DYNAMICS AIRCRAFT OF TILTING PROPROTOR IN CRUISE FLIGHT by Wayne Johnson Ames ...
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by Wayne Johnson. Ames Research Center. dynamics johnson ......
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NASA
NASATN D-7677
NOTE
TECHNICAL
!
g= Z )..-
Z
i 2 "_"
OF TILTING
DYNAMICS
IN
AIRCRAFT
CRUISE
PROPROTOR FLIGHT
by Wayne Johnson Ames
Research
Center
and U.S. Army Moffett
Air
Field,
Mobility Calif.
R&D
Laboratory
94035 _._._9"_
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION •
WASHINGTON, D. C. •
MAY 1974
1.
Report
4.
Title
2.
No.
D-
Accession
No.
3.
Recipient's
Catalog
5.
Report
Date
MAY
197_
No.
7677 and
Subtitle
DYNAMICS
7.
Government
OF
TILTING
PROPROTOR
AIRCRAFT
IN
CRUISE
FLIGHT
Performing
Organization
Code
8.
Performing
Organization
Report
Author(s)
6.
No.
A-5032 Wayne
Johnson 10.
9.
Performing NASA
Organization
Ames
Nameand
Research
Work
Unit
No.
760-63-03
Addre_
Center
11,
Contract
or
13.
Type
14.
Sponsoring
Grant
No.
and U.
12.
S.
Army
Air
Moffett
Field,
Sponsoring
A_ncy
National
D.
Supplementary
16.
Abstract
A
nine
istics
of
the
rotor
proprotor various
wing
from
tests
of
-
Words(Suggested
wing
Tilt-rotor
the
by
the
on
the
of model
the
the is
Period
Covered
Note Agency
Code
-
gimballed,
comparisons
a
with
the
the
the
stiff-inplane theoretical
Distribution
the The
and
show
for
The
a
of
influence
used
freedom.
rotor results
are
influence
modeling of
dynamics character-
behavior
and
briefly.
degree
the
basic
aeroelastic
treated
lag
The
rotor,
including
rotor
18.
Author(s))
rotor
are
of
wing.
two-bladed
discussed, of
investigations
cantilever
aircraft
influence
proprotors
for a
resulting
flutter,
derivatives
theoretical
presented;
and
whirl
developed
flight
aerodynamics
and
is
axial
classical
stability of
model
inflow
inflow
full-scale are
high
of
aerodynamics, two
rotor
high
the
elements
and
Rotary
Administration
theoretical in
problems on
blade
Key
Space
and
20546
operating
The
inplane
17.
and
Report
Technical
degree-of-freedom
discussed.
of
Addr_s
of
Notes
proprotor
the
and
C.,
Laboratory
94035
Aeronautics
15.
a
Calif., Name
Washington,
of
R&D
Mobility
the
results
hingeless,
good
soft-
correlation.
Statement
dynamics aircraft Unclassified
-
Unlimited
Proprotor CAT.
19.
S_urity
20.
Classif.(ofthisreport)
Security
Classif.(ofthis
21.
_1
Unclassified
Unclassified
"For
sale by the National
Technical
Information
No.
of 253
Service,
Springfield,
Virginia
22151
Pages
22.
Dice" _,6.50
02
TABLE
OF CONTENTS
Page NOMENCLATURE SUb_ARY
..............................
V
.................................
INTRODUCTION SECTION
..............................
1:
BASIC
THEORY
Four-Degree-of-Freedom SECTION 2: Equations The Rotor
FOR
PROPROTOR
Model
DYNAMICS
............
.....................
THEORETICAL MODEL FOR A ROTOR IN HIGH INFLOW ........ of blotion and Forces for the Rotor ............. Aerodynamic Coefficients ..................
27 27 40
SECTION 3: BEHAVIOR OF ROTORS IN HIGll INFLOW ............. Elementary Dynamic Behavior ..................... Whirl Flutter ............................ Two-Bladed Rotor. .......................... Aircraft Stability Derivatives .................... SECTION
4:
NINE-DEGREE-OF-FREEDOM MODEL CANTILEVER WING ......................
Wing Equations Air Resonance SECTION
5:
Proprotor The The SECTION
6:
Dynamic
OF •
,
THE •
THEORY
°
.
°
•
AND °
•
Characteristics
ON A 106 106 132
Stiff-Inplane Soft-Inplane
COMPARISONS
WITH
Rotor Rotor OTIIER
COMPARISON °
.
•
•
,
•
*
FULL-SCALE °
•
°
•
.................. ..................
INVESTIGATIONS
REMARKS
...........................
REFERENCES
............................... ................................
•
WITH
..................
CONCLUDING
FIGURES
90 93
of Motion ....................... ............................
RESULTS TESTS
Gimballed, Hingeless,
FOR A PROPROTOR
60 60 76
...........
•
•
°
.
•
.
135 135 145 151 156 159 161 165
iii
NOMENCLATURE
for less
Conventional helicopter notation is followed in this report, the rotor force and moment coefficients. Quantities are made with
p,
_,
and
R
(air
a
rotor
blade
A
rotor
disk
d
rotor
blade
ed
blade
section
drag
coefficient
blade
section
lift
coefficient
c£
cp Cq I Cq 2
section
density,
area,
rotor
two-dimensional
rotational
lift
curve
speed,
and
for example, dimensionrotor
radius).
slope
_R 2
chord
wing
chord
wing
torsion
wing
vertical
wing
chordwise
structural bending
damping structural
bending
Cx
pylon
yaw
cy
pylon
pitch
CH
rotor
vertical
structural
damping
structural
damping
damping
structural
damping
H force
coefficient, o_R2(C_R)
2
½
%
rotor
%
rotor
ce
rotor
lateral
moment
coefficient,
O'eR3 (riB) 2
longitudinal
moment
coefficient, P _R3 (f'a_') 2 P
power
coefficient, o_R 2 (f).R) 3
cQ
rotor
torque
coefficient,
Q
o_R 3 G'_') 2 T CT
rotor
thrust
coefficient,
onR 2 (92) 2
v
Y Cy
rotor
side
force
coefficient, p_R2(92)
D
blade
section
drag
force
per
EZ
section
f
aircraft
Fr
blade
section
radial
Fx
blade
section
inplane
Fz
blade
section
out
gs
structural
damping
coefficient
h
rotor
mast
height,
wing
tip
spar
hEA
rotor
mast
height,
wing
tip
effective
H
rotor
vertical
modulus/moment
2
unit
length
drag
area
product
equivalent
parasite
aerodynamic
force
aerodynamic
of
plane
force;
also
per
force
aerodynamic
rotor
to
unit
per
unit
force
rotor
length length
per
unit
length
hub
elastic
aerodynamic
axis
to rotor
hub
coefficient
CT
Zb
characteristic inertia support inertias
Io
f
R r2m
wing
lqw
torsion yaw
pylon
pitch
generalized
moment
bending
pylon yaw freedom
of
moment
used
to normalize
rotor
moment model)
of
of
mass
inertia
generalized
pylon pitch moment freedom model)
vi
bending,
dr
pylon
wing
of blade
inertia mass
inertia
of
inertia
[including
[including
rotor,
rotor,
for
four-degree-of-
for
four-degree-of-
and
R IB
f
nB2m
dr,
nBrm
dr
blade
flap
blade
lag
inertia
0
R
IBa o
R I
f
n
2m dr,
inertia
0
R f
_ rmdr
Kp
wing
torsion
spying
constant
Kql
wing
vertical
Kq2
wing
chordwise
Kx
pylon
yaw
Ky
pylon
pitch
spring
Kp
rotor
blade
pitch/flap
L
blade
section
lift
force
m
blade
section
mass
per
mp
pylon
mass
M
rotor
flap
M
Mach
bending
spring
bending
spring
constant
spring
constant
constant
moment
constant coupling, per
unit
tan
unit
63
length
length
aerodynamic
coefficient
number
R
Mb
f
_0
m dr,
blade
Mx
rotor
lateral
My
rotor
longitudinal
MF
blade
flap
mass
(yaw)
hub
moment
(yaw)
hub
moment
moment
vii
blade
Mtip N
tip
lag
Mach
number
moment number,
of
92
divided
by
the
speed
of
sound
blades
\Be N_
-YM
P
wing
torsion
ql
wing
vertical
q2
wing
chordwise
Q
rotor
torque;
also
r
blade
radial
station
r e
effective
R
rotor
blade
R
rotor
radial
S
Laplace
sgn
direction of rotation for counterclockwise
Sw
wing
sB
degree
of
bending
freedom degree
bending
of
degree
rotor
freedom
of
torque
freedom and
Iag
moment
" _
radius force
variable
aerodynamic
in transfer
bending/torsion
nsm
dr
n_m
dr
of
coefficient functions
rotor
inertial
on right
coupling,
wing:
o T
rotor
uG
longitudinal
Up
blade
section
out
uR
blade
section
radial
UT
blade
section
inplane
viii
thrust;
also
rotor
aerodynamic
aerodynamic
gust
of plane
velocity
velocity
velocity velocity
+i
mPzPEAYTw
R f
coefficient
radius
o
S_
aerodynamic
coefficient
for
clockwise
and
-I
U
blade
V
rotor-induced
section
resultant inflow;
speed divided
rotor or aircraft
X
vertical
Xp
rotor shaft vertical
Xw
wing chordwise
Y
lateral
YP
rotor wing
ratio
(forward
velocity
displacement
displacement
displacement
sweep station
wing
Y
rotor
Z
longitudinal
wing
spanwise
length
(wing semispan)
station
side force axis
ZEA
wing
zp
rotor
ZPEA
pylon center-of-gravity axis
Zw
wing vertical
displacement
blade
angle of attack
tip elastic
axis vertical
shaft longitudinal
section
rotor blade mean vertical
shaft
yaw
rotor
shaft
pitch
o_z
rotor
shaft
roll
g
blade
flap
angle
8-1
low-frequency
shift due to dihedral
displacement location,
angle of attack
aerodynamic
rotor
_y
the inflow
tip speed)
forward
shaft lateral
Yw
ax
Up2) I/2
axis
cantilever
_G
+
axis
Yrw
(1
(UT2
when dimensionless,
by rotor
V
YBw
velocity,
gust
angle
rotor
velocity
at
angle angle
flap
pivot at
at
pivot pivot
mode
forward
of wing
tip effective
elastic
B+I
high-frequency
_G
lateral
SO
rotor
coning
_IC
rotor
longitudinal
flap
B1S
rotor
lateral
degree
Y
Lock
rotor
flap
aerodynamic
gust
degree
flap
of
mode velocity
freedom degree
of
freedom
of
freedom
pacR 4 number,
small
Ib
change
in a quantity
sup A
component
of
perturbation
of Up
independent
6uPs
component
of perturbation
of up
proportional
suR
perturbation
surA
component
of
surs
component
of perturbation
_w
wing
dihedral
angle
wing
angle
of
attack
_w k
wing
sweep
angle
63
rotor
blade
blade
lag
l _w
or
uR
(independent
perturbation
of r to r
of r)
of u T proportional of
uT
to r
independent
of
r
2
damping
pitch/flap
coupling,
Kp
63
angle
ratio
of
low-frequency
oscillation,
rotor
lag
fraction
high-frequency
_0
rotor
collective
lag
rotor
cyclic
lag
degree
of
freedom
_lS
rotor
cyclic
lag
degree
of
freedom
nB
blade
flap
n_
blade
lag
_W
wing
rotor
mode mode
bending
critical
damping
lag mode (or rotor
shape shape
mode
of
mode
_+i
X
= tan
shape
speed
perturbation)
degree
of
freedom
@
blade
pitch
@w
wing
torsion
@ o
rotor
collective
elC
rotor
lateral
@lS
rotor
longitudinal
angle angle pitch cyclic
input
pitch cyclic
input pitch
input
eigenvalue
_B _8e
blade
flap
rotating
effective
flap
blade
rotating
%
wing
P
air
lag
torsion
natural
frequency,
including
natural
mode
frequency pitch/flap
coupling
frequency
shape
density
rotor
Nc _-_
solidity,
time
constant
blade
inflow
rotor
blade
of
a real
angle,
azimuth
root,
-i -_-
tan -l U_p_p UT angle,
dimensionless
time
variable
frequency rotor
rotational
speed
Subscripts 0
trim
@
blade hub
pitch inplane
blade hub
velocity
flapwise
out-of-plane
blade
lagwise
velocity velocity velocity
xi
N Oj _j
n8,
rotor
nonrotating
degrees
0
collective
1C
cyclic
rotor
mode
1S
cyclic
rotor
mode
/7/
blade
index,
rotor
m
of
freedom
mode
= i,
.
, N
Superscripts
normalized m
blade
Derivatives
d
d
d
d
xii
index,
(usually m
=
i,
by .
dividing , N
by I b
N or _Ib)
DYNAMICS OF TILTING PROPROTOR AIRCRAFTIN CRUISEFLIGHT WayneJohnson AmesResearch Center and U.S. Army Air Mobility R&DLaboratory SUMMARY A theoretical model is developed for a proprotor on a cantilever wing, operating in high inflow axial flight. This theory is used to investigate the dynamic characteristics of tilting proprotor aircraft in cruise flight. The model, with a total of nine degrees of freedom, consists of first modeflap and lag blade motions of a rotor with three or more blades and the lowest frequency wing bending and torsion motions; rotor blade pitch control and aerodynamic gust excitation are included. The equations of motion for a fourdegree-of-freedom model (lateral and longitudinal tip path plane tilt, pylon pitch and yaw) are obtained, primarily to introduce the methods and formulation to be used in deriving the rotor and cantilever wing equations. The basic characteristics of the rotor high inflow aerodynamics and the resulting rotor aeroelastic behavior are discussed. The problems of classical whirl flutter (a truly rigid propeller on a pylon) and the two-bladed rotor are discussed briefly. The influence of the proprotor on the stability derivatives of the aircraft is considered. The theoretical dynamic behavior of two full-scale proprotors is studied, and comparisons are madewith the results of tests of these rotors in the Ames40- by 80-Foot Wind Tunnel and with the results of other theories. These studies show the sensitivity of the theoretical results to several features and parameters of the proprotor configuration and to various elements in the theoretical model. In particular, these studies demonstrate the important influence of the rotor blade lag degree of freedom on the dynamics of both stiff inplane and soft inplane proprotor configurations, the dominance of the section lift curve slope (c_) terms in the high inflow aerodynamics of a rotor and the importance of a good structural model of the rotor blade and the wing in predicting the dynamic behavior of a proprotor. The comparisons also establish the theoretical model developed as an adequate representation of the basic proprotor and wing dynamics, which then will be a useful tool for further investigations. INTRODUCTION The tilting proprotor aircraft is a promising concept for short-haul V/STOLmissions. This aircraft uses low disk loading rotors located on the wing tips to provide lift and control in hover and low-speed flight; it uses the samerotors to provide propulsive force in high-speed cruise, the lift then being supplied by a conventional wing. Such operation requires a 90° change in the ing the rotor
rotor shaft
thrust axis.
angle, which is accomplished by mechanically The rotor is vertical for helicopter mode
tilt-
operation landing and takeoff, hover, and low-speed flight - and is tilted forward for airplane mode, high-speed cruise flight. Thus the aircraft combines the efficient VTOLcapability of the helicopter with the efficient, high-speed cruise capability of a turboprop aircraft. With the flexible blades of low disk loading rotors, the out-of-plane and inplane (flap and lag) motions of the blades are significant, so the blade motion is as important an aspect of tilt rotor dynamics as it is for helicopters. Whenin the cruise mode (axial flight at high forward speed), the rotor is operating at high inflow ratio (ratio of axial velocity to rotor tip speed); this introduces aerodynamic phenomenanot encountered with the helicopter rotor, which is characterized by low inflow. _le combination of flapping rotors operating at a high inflow ratio on the tips of flexible wings leads to dynamic and aerodynamic characteristics unique to this configuration, and which must be considered in the design of the aircraft. The combination of efficient VTOLand high-speed cruise capabilities is very attractive, so it is important to establish a clear understanding of the behavior of this aircraft and to formulate adequate methods for predicting it, to enable a confident design of the aircraft. Experimental and theoretical investigations have been conducted over several years to provide this capability (refs. 1 to 30). This report develops a model of the aeroelastic system for use in someinitial studies of the system character and behavior. Of particular interest are the features specific to the configuration: high inflow aerodynamics of a flapping rotor in axial flow and the coupled dynamics of the rotor/pylon/wing aeroelastic system. Therefore, this work concentrates on the proprotor in airplane configuration: axial flow and high inflow ratio. In addition, rigid body degrees of freedom of the aircraft are not considered, only the elastic motion of a cantilevered wing. Manyfeatures of tile coupled wing and rotor motion can be studied with such a model, theoretically and experimentally, with the understanding, of course, that the model must eventually incorporate the entire aircraft. An introduction to the problems characteristic of a high inflow proprotor is provided by the following discussion (found in tile early proprotor literature, e.g., refs. 3 and 8). Consider the behavior of the rotor in response to shaft pitch or yaw angular velocity, with the rotor operating in high inflow axial flight. A momenton the rotor disk is required to precess it to follow the shaft motion. With an articulated rotor (a rotor with a flap hinge at the center of rotation), this momentcannot be due to structural restraint between the shaft and the blade root, so it must be provided by aerodynamic forces on the blade. For example, pitch angular velocity of the shaft will require a yaw aerod)mamic momenton the disk to precess it to follow the shaft. The aerodynamic momentis due to incremental lift changes on the blade sections; the componentnormal to the disk plane provides the yawing momentrequired. For high inflow flight, this incremental blade section lift also has a large inplane componentand, as a result, the momentto precess the disk is accompanied by a net inplane force on the rotor hub. This force is directed to increase the rotor shaft angular velocity, so it is a negative damping force that increases with the inflow ratio. There is also the usual rotor positive damping due to tip path plane tilt of the thrust vector, plus the dampingdue to the hub momentfor a hingeless rotor. If the inflow is high enough, the negative inplane force (H force) damping can dominate. The rotor and aircraft can be designed so that the velocity for any instability is well above the
flight regime, but the high inflow aerodynamics are always important in the analysis and design. The behavior of the proprotor in high inflow (as outlined above) implies the following characteristics: decreased rotor/pylon/wing aeroelastic stability since the negative H force damping of the high inflow aerodynamics can reduce the dynamic stability at high forward velocity; decreased damping of the aircraft short period modes, again due to the negative H force damping contribution of the rotor; and large flapping in maneuversand gusts. (The last arises because the momentto precess the rotor to follow the shaft is due to the flapping motion of the blades with respect to the shaft; a given shaft velocity requires a fixed componentof the section aerodynamic force normal to the disk, which meansthen that increased incremental lift is required at high inflow and thus more flapping since flapping is the source of the lift.) These features were first delineated in the studies with the XV-3 aircraft (refs. 1 to 3), the first experimental tilting proprotor aircraft. Investigations of the concept and its problems with the XV-3 provided the initial impetus for further theoretical and experimental work with the configuration, much of which is still in progress. The work with proprotor dynamics has its basis in propeller/nacelle whirl flutter investigations (refs. 4 to 7); however, the flapping motion of the rotor introduces manynew features into the dynamics. Experimental and theoretical work has been done by several organizations in the helicopter industry on the various features of tilting proprotor aircraft dynamics, aerodynamics, and design (refs. 8 to 24). This work has culminated in tests of full-scale, flight-worthy proprotors (refs. 25 and 26) and preliminary design of prototype demonstrator vehicles (refs. 27 and 28) as part of the current NASA/Army-sponsoredtilt rotor research aircraft program. However, in the literature there is little concerning the details of the analysis of proprotor behavior. There are someearly reports on very simple analytical models (e.g., refs. 8 and 18), and somerecent reports on the most sophisticated analyses available (refs. 29 and 30). Further exploration of the basic characteristics of the proprotor dynamics is therefore desirable. The objectives of this report are to establish a verified method to predict the dynamic behavior of the tilting proprotor aircraft in cruise flight; to develop an understanding of the dynamics of the vehicle and of the theory required to predict it; and to assess the applicability, validity, and accuracy of the model developed. The model of the wing/rotor system developed here will be useful for future investigations as well as for these initial studies. The primary application of the theory in this report is a comparison with tests in the Ames40- by 80-Foot Wind Tunnel of two full-scale proprotors. The analysis begins with a treatment of the four-degree-of-freedom case: pylon pitch and yaw plus rotor longitudinal and lateral flapping (i.e., tip path plane pitch and yaw_. With this derivation as a guide, the equations of motion are derived for a rotor with flap and lag degrees of freedom and a sixdegree-of-freedom shaft motion. The high inflow aerodynamics involved are discussed, followed by someelementary considerations of the rotor behavior in high inflow. Next, the special cases of classical whirl flutter (no blade motion degrees of freedom) and the two-bladed rotor are considered briefly; the implications of the basic rotor behavior concerning the aircraft stability are investigated. After these preliminary discussions, the development of the rotor and cantilever wing model is resumed. The equations of motion for a
cantilever wing with the rotor at the tip are obtained and combinedwith the rotor equations of motion to produce a nine-degree-of-freedom model for tilting proprotor aircraft wing/rotor dynamics. This model is applied to two proprotor designs, in order to examine the basic features of the rotor and wing dynamics• Finally, the results of the theory are correlated with those from full-scale tests of these two proprotors in the 40- by 80-Foot Wind Tunnel. The author wishes to thank Troy M. Gaffey of the Bell Helicopter Company and H. R. Alexander of the Boeing Vertol Companyfor their help in collecting the descriptions of the full-scale rotors given in table Ill and figures 14 to 17.
SECTIONl:
BASICTHEORY FORPROPROTOR DYNAMICS Four-Degree-of-Freedom Model
Consider a flapping rotor on a pylon with pitch and yaw degrees of freedom operating in high inflow axial flight. Eventually, at least a few more degrees of freedom must be added to this model for both the rotor and the support. This limited model is examined first, however, to demonstrate the methods used to derive the equations of motion, and because this case is studied in the literature. The model is shown in figure I. The pylon has rigid-body pitch and yaw motion about a pivot, with the rotor forces acting at the hub forward of the pivot. The pylon degrees of freedom are pitch angle a_, positive for upward rotation of the hub, The rigid-body pitch about the pivot. the mast height)
and and
yaw yaw
angle _x, positive motion has inertia,
for left damping,
At the hub, a distance h forward of is a rotor with N blades• The rotor
rotation of the hub. and elastic restraint
the has
pylon pivot (h is clockwise rotation
when viewed from the rear, with azimuth angle _ measured from vertically upward The azimuth position of the mth blade, m = 1 9 N is _m = _ + _ where A_ = 2_/N is the angle between succeeding blades. The degrees of freedom are the out-of-plane fl(m) for each blade, defined positive
rotor
motion given by the flapping angles for forward displacement of the blade
tip from the disk plane (upward in helicopter mode, which is the usual helicopter convention). The blade out-of-plane deflection is assumed to be the result of rigid-body rotation of the blade about a point at the center of rotation (by the angle 8(m)). The dimensionless rotating natural frequency of the flap motion is allowed to be greater than i/rev so that blades with cantilever
root
constraint
may
be
treated
as
well
as
articulated
blades
(which
have
an actual hinge at or near the center of rotation)• The mode shape for the flap motion is assumed proportional to the radial distance r, that is, rigidbody rotation. The net forces exerted by the rotor on the hub from all N blades are rotor thrust T, rotor vertical force H, and rotor side force Y. It is
assumed
in the
derivation
of
the
equations
of
moti0n
that
an
engine
governor
supplies the torque required to hold the rotor rotational speed _ constant during any perturbed motion, and that the pivot supplies the reaction to the rotor thrust T. The pivot also reacts the rotor vertical and side forces so that the only pylon motion is pitch and yaw about the pivot. natural frequency greater than I/rev, as with cantilever root with a flap hinge offset or spring, blade flap motion results the
hub.
The
rotor
pitch
moment
on
the
hub
is My
and
the
With a flap restraint or in a moment on
rotor
yaw
moment,
M x.
The rotor is assumed to be operating in purely axial flow in the equilibrium, unperturbed state, at velocity V. The inflow ratio V/93_ (which may be written simply V, with the nondimensionalization implied) is assumed to be of order i. Only rotor aerodynamics are considered; any pylon aerodynamic forces are neglected. Equilibrium of forces and moments gives the equations of motion: flap moment equilibrium for each blade and pylon pitch and yaw moment equilibrium (about the pivot). The linearized equations of motion, that is, for small angles of the blade and pylon displacement, are then: mth
blade
(m =
., N):
i,
zb['_ (m) + _S2B(m) - (_y
(1)
2_x)C°S Cm + (_x + 2_y)sin era]
Yaw :
Zxax + cxa x + _x_x = Mx-
(2)
hy
Pitch:
(3) where flapping flap
moment
motion
aerodynamic
of
of mth flap
inertia blade
moment
of with
on
the
the
blade
respect
to
the
hub
blade
rotating natural frequency of flap motion (I/rev for blade with no hinge spring or offset; greater than cantilever blade)
cy, c x
pitch ing
and the
yaw moment of inertia of the pylon mass of the rotor (as a point mass
pitch
and
yaw
damping
pitch
and
yaw
spring
restraint
of
pylon
motion
an articulated i/rev for a
about the pivot, at the hub)
about
pivot
includ-
These equations are now madedimensionless with p, _, and R;
the
inertias
are
normalized by dividing the flap equation of motion by I b and the pylon equations of motion by (N/2)I b. The normalization of the pylon inertia, damping, and spring constants (division by (N/2)Ib) are denoted by a superscript ,; for example, Iu* = Iy/(N/2)I b. The rotor o = Nc/_R -are introduced; the Lock to inertia forces on the rotor blade, blade area to disk area. Also notice hub force H may be written in terms
H/p_2R
_
paoR 4 _R
(;_12) Fhlp£S and, then
similarly, become
for
the
other
Lock number ¥ = oacR4/Ib number represents the ratio and the solidity is the that the normalized and of the rotor coefficient:
Q_ forces
2
H
2CH
No a p_£2 (_£) 2
and
and solidity of aerodynamic ratio of total dimensionless
moments.
The
oa
equations
of motion
M%,7
g(m)
+ vS2S(m ) _ (a U
2ax)c°s
?m + (ax
em=
+ 2du)sin
Y ao
(4)
....
These terms
equations in the
are flap
[m%
straightforward moment equilibrium.
_c.]
except perhaps Blade flap
for with
the pylon acceleration respect to space is
composed of f3(m), flap with respect to the hub plane, plus %t and ax, give the tilt of the hub plane; hence the Ky and K_ contribuiions to wise acceleration. The remaining terms are due to'_Coriolis acceleration; blade has a velocity 2r in the hub plane, which has an angular velocity a x cos _m + azd sin _m due to pylon motion, and the cross-product gives a flapwlse Coriolis acceleration of the blade. In the the dimensionless aerodynamic flap moment MFm/pf?2R 5 is written simplicity; that is, the nondimensionalization is now implicit MFm. This practice is followed in the following equations.
new
Now introduce a degrees of freedom
f3o -
N 1 _, 77 _
coordinate as
6
transform
(m)
of
5nc
the
2 = 77 _,
m= 1
6ns
-
N2
k m=l
6
Fourier
flap as in
type,
f_(m)c°s
n_ m
57tI2
-
Nl E iv m=l
of these equation, MFm for the notation
defining
the
n_m
r_l= 1
B (m) sin
which the flapthe
(;7) _(m)
(-1
)m
so
that
B(m)
= B0
+ Z
(Bnc
cos
nOm
+ Bns
sin
nO m)
+ 8N/2(-1)
(6)
m
n
The
coning
angle
is
B0;
BIC
and
BIS
are
tip
8N/2 is the reactionless flapping mode. The (N - 1)/2 for N odd, and from l to (N - 2)/2 freedom appears only if N is even.
path
plane
tilt
coordinates;
and
summation over n goes from 1 to for N even; the 8N/2 degree of
The quantities 8o, 8nc, Bns, and BN/2 are degrees of freedom, that is, functions of time (which, when dimensionless, is the rotor azimuth angle _) just as the quantities 8(m) are. These degrees of freedom describe the rotor motion as seen in the nonrotating frame, while the 8(m) terms describe the motion in the rotating by a conversion of the the nonrotating motion with the
1 _-_(.
.),
frame. This equations of
coordinate motion for
frame. This is accomplished summa'tion operators:
2 _Z('
m
")c°s
n_) m,
The
usefulness
by
2 _ Z(.
m
transform B(m) from
must be accompanied the rotating frame
operating
.)sin
on
n@m,
m
of
the
Fourier
the
equations
1 _(.
to of
.)(-1)
m
m
coordinate
transformation
lies
in
the
simplifications it produces in the equations of motion. The above equations of motion have periodic coefficients because of the nonrotating degrees of freedom in the rotating equations of motion and vice versa; the periodic coefficients only appear explicitly so far with the pylon inertia terms in the flapping equation, but there are actually many more in the aerodynamic forces in all the equations. Since the Fourier coordinate transform converts the rotor degrees of freedom and equations of motion to the nonrotating frame, the result is constant coefficients for the inertia terms, and also for the aerodynamic terms for axial flow through the rotor (as considered here). In addition, only a limited number of the rotor nonrotating degrees of freedom couple with the pylon degrees of freedom; in this case, only the BIC and BIS degrees of freedom couple with ay and are coupled from the pylon motion and Thus the transformation reduced a set
_x. The other rotor degrees of freedom represent only internal rotor motion. of N + 2 equations with periodic coef-
ficients to four equations (considering only those influenced by the pylon motion) with constant coefficients. The rotor behavior for this problem is basically part of the nonrotating system, so the transformation which converts the rotor degrees appropriate one.
of
freedom
and
equations
of
motion
to
that
frame
is the
Operating with (II_)_-_.(.
.), (21_)_(.
m
(2/N)_'-_
(.
.)sin
.)cos Cm, and
m
_m on
the
blade
and
tip
path
flapping
equations
gives
the
motion,
assuming
nonrotating
m
equations
for
coning
plane
tilt
that
N = 3;
MF o _0
+ _8280
: Y ac --
"" 810
+ 2_IS
+ I(_B 2 - i) 81C
- _y
+ 2_ x
: y
MEIC ao
81S
_ 281C
+
+ _x
+ 21 y
= _
MEIS ac
(_8 2 _ i) BIS
(7)
where 1 m
MFI C
2 = -_
___
Mpm
cos
_m
sin
*m
m
2
MFiS = N
_E_.MFm m
The Note
pitch
and
yaw
moments
that
the
transformation
on
the
rotor
disk
introduces
are
MFI C and
Coriolis
and
MFIs,
respectively.
centrifugal
acceleration
terms into the 81C and BIS equations. The equation for 80 does not couple inertially with _y and ax, nor will such coupling be found in the aerodynamics; hence it may be dropped. A set of four coupled equations remains for the degrees of freedom that describe the rotor tip path plane tilt and the pylon pitch and yaw motion: 81C, and 81S remain as
BIC , 81S, _, and ex" If N > 3, the above. To these are added equations
equations of motion
for for
80 the
degrees of freedom 8?C, 82S, .... 8nc ,Sn_ , and 8N/2 as appropriate; like the 80 equation, the_e equations are not coupled with ey and ex, so they may also he dropped from the set, since they represent only internal rotor motion. The four-degree-of-freedom model then is sufficient to represent the coupled rotor/pylon motion for the general case of a rotor with three or more blades. The exception is a two-bladed a later section.
and
The equations _x) are then
of
motion
rotor,
for
the
N
= 2, which
four
degrees
is considered
of
freedom
separately
(81C,
81S,
Sy,
in
o1,,c).. [i2o2],1c). [i °_1 o 1// 18
1
+-
:_y_ o/_
0
o Zx'U\_x
0
0
+
"¢B
,¢8 2 0 0
The
rotor
the
equations.
1
2 _ 1
0
Cy*
0
0
_Cy
ax*
x
ol{ .A /
\
°lib'q:
|
4
oi_i
00 0
Ky* 0 0
forces
(right-hand
aerodynamic
0
U%/_a+ h(2an/oa) I (8)
Kx,J_x/
_2CMx/aa
side)
introduce
- h(2ay/aa)!
much
more
coupling
of
The hub pitch and yaw moments due to the rotor, My and M x, might be found by integrating the forces on the blade (as is done for the other forces on the hub), but it is simpler to express them directly in terms of the rotor flapping motion. The source of the hub moment is the bending moment at the blade root due to nonrotating moments:
flapping, frame and
M m = Ib(`082 -l)B (m). Transforming the moment summing over all N blades gives the hub pitch
= E
C-Ib
_
(`082
1)B(m)cos
Cm ]
=
_
__ib(`08N
2
_
into the and yaw
I)BI C
m
(9)
Mx
[Ib ('082"
= E
1)8(m)
sin
Cm]
= _-Ib(`0 N
82
_ 1) 81,9
m
where
the
definition
of
the
tip
path
plane
coordinates
BIC
and
SIS
has
been
applied; `08 is the rotating natural frequency of the flap motion. If rotor blade has a flap hinge at the center of rotation, then the only restraint of the blade is due to the centrifugal forces, resulting in
the spring v 8 = l;
in
81C
that
case,
no
moment
on
the
hub
is produced
by
tip
path
plane
tilt
and
BIS (except for the torque terms), as required for a hinged blade. With hinge offset, hinge spring, or a cantilever root, the natural frequency is greater than i/rev and so tip path plane tilt produces a hub moment. Dividing by y(N/2)I b
gives
aa
=
2CMx c_a
`082 - i y BIC
(10)
`082 - I -
y
81S 9
the
Rotor aerodynamicsConsider now the rotor aerodynamics. aerodynamic environment of the rotor blade section, and
the
section
velocities
and
forces.
A hub
plane
reference
the
frame
Figure 2 shows definition of is used,
that
is, a coordinate frame fixed with respect to the shaft and tilting with pylon pitch and yaw (ay and ax). All forces and velocities are resolved with respect to the hub plane coordinate system, and the blade pitch angle and flap angle are measured from the hub plane. Tile velocities seen by the blade section
are
uT
(in
the
hub
plane,
positive
in
the
blade
drag
direction),
up
(normal to the hub plane, positive rearward through tile disk), and uR (in the hub plane, radially outward along the blade). The resultant of up and uT in the blade section is U. Tile blade pitch angle, 0, is composed of collective root pitch, built-in twist, and any increment due to control of the perturbed blade motion, lqle inflow angle is _ = tan -I up/_T, and the section angle of attack, a = @ - _. The aerodynamic forces on tile blade section are lift L, drag _,, and radial force F r. _ISe latter is positive outward (in the same direction as positive UR) and has contributions from the tilt of the lift vector
by
blade
flapping
and drag are resolved forces F s and F x.
the
The lift
and
from
with
section aerod>mamic and drag coefficients
L
the
respect
lift as
radial
drag
due
to
hub
plane
and
= _ po(_T2
the
drag
to u_.. into
forces
are
= _
U2_£
+ Up2)C£
The
normal
expressed
section and
in
lift
inplane
terms
of
(11) I .0 = -2 pc(ulp2
+ _p2]ec f = gc
U2cc[
Working with dimensionless quantities from this point on, has been dropped in the last step in equations (ll). The functions of the section angle of attack antl Mach number:
the air density p coefficients are
c _ = c _ (a,M)
,s _
(c, ,_r)
where Up a
: O -
tan-t uT
M
U 2
and Mti p is the section forces 10
tip Mach number, resolved into the
= MtipU
=
uT2
Pd_ divided hub plane
+
Up 2
by are
the then
speed
of
sound.
The
Fs
Lu T
U
Lup
+ Du T
-
DUp
l (12)
u Fr
-
U
j _Fz
The radial force F r has terms due to radial drag and due to the tilt of F a by the flap angle 8. _le radial drag term in F r is derived assuming that the viscous drag force on the section has the same sweep angle as the local section velocity. Such a model for the radial drag force is only approximate, but is adequate for proprotors since this term is not important in high inflow aerodynamics. Substituting for L and D, and dividing by ac, where a is the two-dimensional section lift curve slope and c is the section chord, yields
Fz -ac
= U lu T C2aL
Up
Cd)
(13)
P _
ac
Fr a-c -U = Uu_
cd 2a
F_i +UT 8 a--c
I
l_e net rotor aerodynamic forces are obtained by integrating forces over the span of the blade and summing over all N blades. required are thrust, rotor vertical force, rotor side force, and
the The flap
section forces moment:
1
F
dr
+ sin
$m
/ o
Fx
d
(14)
F
r
dr
- cos
_m
f
F
x
d
0
l
ii
or,
in
coefficient
form,
CT
i
_a
N
}__fi m
(:"
2Cll
-,_ 2
_a
Fz dr
o
__._. cos m
_m
aT dr
+
sin
Sm
0
:"4" aT
0
(15) 2Cy
2
aa
N
( {'. sin
m
--- = MF (A_.
and
for
the
flap
fo
1
r
equations
Sm
--r-r dr ac
-
cos
Sm
i ) _xx dr ac
0
_ dr ["_ ac
of
motion
MFo
'E
= -N
_Vm
m
'V.
MF1C : _
;fFm cos
tm
MFm sin
$m
m
MFI s = _ m
The
net
blade
forces
required
then
s,. _ZZac dr
=
are,
U
if
T
one
substitutes
2{--_ - Up
o
for
Fz,
Fx,
and
Fr:
..) v_ a
dr
(16)
___xdr 0
=
rU
+ u_
0
(Eqs. 12
(16)
continued
on
next
page.)
s':s'{."') _
dr =
0
V
S,.r S, ao
dr
expressions
in
Uu R _-_ dr
To
evaluate
-
S
1
Fz r ac--dr =
s'(. rU
T-2a-
Up_
o
equations
(16)
the
blade
forces,
(16)
-kz ac dr
o
give
plane (thrust) and its moment about in the hub plane (blade drag force),
velocities has a trim
dr
o
o
The
+ Ur TJ
.. f,.
=
0
s
p _
0
the
net
.) dr
blade
force
normal
to
the
the hub (flap moment), the net blade and the net blade radial force. the
blade
section
seen by the blade section are required. component and a perturbation component,
pitch
angle
Each velocity the latter due
and
hub force
the
component to the blade
and pylon degrees of freedom. When the differential equations of motion are linearized, the perturbation components of the velocity are assumed to be small. The trim velocity components for operation in purely axial flow are
uT
=
Up=V+v UR=0 The velocity u T is due to the rotation of the blade; the rotor rotation speed is included here to show the source of this velocity, but it is usually dropped when dimensionless quantities are used. The inflow Up is composed of the forward velocity V plus the induced inflow v; the latter given by momentum theory as
v
=
-Y12
+
I ([7/2)2
+ CT/2
(17)
or
Y + v
= I'/2
+ /
(Y/2)
2 + CT/2
V + OTI2V where the last approximation is valid for large inflow V (really, the inflow ratio V/92, since it is dimensionless). The induced inflow will, in fact, be very small, u/V 3) do not involve any coupling with the shaft motion or with the blade pitch control or gusts (assuming conventional 17
swashplate control inputs and uniform remain internal rotor dynamics.
gusts);
hence
these
degrees
of
freedom
From helicopter rotor aerodynamics, the tilt of the tip path plane (BIC or BIS) is expected to tilt the rotor thrust vector and hence give an inplane force on the rotor hub. The tip path plane tilt terms in CH and Cy are (from eqs. (23)):
A
A-
--
-_
c_a
-
7a
=
cla
The first terms are the inplane forces due thrust vector by the blade flapping. They of the tilt of the rotor thrust by tip path thrust vector remains perpendicular to the
+ l!
61C,
+ H
B1S
to radial tilt of the blade are only half that expected plane tilt, assuming that tip path plane. The other
in H_. Rotor tip path plane tilt B1C or B1S, steady in causes a flapping velocity in the rotating frame. This changes the blade angle of attack and so tilts the blade the chordwise direction (like induced drag). The inplane velocity,
H_,
may
be
mean because the half is
the fixed system, flapping velocity mean thrust vector in force due to flapping
written CT
where the first term is the tilt of the blade thrust, and HB* is due to the rotor inflow. Thus the inplane hub forces due to tip path plane tilt are, combining that due to direct radial tilt of the blade thrust by B, and that due to chordwise tilt of the blade thrust by 8"
9:-
_
_a
_a
The first term plane tilt, as
tilt
term.
of is
IC
12
the the
rotor thrust due to tip path inplane force due to the inflow
on B. l_e inflow term H_* is negative, so it decreases the to tip path plane tilt. For low inflow, the effect of H_* large inflow (as considered here) it dominates the thrust It
is,
in
fact,
the
negative
an important feature in high inflow rotor on the blade flapwise velocity to produce source; hence the angular velocity of the 18
*I
\Tg
is the inplane component expected, and the second
term of H_ acting inplane force due is small, but for vector
z
|_ 7 / oC _-a
H
force,
already
mentioned
as
aerodynamics. Notice that H_ acts an inplane force, regardless of the tip path plane (with respect to the
hub
plane)
or
the
corresponding
shaft
term
Substituting obtains
(the
from
now
the
For
equations
1
0
-i
0
0
1
0
I
0
0
I
*
hub
the
the
of
1)lane)
hlade
rotor
motion
B i (7
also
radial
Forces For
produces
a
hul)
force,
with
no
force.
and
the
moments
into
equations
four-degree-of-freedom
(S],
one
model:
" "
_1_,,
0
U 0
0
0
Zx *
-y,'_:_
2
- 2
2 +y ;!..',:r lJ
yM_
- y,Vk
- y,',7_
BI,:
[
:?) * +h 2 y (//b +Rp ) /
o
-_vuA
C' *+_.2y
?'1_" /
[//U +/fU ]
(2o] -y(_+
-Y_
'J B 2 _ I+}'J/,y?,l{)
y.V_
'oB2-1+EpyM
],, 0
0
I¸)
Bl.l
v2_l+;,x{2"-"+ 8 " \av
,./_*)
Kp;zyl/e
,< *-ky (7+_,)
:t
+l/
-
;',*-;:y(Y+,'_(./
0
Y)"fO
•
0
shaft /4_ aft duced
of of
the
angular due of
to the by
the flap
rotor
hub hub, the
ve]ocity and shaft
and
due tilt;
MB, tip
produced to the positive
::r. +;"
]
"_
2
-
1) (35)
oC _V(" -M x oa
torque
The inertia acting
on
contributions the hub are
Hinerti
to
a
-
-
Yinertia
the
rotor
2 _IS M
2 S_¢IC
drag,
side
force,
thrust,
and
_ NMbC p+ h U) -
NMb(Yp-
h_ x) (36)
Tinerti Qinertia
a
= -NSso_ 0 - NMb_ P =
-NI¢oa¢O
+ NIoaz
The the nal
drag and side forces are the net inplane acceleration of the rotor due to motion of the shaft and blade; similarly, the thrust is the net longitudiacceleration of the rotor; and the torque, the net angular acceleration.
The
new
inertia
constants
are 1
/= _
Io
r2m
dr
4 1
Mb=j_
0
mdr
Therefore, NI o is the moment of inertia of the entire rotor about the shaft, and NMb, the mass of the entire rotor. When normalized (divided by Ib], Io* is nearly l (exactly 1 if I o is used for I b) and Mb* is around 3 for a uniform mass distribution (Mb* is greater than 3 for usual rotors). In coefficient form, by
dividing
NIbY,
these
the
side
forces
and
drag
forces
by
(N/2)Iby
and
the
thrust
and
torque
are
2CH
S
..
2
- ..
--
+
\ _a----/inertia
_a--/inertia
y
-
_lS
y
_IC
1
- _ Mb*(XP
- _ Mb*
°%_)
-
_" ( 37 )
( C_alinertia
$6o --Y
=
.... Bo - --Y
I_ Oa C_alinertia
-
zp
Io * _o
y
÷
5'
Rotor aerodynamicsThe analysis follows that of the previous section; the section aerodynamic environment, with the conventions for forces and velocities, remains as shown in figure 2. With the present degrees of freedom and shaft motion, the perturbation velocities are 6u7
= "(&z
-
_)
-
_(ay
+ (V + _,)(ay sin + (!}p
Sap
cos
Cm - xP
: r6_TA
÷ 8UTB
= r(_
-
&y cos
= nSapB
÷ 8u_
_u R = h(-&_ + V(-8
cos G sin
%
sin
%
• &x cos
¢m ÷ ax cos _m) sin
_m ) + V(B G cos
_m + aG sin _)
Cm]
(3S) em + &x sin
%)
+&x
+ (V + :q(ay
sin
_m + aG
%) cos
Om)
+ (Vu g + Zp)
-
(_p
sin
cos
%
_m + xv.
-
ax cos
sin
%)
_m)
33
_md the coupling
blade Kp:
pitch
perturbation
is
as before,
including
control
and
pitch/flap
= e - Kps The and
trim velocities uR = 0.
are,
again,
for
equilibrium
The rotor motion contributions to the 6_pB ) assume that nB= n¢ = r for the flap mation is satisfactory for the aerodynamic and lag are nearly this anyway, even for a imate mode shape is correct at and near the dynamic moments
loading occurs. on the blade so
The that
mode
(39)
shapes
axial
flow:
u T = r,
up
= V +v,
velocity perturbations (in _UTA and and lag mode shapes. This approxiforces. The first modes of flap cantilever blade. Also, this approxtip, where the most important aero-
nB =
n_ = P are
used
in
the
aerodynamic
l
2 ]
X
The
use
of
this
mode
shape
for
the
aerodynamic
greatly
simplifies
the
aerodynamic coefficients involved or, at least, reduces the number of coefficients required. With the correct n_ and nC, separate coefficients are required for blade motion and shaft angular motion, and for the lag moments and torque moments on the blade. With n_ : n_ : r, only _ (to some power) appears in the integrands of the aerodynamic coefficients, never n or n 2. hence the evaluation of the coefficients is also simplified. The expressions for the section aerod_mamic forcos (L, D, and Fr, their decomposition into the hub plane F_ and Fx% and the rotor forces and moments (T, Y, H, Q, and MF) in terms of the net rotating forces on the blade are the same as in the previous as linear combinations
0
1
section. Again, the net o£ the velocity and pitch
blade forces perturbations:
Fz
--
=
1 F_
r
--_{r
r J_O 1 F aa
34
=,
ac
dr
_o
+
_i'_1_t_T
= R u6`_}_,-
s
_ -CT ca
+ Q_8"T A
+ _?_uPA
+ O_u>B
+
C.!e_
e
may
be
expanded
where M
is the flap moment; thrust; Q, the blade
blade scripts
denote
the
source
II, the blade torque moment; of
the
force
inplane and R,
or
force (drag direction); the radial force. The
moment;
subscript
o indicates
values; subscript V, hub inplane velocity (speed); C, blade rotational (lag damping); B, flapwise velocity (flap damping); _, hub longitudinal (inflow); and O, blade pitch control. The coefficients may be grouped inplane and the
and out-of-plane :4 and T terms
have
may be grouped as.inplane subscripts v and _ have have similar behavior. particular
group
forces, similar
so
the If behavior.
and
and out-of-plane similar behavior, The only difference
(say,
the
Q terms have Alternatively
velocities and those between
out-of-plane
forces
f', the subtrim velocity velocity as
similar behavior, the coefficients
so the coefficients with subscripts g and the coefficients within
due
to
out-of-plane
with ), a
velocities:
M_, Mt, T_, and T)_) is a factor of r more or less in the spanwise integration (the difference between the force and moment, and between the translation and rotational velocities), hence just slightly different numerical constants. The behavior of the coefficients with a variation in the parameters (in particular, with forward velocity V) is basically the same within a group; that is, it is determined primarily by whether an inplane or out-of-plane force is involved, and whether an inplane or out-of-plane velocity or blade pitch control is the input. The fundamental set of coefficients is considered to be the :4 and :J terms with subscripts'_, B, and @ - one each of inplane and out-of-plane typestogether with T O and Qo for the trim values. Then the behavior of all other coefficients may be inferred from a knowledge of the behavior of this set. Again,
the
inplane
force
due
to
flapping
velocity
is written:
CT • to
show The
forces.
explicitly
the
blade
forces
The
aerodynamic
contribution can
now
due be
to
summed
coefficients
operates only on the blade rotor nonrotating degrees equations of motion are
the
over are
velocity of freedom,
(41)
+ II(_* thrust all
vector
N blades
independent
perturbations. the flap
of
tilt. to
_m,
find so
the the
With the definitions lag moments required
and
net
rotor
summation of for
the the
MF0 ao
-Mo +M_(a
- _0) +:4_o
+ Mx(Vu O + _p)
+ Me(eo - Kp6 o)
MEIC
M [-hax
ac
+ (V + v)a x + VSa + ::r]
+ M[(-_IC
-
+ /de (@IC
{1S )
+ M_(_IC
+
_IS
aU)
(42)
Kp61C)
MFIs ac
-
M
[-h& Y
+
(V + V)ay
+
Va o -
]cp]
. M_(-_iS÷ _1c) * Mg(glS- BIC + M8(@IS
- Kp81s)
35
and, similarly, for the lag moments, The aerodynamic contributions to the torque on the hub are
°a/acro
: (H_ + R)
with the M rotor drag
terms replaced by Q terms, force, side force, thrust
and
[-h&Y + (W + v)_ U + V_a - }P]
•
.
•
+ s_(- _div
31)
with
increase in 2a; a = 5.7
stalled
value
for
:
0.26(0.9
a lower
flow,
}
M)
c_i
(56a)
n as
appropriate for current proprotor sections, and With a compressibility obtained from section tests on rotating blades. The compressibility increment has a critical Mach number of 0.9 at zero angle of attack; critical and its
its
+ _c d
- 0.9) of
M2) -1/2
2a
c_ 2-a
(1
2
term drag above the
ed. The drag coefficient is used for the two-
]_[ > as,
the
following
49
where,
typically,
effects
are
possibly
CZs
not
c_a
= 1.0
included.
= 0 there,
c
= a_s
Cd
= Cds (sin
and
Cds
A very so the
cients dominate the behavior the blade is stalled).
sgn(a)
=
a) 2
2.0.
Combined
important
c£,
Cd,
even
(56b)
I compressibility
influence
of
Cda , CZM , and
CdM
in high
inflow
stalled terms
(if a large
and
flow in
the
enough
is
stall that
coeffi-
portion
of
With these analytical, semiempirical expressions for the section aerodynamics, the influence of the drag and lift terms and of stall and compressibility on the rotor aerodynamic coefficients is examined. For the design of a specific rotor and the prediction of its behavior, the section characteristics appropriate for the actual blade sections should be used. For the present work, it is desired only to check the relative importance of these effects so approximate lift and drag coefficients are satisfactory. The influence of these effects on the rotor coefficients is shown in figure 5, for H@, HU, H_, M0, 3_, and '_!li. _e coefficients were calculated using the exact expressions (eqs. (50)), with the above approximations for the section aerodynamics, for two rotors. I _le collective required to give the rotor thrust for equilibrium cruise at a given V is used. In figure 5, the results for these two rotors are compared with terms. The coefficients with only (fig. 4); the approximation is the pendent the two
of the rotors
section characteristics. so far as the behavior
they have different tip blades have a different
speeds. resultant
tion. The tip resultant 5(b) for the two rotors. The
exact
the coefficients found using only the c£a the cz_ terms are given by equations (31) same for the two rotors since it is inde-
Math
coefficients
of
most important difference coefficients is concerned
between is that
Therefore for a given forward speed V/92, the Mach number 3{ = M_ ._ (r 2 + V2) 172 at a sec-
number
in
The the
M
figure
= _/ti_(]
5 show
+
1,2)I/2 is
a significant
shown
in
difference
figure
from
the
coefficients based only on the cLa terms; the difference is particularly large when the tip critical b_ach number (0.9 for _ = 0 with the section characteristics used) is exceeded. The following conclusions are reached then: the terms in the rotor aerodynamic coefficients give the basic behavior, at c£a least terms
so in
long as the section critical Mach number is not exceeded; the coefficients are not negligible, however, and should
to properly
evaluate
the
behavior
of
a real
at high section a or :V. When the section required, actual section characteristics representative expressions. Three described:
methods
ISpecifical]y, chapters, 5O
hence
the
for
evaluating
this labels
is
for "Bell"
the
the
especially
when
operating
aerodynamics other than cga should be used rather than
rotor
two
and
rotor,
the other be included
aerod)mamic
full-scale
"Boeing"
in
rotors figure
coefficients
examined 5.
are
have
in
later
been
(a)
Approximations based on evaluating the integrands at an equivalent radius. Approximations based on the retention of only the c_a terms,
(b)
with c_a/2a = 1/2; the integrals the coefficients as functions of (c)
The used
based tions
the
the method
on just here.
based on expository
c_a terms (c) should
the
equivalent development,
correctly be used
the These
cz terms coefficients
which is of primary approximation for (c), the coefficients
interest calculating based
here. the on the
only
influence
of
to
giving
Approximations based on analytical expressions for the blade section aerodynamics in the exact coefficients; representative stall and compressibility effects are included, but no specific section is modeled.
coefficients only for
(b) treats required
may be evaluated, V alone.
check
.
the
radius never
and exactly, rather than
(method (b)) include
.
approximation (method in the calculations. and if method
will normally be the basic behavior
.
In fact, this dynamic behavior exact expressions
the
terms
The method used here allows the coefficients, including the influence
the (a).
other
than
level (as
other The used with
terms are coefficients in the inflow
is usually shown later). (eqs. (50)),
c_
(a)) are Method
calcularatio,
a
good Method used here
is
.
derivation of rotor aerodynamic of lift and drag and of stall
and
com-
pressibility, with no more difficulty than a derivation that includes only cza terms. Therefore, a good representation of the rotor aerodynamics is available if one chooses to use it (and if enough information on the section aerodynamics is available). Evaluation of the coefficients in method (c), or even including tabular data for the actual bladesections used, requires numerical integration over the span, but that is no problem for numerical work. There is only one real complication in evaluating the coefficients by the exact expressions: the trim angle-of-attack distribution is required, which means that the blade collective pitch at the given operating state must be known. Hence a preliminary solution of the rotor performance to find the collective pitch is required before the coefficients can be evaluated for the dynamics. In contrast, with only the c_ terms (method (b)) only V/fZ_ is required to evaluate the coeff_clents. Discussion of discussed here; in cients are derived.
the coefficientsSome properties of the coefficients are particular, certain useful equivalence among the coeffiA helicopter rotor in hover (low inflow axial flight)
exhibits equivalence of control plane, that is, these inputs produce the same exceptions). This behavior translates aerodynamic coefficients. behavior of the rotor, and examined now. Consider tilt from
(B1C, equation
the
_IS), (42),
The influence on the basic
longitudinal hub
hub plane, and tip path forces on the helicopter into certain equalities
plane til_ this moment
moment (_y, is
on _x),
of set
the and
plane tilt; (with certain among the rotor
high inflow operation on the of coefficients in general is
rotor control
disk
due
plane
to tilt
tip
path
(@IC,
plane @IS);
51
v y
- 81C
IS
+ ax ) + MS@IS
Hub plane tilt aW gives an inplane component of V + v, hence a flap moment through the speea stability coefficient MU. Tip path plane tilt 81C (with respect to the hub plane) gives a flapping velocity in the rotating frame, hence a flap moment tilt 017 produces a M o. For low inflow,
through the flap moment
flap damping through the
coefficient M_. Control plane pitch control power coefficient
1
Me =
=
so the flapping produced by blade pitch is BIC/@IS = Mo/-M_ = 1 (if _6 = i). That is the familiar result of helicopter hover control (wlth a low inflow rotor): the tip path plane remains parallel to the control plane. In high inflow radius
so
operation, however, approximation,
M@
and
that
-M R are
not
M0
1 = 8 cos
M_
=
equal.
Based
on
the
equivalent
6
cos ¢ 8
M@
1
-M_
cos2
¢
Pitch control power M@ increases with V, while the flap damping M_ decreases; the ratio then increases with V. For a Y/_ up to 1 or so, there are no drastic changes in the magnitudes of the coefficients, but the effect is important.
Based
on
just
the
c£_
Ms =
-M R =
The integrand M@; therefore, cos 2 t (as due The
to B is flapping
of M_ is a the ratio above). This not the same term comes
terms,
the
I rurU
coefficients
_ c_a
a dr I r 2 UT U2 2 c£2a
as that from _Up
= 80
due to @ when = r8 so that UT6U P
-
88
U2 gives
the
additional
dr
factor uTr/U2 = r2/(r 2 + g 2) smaller of the coefficients is of the order high inflow effect results from the
8a
which 52
are
factor
uTr/U
2
in
M_.
the
inflow
than that of re2/(re 2 + V 2) = fact that the 8_
angle
t
is
large.
Consider (in an
now
inertia
so
that
to
space),
the
the
hub
frame),
longitudinal hub
moment
that
plane
in
terms
of
flapping
with
respect
to
space
is,
moment
tilt,
and
81CI
=
BIC - _y
81SI
=
81S
on
the
control
+ ax
rotor
disk
due
plane
tilt
is
to
flapping
(with
respect
MEIS ao
Flapping
with
-
[-M_
respect
+
(V + v)Mlj]C_y + M_(BIS
to space
acts
through
M_
I -81SI ) + M081S
to
give
a hub
The moment due to hub plane tilt _y is [-M_ + (V + v)M_], control plane tilt is M@. In high inflow, these moments while both _y and @IS reference frame also. of
the
coefficients
tilt The
(eqs.
the control difference (S0)),
-M_ + (V + v)_p
it
plane, is, in
follows
r
=
only fact,
moment
as
usual.
while that due to are not equal, for
_y tilts the small. From
hub the
plane definitions
that
rU
+
2a a
V
2aJ
U
v--_
= M@
/,
+
cd)
rU
-_a + r _
= M0
+ --sa
dr
Hence (57) -M_
Now all
M@ ~ 1/(8 cos @), so V, both high and low
(V
+
that M@ inflow.
+ v)M
>> CQ/(_a (which So for all V,
+ (v which low both
means
that
inflow, equal
(V 1/8.
Similar vertical
hub
hub
plane
+ v)M_
results force
due
is
tilt of
and
order
may be obtained to flapping,
the
order
of
plane
tilt
are
equivalent.
while
M_
VCT/C_a)
for
- M0
control V2
is of
small,
for the hub plane
inplane tilt,
and
MO are
forces. and control
of
Consider plane
For order
1 and
the tilt:
53
_a
Tip
path
plane
tor, while forces due
_1S
p)ay
tilt
fllC gives
blade pitch to flapping
an
inplane
-
_x )
hub
_a +
force
by
IC + WeOIs
tilt
control has no such effect. The and pitch, [!_* and }Io, are equal
of
the
thrust
vec-
corresponding hub for low inflow where
l:O = -ll_* ~ V 4 In high inflow, these forces, like the basis of the equivalent radius
£0
the flap moments, approximation,
- 4 cos
t
sin 6
¢
_Y_* so
-/:'
Both
forces
(COS. %5) -2 coefficients
increase :} =
(rU'are
_,'ith +
V2)/>,.
2_e
V,
but
•
On
0
longer
equal.
On
1 cos?
*
8
::0 the
t
increases basi c_2a
large
_-
V, compared
dr r
(59)
2aJ
with
M_
or
H_;
for
lligh inflow, V/P_ of order I, influences the rotor aerodynamic coefficients substantially.. It follows then that the features of high inflow aerodynamics are an important factor in the aeroelastic behavior of the rotor and wing system. In summary, the combinations of the coefficients derived are
-M_ + (V + v),V
-itS*
+ (v
+ v)(H
+ _
--- _' O
) _ il
M
The first two approximations are valid for high inflow; to these may" also be added H_ Performance considerationscoefficients for the analysis of tion of the proprotor performance. 5O
The the
(60)
_ -/I"
all V, while the last _ H_* for high inflow.
evaluation proprotor First, to
is only
of the rotor aerodynamic dynamics requires a consideraobtain the rotor collective
for
pitch value requires state, for example,
a solution for the performance cruise flight (CT given by the
in a specified airplane drag)
operating or autorota-
tion (CQ = 0). The rotor collective pitch is needed only to evaluate the complete expressions for the coefficients, however (eqs. (S0), described as method (c) previously). Second, the total axial velocity V + v, a major parameter in the coefficients, includes the induced inflow v, which is related to the
rotor
thrust
expressions (CT and induced
rotor namic
and
operating
required
CQ) to inflow.
find
for the
an
state.
_¢o
elementary
collective
topics
analysis
pitch;
and
are of
an
now
the
considered:
proprotor
evaluation
of
the
performance the
rotor-
In the previous analysis, expressions were obtained (eqs. (50)) for the thrust and torque coefficients in terms of the blade section aerodyforces. Using the identity of the power and torque coefficients for the
rotor
iCp = CQ
since
P
= Q_),
these
expressions
are
f Op_ c_a
where
U2 = r2 +
expression
2av vv
/
If uniform
iV + v)
(61)
induced
inflow
v,
is assumed,
the
power
is
U3 dr Cp
which flow.
=
IV + v)O T +
is the usual result for the power The first term is the sum of the
i62)
I _o2d
required induced
by a rotor operating in power loss and the useful
axial work
done: Cp.
=
(V + v)C T
_
(V + CT/2V)C
2
(63)
%
For high inflow, the induced discussion below); then CPi
loss term vC T _ CT2/2V is proprotional to C T,
operation where proportional to
inflow
The section value
second drag
of
od
the induced CT 3/2 term
in
coefficient is
used),
equation is
is
(62)
constant
important
is over
the the
and
rotor span
is negligible in contrast to for
which
profile (or
if
CPi
power an
(see hover is
loss;
effective
if
the
mean
then
57
UOd[
CPo
=
fl
_ad8
/1
o) 213/2_r
[r 2 + (v +
2
+ (v + _,)2
1 + 7
+ 5 (v + v) _ _n 1 + /1 + (V + z,) "_ V + V
ac d --g(70
[l
+ 3(V
+ u) 2]
(64)
low
V
,
-_
(6.27)
V + g) =
(473)
high
1
gCcZ
The two rotor operating with the rotor providing flight, and autorotation the shaft). The latter the proprotor and wing is
then
to
find
for a given autorotation).
CT
The induced
the
rotor
less than the to the inflow
conditions of primary interest here are: operation the propulsive force required for equilibrium cruise operation (no net power supplied to the rotor through corresponds to the condition in which dymamic tests of are often performed. The performance problem involved
rotor
(thrust
collective required
aerod3mamic
velocity
v.
V
For
pitch for
equilibrium
coefficients high
required
inflow,
cruise
require however,
forward speed of the rotor calculation is not required
(and
an the
the
other
flight)
or
estimate induced
(as shown below), to satisfactorily
of
coefficient) CO
the
(zero
for
rotor
velocity
is much
so great evaluate
attention the coef-
ficients. 3q_e assumption of uniform inflow is adequate then, and it may, in fact, be possible to neglect the induced inflow entirely. The rotor CT required in cruise flight is obtained by equating the rotor thrust (for two rotors) with the aircraft drag, and expressing the drag in terms of an equivalent the thrust
flat-plate area required of the
f for rotor,
the
CT where
A
is
the
disk
the induced inflow; thrust C T is:
area the
of
the
usual
aircraft
(I/2)0V2f).
Therefore,
for
I ,, r,_2 - 4A
rotor.
result
Y + v = vI2
58
(L =
Momentum (ref.
32)
theory for
axial
+ /(v12) 2 + 0/2
can
be
flow
used
to
operation
estimate at
(6s)
Substituting
for
CT yields
v
/f
+ (:72.4)
g or,
since
f/d
and
hence
CT/V
2
is
small, 7) ~
CT
V TyTical follows
values then
For
proprotor
the
of that
l
2V 2
equilibrium
When
cruise
flight,
the
rotor
is
operated
O, which for the is
in
requires profile
For the windmill
f/.4 _ 0.0033; much less than 7;/Y
is
a
it 1.
reasonable
the
performance
CT = -Coo/(V + _) C@ -- lacd/2)V,
aod 2
requirement
_ -C(2o/V. the thrust
With the required
high in
V2
proprotor in axial flight at high inflow, autorotation occurs in the brake state (i.e., at V > 2/]CTI/2, or ICT/2V2! _ Iv/V1 < I/4); hence
momentum The same
theory may conventions
required in result is
again be used are used for
autorotation
to estimate the induced the directions of V and
is negative,
Substituting
for
Typical
values
of
_ 0.0004. than in
The
the
effect
required
of
the
v/Y
maximum _
v
CT
V
2V 2
drag
induced
aCd
_
CT/2V
2
(valid
the
momentum
theor,v
(6(_]
inflow
= _
of
tan
6a for
-1
= both
0
4
coefficient
the aerodynamic by considering
value
previously];
(ref. -o). the C T
+-T
Hence the typical induced cruise flight. Again, the
_a
a
given
inflow CT_ (so
C T yields
proprotor
required to evaluate may be investigated
has
as
v =7+
V+v
result
give indeed
neglecting
autorotation,
that power,
CT-
which
and radius which is
then.
is that Co = inflow result autorotation
OOdo/2 smaller
("
8 A
proprotor aircraft drag v/V = 0.0017 typically, in
approximation
- 1
2
on
and
the
V + v
v
ar
V
powered
yield
v/g _ -0.0002, inflow may be
blade
coefficients the change in
(1/2)v/V.
solidity
inflow induced
load
which is neglected.
distribution
for the angle of
(that
dynamics analysis), attack due to v/Y:
V/fm 1
+
Use and
(V/far) of
the
autorotation
2 momentum
theory
operation
in
59
high inflow) and in angle of attack:
the
mean
angle
of
attack
_ot
~
typical values of high infiow (V of
rotor order
the
maximum
change
a
fraction
ua
a For for
_ = 6CT/a a yield
24V2
solidity, 1).
_a/a
= O.02V
-2
which
is
small
Numerical calculations were performed to verify that V + v _ V is a good approximation in the calculation of the aerodynamic coefficients. The important consideration in the dynamics analysis is that the performance calculation and the calculation of the aerod?mamic coefficients be consistent, either neglecting the induced inflow v/V or using the same estimate of v/V for both calculations. Any error in estimating the rotor performance or the collective pitch required is not relevant to the dynamics analysis. The aerodynamic coefficients or C0 :ire lstribution d.
_
that correspond to obtained, the only over the blade'.
SECTION
the operation error being
3:
BEIIAVIOR
a
of the small
OF ROTORS
rotor change
IN
at a given value of in the angle-of-attack
tIIGII
CT
INFLOW
In the next four chapters, several topics on the behavior of high inflow proprotors are investigated, based on the equations of motion derived previously. The development of the proprotor and cantilever wing model is resumed chapters
in section 4. in this section.
The
reader
interested
Elementary
Dynamic
in
that
topic
may
skip
the
four
Behavior
Some aspects of the dynnamic behavior t_ical of proprotor aircraft are examined. First, the fundamental stability of the blade motion is examined through the eigenvalues of the uncoupled blade motion. Then the influence of the transformation on the eigenvalues proprotor and wing tions
are
useful
The equations of into longitudinal motion, in fact, study of to shaft lateral/vertical since that the aircraft
60
to nonrotating degrees of freedom and equations of motion of the rotor is examined. The actual coupled motion of the system is considerably more complex, but these considerain
the
interpretation
of
the
motion and the hub forces for and lateral/vertical groups couples these groups, but it
the dynamics motion, gust,
to neglect or blade
that pitch
coupling, control
systems. Attention is response is useful in evaluating stability.
directed the
results
for
the rotor (eqs. (44) is useful in
and the
the
examine the longitudinal
to the influence
complete
model.
were found to separate to (48)). The wing for a preliminary rotor or
response in the
low-frequency response, of the proprotor on
Blade single
stability-
blade.
The
Consider
equation
of
IB* _ -
The
roots
(eigenvalues)
the
of
uncoupled,
motion
_M_
+
this
(in
the
(I_*v82
equation
shaft
fixed
rotating
flapping
frame)
motion
of
a
is
+ KF_M0) B : YM0@
(67)
are
/ (68) X - ,(M_
_+ i
/v
2 + Kp
....
2I B* For low inflow, result for the
-M_ = M@ = 1/8, and equation flapping motion of a hovering
(68) reduces rotor. The
then to the flap damping
tive, M_ < 0, so the real part of X is negative and the flap motion is As V/fag increases, the flap damping -N_ decreases and the pitch control M@ increases. Then the real part and the stability of the flapping
Pitch/flap
coupling
force
M@,
Kp
introduces
which
changes
2 vB
= e
Negative VBe. Again,
pitch/flap
coupling,
a
the
vB2
> 0,
Consider
the
uncoupled
lag
lag
for
this
motion
damping
Q_
is
+ XQ_
*_
term
flap
through
natural
the
frequency:
increases
(69)
the
increases of order
with
the
+ I{*_
2{
effective the l or
rotating
flap
frequency
effectiveness less. equation
of
of
Kp.
motion
there
is a
proprotor dynamics. for the flap damping,
= 0
(70)
are
positive,
cient Q_ increases with For low inflow, the lag however,
spring
c_ in_low.
TMO -IB*
+ Kp
v¢ 2
2iB,
The
flap
(at least the extremely high
form):
I
roots
motion
negative for even
effective
Increasing V/FaR increases M 8, and so the influence is not great for g/gR
(homogeneous
The
fp
stable. power
of _ decreases in magnitude as V/_R increases, motion decreases. The change is not great
for V/f_R of order l, however; and M'@ is always contribution is) so the motion remains stable
aerodynamic
usual is posi-
Q_
V/FaR, hence aerodynamic
significant In high namely,
> 0,
-
so the
motion
is
the stability of the damping is very low.
increase
in
lag
damping,
stable.
The
lag motion For high which
is
coeffi-
increases. inflow, important
inflow, the source of lag damping is the the lift change due to angle-of-attack
same
in as
61
perturbations damping
are
The aerodynamic
(i.e., of
flap
the
and forces
the same
lag in
o z c_ terms). . order in high
motions of high inflow;
aerodynamic coefficients all the coefficients).
M_ and
the
blade specifically,
Q_,
of motion for the by tile transformation
Coriolis terms B1S motions.
(from
are
tip
introduced,
The eqs. homogeneous (44)):
are
both
strongly they
are
order
1
of
path plane tilt to the nonrotating
of
flap
damping
coupled coupled in
and
by
high
lag
the by the
inflow
cross (like
Consider the blade uncoupled flap motion, The coning mode 8 o has an equation of of tile blade in tile rotating frame, so same as those given previously. The
which
equations
are
which
Nonrotating system eigenvaluesas observed in--the nonrotating frame. motion (eqs. (47)) identical to that tile eigenvalues of its motion are the equations modified
Therefore, inflow.
have
motion
the
in
coordinates, frame; effect
of
Laplace
form
_IC centrifugal
and
_IS, and
the
81C
and
and
81S
are
coupling
for
81C
are
I_ *8 2-yM_o +/8 * (vB 2_ i ) +Epy?_'] O : 0
- (2_B _ql
the The
+
is
1/rev),
_ -i frequency result is that
has
the
greater the
following
than
resonance
the
influence
ql
on
frequency
increases
is less than the ql the rotor lag motion
the
the
stability:
(roughly, ql
mode
when
the
when
damping,
frequency, it decreases significantly reduces
while,_hen the damping. the wing vertical
bending mode damping at the higher speeds. The speed at which the ql mode becomes unstable is not changed much, however, which indicates that the high inflow instability mechanism is not greatly influenced by the lag motion. The reduction in damping at high speed is then more importantly accompanied by a great reduction in the rate at which the damping decreases, which is very beneficial. The rotor lag motion thus makes the high inflow instability less severe. The flap and lag modes in high inflow are highly coupled by the aerodynamics. Eliminating the _IC and _IS motions therefore greatly influences the behavior of the B ± 1 mode, as shown in the root loci of figures 18 and 19. The rotor flap motion is expected to be important to the proprotor and wing dynamics. The above comparison shows that the rotor lag motion can be equally important. aerodynamic The wing case the a
146
influence
of
aerodynamics on is autorotation wing
small
is
The influence of the lag motion is a combination of forces and inertia coupling with the shaft motion.
the
aerodynamic influence
most
on
important
the
rotor
the system operation,
speed
perturbation
stability including
forces
decreases
the
the
damping,
which
q2
effect.
Powered
degree
is shown the wing ql
and
high
freedom
and
in figure 20. aerodynamics.
p mode
indicates
operation
of
the
(including
the
but
C£_
the
the
The basic Eliminating
damping,
that
inflow
has
wing
wing
only
damping
aerodynamics again) is considerably stabilizing for all the wing modes, over the autorotation case. The powered model considers the hub rotating at a constant speed, so the _ modebecomesthe elastic motion of the blades about the hub, with spring restraint v . The powered operation has little influence on the _ ± l and _ ± l modes, or_on the wing modefrequencies. The influence of the complete expressions for the rotor aerodynamic coefficients, that is, including the c_, Cd, Cda, C_M, and CdM terms as well as
the
reduces
c_a the
terms,
is
shown
predicted
in
figure
stability
of
21. the
The
wing
better
modes,
rotor for
aerodynamic
both
model
autorotation
and
powered cases. The details of the coupling of the high-frequency rotor modes fl + 1 and _ + 1 are also changed somewhat. The complete rotor aerodynamic coefficients were calculated both by use of the correct collective pitch from a performance analysis (the collective pitch required for C_ = 0 for autorotation, or for the C T needed in equilibrium cruise for powere_ flight), and by use of an approximate collective value based on the inflow at 75 percent radius (@0 • 75 = tan-l(V/_R)/(3/4) + 1 •25 °) . The performance calculation is very sensitive to the collective pitch used, but figure 21 shows that the dynamics behavior is not; the approximate collective used is, in fact, within 1 or 2 ° of the correct value for both autorotation and powered flight at high speed, and so evidently is an adequate representation of angle-of-attack distribution. The complete expressions for the aerodynamic coefficients give somewhat different
numerical
values
compared
with
those
obtained
when
only
the
cza
terms
are used (fig. 5), but the general behavior remains the same. An exception is when the drag divergence critical Mach number is exceeded• The helical tip Mach number exceeds the critical Mach number for the blade section characteristics used (Mcrit = 0.9) at about 475 knots (V/_R = 1.33); it exceeds the sonic value (M = l) at about 550 knots (V/_R = 1.55). are limits to the validity of the theory, but the main blade aerodynamic model occur below these only the c_ terms in the rotor aerodynamic studying neither this
limits. It is concluded that coefficients is satisfactory
the basic behavior, and, in fact, is quite too small (low inflow) nor too large [stall
example,
the
range
in which
the
c2a
These points [fig. 21) effects of the better
expressions
using for
accurate so long as and compressibility). are
adequate
is
V is For
approxi-
mately V = 25 to 350 or 400 knots (V/_R = 0.i to 1.0 or 1.1). When one predicts the characteristics of an actual aircraft, however, especially the high-speed stability, the best available rotor blade aerodynamic model should be used, which probably means tabular data for the lift and drag coefficients as a function of angle of attack and Mach number for the blade sections used. Figure
22
shows
the
influence
of
using
the
simplified
on the predicted system sweep terms (except for
stability. The effect is that of the effective elastic axis shift,
through
model
hEA);
the
basic
already
uses
only
the
c_a
theoretical
model
eliminating the wing which is retained
terms
in the
rotor
aerodynamics and has no angle of attack or dihedral. The effect of the blade aerodynamics was discussed previously; the effect of dihedral is expected to be similar to that for sweep; and there is little influence angle of attack generally (either angles at least). Therefore, the A-5032
experimentally or simpler theoretical
theoretically, for model is quite
better of small
147
satisfactory for studying basic proprotor dynamics, giving the samegeneral characteristics as the more involved model. For the design of an actual aircraft, however, a good structural analysis of the wing and pylon motion should be used. Figure 23 shows the behavior of the system dynamics during a rotor rotational speed sweep at 185 knots. The decrease in the wing modefrequencies is almost exactly proportional to m-l, that is, the dimensional frequencies are nearly _onstant during the _ variation. The lag frequency decreases with faster than the (per rev) wing frequencies do. The _ - l modeagain showsa frequency resonance with the ql mode with increased damping when the _ - l frequency is higher, and decreased damping when it is lower than the ql frequency. Some of the damping variation probably results from the high inflow influence. At low _, a resonance of the B + l and p modes occur, which is apparent The
in
both
dynamic
the
frequency
characteristics
and
damping
of
the
of
Bell
these rotor
two on
eigenvalues.
the
quarter
stiffness
wing, at half normal operating rotor speed (_ = 229 rpm), are shown in figure 24, including a comparison with the full stiffness wing results (plotted vs. V/N). The frequencies of the modes are given in figure 24(a) (except for the B, B + I, and _ + 1 modes), and the great increase in the lag frequency that results from slowing the rotor is evident (see also fig. 16(b)). The wing frequencies are fairly well matched between the quarterand fullstiffness wings. However, because of the difference in lag frequencies, the damping for the wing modes is not well simulated on the quarter-stiffness wing (figs. 24(b) and (c)), especially for the ql mode, which, for the fullstiffness wing, encounters a resonance with the _ 1 mode. The influence of the rotor lag motion may be removed from the full-stiffness wing theory by eliminating the _lC and _IS degrees of freedom and, indeed, the ql damping on the quarter-stiffness wing correlates well with that case. With _he increased lag frequency on mode) encounters torsion
the quarter-stiffness wing, the p mode (instead of the a _ - 1 mode resonance, with a corresponding influence
ql on
the
damping.
Figure 25 shows the eigenvalues and eigenvectors for the Bell rotor at the typical cruise condition V/fZR = 0.7, a = 458 rpm, V = 249 knots. This figure is a time vector representation of the modes, so the eigenvector set for a given mode rotates counterclockwise at _ = ZmX and decreases exponentially at a rate given by ReX. The projection of each vector on the real axis gives the participation of the degrees of freedom in the motion during the damped oscillation of the system in that mode. The degrees of freedom not shown for a given mode have a magnitude negligible on the scale used (i.e., less than about 5 percent of the maximum). The autorotation and powered cases show little difference except for the _0 motion, of course, and in the wing mode eigenvalues. The rotor degrees of freedom participate significantly in the wing modes. The B ± l and _ ± 1 modes show the coupling of the flap and lag motions due to the high inflow aerodynamics, but little coupling with the wing motion or with the collective rotor degrees of freedom. If BIC leads BIS in the time vector representation, the plane wobbles in the same direction the
148
lag
modes.
With
the
stiff-inplane
flap mode is progressive (the tip path as the rotor rotation) and, similarly, rotor
(v r > l) and
negative
63
for
(so the
effective v8 < 1), then the 8 - i, 8 + i, and _ + 1 modesare progressive and the _-i modeis regressive, as expected. The frequency response of the Bell rotor to each of the six input quantities is shown in figures 26 and 27 for autorotation and powered flight, respectively. The magnitude of the response of each degree of freedom to the input is shown; the rotor is operating at V/?AR= 0.7, _ = 458 rpm, and V = 249 knots (the sameas for the eigenvectors in fig. 25). The frequency response of the system is a good indicator of the dynamics involved, particularly the peaks in the response that occur at the resonant frequencies if the degree of freedom can be excited by that input. The frequencies of the eigenvalues are also shown (lower right) to identify the resonances. The wing vertical bending resonance (ql) is most important for the cyclic inputs (_G, BG, @lC, and @lS), and the chordwise bending resonance (q_), for the collective inputs (uG and 80)" There are also significant resonances Qith the upper-frequency rotor modes (8 + i, C + 1). The degrees of freedom usually show significant excitation the higher frequencies, especially near resonance with the wing modes, even there is small or negligible steady-state response. The major differences between the powered and autorotation cases are the steady-state response (especially too,
the
and
for the
the
response
collective of
the
The response shown at static response of the
inputs),
which
carries
into
the
low
at if
frequencies,
C 0 motion.
very low frequencies in figures 26 and 27 indicates system to the six inputs. The system generally
separates into a longitudinal or collective inputs u G and 80) and a lateral/vertical or
group cyclic
(variables 80 and group (variables
C 0 and BIC , 81S ,
_IC, and _IS and inputs aG, 8G, @IC' and @IS ). The wing variables (ql' q2' P) couple the two groups, but are excited most by the cyclic group. In autorotation, the static response of the cyclic rotor variables to the cyclic inputs is of order 1 for the flap motion and of order 0.I for the lag motion; their response to the collective inputs is negligible. The static response of the collective rotor variables to the cyclic inputs is negligible; the response of 80 to @0 is small, and to UG, it is negligible. The response of C 0 to the collective inputs is of order i; _o/UG = -i, of course, as discussed earlier (eq. (86)). The static response of the wing variables to the collective inputs is negligible; the response of q_ and p to the cyclic inputs is of order 0.05 and the response of q2 is of order 0.005. For powered flight, there is negligible effect on the response to the cyclic inputs compared to autorotation, but the response to collective inputs (80, UG) increases significantly. The static response of the cyclic flap motion to the collective inputs is then of order 0.05, the response of the cyclic lag motion is of order 0.005, the response response ables
of ql is of order 0.i, of p is of order 0.01.
(80 and
C0)
to
the
the response The static
collective
inputs
of q2 is of order 0.05, and the response of the collective variin
powered
flight
is of
order
0.2.
Consider a comparison of the predicted dynamic characteristics for the Bell rotor with experimental results from the full-scale tests in the 40- by 80-Foot Wind Tunnel and with the results of the Bell theories. Full-scale experimental data are available for the frequency and damping ratio of the wing modes. The data are limited by the tunnel maximum speed (about 200 knots) and by the use of an experimental technique that gave primarily only the damping ratio for the wing vertical bending mode. The data were obtained by use 149
of an aerodynamic shaker vane on the wing tip (evident in fig. 12; the same technique was used for the Boeing rotor, fig. 13). The vane was oscillated to excite the wing motion desired; when sufficient amplitude was obtained, the vane was stopped and the system frequency and damping were determined from the subsequent decaying transient motion. This configuration is best suited for excitation of the wing vertical bending mode (ql). Figure 28 shows the variation of the system stability with velocity at the normal operating rotor speed (£ = 45S rpm), in terms of the frequency and dampingratio for the wing modes. The results of the present theory are compared with the experimental data from the full-scale test, and with the results of the Bell linear and nonlinear theories. Reasonable correlation with both experiment and the Bell theories is sho_. The good correlation of the frequencies predicted by use of the present theory with the experimental data (fig. 28(a)) follows because the wing stiffnesses were chosen specifically to match the measured frequencies (at around 100 knots). The difference between the predicted damping levels of the Bell linear and nonlinear theories is largely due to the neglect of the wing aerodynamic forces in the former. Figure 29 shows the variation with rotor speed _2of the wing vertical bending modedamping for the Bell rotor at Y = 185, 162, and 150 knots. Reasonable correlation is shownwith both the experiment and the Bell theories. For Y = 162 and 150 knots, the predictions from the Bell theories are aw_ilable only
at
normal
operating
rotor
speed
(_ = 458
rpm,
from
fig.
2S(b)).
Figure 30 shows the variation of the system stability with forward speed for the Bell rotor on the quarter-stiffness wing, at half normal operating rotor speed (P_ = 229 rpm). During the full-scale test of this configuration, the available collective pitch was limited to the value reached at about 155 knots (at 2 = 229 rpm). Since the rotor was operated in autorotation, the collective pitch and inflow ratio Y/_b_ were directly ,,_orrelated. The maximum value of the inflow ratio was reached at 1S5 knots, where 7/[I£ = 0.875. Above this speed, the collective was constant, and the inflow ratio was fixed at about V/fZ_ -- 0.840. The increase in velocity above 155 knots was accompanied by an increase in the rotor speed 2 to keep the infl_w ratio at the constant wllue demanded by the collective limit. The theoretical predictions include the actual rotor speed. The predicted frequency and damping with the rotor speed maintained at a constant value (_ = 229 rpm) are also shown in figure
30.
The true values of the inflow ratio V/_.£ for the experimental points above 155 knots are shown in figure 30. Reasonable correlation is shown with both the experiment and the Bell theories. The decrease in the frequencies at high speed is produced mainly by the increasing £. The increasing _ at high speed due to the collective limit significantly reduces the wing vertical bending antl torsion damping, primarily per rev) wing frequencies.
because
of
the
decrease
in
the
effective
(i.e.,
The variation with rotor speed of the wing vertical bending and torsio,_ damping for the Bell rotor on the quarter stiffness wing is shown in figure 31 for V = 150 and 170 knots. Reasonable correlation _.s shown with experiment and the Bell theories.
150
Figure 32 shows the rotor flapping due to shaft angle of attack. The correlation of experiment and theory is shown in figure 32(a). The present theory predicts fairly well the magnitude of the flapping due to shaft angle of attack and also the longitudinal flapping BIC. However, the theory underestimates the lateral flapping BIS by about a factor of 1/2. The results of the Bell linear theory are almost identical with the results of the present theory. The Bell nonlinear theory, however, predicts well the lateral flapping BIS also, as shown in figure 32(a). As discussed in reference 25, the better prediction of BIS with the nonlinear theory is probably due to the inclusion of the influence of the wing-induced velocity on the rotor motion. Further evidence for that conclusion is the single point in figure 32(a) for which the present theory adequately predicts the lateral flapping _IS- That point is from the powered test, which was not conducted on a wing. The lateral flapping BIS is small comparedto the longitudinal flapping BIC, so the present theory does predict the magnitude of the flapping well. The azimuthal phase prediction has the sameorder of error as does BIS, however. Figure 32(b) shows the predicted and experimental variation of the flapping with inflow ratio V/_. The theoretical results are for a velocity sweep at normal rotor speed (_ = 458 rpm), while the experimental results include limited variation o£ _ as well as V, and the flagged points are even for the quarter-stiffness wing. Yet the flapping correlates well with the single parameter that the primary influence is the rotor aerodynamic forces. tion of BIS is again observed; the single point that agrees the powered test point.
V/_, The with
indicating underpredicthe theory
is
Figure 33 shows the variation of the wing vertical bending (ql) damping with V/_R, during velocity sweeps for the Bell rotor on the full-stiffness and quarter-stiffness wings. The full-scale experimental data show a definite trend to higher damping levels with the full-stiffness wing than with the quarter-stiffness wing, and this trend correlates well with the present theory. The difference in damping at the same inflow ratio results from the lag motion. Figure 33(b) shows the frequencies of the _ - l, ql' and p modes for the full-stiffness and quarter-stiffness wings. The full-stiffness wing has a resonance of the _ - 1 and q. modes that increases the q_ damping below the resonance and decreases it a_ove, and produces the peak zn the damping observed in figure 33(a). Slowing the rotor on the quarter-stiffness wing greatly increases the lag frequency and removes it from resonance with ql (instead there is a resonance with the p mode, as shown in figure 33(b) and discussed earlier). Another way to remove the influence of the rotor lag motion - in the theory - is to simply drop the _IC and _IS degrees of freedom from the full-stiffness wing case. Then the predicted wing vertical bending damping is almost identical to that for the quarter-stiffness wing (fig. 33(a)).
The
Hingeless,
Soft-Inplane
A 26-ft-diam, flight-worthy, hingeless, and constructed by the Boeing Vertol Company, Wind Tunnel in August 1972. The configuration consisted
of
the
windmilling
rotor
mounted
on
Rotor
soft-inplane proprotor, designed was tested in the 40- by 80-Foot for the dynamics test (fig. 13) the
tip
of
a cantilever
wing,
151
with the rotor operating in high inflow axial flow. The rotor and wing were described previously. The full-scale test data for the quarter-stiffness wing runs, and the theoretical results from the Boeing theory are from reference 26. The theoretical dynamic characteristics of this rotor and wing are discussed, followed by a comparison with the full-scale test results and the Boeing theoretical results. _e major changes in the dynamic behavior comparedwith that of the Bell rotor are due to the different placement of the blade frequencies. The Boeing rotor has 1 - v_ of the sameorder as the wing vertical bending frequency (as did the Bell rotor), but the soft-inplane rotor with v_ < 1 introduces the possibility of an air resonance instability, that is, a mechanical instability that results from the resonance of the _ - 1 and ql modes. This instability will occur at a definite _ (for resonance) which, in this case, normal operating rotor speed and at low forward speed. At high the lag damping _ becomes ical discussion of the air introducing the possibility motion of the sofc-inplane mode stability.
that
The Boeing rotor has v B - l is very close
the B - l mode at high Y/f_.
is above the enough Y/_7_,
large enough to stabilize the resonance. An analytresonance instability was given earlier. Besides of an instability, at high _ and low V, the lag rotor generally decreases the wing vertical bending
cantilever blades with v B sufficiently above to the wing vertical bending mode frequency.
I/rev so Hence
takes on many of the characteristics of the ql mode, especially In fact, it is usually the B - 1 mode that becomes unstable at
high inflow rather than the ql mode. By the time the B - 1 root enters the right half plane, the mode has however assumed the character of a wing vertical bending mode (this behavior is discussed further in terms of the eigenvectors of the two modes). Thus observed already for the
the high inflow Bell rotor.
instability
mechanism
is
the
same
as
The predicted variation of the eigenvalues of the system with forward speed, at the normal airplane mode rotor speed (_ = 386 rpm), is shown in figure 34: frequency and damping ratio and root locus. The flap frequency is greater than 1/rev and the coupled frequency of the B + l modes increases somewhat with the inflow ratio. The lag frequency is less than 1/rev and the coupled frequency decreases mode frequency increases. quencies to the ql' and The q_1 damping is quite because of the influence
with the inflow The proximity of
ratio. Since the B - i and
v_ < I, the _ _ - 1 mode fre-
even the q2' frequencies is apparent in figure low at low speeds and has a minimum around 200 of the _ - 1 mode, that is, the air resonance
ior. The ql damping increases of the B - 1 and q modes (as 1 tors). The B - l mode damplng
1
34(a). knots behav-
at high V, but there is considerable coupling indicated by the frequencies and the eigenvecdecreases very quickly at high speed and, by
the
time the root crosses into the right half plane at Y = 480 knots (7/_7_ = 1.S4), the mode is really a wing vertical bending instability, that is, the high inflow inflow proprotor and wing instability. This change in the character of the B l and ql modes is Y = 250, 400, and 500
shown in figure 34(d), which presents knots. At low speed, the eigenvector
the eigenvectors on the left is
clearly identifiable as the ql mode, and the eigenvector on the right as the B 1 mode, based both on the frequency of the root and on the participation the degrees of freedom in the eigenvector. As forward speed increases, the
152
at
of
wing vertical bending motion decreases in one modeand increases in the other. The modethat is originally the rotor low-frequency flap mode _B l) becomes unstable just before 500 knots, and by that time this modehas assumedthe character of the primary wing vertical bending mode. Note that the wing vertical bending motion is characterized not simply by the ql degree of freedom,
but
also
The wing is encountered
by
the
motion
of
_IC'
chord (q2) mode damping at V = 510 knots (V/_
_lS'
_0,
and
decreases = 1.64).
p associated with This
speed is an
with
the
mode.
until an instability air resonance
instability, as indicated by the coincidence of the _ - l and q2 mode frequencies at this speed (fig. 34_a)). Wing chord bending produces a lateral motion at the rotor hub forward of the wing tip, which couples with the rotor lag motion. An air resonance instability can occur at even high speed with the wing chord mode because the wing aerodynamic damping of that mode remains small. The instability,
case,
q2 instability occurs at a slightly higher so, in some cases, it may be the critical
The wing torsion (p) mode couples with mainly due to simply a coincidence of
the the
speed than boundary.
rotor coning damping and
the
B - I/ql
(_) mode in this frequencies of
the two modes. These modes have fundamentally different character _0 is in the longitudinal group of variables and p, in the lateral/vertical group) and do not really want to couple. The roots try to cross on the root locus plane (fig. 34Cc)) and instead exchange roles; the coupling is significant only in a narrow region near 300 knots; able. While this coupling does
elsewhere, the not have great
roots are clearly distinguishphysical significance, it is
discussed because a slight change in the parameters or in the model may eliminate the coupling. For comparison with such cases, it is most convenient to plot the damping (fig. 34_b)) as if the root loci really did cross, that is, by joining the corresponding p and _ pieces. This practice is followed in the comparisons that follow. The influence of the rotor lag motion on the system stability is shown in figure 35, which compares the damping of the wing modes with and without the _IC and _IS degrees of freedom in the theory. The rotor lag motion has a large and important influence on the wing modes. The rotor lag motion substantially decreases the stability of the ql and B - 1 modes. The _ 1 mode behavior remains the same when the lag degrees of freedom are eliminated, but the sharp damping decrease (and instability) occurs at a speed about 250 to 300 knots higher, beyond the scale used in figure 35(a). The low damping of the ql mode around 200 knots is shown to be due to coupling with the _ 1 mode, that is, air resonance behavior. The rotor lag motion also decreases the q^ mode stability at high V, another air resonance effect The lag motion s_abilizes the p mode, but that is not really needed. The complete root locus is shown includes The
in figure 35(c), which is to the lag motion (fig. 34(c)).
be
influence
perturbation
of
the
rotor
speed
compared
with
the
degree
root
of
locus
freedom
that
and
the
wing aerodynamics on the system stability is shown in figure 36. The basic case is autorotation operation, including the wing aerodynamics. The rotor speed perturbation degree of freedom generally decreases the stability, that is,
powered
operation
is more
stable,
especially
for
the
ql
mode
where
the
air
153
resonance behavior is much less noticeable. generally increase the stability.
The wing aerodynamic forces
The influence of the complete expressions for the rotor aerodynamic coefficients is shown in figure 37(a) (for clarity, only the autorotation case is shown for the B l and p modes). Generally, the use of the better blade section correct
aerodymamics decreases collective was obtained
autorotation based on the
operation of inflow angle
the predicted stability of the wing modes. The from a performance analysis for powered and
the Boeing rotor. The approximate collective used, at 75 percent radius, was @0.75 = tan-l(v/fa_)/(3/4)
1.0 °. The helical tip blach number reaches the (_ = 0.9) at about 500 knots (V/faR = 1.61) and about 580 knots (V/P_ = 1.86). The conclusions
+
blade critical Mach number reaches the sonic value at are the same as for the Bell
rotor: the basic behavior of the system is described well with only the c_ terms in the rotor aerod>mamic coefficients, but the complete expressions should be used to obtain correct predictions for actual vehicles, particularly for the high-speed stability boundaries. The use of the approximate collective does not influence the dymamics much, although it is, of course, not satisfactory
for
performance
calculations.
Figure 38 shows the influence of the use of the simplified theoretical model on the predicted system stability. As for the Bell rotor, it is concluded that the simpler model is satisfactory for studying the basic behavior, but for the design of a particular vehicle, the best available model should be used. Figure 39 shows the variation of the system eigenvalues with rotor speed for the Boeing rotor at 50 knots. At this low speed, the ¢ - 1 and ql frequency resonance around 530 rpm (fig. 39(a)) results in an air resonance instability in the ql mode (fig. 39(b)). At resonance, there is a corresponding increase in the _ - l damping. The resonance and corresponding instability occur above the normal rotor operating speed (_ = 386 rpm) even with the wing used for the wind-tunnel test, which was softer in bending than the full-scale design. The general decrease in the ¢ ± 1 mode damping with _ results from the low lag damping at low inflow. Figure 40 shows the variation of the eigenvalues with _ for the Boeing rotor at 192 knots. The ¢ - 1 and ql resonance again occurs at about _ = 500 rpm, but this speed is sufficient to stabilize the air resonance motion. Figure 41 summarizes the Boeing rotor air resonance behavior: the wing vertical bending mode (ql) damping variation with rotor speed 9 for Y : 50 to 192 knots. The stabilizing influence of the forward speed is shown. An earlier section derived an estimate for the Y value required to stabilize the air resonance motion. In this case, resonance occurs with the wing vertical bending frequency of about 0.28/rev, and v¢ of about 0.8/rev; with the other parameters required from table 3, equation (203) gives V/_ > 0.268 for the stability requirement. At this speed, resonance occurs at _ = 500 rpm, so the velocity requirement is Y > 108 knots. The use of the equivalent radius approximation for the rotor lag damping Q_ gives instead g/P_ > 0.285 or g > ll4 knots, which is only about 6 percent higher. The estimate compares well with the calculated (fig. 41), better, in fact, than is reasonable used for the air resonance estimate.
154
boundary of about 120 knots to expect from the simple model
The dynamic characteristics of the Boeing rotor on the quarter-stif_less wing, at half normal operating rotor speed (_ = 193 rpm) are shown in figure
42. The frequencies of some of the corresponding modes on the full-stiffness wing are also shown in figure 42(a) (plotted vs. V/N; the 8 + i, _ + l, and fl frequencies are not shown for the full-stiffness wing). The wing mode frequencies are well matched to the full stiffness wing values (per rev), but slowing the rotor increases both the flap and lag frequencies of the blade considerably, the flap frequency to near 2/rev and the lag frequency to near 1/rev, as com-
pared with about vB = 1.35 and v_ = 0.75 at normal _ /see also fig. 17). The lag frequency moves nearer I/rev-and thus the _ - 1 mode frequency is lower for the quarter-stiffness wing (besides the influence on the dynamics, the lag frequency near I/rev also means large vibration and blade loads). With the rotor frequencies so different, the system damping shown in figure 42(b) has much different behavior than for the full-stiffness wing (compare with fig. 34(b)), especially for the q_ system stability with stiffness
wing
at
400 due
rpm and in q_, to the coupllng
the
Figure typical
80
and B - 1 modes. Figure 43 shows the variation of rotor speed _, for the Boeing rotor on the quarterknots.
Air
resonance
at about 500 rpm. The with the fl - 1 mode.
44 shows the eigenvalues cruise condition of V/_
this soft-inplane (v_ < i) and and _ + 1 modes are progressive
effects peak
in
are the
and eigenvectors = 0.7, _ = 386
evident q2
damping
in ql at
at 225
the about rpm
is
for the Boeing rotor at rpm, V = 218 knots. With
cantilever (_fl > l) rotor, the _ and the fl 1 mode is regressive
I, fl + l, as expected.
The frequency response of the Boeing rotor to each of the six input quantities is shown in figures 45 and 46 for autorotation and powered flights, respectively. The magnitude of the response of each degree of freedom to the input is shown; the rotor is operating at V/f_r_ = 0.7, _ = 386 rpm, and V = 218 knots (the same as for the eigenvectors in fig. 44). The steady-state (lowfrequency) response, compared with that of the Bell rotor, shows only the following differences: with the hub moment capability of the cantilever rotor (vB < l), the flap motion with respect to the shaft motion is increased. There is increased lag motion
is reduced, because of
blade inplane restraint (lower _) and there is a change in the phasing of the cyclic rotor response (e.g., fllC and BIS) to the (e.g., Olc and 01S ) because of the change in rotor frequencies. Consider a comparison of the predicted Boeing rotor with experimental results from by 80-Foot Wind Tunnel and with the results experimental data are available vertical bending mode, obtained as used with the Bell rotor.
and the wing the softer azimuthal cyclic inputs
dynamic characteristics for the the full-scale tests in the 40of the Boeing theory. Full-scale
for the frequency and damping of the wing by the same shaker vane excitation technique
Figure 47 shows the variation of the system stability with velocity the normal operating rotor speed (_ = 386 rpm) in terms of the frequency damping ratio for the wing modes. Reasonable correlation of the present with both experiment and the Boeing theory is shown. However, data are available only for wing vertical bending mode damping.
at and theory
155
Figure 48 showsthe variation of the wing vertical bending modedamping for the Boeing rotor with rotor speed _ at V = 50 to 192 knots. These runs were conducted to investigate the air resonance behavior of this proprotor and wing configuration. Reasonable correlation is shown with both experiment and the Boeing theory, except at the higher tunnel speeds. There the data show considerable scatter because the tunnel turbulence made analysis of the transient motion difficult.
Figure 49 for the Boeing rotor speed (2 for the Boeing correlation also shows
shows rotor = 193 rotor
is shown with both experiment and the air resonance behavior in both
SECTION
In this compared are the
the variation of the system stability with forward speed on the quarter-stiffness wing, at half normal operating rpm). Figure 50 shows the variation with rotor speed on the quarter-stiffness wing at V = 80 knots. Reasonable
chapter,
6:
COMPARISONS
the
present
with the published theoretical models
WITH
theory
work of developed
the Boeing theory and
OTHER
and
the
theory. The experimental
_ sweep data.
INVESTIGATIONS
results
obtained
other authors; of primary in the literature.
are
interest
here
llall (ref. 8) discussed the role of the negative H force damping on the high inflow proprotor behavior, reviewed the problems found in the XV-3 flight tests, and reviewed the results of the 1962 test of the ×V-3 in the 40- by 80Foot Wind Tunnel. He presented an investigation of the influence of various parameters on the stability of the rotor and pylon, particularly forward speed, pylon pitch and yaw spring rate, and pitch/flap coupling (_3); this investigation used the full-scale XV-3 test results, model tests that simulated the XV-3 configuration, and analysis results from a theory presented in the Hall derived the equations of motion for a two-bladed rotor on a pylon; was chosen because the analysis was to support the ×V-3 investigation. model then had three degrees pitch _x, and pylon yaw _. with Hall's equations, wi_h
of freedom: The present the following
Present
Hall considered only the teetering blade; to the pylon expressed in 156
flap angle B results (eqs. correspondence
Notation
tx
ax
ty
ME I
(II2)MS
H
H
Y
H cos
the case of uB = i, that therefore, no hub moment
is, due
The the
(teetering), pylon (146) for N = 2) agree of notation:
Hall
-ay
motion in his model. terms of integrals of
paper. N = 2 His
sin
no to
hub spring restraint of flapping is transmitted
aerodynamic forces M B and H were blade section forces F z and F x over
the
span, which agree with the present results except that the radial drag force was neglected. Hall did not, however, expand the rotor aerodynamic forces in terms of the perturbed rotor and pylon motion because he did not derive a set of linear differential equations. Hall solved for the dynamic behavior by numerically integrating the equations of motion; hence he found the transient motion rather than eigenvalues because of the periodic coefficients for N = 2. With this method, the exact, nonlinear aerodynamic forces could be included rather than the linearized expansion. Gaffey, behavior and
Yen, and Kvaternik (ref. ll) discussed the proprotor aircraft design considerations in relation to the wing frequencies, _st
response, and ride quality. flap coupling on the rotor was shown that a cantilever
The influence of the blade and the rotor/wing stability rotor, that is, _B > l, has
frequencies and pitch/ were discussed. It greater stability, and
that v B > 1 reduces flapping significantly but also increases blade loads. Expressions were given for the low-frequency response of flapping to shaft angle of attack (xp/V here) and shaft angular velocity (_ here) in terms of the equivalent radius approximations; the present results _eqs. (96) to (100)) agree with their expressions. Experimental proprotor/wing stability, flapping, loads,
and theoretical vibration, and
data were given gust response.
for
Tiller and Nicholson (ref. 13) discussed the stability and control considerations involved with proprotor aircraft. They found the following influences on the aircraft stability. The proprotors with positive pitch/flap coupling
and
H
an
force,
clockwise
rotation
increased
on
effective
the
right
dihedral
in
wing
produce,
C_B,
the
through
effect
the
negative
increasing
with
forward speed. The proprotor negative damping requires a larger horizontal tail for the short-period mode frequency and damping; the rotor contribution found was on the order of 30 to 40 percent of the stabilizer contribution. Similar thrust (Q%)
results
were
damping and
hub
found
in yaw force
the
wing
value.
for
this
rotor
for
(T_)
during
The
the
contributes rolling
thrust
rotational
vertical
due
tail
significantly
increase
C_p
to rolling
direction)
requirements
by
(T_)
that
to about
The
Cnr.
The
30
50 percent
produces
appreciably
(CnB).
to
adverse
alters
rotor
yaw
the
rotor
torque of
(ACnp
Dutch
< O,
roll
damping and mode shape. The rotor influence on the aircraft stability derivatives found here agrees with the results of Tiller and Nicholson. They also discussed other features of the proprotor configuration that influence the aircraft nacelle
stability
and
contribution
stiffness
on
the
control:
to
C_a,
lateral
the
the
thick
wing,
important
derivatives
the
influence
(particularly,
high
roll
inertia,
of
the
interconnect
Cnr
and
Cnp),
and
in helicopter rotors on the
and cruise mode. lateral dynamics
They point out is more complex
that than
the longitudinal matter of enough
dynamics, vertical
that meeting effectiveness.
requirements
largely
Young
and
Lytwyn
(ref.
18)
developed
model (BIC, 81S, _, and _x) for studying optimum value for the flap stiffness for
a
the
is
four-degree-of-freedom
proprotor pylon/rotor
dynamics. stability
shaft the
control features influence of the
but tail
the
the that
on
a
theoretical They found at about
an
157
vB = I.I. An approximation to this result was obtained by setting to zero the term that couples the rotor with the pylon; that is, in the present notation,
(' B2 _ 1) r
16
pc r +
\
=o
There is then no moment about the pivot due to tip path plane tilt, which greatly increases the rotor/pylon stability. This optimum was discussed in a previous section and was also the subject of the discussion of reference 18 by Wernicke and Gaffey. Young and Lytwyn presented several results for the whirl flutter case (a truly rigid propeller on the pylon), which were also discussed previously. Yotmg and I,ytw>m found the power-on case to be less stable than the is,
windmilling the influence
present namic factor degree
case; of
they were considering, the c_ terms in the
rotor
however, the aerodynamic
results confirm that the use of tim better coefficients decreases the predicted stability. in windmill operation (autorotation) is the of freedom, which makes the windmilling case
power-on ?_'-bladed
case. rotor
The theoretical (N -> 3) on an
model elastically
considered restrained
by
Notation
Young
61 C
Young pylon
essentialiy the rotor
equivalent aerodynamic
Present
158
Lytwyn pitch
was and
an yaw
flapping (n6 = r), but possible. Only the so the system reduces model was
Lytwyn
Cy
the rotor different
to the coefficients
and with
¢r
%r of
that The
13q
-%j
the derivation was considerably
O,
6C
1G
Although equations
and
#
calculation of the aerodyThe really important rotor speed perturbation much less stable than the
degrees of freedom. The blade :notion allowed was rigid elastic blade restraint was included so that _ > 1 was rotor tip path plane tilt couples with the pylon motion, to four degrees of freedom. The same four-degree-of-freedom considered here (fig. 1), with the corresponding notation: Present
case of _ coefficients.
aerodymamic from that
present
result.
coefficients used here, The
corresponding
and
Lytw_
is Notation
Young
the
2CT/_a
MTO
2C4/_a
MHO
2Hn
MHT
2N u
MTT
-2M_
MTp
in the final
linear form is
notation
for
Young and the blade
Lytwyn span.
equations
(26).
evaluate The set
these coefficients assuming of four equations of motion
constant obtained
_ and c_ correspond
Descriptions of analyses typical of the most sophisticated for calculating the dynamic characteristics of tilting proprotor be found in references 29 and 30. These are, in fact, the most
over to
currently aircraft complete
used may
descriptions available in the literature for the proprotor Their use lies primarily in the development and support of
aircraft analyses. the design of
specific aircraft. More exploratory investigations
elementary models remain of proprotor dynamics.
for
Descriptions of the considerations involved,
tilting may be
valuable
proprotor aircraft, found in references
general
and the design 10, 15, 22, 27,
and
and
28.
The XV-3 flight test results are described in references 2 and 3, and the ×V-3 tests in the 40- by 80-Foot Wind Tunnel are described in references l, 8, and 9. Recent tests of full-scale proprotors in the 40- by 80-Foot Wind Tunnel are described in references 14, 15, 25, and 26. Some experimental data from small-scale model tests are also available (refs. ii, 15, 16, and 24, for example).
CONCLUDING
A
theoretical
model
has
been
REMARKS
developed
for
a proprotor
on
a cantilever
wing, operating in high inflow axial flight, for use in investigations of the dynamic characteristics of tilting proprotor aircraft in the cruise configuration. The equations of motion and hub forces of the rotor were found including the response to general shaft motion. This rotor model was combined with the equations of motion for a cantilever wing. In further studies, however, the rotor model could easily be combined with a more general vehicle or support model, including, aircraft. The general
for example, behavior of
the rigid-body the high inflow
degrees of freedom of the rotor has been investigated
and, in particular, the stability of the proprotor and cantilever wing configuration. The effects of various elements of the theoretical model were examined, and the predictions were compared with experimental data from windtunnel tests of two full-scale proprotors. From
the
theoretical
results
comparisons with the full-scale, the nine-degree-of-freedom model tion of the fundamental proprotor
for
the
two
full-scale
rotors,
and
wind-tunnel test data, it is concluded that developed here is a satisfactory representadynamic behavior. The model consists of
first mode flap and lag blade motions of a rotor with three or more blades, and the lowest frequency wing modes. The limitations of the present theory are primarily the structural dynamics models of the rotor blades and the wing and the neglected degrees of freedom of the proprotor aircraft system. For the rotor, it was assumed that the blade flap and lag motions are not coupled, that is, are pure out-of-plane and pure inplane motions, respectively. The model neglected the higher bending modes of the blades, and the blade elastic torsion degrees model used only
of an
freedom were neglected entirely. elementary representation of the
For the support, structural modes
the of the 159
wing. The model was limited to the cantilever wing configuration, neglecting the aircraft rigid-body degrees of freedom as well as the higher frequency modesof the wing and pylon. The present model does incorporate the fundamental features of the proprotor aeroelastic system. Hence these limitations of the model are primarily areas where future work would be profitable, rather than restirctions on its current use. Froma comparison of the behavior of the gimballed, stiff-inplane rotor and the hingeless, soft-inplane rotor, it is concluded that the placement of the rotor blade natural frequencies of first modebending - the flap frequency v8 and the lag frequency v_ - greatly influences the d_namics of the proprotor and wlng. Moreover, the rotor lag degrees of freedom was found to have a very important role in the proprotor dynamics, for both the soft-inplane (v < 1/rev) and the stiff-inplane (v > 1/rev) configurations. The theoretical model developed here has been established as an adequate representation of the basic proprotor and wing dynamics. It will then be a useful tool for further studies of the dynamics of tilting proprotor aircraft, including more sophisticated topics such as the design of automatic stability and control systems for the vehicle. AmesResearch Center National Aeronautics and SpaceAdministration Moffett Field, Calif., 94035, Dec. 26, 1973
160
REFERENCES
Koenig,
,
D.
G.;
Grief,
Investigation Convertiplane.
,
.
.
.
.
.
H.;
and
Kelly,
M.
W.:
Quigley, H. Stability
C.; and Koenig, D. of a Tilting-Rotor
C.: A Flight Convertiplane.
Reed,
Ill;
R.:
W.
H.
and
Bland,
Precession
C.: Test
S.
An
Reed, Wilmer H. IIl: Propeller-Rotor Review. J. Sound Vibration, vol.
W.
Earl,
Pylon pp.
Jr.:
Soc.,
Edenborough,
Review
of
Prop-Rotor vol.
Ii,
TN
II. Kipling:
Stability.
at High
2, April
vol.
_Nirl
of
Aircraft
1966, of
Advance
pp.
Art
NASA
Ratios.
TR
J.
Am.
11-26.
Tilt-Rotor
5, no.
Flutter.
VTOL
Aircraft
2, blarch-April
Rotor-
1968,
97-105.
ll.
Gaffey, T. M.; Yen, J. G.; and Kvaternik, R. G.: Analysis Tests of the Proprotor Dynamics of a Tilt-Proprotor VTOL U.S. Air Force V/STOL Technology and Planning Conference, Nevada, DeTore, rotor 1970,
Sept.
pp.
vol.
Composite Aircraft, Design vol. 14, no. 2, April 1969,
State of the pp. 10-25. and Model Aircraft. Las Vegas,
1969.
J. A.; and Gaffey, VTOL Aircraft. J.
T. M.: The Stopped-Rotor Am. Helicopter Soc., vol.
Variant 15, no.
of the Prop3, July
45-56.
Tiller, F. E., Considerations Soc.,
14.
XV-3
Propeller-Nacelle Whirl 3, March 1962, pp. 333-346.
Wernicke, K. G.: Tilt Proprotor Art. J. Am. Helicopter Soc.,
13.
the
1961.
10.
12.
of
Treatment
D-659,
Propeller-Rotor
Investigation
J. Aircraft,
Tunnel
Whirl Flutter: A State of the 4, no. 3, Nov. 1966, pp. 526-544.
Stability
no.
Evaluation May 1960.
Analytical
NASA
Wind
a Tilting-Rotor
Study of the Dynamic NASA TN D-778, 1961.
IToubolt, John C.; and Reed, Wilmer H. III: Flutter. J. Aerospace Sci., vol. 29, no.
Hall,
Scale of
Limited Flight Center,TR-60-4,
Instability.
Reed, Wilmer If. III: R-264, 1967.
Full
Longitudinal Characteristics TN D-35, 1959.
H.; and Ferry, R. Air Force Flight
Helicopter
.
R.
the NASA
Deckert, W. Aircraft.
Propeller
.
of
Jr.; and Nicholson, Robert: for a Tilt-Fold-Proprotor
16,
no.
3, July
1971,
pp.
Stability Aircraft.
and Control J. Am. llelicopter
23-33.
Wernicke, Kenneth G.; and Edenborough, H. Kipling: Full-Scale Proprotor Development. J. Am. llelicopter Soc., vol. 17, no. i, Jan. 1970, pp. 31-40. 161
15.
Edenborough, H. Kipling; Gaffey, Troy M.; and Weiberg, JamesA.: Analysis and Tests Confirm Design of Proprotor Aircraft. AIAA Paper 72-803, 1972.
16. Marr, Robert L.; and Neal, Gordon T.: Assessmentof Model Testing of a Tilt-Proprotor VTOLAircraft. Mideast Region Symposium,American Helicopter Society, Philadelphia, Pennsylvania, Oct. 26-27, 1972. 17. 18.
19.
20.
Sambell, Kenneth W.: Proprotor Short-Haul Aircraft - STOLand VTOL. J. Aircraft, vol. 9, no. 10, Oct. 1972, pp. 744-750. Young, Maurice I.; and Lytwyn, RomanT.: The Influence of Blade Flapping Restraint on the Dynamic Stability of LowDisk Loading Propeller-Rotors. J. Am. Helicopter Soc., vol. 12, no. 4, Oct. 1967, pp. 38-54; see also Wernicke, Kenneth G.; and Gaffey, Troy M.: Review and Discussion. J. Am. Helicopter Soc., vol. 12, no. 4, Oct. 1967, pp. 55-60. Magee, John P.; Maisel, Martin D.; and Davenport, Frank J.: The Design and Performance Prediction of Propeller/Rotors for VTOLApplications. Paper No. 325, 25th Annual Forumof the American Helicopter Society, Washington, D. C., May 14-16, 1969. DeLarm, Leon N.: Whirl Flutter and Divergence Aspects of Tilt-Wing and Tilt-Rotor Aircraft. U.S. Air Force V/STOLTechnology and Planning Conference, Las Vegas, Nevada, Sept. 1969.
21.
Magee, John P.; and Pruyn, Richard R.: Prediction of the Stability Derivatives of Large Flexible Prop/Rotors by a Simplified Analysis. Preprint No. 443, 26th Annual Forum of the American Helicopter Society, Washington, D. C., June 1970.
22.
Richardson, David A.: Proprotor Aircraft. pp. 34-38.
23.
Johnston, Robert A.: Parametric Studies of Instabilities Associated With Large Flexible Rotor Propellers. Preprint No. 615, 28th Annual Forum of the American Helicopter Society, Washington, D. C., May 1972.
24.
Baird, EugeneF.; Bauer, Elmer M.; and Kohn, Jerome S.: Model Tests and Analyses of Prop-Rotor Dynamics for Tilt-Rotor Aircraft. Mideast Region Symposiumof the American Helicopter Society, Philadelphia, Pennsylvania, Oct. 1972.
25.
Anon.: Advancementof Proprotor Technology Task II - Wind Tunnel Test Results. NASACR-I14363, Bell Helicopter Co., Sept. 1971.
26.
Magee, John P.; and Alexander, H. R.: Wind Tunnel Tests of a Full Scale Hingeless Prop/Rotor Designed for the Boeing Model 222 Tilt Rotor Aircraft. NASACR I14664, Boeing Vertol Co., Oct. 1973.
162
The Application of Hingeless Rotors to Tilting J. Am. Helicopter Soc., vol. 16, no. 3, July 1971,
27.
Anon.: V/STOLTilt-Rotor Study Task II - Research Aircraft CR-I14442, Bell Helicopter Co., March 1972.
Design.
NASA
28.
Anon.: V/STOLTilt-Rotor Aircraft Study Volume II - Preliminary Design of Research Aircraft. NASACR-I14438, Boeing Vertol Co., March 1972.
29. Yen, J. C.; Weber, Gottfried E.; and Gaffey, Troy M.: A Study of Folding Proprotor VTOLAircraft Dynamics. AFFDL-TR-71-7, vol. I, Bell Helicopter Co., Sept. 1971. 30.
Alexander, H. R.; Amos, A. K.; Tarzanin, F. J.; and Taylor, R. B.: V/STOL Dynamics and Aeroelastic Rotor-Airframe Technology. AFFDL-TR-72-40, vol. 2, Sept. 1972, Boeing Co.
31.
Bailey, F. J., Jr.: A Simplified Theoretical Method of Determining the Characteristics of a Lifting Rotor in Forward Flight. NACARep. 716, 1941.
32.
Gessow,Alfred; and Myers, Garry C., Jr.: Aerodynamics of the Helicopter, Frederick Ungar'Publishing Co., NewYork, 1952.
33.
Peters, David A.; and Hohenemser,Kurt H.: Application of the Floquet Transition Matrix to Problems of Lifting Rotor Stability. J. Am. Helicopter Soc., vol. 16, no. 2, April 1971, pp. 2S-33.
34.
Coleman, Robert P.; and Feingold, Arnold M.: Theory of Self-Excited Mechanical Oscillations of Helicopter Rotors With Hinged Blades. NACA Rep. 1351, 1958.
163
164
VUG-_
/
/
/
/
/
/
/ /
/
/
/
/
/
/ /
H
/
/
v -.91---i
/ / /h
yT i Hub
Y Blede
Figure i.- Four-degree-of-freedom for hub forces, pylon motion, blades is shown.
model for proprotor dynamics with and gust velocity; only one of the
conventions N rotor
165
0
4-a
@
_ _
0 °r-t
m @
'..) 0 m
0
0
0
m
m @ o_ 4J .el U 0 _0 > I C"I
%
,,-._
166
V_ G
VBG
V
/
/
/
/ /
/
/
/
/
/
/
/
/ /
/
Xp'ex
/
/
/
/ /
H, Mx
"/
/ /
/
/ Hub ID--
__
yp,ay
_"
J
,..-T, Q y
Zp,_'z
Y, My
Blade
Figure 3.- Rotor model for proprotor dynamics: hub forces and linear and angular motion, and gust velocities; only one of blades is shown.
moments, pylon the N rotor
167
o
% ©
r-_
I o o ,._
1
o
% o • t4-4 m i.-)
CK mt._ Q
o
4--_ ,x:: c_ 4-_ o
,,--_
o %
o
o o .r-.I m %
I
l 0
o %
I 0
r-
168
.4
-
W Only Cza terms Boeing
// l
/'//I"
.3
Mod/_/j .2
._51 _
..f_
_M__=_._ 0
/
(a)
I
I
.4 -
Tip
I
resultant Boeing
,_
_." _._
Bell
--
[
I
i
Mach number
I I .75 I I
I
I
I
I
I 1.O
I
.75
1.0
.2
.I
0
)
I
I
I
1.0 f
J
J
,, I/I ., ,,V/,'/
/
/
/
/://.'..5./
.5
z.z,zz_
HI_
1.0 V/aR
2.0
(a) M M ' M_ (b) [-18, (c) _e6, Figure the cZ
5.-
Influence
Bell and terms.
of Boeing
blade rotors,
section compared
P
aerodynamics with
on
coefficients
rotor
coefficients based
on
for only
the
169
Ky
Ky
\ \ \ \ \ \ \ \ \
I /
/
Hyperbol ic divergence
J \
\
/ /
boundary
/ / /
j
/
/ / / / /
LF
\
/
\
\
L/_ _i KF
Hyperbolic divergence
/
/
boundary
l jr
\N J,//
}m--K
x
J
//]\\\ /
K_
/
/
\
L/._
I" Ltz
/ /
Kx _
/
/
K_ KF
! !
Figure
6.-
Whirl
flutter
(two-degree-of-freedom) boundary.
170
divergence
instability
x
\ \ \ \ \
o h q_
o
4_ I
>-
,--4 .z= o ,_ 0
><
t_ 0 i
•
_D .,-I
N
\
o
o
\
\ \ \
._1 0
0
4_
°_ 0
0 .r,.t
I
,z
171
8 aence Stable
6
Flutter
2
e .,_\
1,11"-I
I
I
2
0
1 I I .5
I
I
1 I
I
I
I
_x
Figure
172
8.-
\\
\\\\\7(k\\\\\\\
,_
]
4
Typical
whirl
I
'
I
I 8
Ib I 1.5
1
1
= "v/"K'x/Ix
flutter
\ \ \ \ \
6
Kx = K x / L 0
\\
stability
boundaries.
I
I
1 2
VI,Q,R
= ll
h=.3,
N II.-_--Ib=2,
7,=4,
C*
=0
15 S = Stable F = Flutter 1 = l/Rev divergence D = Divergence
I0 S
.Q
zJ_ V II
5
F
F1
D 0
5
I0 Kx*=
Figure
9.-
Whirl
flutter
stability
N K x I-_-
]_b
boundaries
for
a
two-bladed
rotor.
173
_.,13 0
g,-
¢D C.)
0 0
G .
°m-I ._-_
,-4
0_-_
> O_
@
>.., ._
X
E 0 At:_ q...q
,_.4 E
N
,< 4--" C
I
d
U ,,It,..
_.'£, %
°_-_ 0
174
,.Cl "!
0 rl-
k\
I N
"1
\
0 °t-4
\ -..\
I
0 E
x
/
:>.. f,-i
/
/ /
0
.,,-I
/ / /
/
> _J
c:l ej I
/ /
.,,-I
/ / / / / /
175
Figure 12.- Bell Helicopter Company full-scale, 25-ft-diam proprotor cantilever wing for d'_namic tests in the Ames 40- by 80-Foot Wind Tunnel. 176
on
Figure 13.- Boeing Vertol Companyfull-scale, 26-ft-diam proprotor on cantilever wing for dynamic tests in the Ames40- by 80-Foot Wind Tunnel. 177
40
30
2o g
0
-I0
Bell Boeing
30
\
2O Q.
\
,7
I0
(b)
Thickness I
I
I
I
0
I .5
I
I
I
I
r/R
Figure
178
14.-
Geometric
(a)
Twist.
(b)
Thickness
characteristics
ratio.
of
two
full-scale
proprotor
blades.
I 1.0
-'11142 6O 5O -- -- Boeing
l
.-N_ I
E 40 - 30 E
--Be,,
-I I I
2O
F I I
I0 I
I
I
I
I
I
I
I
I
I
I0 8 6
% 3 i Z
0
i
2 o ks_ H W
l
-l_
I
\ 0 25
2O
% i
z
15
% o H hi
I 0
Figure
15.-
I
I
I
i
.5 r/R
I
I 1.0
(a) Section
mass.
(b) Section (c) Section
flapwise modulus/moment product. chordwise modulus/moment product.
Structural
characteristics
of
two
full-scale
proprotor
blades.
179
1.2
--
i.l Q.
(a) 1.0 0
i
I
I
I
I
I00
200
300
400
500 R
I
I
I
0
I
I
I
600
700
800
900
fps
I
200
I00
Y
500
I
I
I
I
400
500
600
700
rpm
_,rpm
D
(I.)
229
550
(b)
550 I
I
I
I
0
I 1.0
I
i
I
458(Normal I
V/,O, R
(a) Flap frequency v B. (b) Lag frequency v Figure
180
16,- Blade rotating
natural
frequencies
for Bell rotor.
_) 1 2.0
_., rpm 193
300 386 (Normal _) 55O t.
CZ
(o) I
o
218 knots at normal ,O, V I 1 I I
I
1
I
I
I
2-
_.t
I _
j|
....
b)
1 -
300 386 (Normal ,D,) 550
"
218 knots at normal _, _7 ,I _ I I ,-t,
,1.o
0
"2.0
(a) Flap frequency (b) Lag frequency Figure
193
...............
Q.
17.- Blade rotating
natural
v8. v
frequencies
for Boeing
rotor.
181
3
n
---__
0
o
\.
¢.)
191
o o_
+ 0
_:
I ii
cM
_
.o ii
(_
+ L_
iii
ii
*<
¢..n
o
(/}
+
\
ii
ii
o
(b. _
°_
4-
0 U
°_
_
+
I --
0 I ii
_I• 0
;
,,,.oII
0 e_
t'M 0
ii
.,.< o_-.I o
%
b,-
._
__
+
e,!._
.__.
0
,',,,
\ o
f
/ 192
o
+
_
_
.
I
--
_I$
Blc .I
.01 I
_ic .I
I
.01 i
_0
p
.I
I
.01
^1
A
/fix
I
i
I
q2
ql .I
-
J .01 I
.oi
I
I0
I
I
_I,0.
m
I
l 7 oo.
- _r e_ a. "i- o"
_1. +
+
(o) I
.01 .01
.I
IO
t wl_
(a) Figure V = to
26.-
Bell
249
knots),
input
at
rotor
Vertical in
autorotation
magnitude
frequency
gust
of
response
aG at
input. V/_ of
each
=
0.7
(_
degree
= of
458
rpm,
freedom
_. 193
I
--
_IC .I
,SiS
.01 I
--
.01
,80
.I --
ql .i
-
q2
-
I .01
AAI
,1
I I
I0
_/,0,
T
.01 .01
(b) .I
I I0
_/_
(b)
Lateral
Figure
194
gust 26.-
BC
input.
Continued.
TEE -
(_ Q. (_. +
+
,Sic.I
.01 I
--
_IS
_IC .I -
-
I
.01 I
/
--
_o
_]0 .I --
I
I
I
I
I0
I
1
q2
ql .I --
.01
.I _/_, 1
P
.I
T m.
--
.01 .01
,
T
_ o.. q_.. + o"-- o"
+
,c,,
.I
I
I0
o#9,
(c)
Longitudinal Figure
26.-
gust
u G input.
Continued.
19,5
I
m
_1$
_IC .I -
.01 I
_ic .I
.01 I
_0
h
.I
_o
I .01
q2 ql
E
.I .01 .01 I
.I
T q_
P .I
.Oi .01
I
I0
I
I
_ID.
m
I
I
F_I _--_ T
-- _r _ o. q_L + + oI12. J..J_
(d) I
I .I
I0
(d)
Lateral Figure
196
Z
cyclic 26.-
pitch Continued.
01C
input.
_lC .I
.01 I
_t¢
_t$
.I
.01 I
B
,BO .I
_o
l
.01
AI.,AA
I
ql
.I
.01 .01
I
I0
I
I
,,_/g'/,,
I I
I
T
P .I
(e)
.01 .01
.I
(e)
I
Longitudinal Figure
--
e_ cx _1
+
+
I tO
cyclic 26.-
pitch
@IS
input.
Continued.
197
Bic .I
.01
_is
r
J
--
_tc .I -
_is
.01 I
/_o .I
.01
I
I
I
.I
I
I I0
I
q2
m
ql .II .01 .Oi J
m
_l,g, h
P
.J
I
T ¢o.
--
I .01 .01
_/_
.I
(f)l I
(f)
I0
Collective
pitch
e
input. o
Figure
198
26.-
Concluded,
I
T
_
_
"
m. +
+
J Bic
.I -
,Sis
.01
I
I
I
I
_lC .I
.01 J
J
B
_0
--
q2
--
m
q# ,i .01 .01 J
.I
I
I0
w/_
m
P .I
I
.01 i_j_
.01
(a)
.1
I
t
_-
o._
V/_R
=
,"_
,...n
_.
+
+
I
I0
w/C2,
(a) Figure
27.-
Bell
rotor
Vertical in
V = 249 knots), magnitude input at frequency _.
powered of
gust
aG
operation response
of
input. at each
degree
0.7 of
(_
=
458
freedom
rpm, to
199
I
--
_lS
,B1c .I
I
.01 I
i
J
--
_s I;ic
.1_ .Ol I
_0
ql
--
.I --
_o
.I
q2
.0t
V
.01 .01
I
I0
I
I
oJo_m.¥
¥
_I,0, I
p
-
-
(b)l .I
I
(b)
10
Lateral Figure
2OO
.I
.01 I
gust 27.-
6G
input.
Continued.
l m
,Bis
.01 I
--
_ic .I -
.01 I
_0
B
.I -
_o
-
.01 i
m
ql .I
q2
.01 .01 ]
I0
--
P .I
I
I
QO. J._
-
I_"
_
_"
_..L..._ ¢_0. " + + Q:_,,.._
(c) I
.01 .01
I
I0
_I,Q,
(c)
Longitudinal Figure
27.-
gust
u G
input.
Continued.
201
I
--
,BIC .I
_IS
-
B
.01
_IC .I
.01 I
/30
--
_0
.I
.01 I
B
q2
ql .I .01
.I
.01
I
I0
I
I
wl,O, I
--
I
I
P .I
.01 .01
I
0"
-
(d) I .I
I
I0
_I_,
(d)
Lateral Figure
202
cyclic 27.-
pitch Continued.
OIC
input.
N _..,._
0"
_..
+
+
_lS
.01 I
_1$
]
--
190 .I
_o
.01 I
--
ql .I
Cl2
.01 .01 i
.I
--
I
I0
I
I
"_" (_I_"+
4-
_/.fl, I
P .I
--I
--I
-
0"
°J n
I_"
(e) I
.01 .01
.I
(e)
I
Longitudinal Figure
I0
cyclic 27.-
pitch
OIS
input.
Continued.
203
_tlC.I
_15
1
.01 I
I
_i$
,8o
.1
_o
1
I
ql
.I
q2
-
.01
.01
.I
I
I0
w/D,
I -
I
c_, ,.,m
.01 .01
(f)l I0
(f) Collective Figure
204
pitch
eo input.
27.- Concluded.
qD, _,.m
2
"-----'-
Present theory
_ --
Nonlinear} Linear
-o
Bell theory
Experiment
........
nm_ml
Q
w/D,
I
ej Q O
Q
O
-
q2 ..._w w _"
q, (a)
0
_) 0
I
I
I
I
I
I
I
I
O q2
'05Ic
e
I
0
I
I
I
.10 -
.O5
P
o
(d)
=m"
=ram
=am
I
o
50
,ram
I
I
I00
150
I
I
200
250
V, knots
;I o
I
I
I
I
I .5
I
V/9,R
(a)
Frequency
of
the
modes.
(b) Wing vertical bending damping (ql). (c) Wing chordwise bending damping (q2). {d) Wing torsion damping _v). Figure
28.-
Bell
rotor scale
velocity experimental
sweep at _ = 458 rpm, comparison data and the Bell theories.
with
full-
205
Present t heory _
NonlineorLineor } 0
Bell
theory
Experiment
.05
(a)
185 knots
I
1
I .5
.6
I .4 V/_R
.05
(b)
162 knots
I
I
I •5
I .4 V/D,R
I .3
.05 -
Q
(c)
150 knots I 50O
04oo
,_, I 5
I 6O0
rpm
I .4
I .3 V/,O, R
I 5OO
I 600
I 700
I 800
I_,R, fps
Figure
206
29.-
(a)
Wing
vertical
bending
damping
(ql)
at
185
knots.
(b)
Wing
vertical
bending
damping
(ql)
at
162
knots.
(c)
Wing
vertical
bending
damping
(ql)
at
150
knots.
Bell
rotor
rpm
sweep, and
comparison with Bell theories.
full-scale
experimental
data
I
O iO
0..i
.e
o
o.
._-I
g o o
\\,, _,
"_
_
--
m
---
-.-1W
>
i
_ _ 121.. Z
CE >
'/
x
_.)
II,°t
E
o l:io_ I_._
O LID
_
.
°"_
I
I
O
u0
O
O
o.
u_
_
"O _ O ._ IN"O
O
O
.,-_ _ "_ _ _.,_
o
o. ,_,.O
/
I/
!
O O 0d
_
_
O
_-_ o
_.,_ o _
_ o
o _
_ o o_
_ o
_.._"
1 O
_
C
I
? E
t14
M
o
m
o_ n._
o"
>
o
>_
O o
O If)
,,--4
!
I Od
I
v
I O
O
O
o.
O
207
--
Present theory
....
Nonlinear I Bell theory Linear I
_
....0..0 Experiment 0
.O5 _
_
.,_.
ql
_ _
-C_-'-- O..._.O
0
_" _.,..
0
/
0
.
-- --'_---,--_'_..__
( P a)150knotsj l I I .9 .7
i I
1
I .5
_ ""
J I .5
I V/g,R
ql 0
P (b) 170 knots I 0 200 300
I 400
1 500
I 600
,Q,, rpm I 500
I 400
I 500
1 600
I 700
I 800
_,R, fps t I 1.0
I .8
I
I .6
I
I .4
V/,O.R
(a) Wing (b) Wing Figure
208
31.-
vertical vertical Bell rotor full-scale
bending bending
(ql) (ql)
and and
torsion torsion
on quarter-stiffness experimental data
and
(p) damping (p) damping
wing, with
at at
150 170
knots. knots.
rpm sweeps; comparison Bell theories.
with
I0
o
.5
J_
--
0
0
_
theory r_/O
_
/
/_i
/
/
Bell nonlinear
theory
/
/-
0
•
, ooo
.5
1.0
alBl aa
[
I
0
I
.5
1.0
[
I
J
.5
I.O
aB,c _'_e,./;%n
(a)
aa
0 aBis aa
1.0 Theory
al/_l/a=
/
Experiment
--
- _
oc_
a_ -a B_s/ aa ----
Oa
[] 8Bis
.5
/J/
/ / /
no (b)
_____ _.__
/
Correlation
(b)
Variation wing
Figure
° I
I 1.0
.5 V/_R
0
(a)
/
Detweeli at
Laeury
with inflow ratio _ = 229 rpm).
and Y/_
experiment. (flagged
symbols
are
for
quarter-stiffness
32.- Bell rotor flapping due to shaft angle of attack; comparison full-scale experimental data and Bell theories.
with
209
Experiment 0 •
Theory ' Full stiffness wing __Quarter stiffness wing .....
Full stiffness
wing, no t_lC ' t_lS
// .O5
/'/. Q
0
/"_
o° OO
Ca)
__
Full stiffness wing Quarter stiffness wing
_(;-I
2
_J
ql
(b) 0
I
(a) Comparison with _111Lscale (b) Frequency of the modes. Figure
210
I 1.0
.5 V/,GR
experimental
data.
33.- Bell rotor wing vertical bending (ql) damping, velocity full-stiffness and quarter-stiffness wings.
sweeps
on
3
--
B+I
_+I
P
B
,8 p
q2
f
,8-i
ql
0
(a)"
I
I
I
I
I
.15
_,+I .I0
.O5
ql ql
0
I
I
I00
200
"
I
I
300
J
400
500
600
V, knots I
I 1.0
I .5
0
J 1.5
V/,O,R
(a) (b)
Figure
34.-
Boeing
rotor
Frequency of the Damping ratio of
velocity
sweep,
_ =
modes. the modes.
386
rpm;
predicted
eigenvalues.
211
\\f3-i \ \
/
f
/ /
\
.IO
/
\\11
/1 I\ I
\ \
/ /
\
\
/ /
\
/
\ \ \ \
C .15,-
/ /
/
/
/
.IC i//////
05
60O 0
IO0
200
30{) V, knots
0
500 J 1.5
_L 1.0
1 .5
L
400
V/_,R
(a) Wing (b) Wing
vertical chordwise
bending bending
(ql) and rotor flap (B - I) damping. (q2) and torsion (P) damping.
_IC Figure
214
35.-
Boeing
rotor
velocity and 51S
sweep, rotor
_ = 386 rpm; lag motion.
with
and
without
_
,B+I
--...-____.
_+1
oJ
q2 f r
ql
/3-1
we-I (o)"
I
ql
I
I
I
I
500
I 600
/3+1 .10 -
.05
q2
ql
(b) I I00
0 [ 0
I 200
I 300 V, knots
I .5
I 400
I 1.0
1.5
V/_,R
[a) Frequency of the (b) Damping ratio of Figure
34.-
Boeing
rotor
velocity
sweep,
modes. the modes.
_ = 386
rpm;
predicted
eigenvalues.
211
/_+1
Velocity
sweep
Nc_OO OOOo OOOOOO --
(M
V,
q2 off
I
,I
I
I
I
I
I
I
= 0
-.05
for
q I,
q2
(c) Root Figure
212
34.-
locus. Continued.
;Re X
IO q¢ IO (D
knots
,k = -.005
+ i.351
V = 250
knots
;k = -.124
+ i .;)50
P
= /3_s
q=_B=s
/_lC
;,s
_,c
X = -.022
X'_
+ i.334
_:,c
V = 400
knots
k = -.034
+ i.292
-- BIS
B=c
X = -.037
,.BIC
+ i .343
V = 500 knots
X = .004
+ i .286
P
= _ls
(;o
" B_s
(;Ic
r¢;'s
.Bic /_lC
C (d)
Boeing
rotor
eigenvectors modes.
at
_ = 386
for
wing
rpm;
variation
vertical
Figure
34.-
bending
with (ql)
velocity and
rotor
of
predicted
£1ap
(B -
1)
Concluded.
213
.15 -
Without
_lC, _lS
/ i
With _IC,_IS
_,/9-1 T '_
/
/
.I0 -
.I
',,,/
/
.05 -
/
\
//
\
/// ____
(a)
ql
I
0
\
1 I
I
\
I
t
\_ I
I
I
"q_
300
400
.15 -
/ / / / /
/ .I0 _
//
I///
.05
(b] 0
I
I
I00
200
I 600
500
V, knots I 0
I .5
I 1.0
I 1.5
V/,Q,R
Figure
(a) Wing
vertical
(b) Wing
chordwise
35.-
Boeing
rotor
bending bending velocity and
214
_IS
(ql)
and
(q2) sweep, rotor
and
rotor
flap
torsion
_ = 386
(p)
rpm;
lag motion.
(B
I)
damping.
damping.
with
and
without
_ic
ImX
Ve loci ty sweep _o (MOOo0oo0 0000 V,
Without
l(C)
I
I
I
I
I
knots
_IC,
_;IS
ReX 0
-.05
for
ql,
q2'
p
(C) Figure
Root 35.=
locus. Concluded.
215
wing aerodynamics
Powered ql
wing aerody
nomlcs
(a)
0 .15
.10
wing aerodynamics Basic. ered
.O5 No wing aerodynamics Paw (b) 0
2OO V,
t
knots
I
0
I 600
4OO
.5
I
I
1.0
1.5
V/D,R
(a) Wing (b) Wing
vertical chordwise
bending bending
(ql) and rotor flap (8 - I) damping. (q2) and torsion (p) damping.
Figure 36.- Boeing rotor velocity sweep, (autorotation and wing aerodynamics), cases.
216
_ = 386 powered,
rpm; and
comparison of basic no wing aerodynamics
.15
-
Correct --
.10 ....
collective
_ Complete
Approximate col lect ive
expressions
/ coeff for rotor aerodynamic icients
Only CL, ' terms coefficients
/
in rotor aerodynamic
/ / /
/ /
/
/
Powered
.O5
/
Autorotation
/ /
ql
/ / /
Autorotation /
(a)
.15
I
I
-
I I I I I I I
.10
P
I
-
/ Autorototion
I
.05 Autorotation Powered
(b)
l I00
0
._.
I
I
200
300
q2
400
500
V, knots I
I
0
.5
I
I
1.0
1.5
V/,Q,R
(a) (b) Figure
37.-
Wing Wing Boeing
vertical chordwise rotor
expressions
bending bending velocity for
the
(ql) and rotor flap (8 - 1) damping. (q2) and torsion (p) damping. sweep, rotor
a
=
386
aerodynamic
rpm;
influence
of
the
complete
coefficients.
217
0 0 (,,O
u
0
0 0 L_
0 0
",
_
m
•_
o 4-J
• r"l
_-J
m
ID
o
_ m
o
_
• ,...i k
8 °
o
_o.
,,
N'--'
r_-
m
_
> 0 0 o4 m
>
o > 0 0
.l..J .l.a o _ ..---,
• ,-I
.,-I
(XI
o
I
1
I
--0 0
--
--
O
_ m e-_ ,--_ 1.4 _ o _
O
p_u
218
O
3
_
_+_
I
0 .15
/
/3+I
.I0
\ q2
ql 0 _
_+I
(b)
I 2OO
I
I .3
V I 400 ,0,, rpm
I .2
I
I
I 6OO 1 .I
V/Q,R
I
I
200
I
1
400
I
I
600
I 800
9,R, fps (a) Frequency of the (b) Damping ratio of Figure
39.-
Boeing
rotor
rpm
modes. the modes. sweep,
V = 50 knots.
219
_l,O,
.10 -
.05
O,
(b) I
I
Vl
200
I
I 600
400 ,Q, rpm
II
I
1.2
1.0
I
I
I
n
I
I
_
.6 V/,O,R
I
2OO
L
.8
I
I
400
I .4
I
600
I 800
_,R, fps
Figure
220
(a)
Frequency
(b)
Damping
40.-
Boeing
of
the
ratio
rotor
modes.
of rpm
the
sweep;
modes. V = 192
knots.
0 0
.r.I
0
.,.-4
m
.._
0
,-q
,._
U
C_
t_ ¢)
0
bSO I=Ln II
0 ._
4-D
e-,._
o o
m
o
0
0
._
o o k
°_ o
o. !
.T't
221
3
-
B+I Ouerter stiffness ....
2
Full stiffness
wing
wing
-
,8 _I,_
.10 -
.05
,8+1
(b)
I
0
I
I00
200
3OO
v, knots I 0
I .5
I 1.0
1 1.5
V/Q,R
(a) Frequency of the (b) Damping ratio of Figure
222
42.-
Boeing
rotor
velocity
sweep,
modes. the modes.
quarter-stiffness
wing,
_ =
193
rpm.
_
_
\ q2
I
0
I
.15 -
/_+1
q2
(b)
I I00
71 200
I :500
I 400
I 500
Q,, rpm I I 1.0.8
I .6
I .4
I .2 V/Q,R
I I00
I 300
I 500
I 700
Q,, fps (a) Frequency (b) Figure
43.-
Boeing
rotor
Damping rpm
of
the
ratio
sweep,
of
modes. the
modes.
quarter-stiffness
wing,
V = 80
knots.
223
o
0
+ 0
u u
--to)
_
ii
oO
ii
co
o I!
0 o
_
i_ f,..,
_ i_. "0. ii
pr)
o_
"_
OJ
I"
° 00
I!
+
.,< I-,
Ii
II
o'} C_
I
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