N A S A C O N T R A C T O R
R E P O R T
N A S A CR-2322
ANALYSIS OF STALL FLUTTER
OF A HELICOPTER ROTOR BLADE
by Peter Crimi
AVCO SYSTEMS DIVISION
Wilmington, Mass. 01887
for Langley Research Center
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. NOVEMBER 1973
1. Report No.
4. Title and Subtitle
ANALYSIS OF STALL FLUTTER OF A HELICOPTER
I 2. Government Accession No.
3. Recipient's Catalog No.
5. Report Date
6. Performing Organization Code
Peter C r i m i
9. Performing Organization Name and Address
AVCO Systems Division
Wil mington, Massachusetts 01887
8. Performing Organization Report No.
10. Work Unit No.
11. Contract or Grant No.
13. Type o f Report and Period Covered
2. Sponforing Agency Name a?d Address
National Aeronautics and Space Administration
Washington, D.C. 20546
7. Key Words (Suggested by Author(s))
He1 icopt er rotor, Ae roe1 a st icity ,
Dynamic stall, Torsional stability
14. Sponsoring Agency Code
18. Distribution Statement
Unclassified - Unlimited
5. Supplementary Notes
The contract research effort which has lead to the results in this report was
financially supported by USAAMRDL (Langley Directorate).
This is a final report.
A study of rotor blade aeroelastic stability was carried out, using an analytic
model of a two-dimensional airfoil undergoing dynamic stall and an elastomechanical
representation including flapping, flapwise bending and torsional degrees of freedom.
Results for a hovering rotor demonstrated that the models used are capable of
reproducing both classical and stall flutter.
occurrence of stall flutter in hover was found to be determined from coupling between
torsion and flapping. Instabilities analogous to both classical and stall flutter were
found to occur in forward flight. However, the large stall-related torsional
oscillations which commonly limit aircraft forward speed appear to be the response
to rapid changes in aerodynamic moment which accompany stall and unstall, rather
than the result of an aeroelastic instability. The severity of stall-related instabilities
and response was found to depend to some extent on linear stability.
linear stability lessens the susceptibility to stall flutter and reduces the magnitude
of the torsional response to stall and unstall.
The minimum rotor speed for the
For sale by the National Technical Information Service, Springfield, Virginia 2215i
ANALYSIS OF STALL FLUTTER
OF A KELICOPTER ROTOR BLADE
By Peter C r i m i
Avco Systems Division
A study of rotor blade aeroelastic s t a b i l i t y was
carried out, using an analytic model of a two-dimensional
a i r f o i l undergoing dynamic s t a l l and an elastomechanical
representation including flapping, flapwise bending and
torsional degrees of freedom. Results f o r a hovering rotor
demonstrated that the models used are capable of reproducing
both classical and stall flutter. The minimum rotor speed
for the occurrence of stall f l u t t e r i n hover was found t o
be determined from coupling between torsion and flapping.
I n s t a b i l i t i e s analogous t o both classical and stall f l u t t e r
were found t o occur i n forward flight. However, the large
stall-related torsional oscillations which commonly l i m i t
a i r c r a f t forward speed appear t o be the response t o rapid
changes i n aerodynamic moment which accompany stall and
unstall, rather than the result of an aeroelastic instabil-
i t y . The severity of stall-related instabilities and re-
sponse was found t o depend t o some extent on linear stabil-
i t y . Increasing linear s t a b i l i t y lessens the susceptibility
t o stall f l u t t e r and reduces the magnitude of the torsional
response t o stall and unstall.
Aeroelastic s t a b i l i t y of a helicopter rotor blade is
a multifaceted problem because of the extreme variations of
the aerodynamic environment within the flight envelope of
the aircraft. In hovering f l i g h t , a blade can undergo
classical binary f l u t t e r (Ref. 1) or stall f l u t t e r (Ref, 2 ) .
In forward flight, the linear Instability experienced by
systems with periodically varying parameters can occur
( R e f , 3). While these types of instability are not normally
encountered with blades of current design, due t o the rela-
tively low disc loading and weak coupling of translational
and rotational degrees of freedom, they are certainly not
precluded from new designs, particularly those intended t o
extend present performance capabilities.
concern, however, i n both design and operation, is the
occurrence of large-amplitude torsional oscillations and
excessive control-linkage loads associated with blade stall
on the retreating side of the rotor disc at high forward
speed o r gross weight, effectively limiting aircraft per-
formance. This problem has prompted a number of recent
studies of dynamic s t a l l and the effects of stall on blade
dynamics (Refs. 4-8).
O f immediate
While stall has been identified as a causal element of
the problem, the nonlinearity of the s t a l l process, coupled
with the unsteady aerodynamic environment, has precluded an
analysis t o the depth required t o gain a thorough under-
standing of the mechanisms invol.ved.
not been clear whether the blade undergoes a true aero-
elastic instability, a simple forced response, or some
hybrid phenomenon which takes on the character of one o r
the other extreme, depending on flight conditions and blade
In particular, it has
S t a l l f l u t t e r for axial flight is amenable t o analysis
by empirical methods similar t o those developed for analyz-
ing s t a l l f l u t t e r i n cascades (Ref. 9 ) . The f l u t t e r
mechanism f o r that case has been identified as deriving
from the extraction of energy from the free stream by the
periodic variation of the aerodynamic moment.
methods applied t o the forward-flight problem (Refs. 10
and 11) have been inconclusive, however, the primary d i f f i -
culty possibly being i n applying empirical methods without
a clear definition of the underlying mechanism of the problem.
A method was recently developed for analyzing dynamic
stall of an a i r f o i l undergoing arbitrary pitching and
plunging motions which provides an ideal tool for analyzing
the stall problem i n forward flight. The method, which is
described i n detail i n R e f , 7, employs models for each of
the basic flow elements contributing t o the unsteady stall
of a two-dimensional a i r f o i l . Calculations of the loading
during transient and sinusoidal pitching motions are i n
good qualitative agreement with measured loads.
overshoot, o r l i f t i n excess of the maximum static value,
as well as unstable moment variation, are in clear evidence
i n the computed results.
This study was directed t o analyzing the aeroelastic
stability of a helicopter rotor, particularly a s it relates
t o s t a l l , using the method of R e f . 7 t o compute aerodynamic
loading. The representation of the elastomechanical system
includes flapping and flapwise bending degrees of freedom
as well as torsion.
t o perform the calculations is given i n Appendix A .
A l i s t i n g of the computer program used
blade semichord, m
mean l i f t coefficient, r a t i o of t i m e average
of 1 t o
p a 2 R2 b
l i f t coefficient, C 1
moment coefficient refe red t o quarterchord,
P U 2
blade chord, m
= C1 /(
p U2 b )
mode shape of first uncoupled torsional mode,
unit t i p deflection
mode shape of first uncoupled flapwise
bending mode, unit t i p deflection
t i p deflection due t o flapping, semichords
t i p deflection due t o bending, semichords
translational coordinates of 2-D system
( i = 1, 2), semichords
moment of inertia of 2-D system about pitch
axis, kg - m
blade moment of inertia about elastic axis
per unit span, kg - m
translational spring stiffnesses of 2-D
system ( i = 1, 2), N/m2
torsional spring stiffness of 2-D system, N/rad
l i f t per unit span a t aerodynamic reference
offsets of sprin s from pitch axis of' 2-D
system ( i = 1, 2
t o t a l blade mass, kg
blade mass per unit span, kg/m
aerodynamic moment per unit span a t aerodynamic
reference radius, N
masses of 2-D system, kg/m
rotor radius, m
inner radius of blade l i f t i n g surface, m
aerodynamic reference radius, m
instantaneous free-stream speed at aerodynamic
reference section, m/sec
reference speed, Uo = ! 2
distance a f t of e l a s t i c axis of blade
section mass center, m
distance a f t of pitch axis of mass center
of mlJ m
generalized coordinate of 2-D system, equivalent
t o hp , semichords
generalized coordinate of 2-D system, equivalent
t o h e(, semichords
angle of attack, deg
flapping hinge offset, m
collective pitch angle, deg o r rad
blade t i p torsional deflection, rad
angle of zero restraint of 2-D system torsion
advance ratio, r a t i o of forward speed t o fl R
free-stream density, kg/m3
dimensionless t i m e ,
7 = Uo t/b
blade azimuth angle measured from downwind
direction, deg o r rad
rotor rotational speed, rad/sec
dimensionless rotor speed, a* = nR/(%o b )
f l u t t e r frequency, rad/sec
frequency of first uncoupled, nonrotating
torsion mode, rad/sec
frequency of first uncoupled, nonrotating
flapwise bending mode, rad/sec
In the f l u t t e r analysis, only leading-edge stall was
considered, so the following relates specifically only t o
that type, even though the basic method can treat trailing-
edge stall as well. When the a i r f o i l is not stalled, the
flow elements represented are (see Figure la): (1) the
laminar boundary layer from the stagnation point t o separa-
tion near the leading-edge, (2) the small leading-edge
separation bubble; and, (3) a potential flow, including a
vortex wake generated by the variation with time of the
circulation about the a i r f o i l . When the a i r f o i l is stalled,
as indicated i n Figure l b , the flow elements are: (1) the
laminar boundary layer, (2) a dead-air region extending
from the separation point t o the pressure recovery point;
and, (3) a potential flow external t o the a i r f o i l and
dead-air region, again including a vortex wake.
analytic representations of these elements are described
Details are given i n R e f . 7.
i s t i c s and motions, together with the distribution of
pressure i n the dead-air region i f the a i r f o i l is stalled,
the flow and pressure over the a i r f o i l must be determined
t o compute the integrated load and analyze the boundary
layer. The problem was formulated by imposing linearized
boundary conditions of flow tangency and pressure, using
a perturbation velocity potential derived from source and
vortex distributions. The resulting coupled set of singular
integral equations is solved by casting the singularity
distributions i n series form and solving for the unknown
coefficients by imposing boundary conditions at prescribed
the a i r f o i l section character-
the individual elements of the bounda.ry layer flow as they
affect dynamic stall could not be established i n advance,
the representation in Ref. 7 was made as general as possible.
The method of f i n i t e differences for unsteady flows with
variable step size i n both streamwise and normal directions,
was employed, with the error i n each finite-difference
approximation the order of the square of the step size.
It was determined from preliminary calculations performed
for t h i s study that, at least for leading-edge stall,
results are virtually unaffected by assuming quasi-steady
flow in the boundary layer. That assumption was therefore
employed for a l l f l u t t e r computations, t o take advantage
the relative importance of
( 2 LEADING-EDGE BUBBLE
( 3 ) AIR FOIL AND VORTEX WAKE
LAMINAR BOUNDARY LAYER
(a) ATTACHED FLOW
LAMINAR BOUNDARY LAYER
AIRFOIL AND VORTEX WAKE
( b ) LEADING -EDGE STALL
Figure 1 FLOW ELEMENTS
of the resulting substantial savings i n computer storage
requirements and computing time.
dead-air region is t o define the streamwise distribution
of pressure i n that region, given the locations of the
separation and recovery points and the pressure at the
recovery point. The dead-air region is assumed t o consist
of a laminar constant-pressure free shear layer from sepa-
ration t o transition, a turbulent constant-pressure mixing
region, and a turbulent pressure-recovery region. The
laminar shear layer is analyzed by the method of R e f . 12,
assuming quasi-steady flow. The turbulent nixing and
pressure-recovery regions are analyzed using the steady-flow
momentum integral and first moment equations.
parameters i n these regions are assumed t o be universal
functions of a dimensionless streamwise coordinate, with
those functions derived from an exact viscous-inviscid
interaction calculation. Matching of approximate solutions
for the mixing and pressure-recovery regions at t h e i r inter-
face completes the analysis.
function of the model of the
unstalled a i r f o i l is analyzed using the same basic relations
employed for the dead-air region. Given the boundary-layer
parameters a t separation, the length of the bubble and the
amount of pressure rise possible, for that length, i n the
pressure recovery region, are computed. That pressure
rise is compared with the rise i n pressure i n the potential
flow over the length of the bubble.
greater than the former, the bubble is assumed t o have
burst, and the stall process is initiated.
leading-edge bubble on an
If the latter is
Loading Calculation Procedure.-Calculations
by forward integration i n time, using the blade motions
derived by integrating the equations of motion of the
If, a t a given instant, the
a i r f o i l is not stalled, the potential flow is computed,
and the boundary layer and leading-edge bubble are analyzed
t o check for bubble bursting. If the a i r f o i l is stalled,
the pressure distribution in the dead-air region is com-
puted, t h e potential flow evaluated, and the boundary layer
is analyzed t o locate the separation point. The last two
steps are repeated iteratively u n t i l assumed and computed
separation points agree. Rate of growth of the dead-air
region is determined from an estimate of the rate of fluid
entrainment derived from the potential-flow solution.
Unstall is determined by first postulating its occurrence
and analyzing the leading-edge bubble which would then
form to ascertain whether that event did in fact occur.
During unstall, the dead-air region is washed off the
a i r f o i l at the free-stream speed,
The equations of motion for a rotor blade with flapping,
flapwise bending and torsional degrees of freedom can be
written i n the form (Ref. 3)
d e l + b M @ e
2 R G d z ' + $ , F+
- b E 2
are t i p displacements due t o flapping
i n semichords, 81 is torsional
displacement at the blade t i p and the frequencies* are the
following functions of rotational speed:
The i n e r t i a l and centrifugal-force coefficients are
*Barred quantities are dimensionless frequencies, Uo/b
being reference frequency; e .g.,
a. = 51 b h 0 .
(r - 6 ) f6 fe m xm dr
The complexity of the aerodynamic representation pre-
cludes evaluation of the generalized forces Fp , Fb and
Fe by the usual s t r i p approximation.
however, t o retain both translational degrees of freedom In
the investigation of the forward-flight problem, so a
simple two-dimensional model of the dynamics could not be
used. Therefore, a two-dimensional a i r f o i l suspended i n
such-a way as t o have three degrees of freedom was analyzed.
I n e r t i a l and stiffness parameters were assigned t o make the
coupled natural frequencies of the two-dimensional system
match those of the rotor blade.
It was f e l t essential,
The system analyzed is shown schematically i n Figure 2.
The matching of the two-dimensional system with the blade
dynamics proceeds as follows.
are first defined t o correspond t o those of the blade.
Clearly, angular displacement e
torsional displacement a t the b 1 ade t i p .
of flapping and bending, Z p
Three generalized coordinates
should correspond t o blade
Ai hl + Bh2,
A2 hl - Bh2
- 2 ,
0 2 -
OPl - -
A 2 = -
" d -
Figure 2 TWO-DIMENSIONAL ELASTOM ECH AN ICAL SYSTEM
i = 1, 2.
With the above definitions, Zp + z
correct translational correspondenck!.
shown that the uncoupled natural frequencies of the two-
dimensional system match those of the blade, provided
= -h , t o give the
It &an further be
0 - 2 satisfy
w 1 + (1 + m2/m1)
o2 = o8 +
By comparing the generalized masses of the two systems, it
The last relation, together with Eqs. (1) and (2), fixes
m 2 4 :
Equating the corresponding coefficients of the characteris-
t i c equations of the two systems provides three additional
relations, which can be solved f o r the coupling parameters
x, lsl, Is2. That calculation is outlined i n Appendix B.
To complete the matching, quasi-steady approximations
t o the damping terms of the flapping equations are equated
with the result that
ml R/(-A~) = 4 -
R2 [l - (r0/R)4]
selected t o be .75R.
a rR = Uo. The aerodynamic reference radius r R was
The angle of zero restraint i n torsion was varied
periodically t o approximate the effects of cyclic pitch
variation i n forward flight, according t o the formula
8 = 8,
[l - 2 (R/rR)
p s i n $ ]
This variation gives nominally constant l i f t .
The equations of motion were solved by integrating
analytically, using linear extrapolations t o approximate
the variation of l i f t and aerodynamic moment over the
interval of integration. This scheme was found t o give
satisfactory results, provided the t i m e interval of integra-
tion is no longer than about one f i f t h of the period of the
coupled mode having the highest natural frequency.
RESULTS OF COMPUTATIONS
Vibrational and aerodynamic characteristics of the
blade analyzed were selected t o correspond t o those of the
model rotor blade described i n iief. 2. That blade is un-
twisted, of constant chord, with offset flapping hinge.
Pertinent dimensionless parameters of the model blade are
listed i n Table 1.
BLADE PARAMETERS FOR NOMINAL CONFIGURATION
3 A 9
Two elastomechanical configurations i n addition t o the
nominal one were analyzed.
with a l l other parameters as listed i n Table 1.
The third configuration had x d b = .108, with the remaining
parameters as listed i n Table 1.
One of these had
w ~ o / a ~ o
The bending mode shape, which was computed by a
finite-element method, was found not t o vary appreciably
over the range of rotational speeds of interest. The mode
shape for a$ /a
was used for a?1 computations. The torsional mode shape
f o r the nonrotating blade, also shown i n Figure 3, was used
t o evaluate torsional inertia parameters.
= 1.26, which is plotted i n Figure 3,
0 . 5
Figure 3 BENDING AND TORSION MODE SHAPES
The test blade had a NACA 23012 section. The
variation of s t a t i c lift and moment coefficients with angle
of attack for this section were computed from a series of
transient pitch calculations, and are shown i n Figure 4,
together with the measured section characteristics, from
R e f . 13. The aerodynamic model is seen t o give nearly the
correct maximum lift, but a t a slightly lower angle of
attack, and, as indicated from the variation of Cm 4 4 , the
computed center of pressure is somewhat further aft than
that of the actual a i r f o i l section below the stall angle.
Stability i n Hover
I n i t i a l calculations were performed for hovering flight,
with the nominal configuration, t o allow a direct comparison
with the test results of Ref. 2.
varied parametrically, with the collective pitch at a value
well below the stall incidence.
instability was encountered at
O f / 0 8 = .803.
The variation of bending, flapping,
and torsional aisplacements w i t h azimuth angle a t f l u t t e r
onset are shown i n Figure 5. By way of comparison, tests
(Ref. 2) yielded classical flutter at about
aeo = .72.
First, rotor speed was
A classical bending-torsion
E nR/( c3e0 b ) = 5.3
a* = 7.1
It should be noted that since the system stability was
analyzed by direct simulation, a precise point of linear
instability was not computed.
of a linear instability, both far hover and forward flight,
were obtained by successively increasing o r decreasing rotor
speed, i n small steps, u n t i l the transient response changed
from convergent t o divergent, or visa versa. The maximum
error i n the value of f l u t t e r speed, for the results pre-
sented here, is estimated t o be about three percent.
The values of
S2* at onset
Susceptibility of the system t o stall f l u t t e r was in-
vestigated next. It was found that a torsional l i m i t cycle,
a t approximately the highest coupled natural frequency of
the system, could be triggered for
Computed blade motions for stall f l u t t e r a t
are shown i n Figure 6.
S2* as low as 3.4.
S2* of 3.5
S2* below 3.4, a l i m i t cycle could not be set up,
regardless of the i n i t i a l conditions or the collective pitch
angle. Severe oscillations involving repeated stall and
unstall could be made t o occur by imposing a large i n i t i a l
bending deflection. However, the flapping response modu-
lated the torsional response, and caused continuous s t a l l
and/or unstall of the blade over a significant portion of
COMPUTED-Re, - 2 x I 0 6
( REF. 13)
0.2 0.4 0.6 0.8
Figure 4 AIRFOIL SECTION CHARACTERISTICS FOR NACA 23012
AZIMUTH 9 ,degrees
Figure 5 DISPLACEMENT TIME HISTORIES AT CLASSICAL FLUTTER ONSET
s1*=s.3,eO= 11 deg,p=o
I I I I
m . .
AZIMUTH,+ , degrees
Figure 6 DISPLACEMENT TIME HISTORIES FOR STALL FLUTTER
sZ* 3 3.5,8, = 15.0 dw,
. ' .
a revolution, due t o the large plunging rate generated by
the flapping motion. An example of this occurrence is
shown i n Figure 7. Thus, while s t a l l f l u t t e r involves only
the rotational degree of freedom, the results obtained
indicate that the minimum speed for its occurrence is deter-
mined by coup1ir;g with a translational degree of freedom.
Results for the hovering case are summarized i n
Figure 8, which compares computed and measured f l u t t e r
speed and frequency, plotted against collective pitch angle.
No upper l i m i t i n collective pitch angle for the occurrence
of stall f l u t t e r was calculated, since that l i m i t would
depend strongly on i n i t i a l conditions, and so would be
arbitrary. Quantitative differences between the computed
and measured s t a b i l i t y boundaries of Figure 8 can be attrib-
uted i n large part t o the use of a two-dimensional aerodynamic
model, which cannot precisely reproduce the aerodynamic
coupling between the rotational and translational degrees
From the basic similarity of the computed and measured
s t a b i l i t y boundaries and the character of the computed in-
stabilities (Figures 5 and 6) it can be concluded that the
aerodynamic and dynamic models formulated are capable of
reproducing both classical and stall f l u t t e r as experienced
by a rotor blade, and so can be employed t o investigate the
Stability i n Forward Flight
The nominal configuration was analyzed next for an
advance r a t i o of .1. Computations were carried out i n the
same sequence as for hovering. First, the rotational speed
at which classical f l u t t e r occurs was determined.
stall-related instabilities were investigated.
A linear bending-torsion instability of the Floquet
type (Ref. 14) was encountered a t
motions as a function of azimuth angle a t f l u t t e r onset
are shown i n Figure 9. The torsional and bending displace-
ments are seen t o display the aperiodic character typical
of t h i s type of instability.
steady-state response t o the cyclic pitch variation.
8" = 5.2.
The flapping motion is the
found t o occur for-
tive pitch angle greater
5 1 '
= 4.8 are shown i n
ment t i m e history, while
An instability analogous to stall f l u t t e r i n hover was
as low as about 4.4, with collec-
than 12 deg. Blade motions for
Figure 10. The torsional displace-
not s t r i c t l y periodic, is nonetheless
a ' 2
2 1 I
I I I 1 I 1
I I I
1 II I
AZIMUTH, 9 ,degrees
Figure 7 BLADE RESPONSE BELOW STALL FLUTTER BOUNDARY
3.1,Oo' 15.0 dq, p = O
COLLECTIVE PITCH ANGLE,eo ,deg
Figure 8 FLUTTER SPEED AND FREQUENCY VARIATION WITH COLLECTIVE
PITCH ANGLE FOR A HOVERING ROTOR
AZIMUTH, JI , degrees
Figure 9 DISPLACEMENT TIME HISTORIES AT LINEAR INSTABILITY ONSET
SZa = 5.2, Bo = 6 deg. p = 0.1
I R e v - l R e v - I Rev--I
Figure 10 DISPLACEMENT TIME HISTORIES FOR STALL FLUTTER
a* = 4.8, Bo= 13 deg, p = 0.1
brought a.bout by successive stall and unstall.
positions a t which those events occur are marked by (S) and
(U), respectively, on the $ -scale.
The blade motions f o r the type of instability shown i n
Figure 10 are not of the same character as those of particu-
lar concern i n the limiting of helicopter performance, i n
that the excessive torsiona.1 displacements shown i n Figure
10 persist over a complete revolution of the blade.
control load t i m e history, taken from f l i g h t test ( R e f . 6),
shown i n Figure 11 i l l u s t r a t e s the type of stall-related
blade motions usually encountered at a thrust level o r
forward speed near the upper l i m i t of an a i r c r a f t .
oscillations i n the control loads, presumably deriving from
blade torsional oscillations, are seen from Figure 11 t o
persist only between about
rather than throughout a complete revolution of the blade.
= 270 deg and
$ = 400 deg,
A torsional displacement time history closely resembling
the variation of control loads i n Figure 11 was obtained
less than 4.4, f o r collective pitch angles between
12 and 13 deg.
Results for two typical cases are shown i n
Figures 12 and 13. The occurrences of s t a l l and unstall
are indicated on the abscissas. The large oscillations i n
torsion are clearly related t o s t a l l , but their persistence
is not the result of successive s t a l l i n g and unstalling,
as would be the case for true stall f l u t t e r . The blade
appears t o be responding t o the sudden changes i n aerodynamic
moment at stall onset and unstall, as can be seen by compar-
ing the variation of moment coefficient shown i n Figures 12
and 13 with that of torsional displacement, and noting the
azimuth positions at which stall and unstall occur. There
is some cyclic stall-unstall within the stall zone evident
in the results, particularly at the higher rotor speed
= 4.15, Figure 13). However, the major contributors
t o the oscillations appear t o be the i n i t i a l and f i n a l
pulses associated with stall and unstall upon entering and
leaving that zone. There are, i n general, two cycles of
torsional oscillation of excessive amplitude after the blade
unstalls the last t i m e on a given revolution. The response
can be regarded as transient, on a localized t i m e scale, o r
forced, when viewed on a scale of several rotor revolutions.
The severity of the response is apparently due i n part t o
the suddenness of load changes a t stall and unstall, and
partly t o the relative lack of aerodynamic damping i n pitch,
particularly when the blade is not stalled.
If the collective pitch angle is increased, the blade
does undergo stall f l u t t e r , as seen from the time history
plotted in Figure 14. 'These results are for the same rotor
I O 0
AZIMUTH, ‘b t dOgr0-
Figure 11 VARIATION OF PITCH LINK LOAD IN FLIGHT
TEST OF CH47 AT 123 KNOTS
(from Ref. 6)
i ! ? 0
Figure 12 DISPLACEMENT AND MOMENT TIME HISTORIES FOR EXCESSIVE
- 0.20 0
AZIMUTH, 9 ,degrees
Flyre 13 DISPLACEMENT AND MOMENT TIME HISTORIES FOR EXCESSIVE TORTIONAL RESPONSE
= 4.15, Bo = 12 deg, 1 . 1 = 0.1
= : 0.50
Ii i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i l
Figure 14 DISPLACEMENT TIME HISTORIES FOR STALL FLUTTER AT LOW ROTOR SPEED
a* = 3.89,8, - 14.3 deg, p = 0.1
speed as those of Figure 12, but with 8, increased from
12 deg t o 14.3 deg.
over the whole revolution of the blade f o r t h i s case.
Successive s t a l l and unstall persists
It could be a;.gued t h a t the blade torsional oscillations
of Figures 12 and 13 are s t i l l a manifestation of stall
flutter, even though successive stall and unstall is not
taking place, since the aerodynamic moment ?an undergo
unstable variations when the blade remains stalled through-
out a cycle (Ref. 4). It may, i n fact, be the case that
the large deflections do result partly from that effect, so
choosing t o term them as simply a response may be somewhat
misleading. On the other hand, the solutions are distinctly
different from what is definitely stall f l u t t e r obtained
both i n hover (Figure 6) and i n forward flight (Figures 10
and 14) so that label would seem t o be even less appropriate.
Further, the persistence of the oscillations after exit from
the stall zone is clearly symptomatic of a response, so, f G r
lack of a more precise term, solutions of the type shown i n
Figures 12 and 13 are identified in what follows as exces-
Linear Stability Boundaries
The value of
was determined for the three configurations considered,
for advance ratios of 0, .1, .2, and .3.
advance r a t i o and torsion-bending frequency r a t i o on linear
s t a b i l i t y are shown i n Figure 1-5, where
for two different frequency ratios.
advance r a t i o is seen t o cause some decrease in f l u t t e r
rotational speed, with most of the decrease occurring
between advance ratios of .1 and .2.
crease i n frequency ratio, from 3.69 t o 2.5, caused only
about a 4 percent reduction i n f l u t t e r speed over the range
of advance ratios considered. The insensitivity t o frequency
r a t i o can be attributed t o the large chordwise mass imbalance,
which produces the same effect i n classical binary f l u t t e r
of a wing (Ref. 15).
at the onset of linear instability
The effects of
The substantial de-
The effect of chordwise mass imbalance on linear s t a -
b i l i t y is shown i n Figure 16, where
is plotted against
A s one would expect, the reduction in Xm, and
hence i n the coupling between bending and torsion, causes
a substantial increase i n the f l u t t e r rotational speed.
52* at f l u t t e r onset
P for values of Xm of .216 and .lo8
ADVANCE RATIO, p
Figure 15 EFFECT OF ADVANCE RATIO AND
TORSION-BONDING FREQUENCY RATIO
ON LINEAR STABILITY - X d b -0.216
, Xm/b =0.216
-. - -- Xm/b =0.108
0 . I
ADVANCE RAT IO, p
Figure 16 EFFECT OF Xm ON LINEAR STABILITY -
0 6 0 / ~ ~ o
S t a l l Flutter and Response Boundaries
The effect of forward speed on stall-related instabili-
ties for the three configurations was investigated by
systematically varying the collective pitch angle and
advance ratio, with
equal t o 3.89.
the results t o rotor performance, a mean l i f t coefficient
CL is defined, according t o
I n order t o r e l a t e
1 is the time-averaged l i f t per unit span a t the
aerodynamic reference radius.
good approximation, directly proportional t o the thrust
coefficient (see Ref. 16). The two-dimensional aerodynamic
model does not provide a good measure of EL when the rotor
is partially stalled, so EL was computed assuming it varies
linearly with the collective pitch angle, using the formula
This coefficient is, t o a
CL = a( p )(eo + .0217)
The slope a
rad were obtained from calculations of EL for the nominal
configuration with stall precluded.
with p is shown i n Figure 17.
and zero-lift collective pitch angle of -.O217
The variation of a
The results obtained for the nominal configuration are
summarized i n Figure 18 as a plot of EL vs
is increased at a given
encounter a region of excessive response, of the type dis-
cussed previously, and then, for
region where stall f l u t t e r occurs.
ratio has the effect of suppressing the tendency f o r s t a l l
= .2, stall f l u t t e r occurs at EL = .85,
but a further increase i n results i n excessive response
could not be triggered a t a l l . A s a result, stall f l u t t e r
is confined t o a region somewhat as indicated by the shaded
area i n Figure 18.
cc . A s thrust
Cr , the rotor is seen t o first
~1 of .2 or less, a
Cr' = .3 a l i m i E -cycle type of oscillation
The suppression of Stall f l u t t e r at high advance r a t i o
is apparently caused by an effect similar t o the one en-
countered at low rotor speed i n hover, whereby the flapping
motion prevented a l i m i t cycle from occurring.
seen from the blade motions obtained f o r
This can be
p = .3 and
ADVANCE RATIO, p
Figure 17 VARIATION OF a = dcL/dO, WITH
0 - 0 . 9
0 . 8
TYPE OF SOLUTION
STABLE (NO STALL)
0 . 3
ADVANCE RATIO, p
Figure 18 STALL STABILITY BOUNDARIES FOR a* = 3.89, we0/o~,'3.69
AND Xm/b = 0.216
CL = .78, plotted i n Figure 19.
as the blade enters the stall zone on the retreating side,
it appears that a l i m i t cycle i s being set up, with repeated
stall and unstall cccurring. However, at about
deg, the flapping mlJtion has b u i l t up i n response t o the
large cyclic pitch Ghanges, producing a negative plunging
rate sufficient t o keep the blade unstalled over the remain-
der of its passage on the advancing side. Then, when the
blade again enters the stall zone, the large positive flap-
induced plunging rate precludes unstall u n t i l exit from
the stall zone at about
= 670 deg.
blade subsequently undergoes excessive torsional response,
rather than stall f l u t t e r .
A s a result, the
O n the first revolutlon,
The effect of torsion-bending frequency ratio on s t a l l -
related i n s t a b i l i t i e s can be seen from Figure 20, where CL
is plotted against
of excessive torsional response occurred with t h i s config-
uration f o r an advance r a t i o of .2 or less. Instead,
limit-cycle type oscillations were set up, with almost no
evidence of suppression by the flapping motion, even at
relatively high values of CL with
however, only excessive response was obtained, similar t o
the results for
1.1 = .2.
The marked deterioration i n stability at the lower
frequency r a t i o is apparently associated with the lessened
linear stability of the system. The configuration with
x d b = -108, which is more stable, i n the linear sense, than
the nominal one, exhibited a trend opposite t o the one re-
sulting from a decrease i n frequency r a t i o . The results
f o r the smaller mass center offset, shown i n Figure 21, are
similar t o those of the nominal configuration, Figure 18,
but the region i n which stall f l u t t e r occurs is sonewhat
reduced, there being no occurrence of stall f l u t t e r a t an
advance r a t i o of .2.
Also, the amplitude of the torsional
oscillations i n the region of excessive response is con-
siderably reduced, as evidenced by comparing the blade
motions plotted i n Figure 22, which are f o r
CL = .95 and x d b = .108, with those of the nominal config-
uration plotted i n Figure 12.
CI = .l,
AZIMUTH, JI , degrees
Figure 19 DISPLACEMENT TIME HISTORIES AT HIGH ADVANCE RATIO -
a* = 3.89,
* 0.78, I.( = 0 . 3
0 . 6
TYPE OF SOLUTION
STABLE (NO STALL)
0 . I
ADVANCE RATIO, p
AND Xm/b = 0.216
Figure 20 STALL STABILITY BOUNDARIES FOR !2* = 3.89, wgo/w#o = 2.5
ADVANCE RATIO, p
Figure 21 STALL STABILITY BOUNDARIES FOR 0'' = 3.89, wgo/w#o = 3.69
AND Xm/b 0.108
d - w
IY ' \
I I I I I I I I I I I I I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
- 0 so0
! ! ? n
' p - ,
1 0 0
AZ IMUTH, 9, degrees
Figure 22 DISPLACEMENT TIME HISTORIES FOR EXCESSIVE TORSIONAL RESPONSE.
An analysis has been performed of the aeroelastic
stability of a helicopter rotor blade i n hovering and
forward flight. An analytical model of an a i r f o i l under-
going unsteady stall and an elastomechanical representation
including flapping, flapwise bending and torsional degrees
of freedom were employed i n the study. The following con-
can be drawn- from the results obtained.
Analysis of aeroelastic stability for a
hovering rotor demonstrated that the aero-
dynamic and dynamic representations developed
are capable of reproducing classical and
stall f l u t t e r .
While stall f l u t t e r is an instability
involving a single rotational degree of
freedom, the minimum rotational speed
for its occurrence, i n hover, is determined
from coupling with a translational degree
In forward flight, the rotor can undergo
a linear instability analogous t o classical
f l u t t e r and a stall-induced f l u t t e r which,
while not manifested by a s t r i c t l y periodic
l i m i t cycle, has the same basic mechanism
for its occurrence as stall f l u t t e r of a
The large stall-related torsional oscillations
which l i m i t forward speed and thrust are
primarily the response t o the rapid changes
i n aerodynamic moment which accompany stall
and unstall, rather than the result of an
Linear s t a b i l i t y is relatively insensitive
t o advance r a t i o for advance ratios as
large as .3.
While excessive response due t o stall occurs
at high advance ratio, stall f l u t t e r is pre-
cluded by the large flap-induced plunging
7. The severity of stall-related i n s t a b i l i t i e s
and response depends t o some extent on
linear stability. Increasing linear s t a -
b i l i t y lessens the susceptibility t o stall
f l u t t e r , a n d reduces the rnagnitude of the
torsional response to stall and unstall.
A l i s t i n g of the FORTRAN coding of the computer program
follows. The program was written i n FORTRAN I V for use on
an IBM 360/75 computer.
PROGRAM TO ANALYZE UYFTEAOY A I R F O I L S T A L L
/ R L 1 / N T I M E , N D I Y C *
/CLCYRL / C L V 6 T M T G CHPAVR
C f l M M @ ~ - / I N P T V B ~ - - ~ l ~ R l 6 4 ) ; -7PV6(64)r FPPRVRf 6419
D E L V B g
AT 0 V R-,
- _ _
O I O S V R ( 6 4 ) r
R V B I 64 1 9
XYVB (64) r
-- -- -
COMMON / I N P U T S /
N O T R L t
F R L r
I N O V I
D X I I
P H l H * ._ !VI
R E A 9
A L P H l r
R ORB 9
A L P H Z v
R V 1 9
I * P L O T O P I PSILOW.
J 9 NOUT
C OMHO N/
X L l 3 0 ) r
Y ( 1 0 0 1 ,
Y L I 3 3 ) 9
U P R I M r
E R 2 9
XU[ 30 19
E R 3 r
Y U I 301 9
CMP4r CMPASI RARG,
ps*I,p--- EM19 HVORv
>SPAI_ S V O R t TT)RF* X l V O R
M A I V
M A I N
M A I N
S E T U P S 1 7
SE TUP S18
SE rup si 9
SE TUP 52 0
SETUP S2 L
SE TU P 52 2
SE TUPSZ 3
SE TU P s2 4
SE TUPSZ 5
SE TUP SZ 6
SE 1UP 52 7
S E T U P S 2 8
SETUP 52 9
SE T U P 53 0
Z Z Z / Z(3)
t . -
. .- -
SE T U P S 3 1
01 MENSION U S A V ( 3 0 0 r 100) r S C A L S (300 1-
D I M E N S I O N U S A V t L
D I M E N S I O N CAMBR (24)
D l l F W V F 4 X G A K D O ) ~ X S I G ( L O O ) r X S I G A ( L O O I r X S IGB( 100) xct 3 - m i m - x m - 7 - - -
-_ - LSBL(3001 9 X B S I G t 100)
I M NS I ON AC AP ( 30 r3 1 t BC AP I 100 e 3 T r A F Z T30 1 *as(
_ - - _ _ -
tTHI CK(24 1
I * S C A L S I 3 0 0 1
M A I H
M A I N
M A I N
M A I N
3-09-30 1 I BSI 3 Or 3-0 1 I A SHZMA I N
R4D = 180.
I L = 8888
H A l N
CALL R F ~ D I N T I L ~ ~
NnTF - CFFcETS--A-R-E
AS A FRACTION OF TOTAL CHOROt X 1 BEING MfASURFD FPOY THE
_ _ _
- - --
Pill I N AS LISTED I N THEORY OF d1NG SECTION3t
RE S URE NF -fs ANEVEN- NUM BER .
- . - - .
I -M-~ l o ; -
UI NF =l
- I N D v = I - N W ~
HR I TE I MOUT e 6 )
ITCH = ALPHl
CFtINDV + MOTR .LE.
AMPLU = 1.33333*
I F ( I N
.- _ _ WRI TE (MOUT t 25) NVOR SVOR t HVORe BAR6 *XIVORt EMI>TORFr SSPA
2 A3 2
3 4 3 C A L L Snrrrrnr tYUt KLI YLtNOfl tNFvRDRB
OF 7875 N=ltNF
AMRR I N) =C A WRR( N ) *C HOB8
7E75 THIC Kf N) =THlCK fNI*TWDBB
WRI TE (MTJT t 4 1
WRI TE ( MDUT q7) AMPLU t FREQU t ALPHl t ALPH2t HEAVE? AROTe FREOFt
W R r n t B I
WRI TF fMOUT*9) fh rCAHBRl N) tTHICKf N) tW It NF)
MX=N SR L+ NZ-1
CALL CORD-XTbtrL ?Nz-i-mm
__ . . __ -
- - - __ ._ __
2) cn TC 343
/ (1. - ROVB**4)
R ARG =R A RG /6 .
EBB* C M b BBt TH 1 CK v CA MBR J
- R 086 rR EB
24 19 MEND =M-1
GO TO 2421
M A I N
M A I N
Y A I Y
M A I V
Y A I N
Y A I N
M A I N
M A I N