design phase, vibration level prediction is required in order: (a)
to evaluate the fatigue life of the blade and wing, (b) to estimate
the ride qualities of the vehicle, and, if necessary, (c) to develop
suitable gust alleviation devices.
Several design compromise concepts, which make the present
analysis distinct from helicopter aeroelastic analysis, are now
stated briefly.
In order to obtain high hover efficiency from the rotor, it
is desirable to achieve low disc loading, in other words to use
large-diameter rotors whose swept discs reach nearly to the fuselage. When the aircraft is operated in high forward speed axial
flight in the airplane mode, the rotor is operating at a high inflow
ratio (the ratio of axial velocity to blade tip speed). This phenomenon is very different from the helicopter rotor operation which
involves low inflow. High inflow operation requires a large builtin angle of twist for efficient cruising. Therefore, significant
coupled out-of-plane (flapping) and inplane (lagging) motion occurs
in the large, flexible and twisted blade. This phenomenon makes
analysis more complicated.
The engines and gearboxes are usually located at the wing
tip to avoid transmitting high power through a long drive shaft.
This leads to low wing natural frequencies and possible resonances
in the low frequency range. Also, the center of gravity of the
pylon and rotor does not usually coincide with the elastic axis of
the wing. Hence, this results in coupled bending and torsion.
1.2
Brief Survey of Past Work
A good review and an elementary description of the dynamic
and aeroelastic problems associated with the tilting proprotor aircraft were given by Reed in Ref. 4 and Loewy in Ref. 5. In this
report no attempt will be made to repeat the reviews given in Rers.
4 and 5. The only references cited here will be those pertinent
to the problem being treated.
The possibility of propeller whirl flutter -- a dynamic instability that can occur in a flexibly mounted aircraft enginepropeller combination -- was first recognized in the late 1930's
It was not until 1960 that it beby Taylor and Browne (Ref. 6).
came a problem of practical concern -- with the appearance of the
turboprop aircraft.
Following the two fatal turboprop aircraft accidents, it
was established that propeller whirl flutter could have occurred
if the nacelle stiffness was severely reduced, for example by a
structural failure. Several generalized studies were conducted
at NASA-Langley. One of them was carried out by Houbolt and Reed
in Ref. 7; it gives an elementary and basic treatment of the equations of motion and propeller aerodynamics for propeller-nacelle
whirl flutter.
Because VTOL configurations have unconventional propellerrotor systems, whirl flutter was a major design consideration on
present proprotor aircraft.
The analysis presented in Ref. 8 is for a two-bladed rotor
free to tilt on a shaft with two nacelle degrees of freedom (pitch
and yaw).
No lag or coning degrees of freedom are considered.
The analytical method was compared with test results for an existing tilting proprotor aircraft (the Bell XV-3) and of subsequentlytested scale models. They showed good agreement.
Young and Lytwyn in Ref. 9 present a very precise analysis
for the whirl stability of a multi-bladed rotor mounted on a nacelle
which has pitch and yaw degrees of freedom. Each blade has one flapwise degree of freedom. The blade mode shape is assumed to be a
rigid body mode shape. It was concluded that whirl stability is
poorest when the nacelle pitch frequency equals the nacelle yaw
frequency, but in this situation nacelle damping is quite effective.
There is an optimum value of flap bending frequency somewhere between
1.1 .nd 1.53 2ar Xighly sLabilized whirl motion.
This analysis neglects (as do Refs. 4 and 8) the effect of
coning on proprotor aerodynamics, and flap bending mode shapes other
than the rigid blade mode used. Also, autorotation flight must be
considered as well as powered flight.
In Ref. 10, Gaffey points out.that a highly coupled blade mode
has substantial flap bending even if the primary mode involves inplane motion. This occurs in the case of a highly twisted blade or a
blade operating at high geometric pitch angles such as a proprotor
blade. The analysis shows that a moderate amount of negative
63
(flapping angle at the blade root gives the pitch angle reduction of
the amount 6 -tan a3 if 63 is positive) has a stabilizing influence
on proprotors subject to flap-lag instability at high inflows.
Preliminary design studies of prototype vehicles (Refs. 11 and
12) as a part of the current NASA/ARMY sponsored tilting proprotor
research aircraft program give some results from dynamic and aeroelastic analyses done by Bell and Vertol.
Johnson (in Refs. 13 and 14) derived the equations of motion
for a cantilever wing with the rotor at the wing tip. He develops
a nine degree-of-freedom model which involves blade flapping motion
and lagging motion (each has one collective and two cyclic motions,
respectively), wing vertical bending, chordwise bending, and torsion.
This model is applied to two proprotor designs and compared with the
results of some full-scale wind tunnel tests. It shows reasonable
correlation between theory and experiment.
in conclusion, it appears that most of the investigative work
has been concerned with whirl flutter. The above review shows that
more knowledge is needed for the solution of tilting proprotor aircraft dynamic and aeroelastic problems.
1.3
Objectives of the Present Study
The objective of this study is to establish a verified method
of predicting the dynamic and aeroelastic behavior of the tilting
proprotor aircraft in order to evaluate the fatigue life of the
blades and wings and also to estimate the ride quality.
The equations of motion for a cantilever wing with a rotating
rotor at the wing tip will be derived as consistently as possible.
The great complexity of rotor blade motion will be included by accounting for blade rotation (i.e., centrifugal and Coriolis forces),
significant inplane motion, and the large twist and high pitch angles
at high inflows.
The resulting system of equations, obtained using modal
analysis, will be applied to the analysis of the two proprotor designs (one is a rigid, soft-inplane type rotor and the other is a
gimballed, stiff-inplane rotor).
Finally, the eigenvalues and frequency response of each proprotor design will be determined to establish their dynamic charac-
teristics; the results of the analyses will be compared in terms of
eigenvalues for normal modes (coupled elastic modes), assumed modes
(uncoupled elastic modes), and rigid body modes, using Galerkin's
method.
SECTION 2
THE EQUATIONS OF MOTION
2.1
Model and Coordinate System for the Analysis
The primary interest of this study is in the dynamic and
aeroelastic phenomena of the wing, pylon, and rotor of the tilting proprotor aircraft in cruising flight. Hence, the dynamical
system considered here consists of a cantilever right wing with
a pylon at the wing tip, and an I-bladed rotor mounted on the
pylon, as shown in Fig. 1. The model will be restricted to the
cantilever wing, since it is sufficient to obtain a basic understanding of the proprotor motion, and many such proprotor models
have been tested in wind tunnels.
Therefore, the aircraft rigid body motions are neglected
and the wing antisymmetrical m
=desare a-l
dropped.
The left
wing motions including the pylon and rotor are given by the
mirror image of the right wing.
The wing is assumed to have a high aspect ratio, so that
strip theory is used for the wing aerodynamics and beam theory
for elastic bending and torsion. Wing sweep and dihedral will
not be considered, but angle of attack and angle of twist (builtin twist) will be considered. The elastic axis is assumed to be
a straight line. The elastic axis coincides with the y-axis as
shown in Fig. 2. The free stream vector coincides with the zaxis. Therefore, the angle between the z-axis and the chordline
of the wing results in a wing angle of attack (positive nose up).
The wing motion (Fig. 2) consists of elastic bending and elastic
torsion. The deflection uw of the wing elastic elastic axis perpendicular to the y-z plane is called vertical or beamwise bending (positive upward). The deflection ww parallel to y-z plane
is termed chordwise bending (positive forward). Torsion pw is
defined as pitch angle change (positive nose-up).
A pylon of large mass and moment of inertia is assumed.to
be rigidly attached to the wing tip. Therefore, the pylon motions
(Fig. 2) corresponding to the wing motions are defined as vertical
displacement r along the x-axis (positive upward), longitudinal
displacement rz along the z-axis (positive
forward), pylon yaw vy
about the x-axis (positive counterclockwise), pylon pitch vp about
y-axis (positive nose-up), and pylon roll v r about the z-axis (positive for clockwise rotation, when looking forward). The above pylon motions are accounted for at the point where the wing elastic
axis crosses the plane, parallel to x-Z plane, which includes the
rotor shaft. The pylon lateral displacement along the y-axis is
neglected as a higher order effect.
The pylon and the rotor shaft are assumed to be parallel to
the free stream in equilibrium flight (r,, rz, Vy, Vp, and vr are
all zero), regardless of the wing angle of attack.
The rotor is located at the distance h (Fia 7 from. the
wing tip elastic axis to the rotor-hub (positive forward from the
elastic axis). The distance h is termed the mast height. The rotor
consists of N blades, whose rotational speed A is defined as positive
if clockwise looking foward. The blade (Fig. 2) has out-of-plane
(flapping) deflection wn, defined positive for forward displacement
from the disc plane (upward in helicopter mode, while the rotor
shaft is vertical), and inplane (lagging) deflection vn, defined
positive for clockwise deflection regardless of rotor direction of
rotation. The lower case letter n means the nth blade, n=1,2,...N.
Blade torsion is neglected here.
The azimuth position in
fined as:
(Fig. 3) of the nth blade is de-
4!.
=
.at
(2.1)
+ V. + A
t is time.
where Tn is measured from the vertical and
phase angle between blades, A*n is defined as
(i)
A
2
N
The
(2.2)
The azimuth position of the nth blade excluding the pylon roll
motion is denoted as
(k
2.2
(2.3)
=..t +4
Basic Formulation for Powered Flight
The governing linear equations of motion are derived in
this subsection. A more complete and detailed derivation is
given in Appendix A.
The equations involve ten unknowns: vn, wn, rx,
r z ,v v,
re-a
w (actually vn and wn repr?7ent N unknownc,
r'Zw '"vW'
pectively, but for convenience they may be treated as one unknown
without inconsistency). The equations consist of three categories:
the blade equations, the wing equations, and matching conditions
between rotor and pylon.
The blade equations are complicated by the pylon motions
which produce the centrifugal forces and Coriolis forces. The
wing equations are derived from beam theory. The rotor is rigidly attached to the pylon for the powered flight case and, therefore, rotor motion is related to pylon motion; this gives the
matching conditions between the rotor and the pylon.
(a) Blade Equations
[(EI)c -o~~
+ (El)
are~~
o.co G
-~(Taxr
+
r
±m{
~+r.+
2 -ar
OAM
(2.5)
(E
running spanwise coordinate for the
blade from the axis of rotation.
0
B
angle formed by the rotor disc plane
and the blade sectional chordline,
usually including built-in angle of
twist and collective pitch.
(EI)
bending stiffness in the blade sectional chordline direction.
(EI)B
m
bending stiffness in the direction
perpendicular to the chordline.
spanwise mass of the blade per unit
length.
T
centrifugal force at r expressed as:
Pz
resultant force per unit length in
the z direction on the blade (positive forward).
P0
resultant force per unit length in
the circumferential direction on
the blades (positive clockwise when
looking fowardY.
( )n
about nth blade.
The blade geometry is shown in Fig. 3.
(b) Wing Equations
2L-
e~
+I(ES)
i-El))
201 [(EI)c
+
+
wLW
+ S.'
CBIW)c
+
9W CE OW1
a
= Fr
-- [{(EI.)c Coe-9
+
1
-(#).,"
JP
COOC
+ CEI..,
- (E I-)
(2.6)
a e4
w CA"
UF(2.7)
r
)
+
where
0w
(EIw)c ,(EIw)B
(GJw)
s1, ~4(2.8)
~
angle between the wing sectional chordline
and free-stream direction.
wing bending stiffness defined similarly
as for the blade.
wing torsional rigidity.
mw
Iw
spanwise wing mass per unit length.
wing mass moment of inertia about the
elastic axis per unit length.
sa
static mass moment of wing segment about the
elastic axis defined as
F , Fz
s = ms
where s is a distance between the center of
gravity and the elastic axis of the wing
and positive if center of gravity is ahead
of the elastic axis.
resultant wing force per unit length in the
x and z directionsrespectively.
My
resultant wing pitching moment per unit
length.
The wing cross-sectional geometry is shown in Fig. 4, and
pylon motions are expressed with wing deflections as
2(2.9)
12
Matching Conditions between the Blade and Pylon
(c)
L
&)"e + CIE Iq)acoO4 1
(EI
r
.4-"t211w
E.s
[{a~
a w 21 g-L
aw~w~~
ALM.%coo O
9.
R
+e
+
fr9
P -wan
0t
~'sl
~Iw9
v
P
r
)a
(2.10)
P.q As j p+ vpPz) d r
'a
+<E'w)J
-D
(rE IW)c -
+r LfJe
C
v)
Af-*a -
J]
o- V
C0
+
I4414 + IvF
oo+w
+ [{IV) ccow +
r
+
U.f(f.
*]4=L
+
t 9WCO
44O
G E
(a
(2.11)
EL
I(Lt
-(
J
--- coo
O
r+
rr
f oc Fz
*0
r),r BF.AZc!1,
+
+
zoo
(
-
kPec.J(
(2.12)
0*
+4 (r, +Y)
P
Y-rPz
00o
e
2nr4ur
-
-rO Ucoo<)
d
cVL
Coo 4dr]
14
(2.13)
el
1j
a
+~[f~Mir vr + rt,)d
P.L f](2.14)
where
Mp
Pylon mass
I
Pylon yawing mass moment of inertia
py
I
Pylon pitching mass moment of inertia
IPr
Pylon rolling mass moment of inertia
("),(*) Time derivative applied to entire formula
in the parentheses
=L
The above equations are derived independently of Refs. 7,
9, and 13. However, if the wing motions are eliminated, the
equations are similar to the basic whirl flutter equations derived
in Ref. 7 or Ref. 9.
2.3
Supplement for Autorotational Flight
Autorotation may be defined as the condition of flight where
there is no restraint of the rotor rotation about the rotor shaft.
Therefore, no rotor torque is transmitted to the shaft, and no pylon
roll motion is transmitted to the rotor. This means that rigid body
rotation of the entire rotor about the shaft will be produced. This
rigid body rotation is designated as vR (positive.clockwise).
The equations of motion for autorotational flight are almost
the same as those for powered flight. The equations: 2.4, 2.6, 2.7,
2.8, 2.10, 2.11, 2.12 and 2.13 are the same. In Eq. 2.5 vr must
be replaced by vR. In lieu of Eq. 2.14 the fact that the rotor
inplane motion is independent of the pylon roll motion in autorotation flight results in Eq. 2.15 as follows:
=
L
m (rP.)ar
:
u
0'
Aa
n
This yields a new degree of freedom.
The angle in Eq. 2.1 is also expressed as
(k
in this case.
l
1(2.16)
(2.15)
SECTION 3
AERODYNAMIC FORCES
3.1
Rotor Aerodynamic Forces
The rotor aerodynamic forces will be evaluated next. The
analysis is almost the same as in the helicopter blade element
theory. The significant difference is that it is impossible to
assume small angles for the angle of twist, the collective pitch
and the angle of attack of the blade operating in the high inflow.
The basic idea is presented by Young and Lytwyn (Ref. 9).
The section aerodynamic lift and drag forces dL, dD yield
the resultant forces Pz and P0 as follows (in Fig. 5),
(
- dD sin
Pz = dL cos $
= -dD cos $. - dL sin $.
P
where
dL = 1/2 pacBU
2
'2
(
2(3.2)
dD = 1/2 p cBU CD
and the angle $i is defined as,
U
sin$. =
1
U
(3.3)
=
cos
U
The blade section inplane velocity (positive counterclockwise direction) is UT and Up is the blade section out-of-plane
velocity (positive for negative z direction). Therefore, the resultant air velocity is expressed as,
U=
In cruise flight,
UT = G
U
VT
O r(3
U
V
+ V
p
P&
(3.4)
.5)
where V is the forward aircraft speed (axial velocity) and v is
the induced inflow velocity. The induced velocity is very small
compared with the forward velocity V in the high inflow operation.
Therefore the induced velocity is neglected in this entire study.
The inflow ratio A is defined as the ratio of the axial
velocity to tip speed of the blade:
V
nrQ
(3.6)
The section effective angle of attack is
S
=-1
.
(3.7)
_
IUri
the angle 0B formed by the disc plane and the blade chordline includes the angle of twist and collective pitch. Therefore
9,
0
+ 19.'ITS
%
where 0AT is the built-in section angle of twist of the blade,
which is zero at 75% span of the blade, and 0 .75 is the collective pitch to be obtained from the performance calculation.
Usually 0 .75 is:
-7
AT
,
(3.9)
V
The first term express the inflow at 75% span and 0 D depends on
the proprotor design to obtain the optimum cruising performance.
Pitch-flap coupling OpF, and blade pitch control 0c will be discussed later. Finally:
g
=
+ Ge
AT+ e73+ +Ps IUT'
(3.10)
Substituting Eqs. 3.2 and 3.3 into Eq. 3.1 yields
p, =+pca.
U\IU,.\
c
PcC.
f= , I --
-
' PCaCDo
U,.1|
UUp
J- pcs a U U
1
(1)
where
sign (f)
IW
=
L
(3.12)
Next the perturbation method is applied to Eq. 3.11 to derive the
aerodynamic forces of the rotor in disturbed motion:
P. =e.>
(t.).suT +
(U)
(.A).Cs
.,
(3.13)
In these equations, ( ) means those values are evaluated
in the equilibrium state, given by the following expressions:
AtMI
(3.1 4a)
a..
-
~
.i30~tfG
CO
-
3O
(POO 2
~ l PCOa MR
where
T4
=
(4 =o, I,2,3)p.
2~
f~t(3.17)
A.
0(P
AT
(3.15)
-75 -)t
r
(3.18)
The perturbation quantities 6 UT, 6Up and 6at will be considered next. The perturbation velocities 6U, and 6Up consist of
rotor and pylon motion and gust velocities.
The velocity at the position of the blade axis in x,y,z
coordinate (Fig. 2, see details in Appendix A) can be written as
+-
t
r Or)
C-0.0p
,~rA4Lck
1.
A; - 4
4 .,
r V
coo
(3.19)
Wind velocities at the blade element in the disc plane U are then
obtained by applying the transformation matrix due to pylon motion
to change the x,y,z coordinate system to blade disc coordinate
system. Wind velocities UxB and UyB are xB'
B components of the
inplane velocity of the blade in the xB-YB coordinate system fixed
to the rotor hub (Fig. 6).
Gust velocity as well as the aircraft
forward velocity is included.
+
U
+
(3.20)
Tangential velocity UT is expressed as
UT =
~ UXS AEM ca +
Us cC-00
(3.21)
L
After higher order terms are neglected,
UTeg-I -.
.\A4
.OO
+f'
w + ry --Y coo
+
Finally the perturbation velocities 6U
oT
0
and 6U
are given as
p
0(
0
6
+
where
(3.22)
r
+(3.23)
(3.24)
Pitch-flap coupling gives a change in blade pitch angle proportional to the flapping angle at the root. It is defined as
ai'
reo
(3.25)
The angle 63 is a design factor to yield the optimum pitch angle
gain to prevent blade motion instability. Blade pitch control
consists of collective pitch control and cyclic pitch control.
(3.26)
+ GICL) CVO4.g + 3S
8G0e)
6tt) A.4eC4s.
Therefore, the perturbation quantity associated with the angle of
attack of the blade is simply derived from Eq. 3.7 as
-
e p +O
(3.27)
3.2
Wing Aerodynamic Forces
Wing aerodynamic forces are derived by the blade aero-
dynamic perturbation method. Aerodynamic coefficients are obtained from strip theory (Ref. 15).
The wing element lift, drag and pitching moment (Fig. 7)
per unit span length can be written as
d L =iP U
dD
=
jp
waw cw
u,. C,,. C,
',.
d (,(3.28)
These values are transformed to the x,y,z coordinate system:
= dL Wo# O
=
P
- a DA
cwa-.. tju -
coof..
d D co+d, - L
-.p .Cs , c
=
-
Mc. (
C,,,,
0r
+
-oA Pc W
p ...a,(Zr*-N
,.-
cow O,)
U
+-j Pce~ a,. t4r Oe-
(3.29)
where
e2.
ea
tafy
+-+4v
+
r.
+ur,,(3.30)
and e is the distance (nondimensionalized by wing chord) between
the elastic axis and the quarter-chord line (positive if the
quarter-chord line is ahead of the elastic axis).
Hereafter the small angle assumption may be used for the
wing aerodynamic forces because of the small angle of attack and
angle of twist. Then it is approximated that sin $w =w
cos $ = 1, etc.
uG and wG are chosen to
Perturbation velocities uwF ww jw'
express the aerodynamic forces. The perturbation equations are
then given by
+
_
(
aF+
FL~(F~z).+
(fF5)
M(
~+
(w
+
P.
'Mia
)M '44+(
-)0U4
-%%)Vpq(3.31)
( L4
This study is concerned only with deviation from the equilibrium state. Then the steady state aerodynamic forces (Fx)o,
(Fz)o and (M ) may be dropped from the equations without inconsistency although the blade steady state aerodynamic forces (Pz)o
and (P0 )0 cannot be neglected since they influence the pylon
motion.
Applying Eq. 3.31 to Eq. 3.29, the following .expressions
for the wing aerodynamic forces due to wing motion are obtained
Tr.
+ -L-
Pc~Voa
FX
0
UA
LAW
0
W-W
J- pcw V2 aw
Rr
{j
-r
4
Cb.
0
P's V a1wolw
1)~4
(3.
.5,cw
z
-jo
T
Va-f O~'wo
32a)
*
cw -VTc.D
w
0
-
P C
v
c l ow(
3 . 3 2 b)
i PCrV(,+
M
aw)
=FV(COtCMOIV.
+cWOxi)
U)
jf,
uw
VAW
L
0
.1
0
4±
u,
ur.
0
~pc:v
2
(c, + a.~
+4
-pcVV CC... + a,
z)
-rq
0
Pc4 VC.,+
Cgq or'w.+ a~,
z)
(3.32c)
where the angle of attack in equilibrium state is given as
YOf = (,
(3.33)
SECTION 4
MODAL ANALYSES
4.1
Variational Functional
In order to apply Galerkin's method the variational functional is derived from Eqs. 2.4, 2.5, 2.6, 2.7, 2.8, 2.10, 2.11,
2.12, 2.13, and 2.14.
IT
If
=
c
E
2Ga+ t
acoo'
k
c
V
-L4j{T(
+Z
au
+
+ Zi;
rf
+(~~2.
rlt is--f.(Ix., 40t
,4a44.ck)~~
W
-6 ~M.f~4
2
014t))
zweJ
I~~
+.1~
-l
42:~+ 2a 4lp
0r
+ z
+
r
o
LL
0,
-I.---M
N4
lr-L
Wt-
,
A4+~
'Si
a;&
+
+r~ZA
A
4 .
I~rMC4YC1.}4V
Pj
'L
+
V
(L
;r
fo
4.2
Free Vibration of the Blade and Wing
In preparation for modal analysis of the basic equations,
the free vibration of the rotating blade and the cantilever wing
will be considered. It is very advantageous to use the natural
frequency of the free vibrating beam to represent the quite complicated stiffness terms of the beam.
The variational functional for free vibration of the rotating blade is derived from Eqs. 4.1, neglecting aerodynamic
forces, pylon and wing motion.
=S -
[
1
(EI
A
Gs 4(Et)a CAWI-)4V
(E 1),
--
Aim- es c
(Ei)
-T(
+z;2
T(
+
4
} ]
lJ )CUPj
(4.2)
Assume a series solution in terms of normal modes which express the
coupled motion of inplane and out-of-plane motion
AYLX
}
(4.3)
Substituting Eq. 4.3 into Eq. 4.2 and applying Lagrange's
equations as follows:
-Olt
(
T) 0
4) in zin
yields
+ {t)co +(EI)a co
(s
(4.4)
EX), 21-- ( E T)aa) A;r mOa cOO (98a
+
A.W4L 41W 1
e
1
1
4ra
i
dV
+TVZV±ltv
jy
Xj=
+
a
(4.5)
where X. denotes the jth rotating undamped natural frequency of the
J
blade. Simplification of Eq. 4.5 will result from use of the orthogonality condition (Ref. 15) of the normal modes which is expressed
as
-t(WZW; + v!v
r={
L
j
j4 L (4.6)
The amplitude of the mode shape is normalized to unity as
L+
(4.7)
Hence, Eq. 4.4 results in
(El)
A
+O
(Es)e
CO-e
+ I(sI)c .0, J9i~
8 + (EI)s Ajoe.L
Jr(
( dSV(
,YaL
j
.)
Y,'l
r
df)
Idwi
1X
!"+
vA + TT'Tj4W44
J*
0at
*IV vri cir y
144.8)
In the similar way the free vibration of the wing will be analyzed,
treating the rotor and pylon as lumped masses at the wing tip. The
wing deflection can be expressed as
UVt
j
The, plonmoionis
escib
y
dU
=r]%c~}(.0
where
-&
(4.11)
The wing coupled modes are normalized as
+)
+ NM+(&#)L+
+
2(
+
+
(4.12)
=1
Hence,
AA +CE Iv)s coan
+ (EAdr
Elw)c --
4
+
w)
Otq
+(EIla
w)
F)
A.,6Ow.C
19W
d
(
(ate 64A
aI
+ Id
ail
j=j
(4.13)
L.*j
where A
4.3
is the jth wing undamped natural frequency.
The Variational Functional Described in Modal Form
The simplified variational functional expressed in terms of
natural frequencies is
TT
'I
(2Xjij)
£~ u!
4]
+
r
+
*415u
'p
.0bI
~(
0
4
4.,
&4.)
IL
C.
J~
&
(CZ
al
e
+
At )(
T
(Zvi
+AW
where W
forces.
e
'eL
C-0640.
+4z!
(.14)
denotes the work on the wing done by rotor aerodynamic
Details are discussed in Subsection 4.5.
The principle of virtual work is applied to Eq. 4.14 to
obtain the modal equation for the blade and wing.
For the blade the modal equation can be expressed as
+
x)
+V
'
-4
O.
0*f
?4V.
-
I~@4bt
2S tfj 11CO4M.
-Wj
+pooc~
(4.15)
where
TT
=
R
-
S.M
(t= o,1)
pi..L[
R
s/I'
([ =0, 1)
Pa
(4.16)
=57
(4.17)
For the wing modal equation can be expressed as
aLu+ z
Li
z
14.
dynmi
foce
where
deot
derved
promic
derve
s
orkfc
from the work W.......Jl
of o in
n
beau
dysnmi forcesi divfculte
4.4
otoirN-Rotaingcoordinate
4.8t
roto
the
bai
app earaetowin
workt thsW
emwilte
te
eidc
coe
force
aero
ofiins
odsrb
h
motion in the non-rotating system instead of the rotating system
(Ref . 16) .
(a) For the hingeless rotor, the Fourier coordinate system
is described in general as
+ OrE(-I)"(4.19)
are cyclic
is the collective motion, Qjc(m) and Q
where Q
motion of mth order, defining tilting or warping of the rotor
plane and Q.j is differential collective motion (only occurring
for even N-bladed rotors).
Since many recent proprotor designs have three-bladed rotors, this study is concentrated on the odd N-bladed rotor model.
Therefore, the last term Q, of Eq. 4.19, the differential collective motion, will be dropped hereafter. In addition to this,
the higher order terms Qjc(2)' Qjs( 2 )' ***jc(m)' js(m) are not
coupled with pylon motion. They represent internal rotor motion
and usually make only a small contribution to rotor motion.
Therefore they are also truncated in this study. Hence,
j
4..
c
-
(4.20)
By inverting the system of Ea. 4.20. the new dearees of
freedom of the non-rotating systen can be obtained:
&j
QJ
Y
2 14
I
Jfl
O-0CI
(4.21)
s --
The same approach may be applied for the rotor generalized force
due to periodic aerodynamic forces.
0
N
(2)
(b)
For the gimballed rotor,
special care must be taken
since it has both collective mode shapes (symmetrical modes with
clamped boundary condition at the root) and cyclic mode shapes
(antisyrimetrical mode shapes with hinged boundary condition),
while in the hingeless rotor case both collective mode shapes and
cyclic mode shapes are the same. Therefore, collective motion is
described with collective mode shapes W0, V0 and cyclic motion
J
J
with cyclic mode shapes Wi, V .
J
J
{
+ 5MC.7J
(4.23)
This expression is consistent with the derivation of equations of
motion up to now, if mode shapes W., V. of the coefficients of the
blade collective motion are replaced by the collective mode shapes
W., V. in Eq. 4.15 and 4.18, and those of cyclic motion are reI i
c
c
Therefore, the derivation
placed by the cyclic mode shapes W., V.
J
J
of equations of motion hereafter is based on the hingeless rotor
case, and those equations can be applied to the gimballed rotor
case as stated before without confusion. The coefficients associated with collective mode shapes are denoted by superscript 0
and those with cyclic mode shapes by superscript c.
The truncation of higher order terms and of the differential
collective motion in Eq. 4.19 is also applied to the gimballed rotor case by the same reasoning as for the hingeless rotor case.
The orthogonality condition for the gimballed rotor case is
different from the hingeless orthogonality condition. The entire
rotor system including periodic functions sinn and cos~n must be
considered. For example, the orthogonality condition between the
collective mode and the cyclic sine mode is described as
38
iR
Rg)
W~.-vv
'0
%
L~~~..dat~
(4.24)
= C
Since $h is the phase angle defined in Eq. 2.3, the summation over
n=l,
4.4.2
...
N of sin*n becomes zero.
Pitch-Flap Coupling in the Non-Rotating Coordinate System
Pitch-Flap Coupling is expressed in Eq. 3.25 as
PF
(3.25)
a&
The substitution of mode shapes into this expression
gives
+44]
Gi, emcA.
OPI:
VC I
00,
CAIJ;3A
K
(QZ 0
+ Qe c
e4
cki+
Q;.
A
CIO,
1,
(4.25)
where
P'= --
T-S
(4.26)
For the hingeless rotor case considered here, K . is always
zero because the slopes of blade mode shapes at the root are zero,
due to the clamped boundary condition.
For the gimballed rotor case, Eq. 4.25 is rewritten as
3
*
(4.27)
where
am.
..
(4.28)
From the definition of the collective mode shapes, K ? is
pi
zero. Therefore, for the gimballed rotor case the pitch-flap
coupling contributes only to the cyclic motion of the rotor, not
to the collective motion.
4.5
Aerodynamic Forces Described in Modal Form
4.5.1
Blade Aerodynamic Forces
. From Eq. 4.17 blade aerodynamic force is described as
2e&
5 (i.Vj
?a + w
d)
(4.29)
The application of Eq. 4.22, accompanied by the substitution Eq. 4.21 into Eq. 3.13 yields
ess
(2jt P)
L G
ID
i-+
0
. 5
+
-++
K "kpQ
cQvj ~
(4.30)
0b + ,0
, +%
-141;
ZO
+(4.31)
+
+p
-
G
+-~~ 477j
(Pr,
+0
+
(4.32
where
GjiZ
( P:e,
= fI
S W V4+
Q.
+ FX. i& a.)
2
E
±
vri~Ax
GE) =b ( ,
C
-
va + Fez
%Pa=
vi4
eCi =u
-j
<m. +
=P.O/
/(
(u;)/.i
k
=(u)
Ff3
+ rZ-?)xa(
.
'14C) >
(4.33)
['(VI
A
n.
]
jLQiJ
/[i~'. ~~']
[~P'r~i~l]
(4.34)
4.5.2
Wing-Loading Due to Blade Aerodynamic Forces
The virtual work done by wing loading due to blade aerodynamic forces is described as
~Wa
V~r]
=V
+ VPS
-.?aA
42.44 +442
-A
(P
x
ceg
vP)
4M+
+ *fePO+
Yfea
f
,
e04.,-
s6s4.
Paq
(4.35)
The application of the blade mode shapes and non-rotating coordinate system gives
0
rAT
SW
o/~'z"
V
i
0
K+[A6]JQ;J
+ Y, I4AJQL
OV
6.
(4.36)
where
K
is
an integral operator defined as
I\
d'.
K
(4.37)
and integrand matrices are
IFt
0
--4
o
-1F
0
[A ']=
(ripFI
-0
-
.Fes
- ~
-
- Ix(-F.+F FI,
'0
-
.
-
,)
--
xFa
- F1
0- 00
0
0
(AW]-
0
o0
o
2129
C0'
0
ivc
, o
-+
e
1. !1:,&,
o
i
7LF
-
o
i-
0
2
0
I~m
-
e
o
I-
(4.38a)
-~Fe~41jo
-
-
-
10
-4---
0I1
.(Fr---r
x
+.
-
o
1
1 0)
(4.38b)
V-WTI.
[4]
Q,rv + F,
0
0
(4.38c)
I
Ira, v~l
.I
±~e')
-~
F ,
0
Fe,
-1K;
Fat+,
ox
R
0
- 0
---K-~ ~
r
-ae
(4.38d)
-- --I o I 0
0
[A~1
F(
I.-
7r4
-
-
1
i
F
-_o_
I
I
0
I o
4--
I-d1F
_i __L~
*
(4. 3 8e)
4.5.3
Wing Aerodynamic Forces
The virtual work done by wing aerodynamic forces is expressed by
=W {U,
TbJg
(4.39)
The displacements uw' ww', pw are described in terms of the mode
shapes as
P. a,
(4.40)
where
r~
%_ P. .%
(4.41)
Then, the substitution of Eq. 4.40 into 4.39 gives
SSa. - -h
[
A~
4441{44/V
OIL
Gus
(4.42)
where
LA~L
A)
[s] [A] 3]dg
L
Am]
[3]41
[S]T [Awl
A
( .5T4.43)
The matrices Aw, Aw and AG are derived from Eq. 3.32 as follows:
-{pcVa.([+c
(Ak]
-o
PGWVCCh.+
11
aZ
4-C,(,
10
c.
p c
V a
-
CbM9fwr
I
0
(4.44a)
0
[Aw>
0
-
P
cV
%C
-
S-p
(C(4+4a.4
OW
L
j (4.44b)
4a
CuVa
o' of,. MPC'oiV
q
I
oXW.
PC;V(C4 Gqd~
OCVa(
+Cw
i
p0.a
+C..
.
t
0
0
0
0
j~I0
4.6
2 Co
~
(4.440)
Final Equations of Motion
In this part the final equations of motion will be derived.
They will be expressed in general format, so that any number of
mode shaDes both for the blade and wing can be chosen to describe
the rotor or wing motion as accurately as required.
4.6.1
Equations of Motion in General Format
The substitution of Eqs. 4.20 and Eq. 4.10 into Eq. 4.15
gives the jth blade modal equations of motion. Note that time
derivatives are normalized by rotational speed as in Eq. 3.24.
For the jth blade mode
00
alJQ 48EM4(Q
4[w5LT~a~
+~L~I[T]a)
48
{)
(4.45)
where
Q
{6}
(61=
j
{O
/
c
(4.46a)
'o
CMJ=O-
2.o
(4.46b)
[M44
(-A-)
I.
4)
o
b
o
-V0
-vici
[Cj]
0
(4.46c)
O
a.
0
0
oV0jWj --j~J
I
(4.46d)
0
0
o
0
2twj
0
o
0
o
2v.
01
oJ
(4.46e)
(4.47)
Cpj +Pejls
and the matrix [T] is defined in Eq. 4.11.
Substituting Eqs. 4.21 and 4.10 into Eq. 4.18 yields the
wing equation.
For the wing mode
{00~ iIwJa}
4[w~j{c4
[T
-i
[ M]
[
K
I
[T]T[cI[I]{Qjj
= IF
(4.48)
where
f
0a &
$l
-5.,
a3
[Wa- NI
L
(
0
j
[W
(A)Z
0
diagonal
matrix
(4.49)
No 0
[K]=
0il
o .
Loading matrix{F) for the wing consists of aerodynamic
forces f due to the rotor, and aerodynamic forces g due to wing
motion itself.
-f.
IT:
+ 32.
=
(4.50)
Similarly, aerodynamic forces are derived.
For rotor aerodynamic forces
(Pj +*
=~z[3. {o +2:[A('J. Q
[T] ja + [A~jj [T] c
Lj
a jI]
fA) ]jY
[A.
where
(4.51)
0 O
Cj 0
Gi
(4.52a)
0O
AG!
G
L0 G%wet
0
I)O
--.
jz
Kpi G7 vi J
(4.52b)
C .
A
Is
0 0 -tGz
-Jw-
0 wi
o
0
0o
0
0
- Pt Q5
(4.52c)
0
=8
o
-Xq
(4 .52d)
OA.q
[A:'Jj =-*Iis
3hz;
A;q
0
0
0
. o
0
(4.52e)
The excitation vector {e} is expressed as
VE
V
(4.53)
e3
0
8,5
For wing aerodynamic forces
From Eq. 4.50
lip-)= P+ 3)
(4.54)
Loading to the wing due to rotor aerodynamic forces fi corresponding to a is expressed from Eq. 4.36 as
K}
[T] {cal
=KTf[ar
K[TieAr
+~
K[TY [Aa2)[T]I al
IQL
K [T]'r
[A]
+,
+
J
(.55
Loading due to wing aerodynamic forces gi is described from Eq.
4.42 as
S
4.6.2
4
I+ A
+ [A?
(4.56)
Equations of Motion for Nine Degrees of Freedom
For a typical case, nine degrees of freedom are considered
here. These degrees are rotor flapping motion (collective flapping motion Q1 0 , cyclic flapping motion Olc and Qls), rotor lagging motion (collective lagging motion 0 2 0 , cyclic lagging motion
2c and Q2B), wing vertical bending motion a1 , wing chordwise
bending motion a2 and wing torsion a
3.
These mode shapes are natural modes, therefore they are
coupled.
The final equations in matrix form are
00
[A] jix
+ [8] J
+[c]
J-[o+]jej
(4.57)
where
V
t'=
Gis
V
020
V
(4.58)
Qa3
The coefficient matrices [A], {B] and [C] are 9 x 9 matrices,
and [D] is a 9 x 6 matrix, as given below.
I I
I I
I I
I
__
1(41
I I 71Q
L!A.Jtt
__
I
_LE T
[AP
I I
'Iii
(4.59a)
(4.59b)
[c3=
(4.59c)
(4.59d)
4.6.3. Equations of Motion for Nine Degrees of Freedom
with Uncoupled modes
In order to have a better understanding of tilting proprotor dynamics, a simple special case was considered.
This case
is based on the assumption that the mode shapes are uncoupled.
Therefore, rotor motion is expressed as
The wing motion is also,
E)=
+
a, +I
a
(4.61)
The substitution of Eq. 4.60 and 4.61 into Eq. 4.59 gives
the following matrices for the same nine degrees of freedom
equation as Eq.
4.57.
(Al3
o
1
o
o
0
o
0
0
0
O
o0
o
0
0
0
0
0
0
0
\
0
0
0
0
0
0
Q
0
o Lo
1
0 Ww
-\Yi o
0
o
o
u -
A
0
o0
-w,,
C
-
0
03,
0
0
-WA
o1n0
o
.
(4.62a)
(4.62b)
94+k 4
C kp Le a~o
Kf.Le
O ~L1
0
0
-XLj
a
0
0
a
o
o
o
l) i iL53
0
W,
A Lq
0
(4-
0
0
ALrjT
a(4
va
o~~~~~
N~iL.
0
0
£L13
o
o
o 0
o
0
Q
0
0
?F
1
IT
+jF[jL
N
1
4
a (4C
04c
(~.
-
~Kp~J~
(4.62c)
Vew
0
0
0
0
F.e
O0
0
-FT
0
L0g
O0
C)
O0
LTi
L~
0
Leg
O
O0
Lai
O0
0
0
+ C.,,
O
CNwe,
0a
Lei
4W,
Ti:.
-- (Y1X
+c,
-1- (2L-r1
N3?,~
L2
..
C433
(4.62d)
New symbols will be defined below, but all these symbols
can be derived by applying the uncoupled mode shapes to the general
equations.
Pylon Motions
a
=
VC
a
.12.
at
g=L
at
b=L
at
d-L
at
S=L
d~
1W
__
(4-63)
at
Therefore the matrix[T]ift Eq. 4.10 is described as
a
0
[T]=
o
v'
0
O
0
0
00
0
(4.64)
Aerodynamic Forces
(a) Rotor Aerodynamic Forces
gix
(,z)
Q1GU
(Pia) WIV A(4.65a)
1
L1t .
61x
-
Rg
-~
. (e
L
(
- -L 6 Ta
r,
i
A
(4.65d)
.X d
<s~ax
L -(~e,{ Fe,. iAL(4.65e)
where in general F,L expresses the forces corresponding to the
direction of flapping motion and lagging motion respectively. Subscripts 6 and ; describe the mode shape contributions to the aerodynamic forces; T and P show the derivatives for the perturbation
expression with respect to inplane velocity UT and out-of-plane
velocity Up; superscript + expresses the moment at the blade root.
FT=J-l'
(Vfo)4Z
+
L.
iI
Le =045
(
YIS(F) XX
.. ) AX
(e)
L
At
[-L
LT
-N
2
I F +
a
+
R
M-*;E3
R
/V
We-
+
LP -
-N
2L
Ia Sc(Fez)x~x
(4.66)
C4
LT
WC
_qC "W
F-T W
-2
r'qC.%..*
C621
=
16%
V
Lf,,,
C42.5
-C
+
-a LT + V=i *' 4]
+
13
we'(4.67a)
CdAl=
o-
CQa.= - N
2
.4-
F
L
f we,
.
(b)
=-4 ~(Fo-(L-)(.
(4.6 7b)
Wing Aerodynamic Forces
ca,
- -
A,(o d j
Ca,,---A;
- 12. V,1
d
(L
=CLa=A
As
'
(4.68a)
C
3
Ai,,4
-1
Caa=
Ca=-
-
,
ArhtL a
Aw
ce=
c.,
=
A .,1 a
CA
C,
As(,1du(4-68b)
-
-3AEA
Ce
vjg
4
A4.,and
A= (4.68c)
, A
where A.
w
IJ
w
, and A
iJ
G..
are components of matrix [A.],
w
[A ],
w
1]
and [AG] respectively in Eq. 4.44.
Pitch-Flap Coupling
The pitch-flap coupling coefficient is expressed as
The rotor collective mode shape has zero slope at the root.
There-
fore, pitch-flap coupling appears only in cyclic motion.
Natural frequencies
X
blade flapping natural frequency
X
blade lagging natural frequency
A
4.7
:
wing vertical bending natural frequency
A2 :
wing chordwise bending natural frequency
A3 :
wing torsion natural frequency
Equations for Gimballed Rotor
As discussed in Subsection 4.4, equations of motion
4.45 and 4.48) are applicable to the gimballed rotor.
(Eqs.
The signi-
ficant difference between the gimballed rotor and the hingeless
rotor lies in the application of the blade mode shapes to the
rotor motion.
The gimballed rotor has independent collective
mode shapes for the collective motion and cyclic mode shapes for
the cyclic motion.
Therefore, the above distinction is found in
the following matrices:
For blade equations
0
(1)2.)
I NIQIj =
-I
0
(C.)
o
- I
0
O W.
~
I
Nij3 =
O0y
0
J
0
-W514
J
(4.70)
0
O
0
lcjO0
O
0
2.0
0
'5)
211,*
O
O
J
For Rotor Aerodynamic Forces
4)
0
EAt]
o
0
o
l
[Act ]
=
o
o04
(4.71a)
( t ) Q (@o
IT.8
o
0
CC.) (C
-Pc
qgvj
(4.71b)
0N*
r-,
(a)-
o
o
-4-1cj
-CC) -S
1:A5;a~.II.
J
-
o
~~IB
<4.
oCO
I9
Je. i
3
0
o
7ld)
~m-~ 0 C(0)()1
0
A
.
(4.71c)
01
0o
0
o
00
o
0
(C)
0
(C
0
o
- j-
o
)J
v;
(C)
0
0
J
0
(4.71e)
LA6c
-±VFa + $V
0
0
0*
=
-iT
0
L)
0
50
ex
Fat+)
Val,
F
2zP.
rl
+
OX
0
(4.71f)
AcQ-4L)z I =
0
CL)
0
(0I
0
4
+
L
rlx
kZ x fir3
pe
14
~j3
:~.VNc
+
.(0
gA
p , ps.) -
(C ~~
v (LI-Q
FIX (-VZQ*
t Fil +jw
.L -
-
c.,.)
(4 .71g)
SECTION 5
NATURAL FREQUENCIES AND MODE SHAPES
OF THE ROTOR AND WING
5.1
Rigid Rotor
In order to obtain the natural frequencies and correspond-
ing mode shapes of the free vibration of the rotating blade, the
equations of motion are derived neglecting aerodynamic forces and
pylon motions in Eqs. 2.3 and 2.4.
a?~ [l
c A-z es + (Eacoo'8a
#aabr
+
(E1
-
E~
4+
(T
8a b
*6wo
- M0(51
These equations of motion are rewritten as a usual eigenvalue problem to find eigenvalues and/or eigenvectors.
Boundary conditions for the rigid rotor are
at the root:
aJ
-0Ur.
(5.2)
at the tip;
0
.
--C*o
=0
(5. 3)
As a typical hingeless rotor, the Boeing rotor is chosen in this
study. Mass, stiffness distributions and built-in angle of twist
are shown in Fig. 8.
Nonrotating natural frequencies of the blade are shown in
Fig. 9. Almost no influence of the collective pitch change on
the natural frequency is found in the case of the rigid rotor.
In Figs. 10a and lob, the first and second natural frequencies of the rotating blade are shown versus rotor rotational
speed and inflow ratio. Inflow ratio is related to the collective
pitch as described in Eq. 3.9. At normal rotational speed (in
this case n = 386 R.P.M.) the natural frequency variation due to
collective pitch change is shown in Fig. 11. Typical mode shapes
up to the 4th mode are shown in Figs. 12a through 12d. Blade mode
shapes are normalized with respect to the blade radius at the
point of maximum deflection in either out-of-plane or inplane
bending.
5.2
Gimballed Rotor
In the case of the gimballed rotor, rotor motion is expressed by collective modes and cyclic modes (in Ref. 17 and 18).
Hence, based on Eq. 5.1, the eigenvalue problem should be solved
with boundary conditions for collective modes and cyclic modes
for the powered flight, respectively.
Boundary conditions for collective modes are defined to
yield symmetrical modes for flapping and lagging. Therefore,
at the root:
(.
a i
b'y
(5.4)
-
at the tip;
=im
0
*-&= 0
(5.5)
Boundary conditions for cyclic modes which consist of antisymmetrical modes for flapping motion and symmetrical modes for
lagging motion are expressed as:
at the root:
-=
(5.6)
ar
at the tip:
O__
o
(5.7)
As a typical case, the Bell design is considered here. Mass,
stiffness distributions and built-in angle of twist are shown in
Fig. 8.
Blade nonrotating natural frequencies for the gimballed
rotor are shown in Fig. 13. In Fig. 13b, the first natural frequency for rigid body flapping motion has non-zero natural frequency due to the rubber hub spring, which is intended to increase
control power and damping.
In Figs. 14 and 15 it is shown that collective pitch change
has a large influence on the natural frequency variation except for
the first cyclic mode natural frequency (rigid body mode). Mode
shapes for the gimballed rotor are shown in Figs. 16 and 17. The
normalization system is the same as that of the rigid rotor.
5.3
Wing
The equations of motion for free vibration of the wing are
the same as Eq. 2.5, 2.6, and 2.7 when the aerodynamic forces F ,
Fz, and M are eliminated from these equations. Boundary conditions are as follows:
(a) at the root:
-=0
0
(5.8)
(b) at the wing tip:
P
d
-4[
L~I)e
*")
Is=+Iw)
A.}
e
--
o.4Aj
+ N HS4C.4 NNMSO
~w
w-{
P
+ NMy
N
(5.9a)
0Z
+2.s
j~
VB
(5.9b)
=0
(5.9c)
+r
*aP
Ir '.)
*6
+U.
4NM
8
+
W)
-iq
rx
o
(5.9d)
't0 4
-{(etic
+ N Iv,
LI(V:Ew),
-
A~%G
+ (Er,,),
-(UJ 4A 44..SirCoO W all
-
C00Ag:
=L
(5.9e)
The relationship between pylon motions and wing deflections is
defined in Eq. 2.8. The rotor and the pylon are treated as
lumped masses at the wing tip in these equations; as in the actual
structural dynamic test, the proprotor blades will be removed and
replaced by equivalent weights. Note that in Eq. 5.9 the blade is
treated as a lumped mass which has equivalent weight and equivalent
blade flapping inertia IB* Thi-s flapping inertia leads to lower
wing frequencies, especially in torsion, than when the blade is
treated as a lumped mass with equivalent mass and without equivalent blade flapping inertia.
The same wing is used for the Boeing and the Bell design;
mass and stiffness properties are shown in Fig. 18. The differences appear in the pylon and blade mass properties shown in Table
1. No built-in angle of twist, dihedral angle, and sweep angle are
considered here.
Natural frequencies of both cases are listed in Table 1, and
mode shapes are described in Fig. 19 for the Boeing case and in
Fig. 20 for the Bell case. Wing mode shapes are normalized with
respect to the wing semispan at the point of maximum deflection
in either vertical bending or in chordwise bending. However,
the mode shape corresponding to the third natural frequency of
the wing, which is predominantly one of wing torsional deflection, is normalized at the point of maximum torsional deflection.
SECTION 6
- PROPROTOR DYNAMIC CHARACTERISTICS
6.1
Introduction
In this section the eigenvalues and frequency response of
the system will be discussed. The results are compared with those
of Ref. 13. The analyses -were conducted for two types of proprotor design: the hingeless rotor Boeing design, and the gimballed
rotor Bell design. The case selected to be investigated is cruising flight in the airplane mode for each design. The data for the
calculation are listed in Table 1. One should notice that the
equations of motion shown in Eq. 4.57 were derived based on massnormalized coupled modes. However, for the sake of aiding physical understanding of the frequency response and eigenvalue analysis, those results are based on physical mode shapes, as presented
in Figs. 12, 16, 17, 19, and 20.
The primary differences between the Ref.
13 analysis and
this report are tabulated below.
This Report
Ref.
13
Calculated
Data
Experimental
Data
Mode Shapes
Wing
Rotor
Coupled Modes
Coupled Modes
Uncoupled Modes
Uncoupled Elastic
Modes for Inertia
Terms; Rigid-Body
Mode for Aerodynamic Terms
Structural Damping
No
Yes
Natural Frequencies
The case studies done in the present study are tabulated
below:
Degree-ofFreedom
Mode Shapes
Blade
Wing
Analysis
9 Degree-of-Freedom
18 Degree-of-Freedom
Rigid Body
Uncoupled
Elastic
Coupled
Coupled
Coupled
Coupled
Coupled
Coupled
Eigenvalue
Eigenvalue Eigenvalue
and Eigenvector
Eigenvalue
Frequency
Response to
uG1 G, wG'
81s
The general symbols Q , Qjc' Qjs, and a for blade and wing
motions are convenient for the theoretical derivation of the equations, but from the physical understanding aspect, it would be
better to choose other symbols. For this section, except for the
eigenvalue analysis, the following system will be used to express
blade and wing motion for the 9 degree-of-freedom system: flapping
and lagging motions which are designated as $ and C, respectively,
and wing motions are ql, q 2 , and p, which express the vertical
bending, chordwise bending, and torsion, respectively.
The correspondence between these sets of symbols is tabulated below:
New Symbols
a
o
0lc
Predominant
Flapping
QQ2o
Motion
Qlc
Q2c
O1s
2s
Qi
Q01
02c
Qlc
Q2 s
Qls
0ls
0
o
Clc
General Symbols
Bell Design
Boeing Design
Predominant
;Lagging
Motion
ls
q,
Wing Vertical
Bending
a1
a
q2
Wing Chordwise
Bending
a2
a2
p
Wing Torsion
a3
a3
6.2
Eigenvalues and Eigenvectors
The system stability characteristics are shown in Figs. 21
through 24.
6.2.1
Addition of Higher Mode Degrees of Freedom
The eigenvalue variation with number of degrees-of-freedom
considered is shown in Fig. 21. The 9 degrees-of-freedom include
Q10' lc'
s
s1' 20' 2c' Q2s, a1 , a2 , and a3
o' s1c'
ls' Co'
Clc' ls' q1 , q 2 , and p) corresponding to the blade's first two
natural frequencies and the wing's first three natural frequencies.
This system has 9 eigenvalues and 9 corresponding eigenvectors.
The 9 eigenvalues are denoted by their frequencies, and the participation of the degrees-of-freedom in the eigenvectors is indicated
below:
Present
Eigenvalue Symbol
Participation
Low frequency Q 1
Q0
Ref. 13
Bell
Boeing
-1
C-
4+1
Collective
Q
High frequency Q
5+1
Q2
Low frequency Q 2
4-1
-1
Q0
Collective Q
Q
High frequency Q2
C+l
8+1
a1
The first wing
frequency
The second wing
frequency
The third wing
frequency
g1
q,
q2
q2
p
p
a2
a3
In the 18 degree-of-freedom system two higher elastic
modes for the blade and three higher elastic modes for the wing
are added to the 9 degree-of-freedom system. The same designation system as used for the 9 degree-of-freedom is employed for
the higher eigenvalues. The results of the 18 degree-of-freedom
analysis show that the addition of more degrees of freedom does
not substantially influence the basic system eigenvalues since
the added degrees of freedom have large natural frequencies in
comparison with the original values (see Table 1). The 18 degree-of-freedom system eigenvalue locations are shown in Fig. 22.
6.2.2
The Sensitivity to Mode Shape
Eigenvalues and damping ratios (the fraction of the critical damping) for three different types of mode shapes for the
blade are shown in Fig. 23. One is the rigid-body mode (with
spring restraint at the root),
another is the elastic uncoupled
mode, and the third is the elastic coupled mode, while the wing
mode shapes considered are restricted to the elastic coupled mode
shapes only.
The results tell that mode-shape types make negligible difference in the frequency of the eigenvalues; however, they influence the damping a lot.
In the Bell design, there is almost no difference between
the rigid mode and the elastic mode in the damping ratio. This is
due to the similarity between the rigid-body mode and the elastic
uncoupled mode. In the elastic coupled mode, a slightly higher
damping is obtained.
In the Boeing design, each mode-shape type gives a different damping. For the first natural frequency mode of the blade,
the rigid-body mode calculation is conservative rather than that of
the elastic coupled mode-shape type, and for the second natural
frequency mode, it is nonconservative. This is because the first
mode has both positive out-of-plane and inplane deflection, while
the second has a positive out-of-plane and a negative inplane
deflection as shown in Figs. 12 (a) and (b).
The corresponding
positive out-of-plane deflection of the first mode increases the
damping, while the associated negative inplane deflection of the
second mode reduces the damping. This explanation also holds for
the Bell case. The damping of the second collective elastic coupled
mode of the Bell rotor is lower than that of the rigid-body mode.
The second collective mode has a positive out-of-plane deflection
and a small positive inplane deflection. However, the rotation
direction of the rotor is different for the Bell and the Boeing
designs.
Therefore, the positive inplane deflection mode of the
Bell rotor is physically the same motion as that of the first
mode of the Boeing rotor. The coordinate system related to this
is shown in Fig. 5, and the contribution to the damping of the
coupled mode shape can be seen in the term G.. in Eq. 4.33.
Ji
In conclusion, the use of the elastic coupling mode affects
the damping significantly. The coupled mode with a combination of
forward out-of-plane deflection (upward in helicopter mode) and
inplane deflection opposing the rotor direction of rotation, or
vice versa, increases the damping over that of the rigid-bodymode calculation.
The coupled mode which has forward out-of-plane
deflection and inplane deflection proceeding with rotor direction
of rotation, or vice versa, decreases the damping. This holds
only for the first
beam bending mode; in other words, the coupled
mode without nodal points between the root and the tip.
6.3
Frequency Response
The frequency response analysis is dealt with in this subsection. These calculations are all based on mass-normalized
coupled modes for both the blade and the wing. However, for convenience in physical understanding, the results of the frequency
response calculation are presented in terms of length-normalized
mode shapes in Figs. 25 through 32.
6.3.1
Frequency Response to the Gust
Frequency responses to the vertical gust uG, the lateral
gust vG, and the longitudinal gust wG for both the Bell and the
Boeing designs are shown in Figs. 25 through 30.
As a whole, the behavior of the frequency response is quite
similar to that of Ref. 13 in spite of the differences in the theoretical model stated in Subsection 6.1.
The detail characteristics of the frequency response are
discussed next.
6.3.1.1
Blade Collective Motion Response (a ,Cr)
Collective motion responses have a close relationship with
the wing motions (vertical bending q1 , chordwise bending, q 2 , and
They have strong resonances with these modes. In
torsion, p).
the case of uG and vG inputs, the static response (w equal to
zero) and the lower-frequency-range response are negligible. However, the response to the longitudinal gust wG has a significant
static and lower frequency range response. Resonances of the collective modes occur in the response to the longitudinal gust input.
Comparing the Boeing design with the Bell design, the Boeing has
a larger response in collective responses to each gust input.
6.3.1.2
Blade Cyclic Flapping Motion Response (Slc and Sls)
For the cyclic inputs (uG and vG) there are not strong
resonances between cyclic flapping motions and the wing vertical
bending motion (ql). In the upper frequency range, the high frequency flapping mode (a+l mode) has a significant resonance appearing in the flapping motion. In the lower frequency range,
there is a resonance of the low frequency flapping mode (6-l mode)
for the Boeing design and the low frequency lagging mode (C-1 mode)
for the Bell design.
For the collective input (wG) the response of the cyclic
flapping motion has resonances with the wing motion modes (q , q '
1
2
and p) in both designs.
6.3.1.3
Blade Cyclic Lagging Motion Response (Clc and
ls)
In the Bell design there is an evident resonance in the
cyclic lagging motion response (Clc and Cls) to the vertical gust
input (uG) in proximity to the wing vertical bending mode (ql).
To the lateral gust input (vG), a low frequency lagging mode
(C-1 mode) resonance appears in the lagging motion response.
In the Boeing design there cannot be seen such obvious
resonances in the response to the vertical and lateral gust inputs.
The static response is much larger than that of the Bell design
due to the soft inplane design (the blade lagging natural frequency
is less than unity).
For the collective input (wG), the responses of cyclic lagging motion have resonances with the wing motion modes (ql, q '
2
and p) in both designs.
6.3.1.4
Wing Motion Response (ql, q 2 , and p)
The wing motion includes resonances in each wing mode.
Although the response magnitude is quite large, it is expected to
become rather small if structural damping is included.
6.3.2
Frequency Response to Control Pitch Angle
Frequency responses to longitudinal cyclic pitch input are
shown in Figs. 31 and 32.
SECTION 7
CONCLUSIONS AND COMMENTS
7.1
Conclusions
This study has been devoted to the development and evalu-
ation of a theoretical model of the proprotor on a cantilever
wing, operating in normal cruising flight. This theory expresses
the wing and blade motions in coupled form, and can include any
number of mode shapes required to express the motions accurately.
It has been applied to an investigation of the dynamic characteristics of the Bell and the Boeing designs.
Based on the theoretical results included in this study,
the following conclusions may be stated:
(a) The choice of mode shape (rigid-body mode or elasticcoupled mode) affects the damping significantly. The
dependency of the damping on the mode shape can be estimated for the first beam bending mode. The blade inplane
deflection opposing the rotor direction of rotation,
accompanied by the forward out-of-plane deflection, increases the damping, comparing it with the rigid-body
calculation. The inplane deflection proceeding in the
rotor direction of rotation decreases the damping. The
mode shape has little influence on the frequencies of
the system.
(b) The results of the frequency response are quite similar
to those of Ref. 13, in spite of the difference in the
mode shapes. The amplitude of the response is slightly
different, since structural damping was not included in
the present calculation, and the mode shapes used were
different.
(c) The analysis of the eighteen degree-of-freedom system showed
that the higher-frequency degrees of freedom have small
influence on the basic degrees of freedom.
7.2
Suggestions for Future Research
A direct and useful extension of the present study would be
the stability analysis of the proprotor aircraft, with respect to
air resonance and flutter. This theory will be very powerful because the eigenvector components can be compared directly without
any adjustment between components. Mach number effects of the
blade should be included for the flutter analysis.
Stresses or bending moments of the wing or blade can be predicted from the motions of the wing and blade obtained from this
analysis. In addition, this analysis may be applied to the development of an automatic control device to alleviate the gust response of the vehicle.
REFERENCES
1.
Brown, D., "Vertol Sees Wide Use of Tilt-Rotor Design".
Aviation Week and Space Technology, January 1, 1973,
pp. 52-54.
2.
Ludi, L.H., "Composite Aircraft Design". Journal of American Helicopter Society, Vol. 13, No. 1, 1968, pp. 1-13.
3.
Wernicke, K.G., "Tilt Proprotor Composite Aircraft, Design
State of the Art". Journal of American Helicopter Society,
Vol. 14, No. 2, 1969, pp. 10-25.
4.
Reed, W.H. III, "Propeller-Rotor Whirl Flutter: A State
of the Art Review". Journal of Sound and Vibration, Vol. 3,
No. 4, 1966, pp. 526-544.
5.
Loewy, R.G., "Review of Rotary Wing/V/STOL Dynamic and Aeroelastic Problems". Journal of American Helicopter Society,
Vol. 14, No. 3, 1969, pp. 3-23.
6.
Taylor, E.S. and Browne, K.A., "Vibration Isolation of Aircraft Power Plants". Journal of the Aeronautical Sciences,
Vol. 6, No. 2, 1938, pp. 43-49.
7.
Houbolt, J.C. and Reed, W.H. III, "Propeller-Nacelle Whirl
Flutter". Journal of the Aerospace Sciences, Vol. 29, No. 3,
1962, pp. 333-346.
8.
Hall, E.W. Jr., "Prop-Rotor Stability at High Advance Ratios".
Journal of American Helicopter Society, Vol. 11, No. 2, 1966,
pp. 11-26.
9.
Young, M.I. and Lytwyn, R.I., "The Influence of Blade Flapping
Restraint on the Dynamic Stability of Low Disk Loading Propeller-Rotors", Journal of American Helicopter Society, Vol.
2, No. 4, 1968, pp. 38-54.
10.
Gaffey, T.M., "The Effect of Positive Pitch-Flap Coupling
(Negative 6 3) on Rotor Blade Motion Stability and Flapping".
Journal of American Helicopter Society, Vol. 14, No. 2,
1969, pp. 49-67.
11.
Bell Helicopter.Company, "V/STOL Tilt-Rotor Study Task II -Research Aircraft Design". NASA CR 114442, March 1972.
12.
Boeing Vertol Company, "V/STOL Tilt-Rotor Aircraft Study
Volume II -- Preliminary Design of Research Aircraft".
NASA CR 114438, March 1972.
13.
Johnson, W., "Dynamics of Tilting Proprotor Aircraft in
Cruise Flight". NASA TN-D 7677, May 1974.
14.
Johnson, W., "Theory and Comparison with Tests of Two FullScale Proprotors". Proceeding of the AHS/NASA-Ames Specialists' Meeting on Rotorcraft Dynamics, Feb., 1974.
15.
Bisplinghoff, R.L., Ashley, H., Halfman, R.L., "Aeroelasticity". Addison-Wesley Pub. Co., 1955.
16.
Hohenemser, K.H., Yin, S., "Some Applications of the Method
of Multiblade Coordinates". Journal of American Helicopter
Society, Vol. 17, No. 3, 1972, pp. 3-12.
17.
Balke, R.W~., Bennett, R.L., Gaffey, T.N., and Lynn, R.R.,
"Tail Rotor Design Part II: Structural Dynamics". Journal
of American Helicopter Society, Vol. 15, No. 4, 1970,
pp. 16-30.
18.
Blankenship, B.L., and Harvey, K.W., "A Digital Analysis for
Helicopter Performance and Rotor Blade Bending Mode", Journal of American Helicopter Society, Vol. 7, No. 4, 1962,
pp. 55-68.
19.
Wernicke, K.G., and Edenborough, H.K., "Full-Scale Proprotor
Development". Journal of American Helicopter Society,
Vol. 17, No. 1, 1972, pp. 31-40.
TABLE 1
DESCRIPTION OF THE BELL AND THE BOEING PROPROTOR
DESIGNS CONSIDERED IN THIS REPORT
BELL
BOEING
ROTOR
Type
gimballed,
inplane
Number of
blades, N
stiff
cantilever,
inplane
soft
3
3
Radius, R
156 in.
150 in.
Chord, CB
18.9 in.
14 in.
Lock number,y
3.83
4.04
Solidity, a
0.089
0.115
Pitch/flap
coupling, 63
-15
Collective pitch,
1.25 deg.
1.0 deg.
Lift-curve slope,a
5.7
5.7
Drag Coefficient,
CDo
0.0065
0.0065
0.7
0.7
458 RPM
48.9 rad/sec
386 RPM
40.4 rad/sec
first, X /|Q
1.02/rev (7.78Hz)
0.827/rev.(5.32Hz)
second,X 2/1S|1
1.34/rev
1.32/rev (8.49Hz)
third, X 3/|1|1
4.35/rev (33.2Hz)
deg.
0
6D
Rotor rotation
direction, ?
Inflow ratio, X
Rotational speed,
I'I
Blade Natural Frequencies
(10.2Hz)
3.40/rev
(21.9Hz)
TABLE 1 CONTINUED
BELL
ROTOR (cont' d)
fourth,X 4 /| fl
10.1/rev(77.lHz)
BOEING
6.77/rev( 43.5Hz)
Collective Natural
Frequency
first,Xo) /|
1.31/rev (10.0Hz)
second,(o)/|
2.12/rev (16.2Hz)
third,
fourth A
/(o)
||
4.93/rev (37.7Hz)
10.6/rev (80.9Hz)
Blade flapping
inertia, IB
105 slug-ft 2
150 slug-ft
One blade weight, MB
133 lb
124 lb
Semispan, L
200 in.
200 in.
Chord, cw
62.2 in.
62.2 in.
Mast height, h
Sweep
Dihedral
51.3 in.
0
0
51.3 in.
0
0
WING
Lift-curve
slope, aw
5.7
5.7
Drag coefficient,
CDow
0.004
0.004
-0.005
-0.005
0.01
0.01
2.0 deg
2.0 deg
Moment coefficient C
mo
Aerodynamic center,
e = xA /C
w
Angle of attack, awo
2
TABLE 1
CONCLUDED
BELL
BOEING
WING (cont'd)
Natural Frequencies
first,A 1/|0|
0.347/rev(2.65Hz)
0.365/rev(2.35Hz)
second,A 2 /|
1Q
0. 622/rev (4.75Hz)
0.653/rev(4.20Hz)
third, A 3/|1|1
1.09/rev (8.32Hz)
1.ll/rev(7.14Hz)
fourth,A4 /| 1
2.37/rev(18.lHz)
2.47/rev(15.9Hz)
fifth, A5 /1f|
3.76/rev(28.7Hz)
3.95/rev(25.4Hz)
sixth, A6/|1Q
10.6/rev(80.9Hz)
12.5/rev(80.4Hz)
1420 lb
2000 lb
164.8 slug-ft 2
250.0 slug-ft 2
190.0 slug-ft 2
250.0 slug-ft 2
42.4 slug-ft 2
30.0 slug-ft 2
PYLON
Weight,
Mp
Yaw inertia, I
Pitch inertia,
Ipp
Roll inertia, Ipr
FLIGHT CONDITION FOR CALCULATIONS, X = 0.7
Cruising speed, V
Cruising altitude
250 kt
sea level
218 kt
sea level
PYLON
4AIN WING
ROTOR
WIND
TUNNEL
MODEL SUPPORT
FIG. 1
CONFIGURATION OF ANALYTICAL MODEL, PROPROTOR ON A CANTILEVER
WING OPERATING IN THE AIRPLANE MODE
GUST
VELOCITIES
CAiJTILEVER
ROOT
RESTRAINT
FIG. 2
PROPROTOR GEOMETRY AND MOTION WITH GUST VELOCITY DEFINITION
AFTER DEFORMATION
P
Az
P
w
B EIc
DISC
PLANE
/
POSITION OF
BLADE SECTION
BEFORE DEFORMAT ION
z
FIG.
3
BLADE GEOMETRY
EIB
C.G. and E.A.
AFTER
kw
MU
EFORMATION
F
4
W
u
E
w
zx
pw
(EI )
(EIW) c
C.G.
z
<.--
E. A.
-w
w
w
C
w
POSITION
SECTION B EFORE DEFOR-.
MATION
FIG. 4
WING GEOMETRY
Z.L.L.
DISC
PLANE
i
(a)
P
For Clockwise Rotation
dL
Z.L.L.
DISC
PLANE
(b)
FIG. 5
For Counterclockwise Rotation
VELOCITIES AT THE BLADE ELEMENT AND RESULTING AERODYNAMIC FORCES WHEN LOOKING OUTBOARD
x
x B3
r
r
r
.
n
UT
9,y
UB
FIG.
6
YB
WIND VELOCITIES IL THE DISC PLANE OF THE ROTOR,
LOOKII4G FORWARD
Z.L.L.
w
w
-*-4
w
aX
6 +P
w +u G
V+w
'
E. A.
+w G
z
F
dD
1/4 CHORD
w
FIG.
7
VELOCITIES AT TIE WYING ELEMENT AND RESULTING
AERODYJAMIC
FORCES WHEN LOOKING OUTBOARD
6.4
3.0
BELL
BOEING
2.0
lb/in
1.0
0 0
0.5
(a)
FIG. 8
r/R
Section Mass Distribution
STRUCTURAL CHARACTERISTICS OF TWO PROPROTOR BLADES
0
300
200
BELL
BOEING
100
(LI)g
10
6
'
lb-in"
0.5
0
(b)
r/R
Section Flapwise Bending Stiffness Distribution
FIG. 8
CONTINUED
1.0
800
_
-----
_
BELL
BOEING
600
-- - *
(EI)
6
10 6
bi
400
200
0
0.5
(c)
r/R
Section Chordwise Bending Stiffness Distribution
FIG. 8
CONTINUED
1.0
DEG.
BELL
BOEING
40
20
-
0
1.0
-10
(d)
Angle of Twist
FIG. 8
CONCLUDED
H
z
4th
30
3rd
10 1-
2nd
lst
0
20
40
BLADE COLLECTIVE PITCH AT 75% SPAN
FIG. 9
DEGREES
(6075)
BLADE NATURAL FREQUENCIES, NONROTATING,
FOR BOEING ROTOR
103
1.21'%*Ik
**
= 193
RPM
N
2
-..
0.
D 0.
0
=
300 RPM
Q=
386 RPM
= 550
(NORMAL
RPM
'
1.0
2.0
INFLOW RATIO, V/|0|R
(a) lst Natural Frequency
FIG. 10
BLADE ROTATING 14ATURAL FREQUENCIES FOR BOEING
ROTOR
104
Q)
N
0
2.0
H
= 193
RPM
= 300
RPM
0
>4
0 = 386 RPM
=
-)J
E-4
550
(N4ORMAL
Q)
RPM
- -l
1.0
(b)
2.0
1.0
0
INFLOW RATIO, V/ j
2nd Natural Frccuency
FIG. 10
2
JR
CONCLUDED
105
4th
6.0
C:
o 4.0
O
3rd
M 2.0
2nd
lst
2.0
1.0
INFLOW RATIO,
FIG. 11
V/jrjp
BLADE NATURAL FREQUENCIES FOR BOEING ROTOR AT
CONSTANT ROTATIOiNAL SPEED (P = 386 RP1)
106
W (OUT-OF-PLANE
W
R
V
R
-------
BENDING)
V(INPLANE BENDING)
1.0
/
/
'V
0.5
ol/
/
0
-.
00
~0
/w
*0
0.5
1.0
r/R
(a)
FIG. 12
First Mode (Frequency
X /
= 0.827)
MODE SHAPES FOR BOEING ROTOR AT X = 0.7
AND 0 = 386 RPm
107
W
R
V
R
1.0
w
0.5
0
0.5
-..
r/R
V
(b)
1.0
2
Second Mode (Frequency X2 /l = 1.32)
FIG.
12
CONTINUED
103
W
R
&
R
1.0
o.5-
0
0. 5
-0.5
r/R
1.0
3
V3
(c)
Third Mode
FIG.
(Frequency X3 / 2
12
CONTINUED
109
3.40)
W
V1.0
R
I
4
\V
0.5
0I
/
W
4
0.5
r/R
1.0
-0.5
(d)
Fourth L1ode (Frequency X 4 /Q = 6.77)
FIG. 12
CONCLUDED
110
100
4th
80
60
40
3rd
20
2nd
1st
0
-
0
20
40
DEGREES
BLADE COLLECTIVE PITCH AT 75% SPAN (075
(a) Collective Mode
FIG. 13
BLADE NATURAL FREQUENCIES, NONROTATING,
FOR BELL ROTOR
ill
H
z
100
(INPLANE)
5th
th
80
(INPLANE)
4th
(INPLANE)
3rd (OUT-OF-PLANE)
20
1-
1
0
I
2nd
(INPLANE)
1st
(OUT-OF-PLANE)
20
40
DEGREES
BLADE COLLECTIVE PITCH AT 75% SPAN (075)
(b)
Cyclic Mode
FIG. 13
CONCLUDED
112
3.0..
229 RPM
C:
0
2.0
z
0
H-
E-4
350 RPM
01
458 RPM (NORMAL
0)
550 RPM
w
1.01E4
£
2.0
2.0
1.0
INFLOW RATIO,
(a)
FIG. 14
V/1QIR
First Collective Mode
BLADE ROTATING NATURAL FREQUENCIES FOR BELL ROTOR
113
4.0
= 229 RPM
0
3.0 L.
0
H
2
350 RPM
E-4
0
01
2.0
Q = 550 RPM
1.0
1.0
INFLOW RATIO, v/|IR
(b)
Second Collective Mode
FIG. 14
114
CONTINUED
2.0
1.1
c:
Q = 229 RPM
O
0
H
O
0
01
0
5
Q = 350 RPM
El4
0
=
458 RPM (NORMAL Q)
Q = 550 RPM
1.0
2.0
1.0
INFLOW RATIO,
(c)
V/IS1IR
First Cyclic Mode (Out-of-Plane lode)
FIG. 14
CONTINUED
115
3.0
0
0
01
z S2.0
= 39
2
RPM
=359 RPM
4458 RPM, (NORMAL
1.01
0
2.0
1.0
INFLOW RATIO,
(d)
V/U2| R
Second Cyclic Mode
FIG.
14
(Inplane Mode)
CONCLUDED
116
Q2)
4th
c:
*0 0
0
E-4
3rd
05.0
E-4
2nd
1st
1J
0
0
1.0
INFLOW RATIO, V /1
(a)
FIG. 15
2.0
R
Collective Modes
BLADE NATURAL FREQUENCIES FOR DELL ROTOR AT
CONSTANT ROTATIONAL SPEED (Q = 458 RPM)
117
12.0V
10.0
0
8.01..
6.0 L..
>4
2nd OUT-OF-PLANE
z
0Y
mz
4.0|-
2.0.-.
2nd INPLANE
n~
OUT-OF-PLANE
lst
nI
2.0
1.0
INFLOW RATIO,
(b)
V/ II R
Cyclic Modes
FIG. 15
CONCLUDED
118
(o)
w(0)
COLLECTIVE OUT-OF-PLANE BENDING
V
COLLECTIVE INPLANE BENDING
(a )
1.0
0.5
0.5
r/R
0
W(0)
1i
-0.5
-1.0
(a)
FIG. 16
-
L
First Mode (Frequency A(o)/|02
= 1.31)
COLLECTIVE MODE SHAPES FOR BELL ROTOR AT
A = 0.7 AND
2 = 458 RPM
119
W(
v(0) 1.0
R
w(o)
2
0.5
0.5
(b)
r/R
1.0
Second Mode (Frequency X o)/|Qj
FIG. 16
120
CONTINUED
= 2.12)
(0)
&
R
(0)
1,
R
0.5
1.0
0
0.5 r/R I
-0.5
3
-1.0
L
-)
(c) ThirdMode (Frequency X3
FIG. 16
121
CONTINUED
/IQI = 4.93)
W(o)
R
(o) 1.0
R
4
0.5
-
r/R
0.5
I
~1.0
I
0
I
-0.5
4
-1.0
(d)
-
Fourth Mode
(Frequency X
FIG. 16 CONCLUDED
122
/o)/| = 10.6)
----
w(c)
-IT-
& v(c)
Rn-
W
(c)
CYCLIC OUT-OF-PLANE
BENDING
V
( c)
CYCLIC INPLANE BENDING
1.0
w(c)
0.5
0.5
r/R
1.0
0.0
v(c)
-0.5
(a)
FIG. 17
First Mode (Frequency X (c)/1
= 1.016)
CYCLIC MODE SHAPES FOR BELL ROTOR AT
A = 0.7 AND Q = 458 RPM
123
l(c)
R
, v(c)
V &
R
1.0
/
2
0.51-
V(c)
2
r/R
0.0
.
1.0
0.5
w2
-0.5r"
(b)
Second Mode (Frequency
FIG. 17
XAc) /11
CONTINUED
124
= 1.34)
w(c)
& (c)
R
R
1.0
(c)
0.5
3
0.5
1.0
0.0
-0.5
(c) Third Mode (Frequency X(c/IQI = 4.35)
FIG. 17
CONTINUED
125
W(c)
v(c)
R
1.0
R
0.5
w(c)
4
0.5
0.0
1.0
/
-0.5
(d)
(c)
~4
L
Fourth Mode
FIG.
(Frequency X c) /
17
CONCLUDED
126
2
=
10.1)
Iw
m
2
(slug ft /ft)
(lb/in)
0.015
3.0
-0.01
2.0
1.0
0.005
L0
0.2
(a)
0.4
0.6
0.8
y/L
1.0
Mass and Cross-Sectional Moment of Inertia Distribution
FIG. 18
STRUCTURAL CHARACTERISTICS OF THE WING (THE SAME
WING IS USED BY BOTH BELL AND BOEING)
x 106 lb-in2
x 106 lb-in2
20000
j
16000
-I 8000
(EI )B
(EI W)
GJ
120001-
6000
(EI )
w C
-
80 00
--
---
I----------
---- ----- -- --- ----- -- -
-
4000
- --.
2000
4000 1(EI )B
0.2
(b)
0.4
0.6
0
0. 8
y/L
10
1-0
Stiffness Distribution:
Vertical Bending Stiffness (EIw B'
Chordwise Bending Stiffness (EIw)c, and Torsional Rigidity GJ
FIG. 18
CONCLUDED
Y
L
y
VERTICAL BENDING
1
CHORDWISE BENDING
$
TORSION
1.0
0.5
0
0.5
(a)
FIG. 19
y/L
1.0
First Mode (2.35 Hz)
MODE SHAPES FOR BOEING WING
129
L i 6 0
L'
L
1.0
0.5
2/
0
0
(b)
0.5
y/L
Second Mode
FIG. 19
130
1.0
(4.18 Hz)
CONTINUED
L
L
1.0
r
0.5
L.
'A
3
/
0.5 y/L
Y3
-0.5
r
-1.OL
(c)
Third Mode
FIG.
131
19
(7.11 liz)
CONTINUED
1.0
.. &
L' L
I
1.0
0.5
0.5
0
1.0
4
(d)
Fourth Mode
FIG. 19
132
(15.9 Hz)
CONTINUED
L' L
& $
1.0
-
%,
/
/
/
/
I
0.5
I
I
I
I
/
/
/
0
0
(e)
0.5
Fifth Mode
FIG. 19
133
y/L
(25.4 Hz)
CONTINUED
1.0
1.0
L' L
0.5
1.0
0
-0.5
-1.0
(f)
Sixth Mode (89.5 Hz)
FIG.
134
19
CONCLUDED
-
Y
. &
-
-
y
VERTICAL BENDING
C
CHORDWISE BENDING
4
TORSION
1.0
L' L
0.5 .
OON'
0
0
(a)
FIG. 20
0.5
1.0
WING STATION y/L
First
Mode (2.65 Hz)
MODE SHAPES FOR BELL WING
135
& $
1.or
/
/
/
/
0.5L
1.0
0.5
WING STATION y/L
(b)
Second Mode
FIG. 20
136
(4.72 Hz)
CONTINUED
1.0
L' L
$3
0.5
/
.5
y/L
0
(c)
Third Mode
FIG. 20
137
(8.3 Hz)
CONTINUED
L
L
0.
0.5
y/L
1.0
0
$4
-0.5
-1.0
(d)
Fourth Mode
FIG. 20
138
(8.1 Hz)
CONTINUED
j
1.0
/\
/ r
/
5
0.5-
0
0.5
0
(e)
Fifth Mode
FIG. 20
139
y/L
1.0
(28.8 Hz)
CONTINUED
L' L
0.5
1.0
0
-0.5
-1.0
(f)
Sixth Mode
FIG. 20
140
(80.8 Hz)
CONCLUDED
IMAGINARY
o
9 DOF
o
18 DOF
a Q
(C+1)
0
2.0
(8)
((+1)
Q (C)
a 3 (p)
1.0
a 2 (q 2 ) q
Q2 (C-1)
a1 (q)
Q1 (a-1)
,
REAL
B
-0.3
-0.2
(a)
FIG. 21
-0.1
Bell Proprotor
INFLUENCE OF ADDITION OF HIGHER DEGREES OF FREEDOM ON EIGENVALUES
COMPARED WITH BASIC NINE DEGREES OF FREEDOM
IMAGINARY
a
9 DOF
o
18 DOF
Q2 (+1)
Q+ (C+1)
10
2.0
Q ()
2 g
a3
6'
Q 0(c
Q1 ()
a2 (q2 )
Q; -
Q(-1)
REAL
-0.2
-0.3
(b)
-0.1
Boeing Proprotor-
FIG.
21
CONCLUDED
a 1 (ql)
tP
IMAGINARY
0
Q+
Q4 0
a6
10.0
Q4 0
Q
@33
eQ+3
5.0
9 a4
OQ3
Q1
S
REAL
a
02
Q0
lif
-0.3
-0.2
(a)
Q2
0
S
-0.1
Bell Proprotor
FIG. 22 EIGENVALUES FOR 18 DEGREES OF FREEDOM
a 10
a2
IMAGINARY
a6
10
0
*Q4
a
0
5
a5
Q3
0o
a3
L2
40
S1
SQ2
REAL
-0.2
-0.3
(b)
-0.1
Boeing Proprotor
FIG. 22
CONCLUDED
al.
0
-0
IMAGINARY
BLADE COUPLED ELASTIC MODE
Q+(+l)
1
BIADE UNCOUPLED ELASTIC MODE
0
BLADE RIGID-BODY MODE
0
Q+(+l)
2
2O
Q ()
Q
20
2.0
Q +
0
0
0
Q0(c)
A
a 3 (p)
1.0
a 2 (q 2)
Q2
a1 (q 1 )
Q1 (-l)
I
REAL
-0.2
-0.3
(a)
FIG. 23
I
I
A130O
-0.1
Bell Proprotor Eigenvalues
INFLUENCE OF BLADE MODE SHAPE TYPES ON PROPROTOR SYSTEM EIGENVALUES
AND DAMPING RATIOS FOR COUPLED ELASTIC MODE SHAPES, COMPARED WITH
THOSE BASED ON RIGID-BODY MODE SHAPES AND UNCOUPLED ELASTIC MODE
SHAPES
+
Q (+1)
0
A
A
Q (+1)
+
C+1)
1
C3 2a
2i~1
Q (+1)
(
SQ, ( +1)
IMAGINARY
I 2.0
BLADE COUPLED ELASTIC MODE
D BLADE UNCOUPLED ELASTIC MODE
0
Q (a)
(Q
Q2
13
BLADE RIGID-BODY MODE
0
QO
W
Q0()
2~
00
6
A2
a 3 (p)
1.0
Q (W
a 2 (q
a 1 (q1 )
Ql (C-l)
A
Q (-1)1
Q1 (C-l)
0Q2 (-1)AQ~(0-1)
00
REAL
-0.3
-0.1
-0.2
(b) Boeing Proprotor Eigenvalues
FIG. 23
CONTINUED
A
DAMPING
RATIO (WC)
1.0
0.8
A
BLADE COUPLED ELASTIC MODE
r,
BLADE UNCOUPLED ELASTIC MODE
o
BLADE RIGID-BODY MODE
0.6 1-
0.4 1-
0.2
0I~
I
I
I
1
1
p
___
0
(6-1)
1
(C) (+1)
(c)
2
(C-1)
2
2
a1
(a) (C+1) (ql)
a2
(q 2 )
Bell Proprotor Damping Ratios
FIG. 23
CONTINUED
a3
(P)
DAMPING
1.0
RATIO (C)
r
BA
0.8 1-
BLADE COUPLED ELASTIC MODE
o
BLADE UNCOUPLED ELASTIC MODE
o
BLADE RIGID-BODY MODE
0.6
14
0.4
0.2
A
g
A
0a
0
a
01
(C-1)
Q*
01
(C) (+1)
(d)
02
a
A.
.
0
o
-0
2
(a-1) (a)
+
(0+1)
ft~
A
a1
a2
(ql) (q2 )
a3
(P)
Boeing Proprotor Damping Ratios
FIG. 23
CONCLUDED
02s
1cls)
a
)(q)
Q1lc
Q
MODE (a
x = -
is 1s)
2c
c
lc)
- 1 MODE)
0.282 + 0.0417i
1s)
,Q2s
a 3 (P)
a 3 (p)
lc 01c)
Q
A
Q
MODE
x = -
(l
MODE
= -
(3 + 1 MODE)
0.202 + 1.95i
MODE)
0.316 + 1.31i
1ls (1s)
(a)
FIG. 24
Bell Proprotor
EIGENVALUES AND EIGENVECTORS FOR NINE DEGREES OF FREEDOM
02c
a 3 (p)
ic
2s
4-
+MDE
18)
Qls
is)
QlcOlc)
-
A = -
a3
a 1 (q1 )
(C + 1 MODE)
0.148 + 2.421
(\ )
2c 1 c
Qls (0 1 )
V
2o
Qc (a
2s ( ls
Qls als)
Q
Q2 MODE
A = -
(C
-
1 MODE)
0.109 + 0.245i
Q2c (clc)
(a)
FIG. 24
Continued
CONTINUED
MODE
x = -
(P MODE)
0.0649 + 2.17i
o
0
Q2s
2s
1s
a (q 1 )
1 s)
a 3 (p)
Q10 (C0
a1 (ql)
a 2 (q )
2
Q1c (alc)
!2c
02c (lc)
a2 MODE
X
=
-
(q 2 MODE)
a
0.0136 + 0.610
x
0)
Q2s
a 3 MODE
= -
(a)
FIG. 24
a 3 (P)
1 s)
(P MODE)
0.0449 + 1.27i
Concluded
CONCLUDED
1c)
MODE (q, MODE)
= -
0.0244 + 0.347i
s (is)
Q~ MODE
a 3 (p)
x = -
Q2 s (als)
(c -
1 MODE)
0.271 + 0.168i
(q1 )
Qlc(cic)
a3
/
>)
Q2c Ulc
.ls (cis)
Q2 s (61s)
is
(a
2s
lc
ic)
010 ( o)
al (ql)
Q0 MODE (C MODE)
+ MODE
X= -
a 2 (q )
2
(C + 1 MODE)
0.251 + 1.99i
(b)
Boeing Proprotor
FIG. 24
CONTINUED
.238 +--0.837i
(Y*-
ilc
Qc
1s 0 l
1lc)
Q2c (01c)
a3 (P)
Q
X
(B -
MODE
=
1 MODE)
0.108 + 0.222i
-
010
Q2s(61s)
-j
Ln
w
Olc
(it)
Qlc (;lc)
Qls
2c(01c)
is)
2c 5ic
(a0)
a3
Q2s (a1s)
Q1s (
=
Q2 MODE (0 + 1 MODE)
A
=-0.6836
MODE ($
Q
S
+ 2.24i
(b) Continued
FIG. 24
CONTINUED
-
MODE)
0.106 + 1.28i
Olc
lc)
a 1 MODE
= -
is
is)
(q 1 MODE)
0.0228 + 0.361i
a 1 (ql)
Q2s (61s)
Q2s (a1s)
a 1 (q
Q2c Wlc)
Q1s
Q2o ('
a 2 (q )
2
a 3 (P)
Q2s
1is)
)
Q2c
lc
Q1s
a
a2 MODE (q 2 MODE)
MODE (p MODE)
3= -
1s
x = -
0.0580 + 1.22i
(b)
FIG. 24
Concluded
CONCLUDED
0.0158 + 0.652i
1.0
(
1-
)
(u G/V
0.1 1.
0.01
L
-I
-__
0. 01
0.1
(a)
FIG. 25
P
A
1.0
10.0
Collective Flapping Motion a9 Response to
uG
FREQUENCY RESPONSE OF BELL ROTOR TO VERTICAL GUST uG
INPUT AT FREQUENCY w
lc
0.1
0.01
1.0
0.1
0.01
(b)
o/Q
Cyclic Flapping Motion alc Response to uG
FIG. 25
CONTINUED
10.0
1.0
Is
(uGly)
0.1
0.01
1.0
0.1
0.01
(c)
W/o
Cyclic Flapping Motion als Response to uG
FIG. 25
CONTINUED
10.0
1.0
Co
(u G/V
U''
0.01
0.01
1.0
0.1
(d)
Collective Lagging Motion C0 Response
FIG. 25
CONTINUED
10.0
1.0
lc
(uGly)
0.1
0.01
0.1
0.01
(e)
1.0
o/.
Cyclic Lagging Motion Clc Response to uG
FIG. 25
CONTINUED
10.0
1.0
1ls
CN
(u G/V)-
C
0.1
0.01
1.0
0.1
0.01
(f)
m/Q
Cyclic Lagging Motion Cls Response to uG
FIG. 25
CONTINUED
10.0
1.0
q,
(uG/V)
C1%
0.1
0.01
0.01
0.1
(g)
1.0
W/Q
Wing Vertical Bending q1 Response to uG
FIG. 25
CONTINUED
10.0
1.0
q2
(uGl/)
0.1
0.01
0.01
0.1
(h)
1.0
m/Q
Wing Chordwise Bending q Response to
uG
2
FIG. 25
CONTINUED
10.0
1.0
P
(uGm
0.1 -
0.01
0.01
0.1
(i)
1.0
Wing Torsion p Response to uG
FIG. 25
CONCLUDED
w/Q
10.0
1.0
(vG/V)
0.1
0.01
0.1.0
(a)
FIG. 26
0.1
1.0
m/Sl
10.0
Collective Flapping Motion So Response to VG
FREQUENCY RESPONSE OF BELL ROTOR TO LATERAL
GUST VG INPUT AT FREQUENCY w
1.0
alc
(vG/V)
0.1 .
0.01
(b)
o/Q
1.0
0.1
0.01
Cyclic Flapping Motion a
FIG. 26
Response to VG
CONTINUED
10.0
1.0
Sls
(vG/V)
0.1
0.01
0.01
0.1
(c)
1.0
w/o
Cyclic Flapping Motion Sls Response to vG
FIG. 26
CONTINUED
10.0
1.0
Co
(vG/V)
0.1
0.01
0.01
0.1
(d)
m/Q
1.0
Collective Lagging Motion
FIG. 26
10.0
0 Response to VG
CONTINUED
1.0
lc
(vG/V)
co
0.1
0.01
0.1
(e)
1.0
w / Q
Cyclic Lagging Motion Clc Response to vG
FIG. 26
CONTINUED
10.0
1.0
ls
(vG
0.1
0.01
0.01
0.1
(f)
1.0
W/
Cyclic Lagging Motion Cls Response to VG
FIG. 26
CONTINUED
10.0
1.0
q1
(vG
0.1
0.01
0.01
0.1
(g)
1.0
W/o
Wing Vertical Bending q1
Response to vG
FIG. 26
CONTINUED
10.0
1.0
q2
(vG/V)
0.1
0.01
0.01
0.1
(h)
1.0
o
Wing Chordwise Bending q 2 Response to VG
FIG. 26
CONTINUED
10.0
P
(vG/V)
0.1
-
0.01
0.01
0.1
(i)
1.0
Wing Torsion p Response to
vG
FIG. 26
CONCLUDED
10.0
1.0
So
(wG/V)
0.1
0.01
0.01
0.1
(a)
FIG. 27
1.0
/
10.0
Collective Flapping Motion 5c0 Response to wG
FREQUENCY RESPONSE OF BELL ROTOR TO LONGITUDINAL GUST wG
INPUT AT FREQUENCY w
1.0
61c
(wG/V)
0.1
0 .01
0 .01
0.1
(b)
1.0
Cyclic Flapping Motion a lc Response to wG
FIG.
27
CONTINUED
10.0
1.0
Sls
(wG/V)
0.1
0.01
L-
0.01
0.1
(c)
1.0
Cyclic Flapping Motion
FIG. 27
m/Q
ls Response to wG
CONTINUED
10.0
1.0
Co
(wG/V)
0.1
0.01
0.01
0.1
(d)
1.0
Collective Lagging Motion
FIG. 27
W/Sl
%9 Response
CONTINUED
10.0
to wG
1.0
lc
(wG/V)
0.1
0.01
0.01
0.1
(e)
1.0
m/S
Cyclic Lagging Motion Clc Response to wG
FIG. 27
CONTINUED
10.0
1.0
1ls
(w G/V)
G
0.1
0.01 0
1
0.1
(f)
1.0
m/Q
Cyclic Lagging Motion r s Response to
1
wG
FIG. 27
CONTINUED
10.0
1.0
(wG/V)
0.1
0.01
0.01
0.1
(g)
1.0
m/S
Wing Vertical Bending q, Response to wG
FIG. 27
CONTINUED
10.0
1.0
q2
(wG/V)
C)
0.1
0.01
0.01
0.1
(h)
1.0
m/Q
Wing Chordwise Bending q Response to
wG
2
FIG. 27
CONTINUED
10.0
1.0
P
(wG/V)
co
0.1
1.0
0.1
0.010.01
(i)
Wing Torsion p Response to wG
FIG. 27
CONCLUDED
10.0
1.0
so
(uGly)
Co
0.1
-
0.01
0.01
0.1
(a)
FIG. 28
1.0
w/o
10.0
Collective Flapping Motion a9 Response
FREQUENCY RESPONSE OF BOEING ROTOR TO VERTICAL GUST uG
INPUT AT FREQUENCY w
10.0
Slc
cc
(uGl/)
0.1
0.01
0.01
0.1
(b)
1.0
w/o
Cyclic Flapping Motion alc Response
to uG
FIG.
28
CONTINUED
10.0
1.0
als
(uG/V)
0.1
0.01
0.01
0.1
(c)
1.0
m/Q
Cyclic Flapping Motion S ls Response to u
FIG. 28
CONTINUED
10.0
1.0
(uG/V)
0.1
0.01
1.0
0.1
0.01
(d)
Collective Lagging Motion C0 Response
FIG. 28
CONTINUED
w/o
10.0
1.0
lc
(uG/V)
00
0.1
0.01
0.01
0.1
(e)
1.0
w/
Cyclic Lagging Motion Clc Response
to uG
FIG. 28
CONTINUED
10.0
1.0
;1 s
(uGly)
0.1
0.01
1.0
0.1
0.01
(f)
o/
.
Cyclic Lagging Motion Cls Response to uG
FIG. 28
CONTINUED
10.0
1.0
q1
(uGl/)
co
cc
0.1
-
0.01
0.01
0.1
(g)
1.0
W/Q
Wing Vertical Bending q, Response
to UG
FIG.
28
CONTINUED
10.0
1.0
q2
0.1
0.01
1.0
0.1
0.01
(h)
o/Q
Wing Chordwise Bending q 2 Response to uG
FIG. 28
CONTINUED
10.0
1.0
P
(uG/V)
0.1
0.01
1.0
0.1
0.01
(i)
Wing Torsion p Response to uG
FIG. 28
CONCLUDED
W/Q
10.0
1.0
I-a
'0
0.1
0.01
0.01
0.1
(a)
FIG. 29
1.0
w/Q
10.0
Collective Flapping Motion a9 Response
FREQUENCY RESPONSE OF BOEING ROTOR TO LATERAL
GUST vG
INPUT AT FREQUENCY o
1.0
alc
(vG/V)
0.1
0.01
0.01
0.1
(b)
1.0
w/1
Cyclic Flapping Motion alc Response to v
G
FIG. 29
CONTINUED
10,0
1.0
1s
(vG/V)
0.1
0.01
0.01
0.1
(c)
1.0
Cyclic Flapping Motion
FIG. 29
5
W/o. 10.0
ls Response to VG
CONTINUED
1.0
(vG/V)
j1
0.1
0.01
0.01
0.1
(d)
1.0
Collective Lagging Motion
FIG. 29
W/Q
0 Response to vG
CONTINUED
10.0
1.0
lc
(vG/V)
0.1
0.01
1.0
0.1
0.01
(e)
W/l
Cyclic Lagging Motion Clc Response to VG
FIG. 29
CONTINUED
10.0
1.0
ls
(vG/V
0.1
0.01
0.01
0.1
(f)
1.0
W/9
Cyclic Lagging Motion Cls Response to VG
FIG. 29
CONTINUED
10.0
1.0
q1
(vG/V)
t-Ij
0.1
0.01
A
0.01
0.1
(g)
1.0
m/S
10.0
Wing Vertical Bending q, Response
to vG
FIG. 29
CONTINUED
1.0
q2
(vG/V
0.1
0.01
0.01
0.1
(h)
1.0
w/o
10.0
Wing Chordwise Bending q Response
2
to vG
FIG. 29
CONTINUED
1.0
P
(vG/V)
0.1
0.01
0.01
0.1
(i)
1.0
Wing Torsion p Response to VG
FIG. 29
CONCLUDED
w/&1
10.0
1.0
ao
(wG/V)
0.1
0.01
0.1
0.01
(a)
FIG. 30
1.0
o/E
10.0
Collective Flapping Motion Bo Response
FREQUENCY RESPONSE OF BOEING ROTOR TO LONGITUDINAL
GUST wG INPUT AT FREQUENCY w
1.0
Sic
(wG/V)
0.1 .
0.01
0.01
1.0
0.1
(b)
W/Q
Cyclid Flapping Motion S1 c Response to wG
FIG. 30
CONTINUED
10.0
1.0
Bls
(wG/V)
0.1
0.01
1.0
0.1
0.01
(c)
Cyclic Flapping Motion
FIG. 30
W/Q
ls Response to wG
CONTINUED
10.0
1.0
(wG/V)
0.1
0.01
0.1
(d)
1.0
w/S
10.0
Collective Lagging Motion to Response
to wG
FIG. 30
CONTINUED
1.0
lc
01
(w G/V
0.1
0.01
0.01
0.1
1.0
W/Q
10.0
(e) Cyclic Lagging Motion Clc Response
to wG
FIG. 30
CONTINUED
1.0
Ul
(wG/V)
0.1
0.01
0.01
0.1
(f)
1.0
W/Q
10.0
Cyclic Lagging Motion Cls Response to wG
FIG. 30
CONTINUED
1.0
q1
(wG
0.1
0.01
0.01
1.0
0.1
(g)
10.0
Wing Vertical Bending q, Response to wG
FIG. 30
9
W/Q
CONTINUED
V
I
1.0
(wG/V)
0.1
0.01
1.0
0.1
0.01
(h)
W/Q
Wing Chordwise Bending q 2 Response to wG
FIG. 30
CONTINUED
10.0
1.0
P
(wG/V)
0.1
0.01
0.01
0.1
(i)
1.0
Wing Torsion p Response to
wG
FIG. 30
CONCLUDED
w /S
10.0
1.0
Slc
els
0.1
0.01
0.1
0.01
(a)
FIG. 31
1.0
m/O
10.0
Cyclic Flapping Motion alc Response to 6ls
FREQUENCY RESPONSE OF BELL ROTOR TO LONGITUDINAL
CYCLIC PITCH 01 INPUT AT FREQUENCY w
1.0
ls
0.1
0.01
0.01
0.1
1.0
m/S
(b) Cyclic Flapping Motion 8ls Response
to 6ls
FIG. 31 CONCLUDED
10.0
1.0
lcV
I-'I.-'
01s
0.1
0.01
0.01
0.1
(a)
FIG. 32
1.0
W/
10.0
Cyclic Flapping Motion alc Response to 0ls
FREQUENCY RESPONSE OF BOEING ROTOR TO LONGITUDINAL
INPUT AT FREQUENCY w
CYCLIC PITCH 0
1.0
i1s
V
0.1
0.01
0.01
1.0
0.1
(b) Cyclic Flapping Motion als Response
FIG. 32
CONCLUDED
10.0
W/Q
to
ls
APPENDIX A
DETAILED DERIVATION OF THE EQUATIONS OF MOTION
A.1
Blade Motion
The blade coordinate system is described in Section 2, and
in Figs. 2 and 3. The position vector rndefining an arbitrary
point on the r-axis of the nth blade with respect to the inertial
x-y-z system becomes
VP
0O
0
+cdy o
coo UP
C-00o y
-A
o
O
0
io
-0o
..
rc
-,,+r,,j1
i
.4
(A.1)
where r is the position vector which expresses translation at the
wing tip as
Y'o6
L
(A.2)
The square of the velocity at an arbitrary point on the raxis is expressed as
+ n)
+
213
(i+
+'o~'~
+
.
r
-,.,
-2.a.4,
- kr.nIrf
-2fJ,
A4,2k.
-2
2r. VO-SL.
;;-x
r
,&e4
-4 Yfl.r I,
2 Y.ft
-
Op,)
-2.
r" -
2
A T'..
-- 2
A-- 4,
.AP,+,
s. W-, {j
c4, + 2L
vs-$
-
+ 2r Y Z
2
2r"fl.
1#',
CAV Cp.,
-Z
4
c.04
+
2
r.f
-
#'--nJ'
2 -L
)) AZ-41 '4,
DAL-
z
+ ivi-r
i44+-,
C.,
trIv
+;e e- k
2 Y,
;,.)),
ca4,
,4,'U..C,
1.
+ 2Y
,
z . 4Y--cPk4
,,a.4
c. n eco -
-2
coo.
j
h
O
+ I
r 1..01. C 4O&
(A.3)
The total kinetic energy TE for N blades is expressed as
TE
where m is the blade mass per unit length.
214
(A.4)
The potential energy UE of N blades is
fL(ET2
UE
~
4~~
b4
E
L(
dC4.IB4-)~
-)
*
01
(..
OW
(A.5)
The virtual work done by the centrifugal forces 6WCF is
described as
S
A.2
WCF
(T
T+
A6
The Work Done by Blade Aerodynamic Forces
Aerodynamic forces acting on the nth blade () nare derived
in the z-y-z coordinate system from the P and P0 components
as
-
(A.7)
Then the virtual work done by the aerodynamic forces of all blades
is described as
6W
-- to
AW,
Af
(A.8)
215
A.3
Lagrange's Equations
In addition to the above, the kinetic energy and potential
energy of the wing derived from the simple beam theory are appended.
Then, Lagrange's equations are given by
C1
b
+ au-=(A.9)
where T, U and Q are respectively, the total kinetic energy, potential energy and generalized forces of the system, and
Finally, the above represents the derivation of Eqs. 2.4,
2.5, 2.6, 2.7, 2.8, 2.10, 2.11, 2.12, 2.13, and 2.14.
216
APPENDIX B
STRESS ANALYSIS OF THE ROTOR BLADE AND THE WING
In this appendix, the method of deriving the stresses in
the rotor blades and in the wing due to the gust are briefly
stated. If the deformation of an elastic structure has been determined, it is a straightforward procedure to determine the
stresses corresponding to this deformation. The mode acceleration
method discussed in Ref. 15 is very suitable because the stresses
can be determined accurately and directly when the deformation has
been computed in terms of displacements of normal modes.
Equation 4.57 is written here again in matrix form as
(A]
+ [j3]
xj = [D]{e}I + I(F1B1
-+ []{I
(
where [A], [B] and [C] are square matrices to define the coefficients of equations derived from mass, stiffness and aerodynamic
forces, and the [D] matrix is the excitation. The generalized
coordinates {x} and excitation input {e} are
0.0
Ga.
Qae
t
1421
G41
4 R(B.2)
QICt
asi
The static force {F} from the lift or drag of the wing and rotor
in steady flight may be included if the total stresses are required.
When only the additional stress due to the gust or control input
varying with time is required, the {F1 matrix may be dropped.
217
In general, the stress at a particular point p in the rotor
wing structure is expressed by
(B.3)
P11'-1
=
where the [AP] matrix is a constant matrix that represents the
stress at the point p due to a unit displacement in the normal
mode.
The static mode displacements are given by setting {R} = 0
and {x} = 0 in Eq. B-l.
lt Ici [D3 je 4[cj' lii
.
(B.4)
Therefore, the static stress becomes
ta
titatc.
=
[A PO
j static
(B.5)
When the rotor and wing are vibrating, the total displacements of
the rotor or wing can be expressed as
Lc' ~=[D) (e) + [C]'
-[C3 LB3)
c)XI
-
U1
(B.6)
Substituting Eqs.B.5 and B.6 into Eq. B.3, the total stress becomes
ICY1 ={?1Ptatic
-
-
[IArjIICI 4[A] 1%1
[Ap][CIF'E(B1PC}:
(B.7)
This result gives the stress in the rotor or wing at any instant
in terms of the static stress and an additional stress due to
the vibration.
218