NASA Technical Reports Server (NTRS) 20020086486: Whirl Flutter Stability of Two-Bladed Proprotor/Pylon Systems In High Speed Flight

19 Pages·1 MB·English

by  NASA Technical Reports Server (NTRS)

FINAL TECHNICAL REPORT FOR NASA GRANT NAG101084:
"WHIRL FLUTTER STABILITY OF TwO-BLADED
PROPROTOR/PYLON SYSTEMS IN HIGH SPEED FLIGHT"
ALFRED E. GESSOW ROTORCRAFT CENTER
DEPARTMENT OF AEROSPACE ENGINEERING
UNIVERSITY OF MARYLAND
COLLEGE PARK, MD 20742
AIAA 2002-1600
WHIRL FLUTTER STABILITY OF TWO-BLADED
PROPROTOR/PYLON SYSTEMS IN HIGH SPEED
FLIGHT
Beerinder Singh °
Inderjit Chopra t
Alfred Gessow Rotorcraft Center
Department of Aerospace Engineering
University of Maryland at College Park, MD 20742
Abstract 1 Introduction
The lack of polar symmetry in two-bladed rotors
Aeroelastic stability of a tiltrotor aircraft in forward
leads to equations of motion with periodic coeffi-
flight has been the subject of considerable research
cients in axial flight, which is contrary to three or
since the 1960's. In high speed axial flight, tiltro-
more bladed rotors that result in constant coefficient
tots are subject to an instability known as propro-
equations. With periodic coefficients, the analysis
tor whirl flutter. Due to high inflow through the
becomes involved, as a result very few studies have
rotor in this flight mode, large rotor forces depen-
been directed towards the analysis of two-bladed ro-
dent on the in-plane motion are generated. These
tors. In this paper, the aeroelastic stability of two-
forces are larger in tiltrotor aircraft than in conven-
bladed proprotor/pylon/wing combinations is exam-
tional propeller driven aircraft due to large blade flap
ined in high speed axial flight. Several parametric
and lag motions in case of tiltrotors. With increas-
studies are carried out to illustrate the special nature
ing airspeed, the rotor forces and the pylon/wing
of two-bladed proprotors and to better undemtand
motion interact with each other to a point where
the mechanism of whirl-flutter in such rotors. The
the rotor forces become destabilizing and the ro-
wing beam bending mode for two-bladed rotors is
tor/pylon/wing system becomes unstable.
found to be stable over the range of parameters ex-
Most of the rt_soarch on proprotor whirl flutter has
amined, a behaviour very different from three-bladed
focused on three or more blades. Several inves-
rotors. Also, the wing torsion mode exhibits a new
tigations of this phenomenon were conducted us-
type of instability similar to a wing torsional diver-
ing pylon pivot models. Reed Ill presented a re-
gence occuring at 1/rev frequency. This type of be-
haviour is not seen in three and more bladed ro- view of these early efforts and highlighted some
tors. The interaction between wing chordwise bend- of the key is.uues associated with this instability.
ing and torsion modes is found to be much greater Johnson [2] developed a comprehensive mathemati-
in the case of two-bladed rotors and, over the range cal model of a tiltrotor including a modal repre-
of parameters considered, these two modes govern sentation of the wing, and for high speed forward
the stability of the system. flight. Analytical results obtained with this analy-
sis showed good correlation with full scale proprotor
*Graduate Research Assistant, Student Member AIAA
tests and the analysis was later extended to include
tAlfred Gessow Professor and Director, Fellow AIAA
Copyright {_)2002The American Institute of Aeronautics and elastic blades and helicopter and conversion modes
Astronautics Inc. All rights reserved of operation. Kvaternik and Kohn [3] carried out
1
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an experimental parametric investigation of whirl- creases in the stability boundary, with the effect be-
flutter for aproprotor mounted on a rigid pylon with ing most dominant for offsets at the outboard part
flexibility in pitch and yaw. Results of this study of the blade. Acree [14] also analyzed the effect of
showed that proprotor whirl flutter can be predicted blade tip sweep on whirl mode stability of the V-22
with linear stability analyses using two-dimensional, tiltrotor with a tip mass extended on a boom for-
quasi-steady aerodynamics for the blade loading. ward of the leading edge to compensate for the aft
Nixon[4. 51conducted parametric studies for aeroe- shift of the tip center of gravity due to sweep.
lastic stability of a tiltrotor in forward flight using a All of the above studies were carried out for three-
finite element based comprehensive tiltrotor aeroe- bladed proprotors, although Johnson [2l presented
lastic analysis. The effects of several design parame- some results for a two-bladed propeller as well. Un-
ters such as rotor flap and lag frequencies, pitch-flap like a rotor with three or more blades, a two-bladed
coupling, wing stiffness and sweep were examined rotor lacks polar symmetry, which leads to propro-
to determine their effect on aeroelastic stability in tot equations of motion containing periodic coeffi-
the high speed axial flight mode. The rotor was cients. Thus, the stability analysis in the case of a
assumed to be in a state of autorotation, i.e., ro- two-bladed rotor is more involved and as a result,
tor torque was not transferred to the wing and the a lot less attention has been devoted to two-bladed
wing vertical bending slope at the tip did not af- configuratic, ns. In June 1998, whirl flutter was en-
fect rotor inplane motion. Johnson [2] showed that countered during a flight test of the Pathfinder air-
this assumption gives conservative estimates of whirl craft, fitted with several two-bladed propellers to be
flutter stability. Lag frequency tuning was found to used on the Helios aircraft. These aircraft, built un-
be a practical method for increasing flutter speed in der the Environmental Research Aircraft and Sen-
axial flight and forward sweep of the wing was found sor Technology (ERAST) program, have lightweight
to be destabilizing (Nixon [4' 5]). The above models and highly flexible wing structures and soft-mounted
rely on experimental data to model the lag frequency propellers. The simplified analysis for two-bladed
variation with blade collective pitch and the distri- rotors, when applied to the Pathfinder configura-
bution of blade flexibility inboard and outboard of tion with thi._Helios propellers, was unable to predict
the pitch bearing. To reduce the dependance on ex- the flutter encountered in 1998. Also, it is unclear
perimental data, Hathaway and Gandhi [6] have re- whether design guidelines established for preventing
cently made some refinements to the model which whirl flutter instability for three or more blades can
allow first principles based modeling of the lag fre- be applied to the two-bladed case also. The peri-
quency variation and blade flexibility distribution. odicity of the equations of motion for a two-bladed
proprotor may introduce new phenomena that re-
In recent years numerous approaches for the im-
quire careful examination and scrutiny. Hence asys-
provement of tiltrotor whirl mode boundaries for
tematic study of whirl flutter of two-bladed propro-
three-bladed tiltrotors have been investigated. Srini-
tors is carried out to determine the effects of various
vas and Chopra [71,Popelka [81et al., Corso [9] et al.
design parameters on their stability.
and, Barkai and Rand {10] investigated the use of
2 Analytical Model
aeroelastic tailoring of composite wing and rotor
blades to enhance whirl mode stability of a tiltrotor
in forward flight. Nixon [11] et al. presented a review
Figure 1(a) shows the coordinate systems used for
of the experimental and analytical efforts for sta-
the rotor and wing. The fixed frame rotor forces,
bility augmentation of tiltrotors through aeroelastic
which are transferred to the wing, are also shown.
tailoring of the wing and rotor blades. Srinivas[12l
Figure l(b) shows the geometry of the wing/pylon
et al. carried out studies on improving whirl mode
system. Flap and lag motions of the rotor blades are
stability of tiltrotors by using advanced geometry
considered and, the blades are assumed to be rigid.
blades with tip sweep and tip anhedral/dihedral.
Acree[13l et al. investigated the effect of the chord- 2.1 Rotor and Hub Structural Model
wise positions of the rotor blade aerodynamic center
and center of gravity on stability. Rearward offsets The equations of motion for a single rigid blade
of the aerodynamic center with respect to the blade in the rotating frame with flap and lag degrees of
elastic axis and pitch axis were found to give large in- freedom have been presented by Johnson [2]. These
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Figure 1, while ax, a_ and a, are pylon rotational
degrees of freedom. M_ rn) and M(Lm) are respectively
the aerodynamic flap and lag moments of the mth
blade. The above equations can be written in matrix
form as,
[I;]_ R + [g;]flR + [liplp + [C;p];"= -y/-1-4F (3)
O,C
l/j] (R WIK_I(R -k [[_p]p ----7 ML (4)
O_
where f_R and (R are vectors of the flap and lag
motions in the rotating frame while lffcontains the
pylon degrees of freedom. For N blades,
(a) Coordinate Systems for Rotor and Wing
_=[Z(, ) Z(_) ... _(_)]T
_R= [_(1) _(2)... _(N)] T
Y
_::[_ u_ _ _ _ _.]r
Other matrices in Equations 3 and 4 are defined in
the Appendix. For coupling with the pylon/wing
system, the above equations must be transformed to
the fixed frame. Also, the pylon degrees of freedom
must be replaced by the wing-tip degrees of freedom.
This is accomplished using the following transforma-
(b) Geometry of the Wing and Pylon tions,
_a = [T.F[fiF (R = [TRF](F /Y= [T^]I,_'T,p
Figure 1: Coordinate Systems for Rotor/Wing and
Wing Geometry where/_, and (F are the fixed frame rotor degrees of
freedom, while I_T_p is a vector containing the finite
element degrees of freedom at the wing-tip. [TRF]
equations include the effects of pylon translation and is an N × N matrix relating the fixed frame rotor
rotation on the flap and lag motions, degrees of freedom to the rotating frame degrees of
freedom, while [T^] is a transformation matrix that
depends on the wing sweep A and relates the pylon
i;(_; (_)+_Z())_+;Z_[-(•_-2.x")._..o._C_+(_x
degrees of freedom to the wing-tip degrees of free-
dom. fiE, _F, [TRF] and [TA] are as defined in the
• *** M ('_) Appendix.
+ 2a_)sinCm]+ S_zp = __'-r (1)
ac
For a two-bladed rotor, the collective flap (rio), dif-
I_(_"(_)+_¢(_))+ • "...... ferential flap (f_l), collective lag ((0)and differential
lag ((1) motions comprise the fixed frame rotor de-
•. ML(m) grees of freedom. In case of a three-bladed rotor, the
- ha_)cos¢.,] - I_;. = _ ac (2) fixed frame motions are collective flap (f_o), cyclic
flap (_1c and _ls), collective lag (_0) and cyclic lag
where/_(m), _(m) and Cm are the flap motion, lag
((x_and _i,).
motion and the azimuthal position of the mth blade,
u_ and v; are the non-dimensional flap and lag fre-
Note that, at, this stage, the number of blades is
quencies, respectively, and h is the distance between
assumed to be arbitrary so that the same analysis
the pylon pivot point and the hub. xp, yp and zp are
can be applied to both two and three-bladed rotors.
pylon translational degrees of freedom, as shown in
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_up : r(¢_- oucosCm + a*xSm"Cm) + _'p
v L
I
FlightSpeed
= r&,p_ + _up_ (6)
t_UR= --_pSin_gm -- XpCOS_)m + h(_xsin_bra -- _u
u V
Hubplane
uR In Equations 5,6 and 7, JUTa, JUTe, 5upA, 5upB and
5uR are independent of the radial position r and
hence they may be factored out of the integrands of
(a) Velocity and Force Components on a Typical
Blade Section the aerodynamic force integrals. The aerodynamic
moments on ablade can then be written in terms of
the aerodynamic coefficients (M#, Me etc.) and the
perturbation velocities (see Johnson[2]). For exam-
xx w: t2 ple, the perturbation flap moment may be written
Y as_
JM(m-"--_)= M,_BTa + McJuTA + MxSun_ + %Jupn
I ae
+ MoJO (8)
where J0 is the perturbation in the collective angle.
The perturbation velocities, 5UTa, JU-pB etc., are
(b) Wing Element Degree,-of-Freedom
functions of /3R, (R, WTip and ¢m for each blade,
and thus the right hand side terms of Equations 3
Figure 2: Blade Section Velocities and Wing Ele- and 4 can be expressed in terms of the system de-
ment grees of freedom and moved over to the left hand
side of the equations.
2.2 Rotor Aerodynamics
2.3 Wing Structural Model
Figure 2(a) shows the velocity components at a blade
section defined with respect to the hub plane. Aero- The wing structural model is based on a finite el-
dynamic forces are based on a linear strip theory ement formulation of an Euler-Bernoulli beam un-
and the rotor is assumed to be in a state of autoro- dergoing beamwise bending, chordwise bending and
tation. The trim procedure is based on changing torsion, as shown in Figure 2(b). The axial degree of
the collective pitch until the thrust becomes zero. freedom is not considered since the wing is assumed
Perturbations in the blade section velocities, for a to be rigid in axial direction.
forward flight speed V, can be written as,
• • . 2.4 Wing Aerodynamics
_U T -= r(ot z -- "Jr (--XpSinl_m q- ypCOS_rn -- h(_ x
Wing aerodynamics is based on a quasi-steady
. V
approximation and the aerodynamic stiffness and
cosCm + v%sinCm )+ -_ (_ cOSCm + otusinCm )) damping matrices are determined and added to
the corresponding structural matrices. Wing sweep
(Figure l(b)) is taken into consideration in formu-
= rSuTA + 5UT. (5) lating the wir,g aerodynamic matrices.
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2.5 Coupling of Rotor/Wing System
Rotating Blade
Couplinogftherotor/wingsystemessentiaclloyn- Freq. (per rev)
In-vacuo
sistsoftwoparts:(1)incorporatinthgerotoraero-
1.02
dynamiacndinertialforcesinthefixedframeinto
thewingmodeal,nd(2)incorporatitnhgewing-tip u_ 1.85
degreeosffreedomintotherotorequationosfmo- see Fig. 3
0
tion(seeSec.2.1and2.2),thisleadstoasystemof
equationwsithperiodiccoefficientIsn.statespace Wing Freq. (per rev)
form,thissystemcanbewrittenas, Beal21 0.42
Chord , 0.70
s Torsion 1.3
=A(¢)17 (9)
Table 1: Frequencies of the Baseline System
where (*) denotes a derivative with respect to the
azimuth, A(¢+2,) = A(¢) is a periodic matrix with 2
period 2, and !7is a column vector of the 2m system
1.9 \
states.
1.8
2.6 Stability Analysis
1.7
The stability of the system defined by Equation 9 is \
examined using Floquet analysis. The Floquet tran- >.1.,.6
sition matrix ¢(2_r), relates the states of the system 1.5
at ¢ = 0 to the states at ¢ = 2_r, 1.4
_(2,) = ¢(2,)17(0) (Z0) 1.3
1.2
Each column of the Floquet transition matrix
(FTM) is determined by integrating Equation 9with 1.10 0.2 0.4 0.$ 0.8 1 1.2 1.4 1.$ 1.8 2
F]_ SpeedV/_R
a unit initial condition for one state and zero ini-
tial conditions for other states. Thus 2m integra-
Figure 3: Blade Lag Frequency as a Fhnction of Ve-
tions need to be performed to obtain the FTM,
locity
where 2m refers to the number of states. An effi-
cient method, developed by Friedmann et.al. [15], for
computing the FTM in a single integration pass is A subscript 'p' for the frequency indicates that only
used in the present study. Eigenvalues of the FTM a principal value of the h,equeney can be determined
are called Floquet multipliers and are denoted by due to the fact that the logarithm of a complex num-
0_ (i----1,2...2m). The system is unstable if any of ber has many branches, giving values for Ai that
the Floquet multipliers have an absolute value larger differ in frequency by integer multiples of the rota-
than unity (see Johnson [16], Nayfeh and Mook[17_). tional speed fL Compared to a constant coefficient
Floquet multipliers are related to the Floquet expo- system, the periodic system is also described by nor-
nents (A_) as, mal modes (i.e, eigenvectors) and the roots ),i, but
for a periodic system the eigenvectors are periodic
(11) functions rather than constants (see Johnson[16]).
Real and imaginary parts of the Floquet exponents
are related to the conventional damping (¢',) and fre- The analytic_d model was validated by comparing
quency (wpi) respectively. the frequency and damping variations with flight
speed, with predicted results obtained by Nixon [4]
¢, = -Re(A,) = - _ln(E0,1) (12) for a three bladed rotor. Also, the whirl flutter
LT[ stability boundaries for a two-bladed propeller were
1 compared with the results reported by Johnson [21.
wp_ = lrn(A_) = -2-.ZOi (13) In both cases, the present results matched with those
of Nixon and Johnson.
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E
2.7
1.St 2.4
2.1
1.8
u_.o°'1.2 u.
1.5
t
o -x = - w 1.2
"_0.8
.-_0.6 C _ C3 0._¢
0.f b J
Z 0.4 Zo 03
0.2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Flight Speed V/QR
Flight Speed VA3.R
(a) Frequency
(a) Frequency
0.3
03
0.2_=
0.25
O.i
0.2
1
0,15
0,15 'B.
"_.
°°: 0,1
005
0
(
Unstable -..0_
C2 0.4 0,6 0.8 1 1.2 1.4 1.6 1.8 2
0.2 0.4 0.6 0.8 1 1.2 1A 1.6 1.8 2 Flight Speed V/_IR
Flight Speed V/_R
(b)Damping (b) Damping
Figure 5: Fn_uency and Damping as a Function of
Figure 4: Frequency and Damping as a Function of
Velocity for Baseline System (Three-Bladed Rotor)
Velocity for Baseline System (Two-Bladed Rotor)
3 Baseline System the pitch bearing (Figure 3). The rotor has a radius
The present model is used to obtain the frequency R=12.5 ft, Lock number "r=3.83, solidity cr=0.089,
and damping characteristics of two and three-bladed pitch-flap coupling Kpa=-0.268 and rotational speed
rotor/wing systems. The baseline rotor/wing system f_=48 rad/s. The wing/pylon system has a span of
has the same properties as the baseline three-bladed 16.65 ft, a chord of 5.16 ft and a pylon mast height
rotor used by Nixonl4]. No wing sweep is considered of 4.28 ft. Three finite elements of length 4.55 ft
in the present study. Table 1 shows the important each are used to model the wing, with one pylon el-
in-vacuo frequencies of this system. The collective ement of length 3.0 ft at the tip. The wing has a
beamwise stillness EIb=3.13 x 10z lb-ft a, chordwise
flap mode has a higher frequency because the blades
act as cantilevers in this mode due to a gimballed stiffness EI_==8.48 x 10zlb-ft 2 and torsional stiffness
GJ=l.62 x 107 lb-ft 2.
hub. The in-vacuo lag frequency varies with collec-
tive pitch (and therefore flight speed) due to the dis- Figure 4 shows the frequency and damping charac-
tribution of lag flexibility inboard and outboard of teristics of the baseline two-bladed rotor. It must be
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notedthatthemodeasrehighlycoupleadndhence
between the wing chord mode c, and the low fre-
theiridentificatioisnbaseodnthephysicfarlequency
quency lag mode, ( - 1, at the point where the ( - 1
andthebehavioorfperiodiecigenvectoInrsF.igure frequency crosses the chord mode frequency (Figure
4,'b','c'and't' refertomodetshatareprimarily
5(b), V/12R -- 0.2). The chord mode damping re-
wingbeamc,hordandtorsionmodersespectively,duces slightly at this point since the damping in the
while'DO',j3]'and'(l' aretherotorcollectivfleap,
- 1 mode is lower. A similar interaction is seen
differentifalalpanddifferentilaalgmodesT.hesame
when the _ - 1 mode frequency crosses the beam
nomenclatuisruesedinallsubsequefignut res.
mode frequency at V/f'tR = 0.4 for the three-bladed
rotor (Figure 5), where the beam mode damping
Showinn
Figure 4 are the frequency and damping shows a peak since, at this point, the damping in
characteristics computed using two methods: (1) us- the (i - 1 mode is higher than that in the beam
ing a full 48 state system with 3 wing elements and mode. Figure 4 presents evidence of similar energy
1 pylon element representing the wing, and (2) us- transfer between the differential lag mode ¢'1 and
ing an 18 state system obtained by applying static the wing beam and chord modes for a two-bladed
condensation to the wing equations and retaining rotor. This oceum at the points where the princi-
only the five wing-tip degrees of freedom, prior to pal value of the (1 mode frequency crosses the wing
coupling with the rotor equations. The reduced sys- chord and beam mode frequencies at V/flR = 0.22
tem captures all the essential modes quite well and and 0.48, respectively (Figure 4(a)). As compared
is computationally much faster than the full system. to a three-bladed rotor, the energy transfer between
Further results in this paper are based on the re- the rotor lag and wing modes is smaller in the case of
duced order system. a two-bladed rotor as indicated by the smaller peak
in the beam mode damping for the two-bladed ro-
Figure 5 shows the frequency and damping char- tor (Figure 4(b)). Also, the valley in the chord mode
acteristics of a three-bladed rotor having the same damping at V/_R = 0.22 (Figure 4(b)) is very small
non-dimensional properties as the two-bladed rotor. for the two-bladed rotor. Unlike a three-bladed ro-
As compared to the two-bladed rotor, two addi- tor the frequencies need not actually cross in case of
tional degrees of freedom are introduced in the three- a two-bladed rotor (i.e., only their principal values
bladed rotor due to the flap and lag motions of the cross) since, for a two-bladed rotor, Floquet mul-
third blade. Thus for a three-bladed rotor, there tipliers govern the interaction between the modes.
axe five important rotor modes (Do, f_+ 1, _ - 1, For the two-bladed rotor the energy transfer occurs
+ 1, _ - 1) compared to three rotor modes(Do,_l when both the real and imaginary parts of the Flo-
and ¢'1) for the two-bladed case. In Figure 5,/_- 1 quet multipliers of two modes approach each other.
and _ - 1 refer to the low frequency flap and lag
modes respectively, while/_ -{-1 and _ + 1 are the In the case of a two-bladed rotor, for V/f'tR > 1.1,
high frequency flap and lag modes and Do is the col- the Floquet multiplier for wing torsion becomes real,
lective flap mode. The collective lag mode _0 is not which causes a split in the damping curve for this
shown in Figures 4 and 5 since it has zero frequency mode, one part increasing and the other decreasing
due to the assumption that the rotor is in a state of to become unstable at V/fIR > 1.25 (Figure 4). This
autorotation. is a unique instability for a two-bladed rotor, similar
to the 1/rev divergence of a two-bladed propeller
A key difference between two and three-bladed ro- (Johnson{2]). The differential lag mode also shows
tor behaviour can be seen in the wing beam mode similar behavior at a high flight speed (Figure 4,
damping for the two cases. For the baseline three- V/_R > 1.9), but remains stable.
bladed rotor, the wing beam mode becomes unstable
for V/fiR > 1.19 as seen from the beam mode damp-
4 Parametric Studies
ing in Figure 5(b). However, it is evident from the
beam mode damping for a two-bladed rotor (Figure
Several paranmtric studies are carried out to deter-
4(b)) that the wing beam mode remains stable for
all flight speeds considered. An explanation of this mine the effect of key natural frequencies of the ro-
behaviour is presented in Section 4.1. tor and wing on the critical flight speed and to ex-
amine the interactions between various modes for a
For a three-bladed rotor, there is an energy transfer two-bladed rotor. The critical flight speed VI is the
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1.8 1,8
Two-Bladed
1.7 7 17 --- Two-Bladed
_1,6 --- Three-Bladed t c __" /'_ OC 1.6 - - Three-Bladed
5 1.5
. --_- | J
1.4
1;
c 0 1
01
,_+1 ', ," 0.9
0.9
J o-8_
°-819 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 z1 2.2
1 1.1 12 1.3 1.4 1.5 16 1.7 1.8 1.9 2 2.1 2.2
vp Factor v¢ Factor
Figure 7: Effect of Lag Frequency (u¢) on Critical
Figure 6: Effect of Flap Frequency (u_) on Critical
Flight Speed
Flight Speed
speed at which the damping for a particular mode baseline value, are shown in Figures 10 and 11 for
becomes zero. For comparison, results of these para- two and thr,._e-bladed rotors respectively. The prin-
metric studies are plotted along with the results for a cipal obser_Ltions (italicized) are enumerated below
three bladed rotor having the same non-dimensional along with their explanations (non-italicized).
1. For a two.bladed rotor, the wing beam mode b does
properties as the two-bladed rotor.
not become ur_table for the covered range of ut3 and
4.1 Rotor Frequencies
v_. This is a key difference between two and three-
The rotor frequencies, _ and re, are independently bladed rotors. It is evident from Figures 6 and 7
varied from their baseline values by factors varying that the wing beam mode is quite critical for three-
between 1.0 and 2.2. It must be noted that extreme bladed rotors, especially for low flap frequency, but
variations of in-vacuo flap and lag frequencies may it is stable for the two-bladed case.
not be practically possible, but they are of theoret- For a three-bladed rotor, the wing beam mode be-
ical importance to establish the trends within the comes unstable due to its interaction with the/3- 1
design range. Also, since the baseline lag frequency and torsion modes (see Nixon(5]). As shown in Fig-
is a function of velocity as shown in Figure 3, the ures 5 and 9, the ( - 1 mode frequency acts as a
factor for u¢ shifts the entire curve by a factor be- barrier betw,:en the/3 - 1 and beam mode frequen-
tween 1.0 and 2.2. Since _t_ and v( are uncoupled, cies at low flight speeds, but when the /3 - 1 fre-
in-vacuo flap and lag frequencies of the blades, an quency crosses above the _ - 1 frequency it drives
increase in these frequencies must primarily effect the beam mode into an instability. As u_ factor is
the /3z or _1 frequencies which are the differential increased beyond 1.4, the beam mode for a three-
flap and lag frequencies of a coupled system with bladed rotor does not become unstable as shown in
aerodynamics. Figure 6. Nixon [5] showed that this is because of
increased int_.'raction between the rotor flap and lag
Figures 6 and 7 show the results of varying ut_ and
modes and due to an increase in the wing torsion fre-
u¢ by the indicated factors from their baseline val-
quency whic_ moves it further away from the wing
ues. Dotted lines show the results for a three-bladed
beam mode frequency.
rotor having the same non-dimensional properties
as the two-bladed rotor. Note that, in these fig- For a two-bladed rotor the differential flap (/3z) fre-
ures, only the most critical modes for whirl stability quency does not interact strongly with the wing
are shown. Figures 8 and 9, respectively show the beam mode trequency for the covered range of uZ
frequency and damping characteristics for two and as shown in Figures 4, 8 and 10. Thus, the wing
three-bladed rotors when v_ is increased by a factor beam mode r,;mains stable for a two-bladed rotor.
of 1.1 from its baseline value. Frequency and damp- 2. As shown in Figure 6 the wing chord mode for
ing characteristics, with vz increased to 1.5 times its a two-bladed lvtor initially becomes less stable as vfl
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2 3 \
2.7
1.8
24
. 1.6 p,1
21
_ Po
'1,4 1.8
\ ",-., a 1.5 _0
u.
.__ c f 12
.oE
r_
Of
0.6 Z
_ 0.4 0.3
Z _p-l_________
o.2_ 0.2 0.4 0.6 o.s 1 1.2 1.4 1.S 1.8 2 0.2 0.4 0.6 Rk0_.BtSp1eedV1.=2R 1.4 1.8 1.8
Flight Speed V/_R
(a) Frequency (a) Prequency
0.3
0._ [_-1
0.25
02..=
0.2. 0.2
,_ 0.15
._ 0.15
0.1 i 0.1
Q
0.05
0.05
(
-0.0
-0.0_ 0.2 0.4 0.8 0.8 1 1.2 1.4 1.8 1.8 2 0.2 0.4 0.6 Ri0_.8,zSp1eedV1.2_R 1.4 "1.6 1.8 2
Flight Speed V/_R
(b) Damping (b) Damping
Figure 9: Frequency and Damping Characteristics
Figure 8: Frequency and Damping Characteristics
foru/_Factor1.1(Three-Bladed Rotor)
for vt_ Factor 1.1 (Two-Bladed Rotor)
is increased and then becomes more stable. Chord mode frequency does not cross below the torsion
mode damping for a two-bladed rotor is affected by mode frequency and also the /3z mode frequency
the placement of the/31, (l and wing torsion frequen- moves further away from the chord mode frequency
cies relative to the chord mode frequency. The/3z for V/fIR < 1.1 as shown in Figure 10. This stabi-
frequency exerts a destabilizing influence on wing lizes the wing chord mode at low flight speeds, but
chord mode damping, whereas wing torsion fre- the increase in its critical flight speed is not signif-
quency has a stabilizing influence on it. For low _ icantly large since the/3z mode frequency interacts
factors (u_ < 1.5), the (] mode frequency falls be- very strongly with the chord mode at higher flight
low the torsion mode frequency for V/flR _ 0.5-1.0 speeds (Figure 10(a)).
(see Figures 4 and 8). In this range, the chord mode In case of a three-bladed rotor the wing chord mode
damping reduces faster and the overall effect is a gains greater stability as u_ is increased, as shown
slight reduction in critical flight speed for u_ fac- in Figure 6 since, for a three-bladed rotor, the in-
tors I.I and 1.2. For u_ factorsabove 1.5,the (] stability in the wing chord mode is caused by its
9
American Institute ofAeronautics and Astronautics