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JOURNAL OF THE AMERICAN HELICOPTER SOCIETY 62, 042010 (2017)
Influence of Dynamic Inflow States on Coupled Rotor Fuselage Modes
Salini S. Nair Ranjith Mohan∗Gopal Gaonkar
Research Scholar Assistant Professor Professor
Indian Institute of Technology Madras, Chennai, India Florida Atlantic University, Boca Raton, FL
This paper addresses how dynamic inflow, while improving the correlation with ground-resonance measurements, could
make the mode identification of the rotor–body–inflow system demanding. Specifically, the paper demonstrates dynamic
inflow effects on the modes of two earlier tested configurations: configuration 1 with a nonmatched stiffness rotor (nonro-
tating flap and lag stiffnesses are unequal) and configuration 4 with a matched stiffness rotor (these stiffnesses are equal).
The experimental model represents a three-bladed hingeless rotor on gimbal support; the blades are rigid, and they have
spring-restrained flap–lag hinges; and the gimbal permits roll and pitch motions. The mode identification is based on
relative participation of different states in the eigenvector, modal frequency and phase information about these states. For
configuration 4, for instance, one of the modes has dominant contributions from multiblade flap cosine and sine states and
appreciable contributions from body roll and pitch states and also first harmonic wake states; this mode is identified as the
regressive flap mode (RFM), and another mode with fairly comparable characteristics is identified as the inflow mode. The
phase differences between these flap states, and between these wake states, are used to distinguish RFM from the inflow
mode. The phase information that is exercised herein represents a novelty of this work.
Nomenclature
A, X system matrix and state vector
ehinge offset
Fperiodic eigenvector
fβe additional frequency due to hinge offset
fβnr,f
βo nonrotating flap frequency and uncoupled rotating flap
frequency
Jdiagonal matrix of the eigenvalues of the system
Mmaximum number of harmonics
Mb,I
bblade mass and moment of inertia about flap/lag hinge
m, n index for harmonics and radial shape functions
Rcg radius of center of mass of blade outboard of flap/lag
hinge
ttime
αm
n,βm
nwake or inflow states, (e.g., α1
2,β1
2are first harmonic
states)
β, ζ blade flap and lead–lag angles
characteristic multipliers
λFloquet exponent, inflow
φ,θ fuselage roll and pitch angle states or modes
ψazimuth (nondimensional time)
()0,()c,()smultiblade collective, cosine, and sine states
∗Corresponding author; email: ranjith.m@iitm.ac.in.
Manuscript received December 2016; accepted June 2017.
Introduction
To date, the fundamentals of ground resonance have been under-
stood, and dynamic inflow is recognized as a simple and effective means
of including wake dynamics and thereby improving the correlation with
experimental data; in fact, it is almost exclusively used (Refs. 1,2). Still,
the mode identification of the rotor–body–inflow system is not yet a rou-
tine foolproof exercise; this was brought to focus by the wide variation
in mode identification in a series of correlations with Bousman’s ground-
resonance database (Refs. 1, 3–5). Reference 1 compares the effect of
unsteady aerodynamics (wake) on damping and frequency of different
modes. Specifically, for one of the configurations, the mode that is iden-
tified as the regressive flap mode (RFM) by Ref. 5 is identified as the
inflow mode by Refs. 1 and 4. Moreover, Bousman also investigated
how a quasisteady aerodynamics theory without dynamic inflow com-
pares to the database; therein, this RFM is identified as the pitch mode
(Ref. 3).
While dynamic inflow improves the correlation, it also adds to the
complexity of the mode identification process in three respects. First,
the inflow states strongly couple with the rotor–body structural states,
and in turn they increase mode-to-mode similarity. In particular, the flap
dynamics of cantilever blades induces strong structural coupling between
RFM and the body roll and pitch modes, and rotor wake further increases
this coupling, and mode-to-mode similarity increases. Second, the inflow
state frequencies are virtually artifacts of finite-state wake modeling and
do not provide any cue for mode identification; thus mode identification
has to be based solely on eigenvector analysis. Third, in the eigenvector
analysis itself, the magnitudes of inflow states of aerodynamic origin
do not seem to permit a physically meaningful comparison with the
magnitudes of the structural states such as multiblade flap cosine and sine
DOI: 10.4050/JAHS.62.042010 C
2017 AHS International042010-1
S. S. NAIR JOURNAL OF THE AMERICAN HELICOPTER SOCIETY
states, and body pitch state. That is, the conventional mode identification
on the basis of the largest eigenvector component could lead to precluding
the inflow mode from the set of modes of the rotor–body–inflow system.
All this—mode-to-mode similarity, inflow state frequencies that pro-
vide no cue for mode identification, and the limitations of identifying
the modes from the largest eigenvector component—should help appre-
ciate why mode identification of the rotor–body–inflow system could be
demanding and why the literature shows such a wide variation in mode
identification (Refs. 1,3–5). Thus, there is a need of a study that provides
a better appreciation and understanding of the underlying reasons for
this wide variation and that also provides a means of unambiguously
identifying the modes. This study attempts to do just that.
Modeling and Methodology
The test model represents a simple model of a soft-in-plane, three-
bladed, hingeless rotor on gimbal support that permits pitch and roll
motions; the blades are very stiff relative to the root flexure so that the
first flap and lag modes involve nearly pure rigid-body blade motions. Ac-
cordingly, the analytical model comprises an offset-hinged, rigid-blade,
lag–flap model with flap and lag hinge-spring restraints. The numeri-
cal values of the model parameters used in the analysis are presented
in Table 1. Rotor aerodynamics is based on blade-element theory with
Peters–He finite-state wake modeling. The nonlinear equations of mo-
tion include the effects of hinge offset, and they are linearized about an
equilibrium position of zero collective (for additional details see, e.g.,
Ref. 2). As for dynamic inflow modeling, key details are presented next;
this helps present the results involving the inflow states without any
notational ambiguity.
Dynamic inflow model
The Peters–He model captures both radial and azimuthal variation of
the rotor wake, and it can be used for hover as well as forward flight
conditions. The model has a finite-state representation (ordinary differ-
ential equations); thus, it is amenable to investigating stability through
Table 1. Parameters of the model considered (Ref. 3)
Number of blades 3
Radius, cm 81.1
Chord, cm 4.19
Hinge offset, cm 8.51
Blade airfoil NACA 23012
Profile drag coefficient 0.0079
Lock number 7.73
Solidity ratio 0.0494
Lift curve slope 5.73
Height of rotor hub above 24.1
gimbal, cm
Blade mass, g 209
Blade flap inertia, g m217.1
Configuration 1 Configuration 4
Nonrotating flap frequency, Hz 3.13 6.63
Nonrotating lag frequency, Hz 6.76.73
Damping in lead–lag, % critical 0.52 0.53
Rotary inertia in pitch, g m2633
Rotary inertia in roll, g m2194
Pitch damping, % critical 3.20
Roll damping, % critical 0.929
Pitch frequency, Hz 2
Roll frequency, Hz 4
eigenanalysis. Azimuthal variation is modeled by trigonometric func-
tions, and radial variation is modeled by Legendre polynomials, which
are referred to as radial shape functions. This paper uses a widely used
approach, the so called consistently upgraded or table method (Ref. 6):
One chooses a specific number of radial shape functions for each har-
monic such that the order of the polynomial is the same as that of the
corresponding harmonic. In the differential equations, the states are asso-
ciated with trigonometric cosine and sine functions and they are denoted
by αm
nand βm
n,wheremand ncorrespond to the order of harmonic and
order of radial shape function, respectively. This paper presents results
from models with number of harmonics M=1 and 3. Note that only
the first harmonic states couple with the fuselage states. Moreover, the
M=2 case does not introduce any additional first harmonic states to
that with M=1. For M=1, there are only three states, ordered as [α0
1
α1
2β1
2]andforM=3, there are 10 states, ordered as [α0
1α0
3α1
2α1
4α2
3α3
4
β1
2β1
4β2
3β3
4].
States of the system
The degrees of freedom of the structural system are roll and pitch
for the rotor support or fuselage, and lag and flap degrees of freedom
for the blade. The blade degrees of freedom are converted to multiblade
coordinates (Ref. 7). There are a total of 12 states for the three-bladed
rotor, 4 states for the fuselage, as well as 3 or 10 wake states for M=1
and M=3, respectively.
The analytical model can be represented in state space form:
˙
X(t)=A(t)X(t)(1)
If multiblade coordinates are used and the wake model has harmonics
(M) equal to 1, the system matrix is constant. But with higher harmonics
(M>1), A(t) is periodic with period T, corresponding to rotor angular
speed.
For a constant matrix A, the frequency and damping are given by
the eigenvalues and the eigenvectors give the modal content. However,
when matrix Ais periodic, a Floquet analysis is required to determine
the system stability, and modal participation can be determined using
periodic eigenvectors. If the periodicity is not significant, then the zeroth
harmonic term will dominate and practically an eigenvector analysis
of the Floquet transition matrix (FTM) would suffice. However, this
paper follows the former approach for M=3 and calculates the pe-
riodic eigenvector and the Fourier expansion of each of its terms; that
is, corresponding to every state of each eigenvector. References 8 and 9
provide additional details of this approach. The Fourier analysis is used
to identify the dominant frequency, and the magnitudes of the Fourier
coefficients are used to determine the states that participate in the mode.
This study also makes use of information contained in the phase of
the eigenvector (or Fourier coefficients in the case of a periodic coeffi-
cient system) to characterize the modes. The following summarizes the
approach:
Equation (1) represents the periodic coefficient linear system, where
A(t) is periodic with period T. The eigenvalues of FTM are denoted by
and the characteristic multipliers, by λ, thus the Floquet exponents are
given by Eq. (2):
λ=1
Tlog[Re()2+Im()2]1/2+i
Ttan−1[Im()/Re()]+i2πn
T
(2)
where nis an integer that needs to be determined as the domi-
nant value from the Fourier expansion of periodic eigenvectors. The
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INFLUENCE OF DYNAMIC INFLOW STATES ON COUPLED ROTOR FUSELAGE MODES 2017
Table 2. Frequency, damping and phase of flap and wake states for different modes at 650 rpm for configurations 1 and 4
Phase (deg)
Damping Frequency Damping Coupled Uncoupled
Figure (rad/s) (Hz) β
c
−β
s
α1
2−β1
2(Nondimensional) Frequency (/rev) Frequency (/rev) Mode
Configuration 1
1−1.81 4.01 247.58 219.20 −0.03 0.37 0.37 Roll
2−1.38 2.13 278.87 308.19 −0.02 0.20 0.19 Pitch
3−4.16 0.52 268.93 271.27 −0.06 0.05 0.13 RFM
4−23.65 22.13 90.08 90.15 −0.35 2.04 2.13 PFM
7−36.90 0.19 114.66 63.34 −0.54 0.02 −Wake
−0.25 2.63 301.64 276.08 −0.0037 0.24 0.24 RLM
−0.32 19.59 115.68 137.18 −0.0048 1.81 1.76 PLM
−0.29 8.21 148.94 106.04 −0.0043 0.76 0.76 CLM
−24.18 11.32 122.39 44.64 −13.37 1.04 1.13 CFM
−10.42 0.00 180.00 0.00 −0.15 0.00 −Wake
Configuration 4
9−4.05 4.48 269.30 235.63 −0.06 0.41 0.37 Roll
10 −2.74 2.35 279.53 292.48 −0.04 0.22 0.19 Pitch
11 −3.66 0.93 266.88 273.76 −0.05 0.09 0.25 RFM
14 −23.58 23.56 90.22 90.33 −0.35 2.17 2.25 PFM
17 −33.89 0.35 291.75 242.18 −0.50 0.03 −Wake
−0.20 2.61 304.94 250.25 −0.0029 0.24 0.24 RLM
−0.37 19.62 131.94 123.02 −0.0055 1.81 1.76 PLM
−0.30 8.23 180.73 152.31 −0.0044 0.76 0.76 CLM
−24.22 12.74 144.90 60.20 −0.36 1.18 1.25 CFM
−10.32 0.00 180.00 0.00 −0.15 0.00 −Wake
periodic eigenvector FisgivenbyEq.(3),whereJis diagonal matrix of
the eigenvalues of the system:
d
dt F(t)=A(t)F(t)−F(t)J(3)
Let Ybe an eigenvector of system matrix A(constant coeffi-
cient) or alternatively the vector of Fourier expansion coefficients
of F(t) (corresponding to different states) at a dominant frequency
n; then magnitude and phase of the ith state are calculated from
Eq. (4):
Yim =|Yi|,Y
ip =arg(Yi)−arg(Y2)(4)
where Y2is the value of the component corresponding to a pitch state in
the eigenvector.
A general practice is that, for each eigenvalue, a name is given, based
on the participation of different states. For example, a state with the maxi-
mum magnitude in the eigenvector (or in case of periodic eigenvector, the
magnitude of the Fourier coefficients) determines the name of the mode.
That is, if in the eigenvector Y,Y2mis maximum, then it is called a pitch
mode. Usually, more than one state will have significant contributions
in an eigenvector and the participating states characterize the mode. At
times, it may not be clear which name one could attribute to because the
structure of the eigenvector magnitude has similarities with two or more
of classical modes (like roll, pitch, etc.). Quite often, frequency of the
mode helps to identify the mode in such circumstances. Results will be
presented to show how mode identification could become problematic in
the presence of inflow states and how the phase information complements
the conventional mode identification metrics of eigenvectors and modal
frequencies. That is, how the phase information will help identify the
modes unambiguously will be demonstrated with the results presented
here.
Results and Discussion
Mode identification: Background
Bousman’s database on ground resonance addresses five rotor con-
figurations for different combinations of (nonrotating) flap and lag stiff-
nesses as well as pitch–lag and flap–lag couplings (Ref. 3). Configura-
tions 1 and 4 cover the bulk of the database; both have no couplings,
whereas configuration 1 has unequal flap and lag stiffnesses (unmatched
stiffness), configuration 4 has equal flap and lag stiffnesses (matched
stiffness). The present mode identification refers to this database for
configurations 1 and 4.
With inclusion of the wake model, the mode-to-mode similarity of
the rotor–body–inflow system could increase significantly and in turn
the mode identification could become more demanding. Therefore, it is
expedient to begin with a broad comparison of wake effects on config-
urations 1 and 4. Configuration 4 has a flap stiffness that is much larger
than that of configuration 1. As a result, configuration 4 could experience
stronger rotor–body coupling, larger body rotations and wake participa-
tion, and also more marked mode-to-mode similarity. This comparison
should also help explain why several related studies (Refs. 1,3–5) show
a wide variation in mode identification for configuration 4 and not for
configuration 1.
Table 2 lays the groundwork for mode identification of both configu-
rations by providing three sets of information: (1) uncoupled frequencies,
(2) damping levels and frequencies, and (3) phase angles between flap
states βcand βsand between wake states α1
2and β1
2. Of the nine columns,
the first and the last show the figure numbers and the identified modes,
the second and the third show the dimensional damping and frequency
from the eigenanalysis, the fourth and the fifth show the phase angles
between βcand βsand also between α1
2and β1
2, and the sixth and the
seventh show the corresponding nondimensional values of damping and
frequency. Nondimensionalization is relative to rotor rotational speed
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S. S. NAIR JOURNAL OF THE AMERICAN HELICOPTER SOCIETY
=650 rpm, the angular speed for which the analysis is done. Sim-
ilarly, the eighth column shows the uncoupled frequencies. As for the
rotating uncoupled frequencies, consider the collective flap mode (CFM)
of configuration 1, for which Table 2 shows a frequency of 1.13 rev
(ninth row and eighth column). The corresponding nonrotating flap fre-
quency fβnr =3.13 Hz (see Table 1). The hinge offset introduces an
additional frequency of fβe =4.73 Hz; (fβe =2eRcg MB
IB)andfi-
nally, the uncoupled rotating flap frequency fβo is found to be 12.23
Hz; (fβo =2+f2
βe +f2
βnr)or1.13 rev. The corresponding uncou-
pled RFM and progressive flap mode (PFM) frequencies are 0.13 and
2.13 rev, which are also shown in Table 2. Essentially, a similar procedure
is used for collective lag mode (CLM), regressive lag mode (RLM), and
progressive lag mode (PLM).
Mode identification
Mode identification is based on three metrics: (1) comparison be-
tween uncoupled and coupled frequencies, (2) modal content—relative
participation of different states in an eigenvector, and (3) phase angles
between βcand βsand also between α1
2and β1
2. Moreover, this mode
identification is supplemented by three additional sets of information: (1)
variation of modal frequencies and both phase angles with (400 ≤
rpm ≤1000), (2) comparison of predicted modal frequencies with the
test data, and (3) comparison of modal participation of rotor–body states
with and without dynamic inflow. Henceforth, eigenvector and modal
content are used synonymously. In the present case, rotor–body dynam-
ics under ground-contact conditions, the collective flap and lag modes
remain uncoupled and thus they are not included; here, the focus is
on roll, pitch, flap regressive, flap progressive, and wake modes. If not
stated otherwise, results and analysis are for a representative rotor speed
=650 rpm.
The first set of eight figures refers to configuration 1, and mode
identification is based on three metrics mentioned before. Figure 1 shows
the modal content for an eigenvalue (−1.8129 +25.2009i,1/s); therein,
roll state φis the most dominant component and it appreciably couples
with flap sine state βs. The modal frequency of 4.0108 Hz (25.2009,
1/s) is remarkably close to the uncoupled roll mode frequency of 4 Hz
(0.37 rev); also see Table 2. Thus, based on these two key features—the
dominance of roll state in the eigenvector and closeness of the frequency
to the uncoupled roll mode frequency—this mode is identified as the roll
mode.
0 c s 0 c s 1
0
2
1
2
1
States
0
0.005
0.01
0.015
0.02
0.025
0.03
Magnitude
−1.8129+25.2009i
Fig. 1. Modal content of roll mode for configuration 1 at 650 rpm.
Similarly, Fig. 2 shows the modal content for an eigenvalue of
(−1.37638 +13.4032i,1/s). It is seen that pitch state θis the most
dominant component and that it appreciably couples with the flap co-
sine state βcas is the case in Fig. 1 between the most dominant roll
state and flap sine state βs. In addition, the modal frequency of 2.13 Hz
(13.4032, 1/s) is fairly close to the uncoupled pitch mode frequency of
2Hz(0.19 rev). (The other modal frequency that is somewhat close to
2.13 Hz is the RLM frequency of 2.63 Hz (0.24 rev).) Therefore, this
mode with the most dominant state represented by the pitch state θin
the eigenvector and with a frequency of 2.13 Hz that is fairly close to the
uncoupled pitch mode frequency of 2 Hz, is identified as the pitch mode.
Figure 3 continues the pattern of two earlier figures; it shows the
eigenvector for an eigenvalue of (−4.1646 +3.2927i,1/s). The modal
frequency of 0.048 rev (3.2927, 1/s) does not help connect with any
of the uncoupled modal frequencies; this is quite unlike the two earlier
studied cases: roll mode in Fig. 1 and pitch mode in Fig. 2. On the other
hand, Fig. 3 shows that βc≈βsand that they represent the most dominant
components of the eigenvector. Therefore, this mode with a relatively low
frequency of 0.048 rev is identified as RFM. Giventhe extreme sensitivity
of RFM frequency to wake, the appreciable difference between the modal
frequency of 0.048 rev and the corresponding uncoupled RFM frequency
of 0.13 rev is understandable.
0 c s 0 c s 1
0
2
1
2
1
States
0
0.01
0.02
0.03
0.04
0.05
0.06
Magnitude
−1.37638+13.4032i
Fig. 2. Modal content of pitch mode for configuration 1 at 650 rpm.
0 c s 0 c s 1
0
2
1
2
1
States
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Magnitude
−4.1646+3.2927i
Fig. 3. Modal content of regressive flap mode for configuration 1 at
650 rpm.
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INFLUENCE OF DYNAMIC INFLOW STATES ON COUPLED ROTOR FUSELAGE MODES 2017
0 c s 0 c s 1
0
2
1
2
1
States
0
1
2
3
4
5
6
Magnitude
10−3−23.65358+139.0345i
Fig. 4. Modal content of progressive flap mode for configuration 1 at
650 rpm.
c
s
2
1
2
1
Fig. 5. Regressive flap mode phase angles for configuration 1 at 650
rpm.
Figure 4 shows the modal content for an eigenvalue of (−23.65358 +
139.0345i,1/s). The modal content is almost exclusively dominated by
βcand βs. Since the modal frequency of 139.0345 1/s(2.043 rev) is
relatively high, the virtual nonparticipation of other states such as the
body states is expected. Thus, given the exclusive dominance of βcand
βs, and the modal frequency that is relatively high, this mode is identified
as PFM. As an aside, this PFM frequency of 2.043 rev correlates with
the RFM frequency of 0.048 rev and with the CFM frequency of 1.04 rev
(Table 2).
Two noteworthy features of RFM and PFM merit mention and elabo-
ration. The first one is that βc≈βsin the modal content; see Figs. 3 and
4. The second one is that the phase angles between βcand βsand between
α1
2and β1
2are close to π/2; see Table 2. For RFM, this is typified by
Fig. 5, which shows that the phase angles between βcand βsis 91◦(for
an isolated flapping rotor the phase difference is 90◦) and between α1
2
and β1
2is 89◦. (As it turns out, for both RFM and PFM, these two phase
angles remain close to 90◦for varying and also for configuration 4;
this is taken up later. Moreover, for PFM, the phase variation is similar
to that for RFM except for sign; hence it is omitted.)
Thus far, the focus of the investigation has been on the mode identifi-
cation of the rotor–body–inflow system. This investigation also showed
y
x
Inflow magnitude
0
1
1
0.05
−4.1646+3.2927i
0
0.1
0
−1−1
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Fig. 6. Magnitude of inflow for regressive flap mode for configuration
1 at 650 rpm ([−1 1] correspond to rotor disk).
0 c s 0 c s 1
0
2
1
2
1
States
0
0.005
0.01
0.015
0.02
0.025
Magnitude
−36.9008+1.16921i
Fig. 7. Modal content of wake mode for configuration 1 at 650 rpm.
that for both RFM and PFM, the phase difference between the wake
states α1
2and β1
2is close to π/2 and that the modal content is character-
ized by α1
2≈β1
2(Figs. 3–5). It is equally instructive to investigate how
inflow as a flow field (rather than as inflow states) varies with azimuth
for RFM and PFM. Also recall that inflow modeling is based on the first
harmonic inflow distribution (M=1); that is, the inflow radial variation
is linear. For ease of presentation, consider RFM. Let a=r1eiθ1and
b=r2eiθ2represent the wake states in the eigenvector. Since, α1
2≈β1
2,it
follows that r1=r2=r, and the azimuthal variation of inflow (complex
valued) is given by λ=acos(ψ)+bsin(ψ) (excluding the constant
component). It is verified that |λ|=r[1+sin(2ψ)cos(θ1−θ2)]1/2.Since
θ1−θ2=π/2, it is also seen that the inflow has a constant magni-
tude for all azimuthal locations for a given radius. The inflow depicted
in Fig. 6 shows this axis-symmetric feature of the inflow flow field for
RFM.
Figure 7 shows the modal content for an eigenvalue of (−36.9008 +
1.16921i,1/s); therein, βcand βsare the most dominant states, along with
appreciable participation from α1
2and β1
2. Regarding the participation of
these four states in the modal content, this mode has some measure of
resemblance to RFM (Fig. 3). It is, nevertheless, distinct from RFM
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S. S. NAIR JOURNAL OF THE AMERICAN HELICOPTER SOCIETY
c
s
2
1
2
1
Fig. 8. Wake mode phase angles for configuration 1 at 650 rpm.
0 c s 0 c s 1
0
2
1
2
1
States
0
0.005
0.01
0.015
0.02
0.025
Magnitude
−4.04835+28.1272i
Fig. 9. Modal content of roll mode for configuration 4 at 650 rpm.
in two key respects: (1) very high damping and very low frequency;
compared to RFM, the damping is nine times higher and the frequency
is three times lower; (2) regarding the phase angles between βcand
βsand also between α1
2and β1
2, it differs significantly from RFM (for
illustration, compare Figs. 5 and 8 and see Table 2); whereas the phase
angles between βcand βsand between α1
2and β1
2are 269◦and 271◦
for RFM, the corresponding values are 115◦and 63◦for this mode
(Fig. 8). To reiterate and conclude: Given these two features—(i) very
high damping and very low frequency and (ii) phase angles differing
significantly from those of RFM—this mode is identified as the inflow
mode. (It is recognized that the structural states βcand βsdo not seem to
permit a direct comparison with the aerodynamic states α1
2and β1
2.These
subtleties are revisited later in conjunction with the mode identification
of configuration 4.)
For configuration 4, the mode identification begins with Fig. 9, which
shows the modal content for an eigenvalue of (−4.04835 +28.1272i,
1/s). Therein, the roll state φis the largest component and the flap sine
state βs, the next largest. Moreover, the modal frequency of 4.48 Hz
(28.13, 1/s) is reasonably close to the uncoupled roll mode frequency of
4Hz(0.37 rev). In addition, this modal content is similar to that of the roll
mode of configuration 1 in Fig. 1. Thus, based on the modal frequency,
the dominance of the roll state in the modal content and the similarity
of the modal content with that of the roll mode of configuration 1, this
mode is identified as the roll mode.
0 c s 0 c s 1
0
2
1
2
1
States
0
0.01
0.02
0.03
0.04
0.05
Magnitude
−2.73826+14.7823i
Fig. 10. Modal content of pitch mode for configuration 4 at 650 rpm.
0 c s 0 c s 1
0
2
1
2
1
States
0
0.02
0.04
0.06
0.08
Magnitude
−3.6633+5.8681i
Fig. 11. Modal content of regressive flap mode for configuration 4 at
650 rpm.
Figure 10 shows the modal content of a mode with a frequency of
2.35 Hz (14.78 1/s), which is fairly close to the uncoupled frequency
of 2 Hz (0.19 rev) for the pitch mode. As seen from the modal content,
the pitch state θrepresents the largest component and the flap cosine
state βcthe next largest. While the modal content is basically similar to
that of the pitch mode of configuration 1 (Fig. 2), it has the distinctive
feature of larger participation of the roll state φ. Since the flap stiffness
of configuration 4 is much larger than that of configuration 1, this larger
participation of roll state φis expected. That it is not RFM is demonstrated
by two phase angles: As seen from Table 2, about 280◦between βcand
βsand about 292◦between α1
2and β1
2. (Recall that for RFM and PFM,
the phase angles between βcand βsand also between α1
2and β1
2remain
close to 90◦.) To sum up, the frequency is fairly close to that of the
uncoupled pitch mode, the pitch state is the most dominant component
in the eigenvector, and both phase angles differ significantly from those
of RFM. Accordingly, this mode is identified as the pitch mode.
Figure 11 presents the eigenvector of a mode with a frequency of
0.93 Hz (5.8681, 1/s). This frequency does not provide any cue for
mode identification in that it does not match with any of the uncoupled
modal frequencies. And the eigenvector has the following characteristics:
(1) the flap cosine and sine states βcand βsare nearly equal, and they
represent the largest component; (2) the pitch state θ, the next largest, is
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INFLUENCE OF DYNAMIC INFLOW STATES ON COUPLED ROTOR FUSELAGE MODES 2017
0 c s 0 c s
States
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
Magnitude
−16.394+13.0885i
Fig. 12. Modal content of regressive flap mode for configuration 4
without wake at 650 rpm.
c
s
2
1
2
1
Fig. 13. Regressive flap mode phase angles for configuration 4 at
650 rpm.
about 90% of βcand βs; (3) the roll state φis not far behind θ,and(4)
the wake states α1
2and β1
2are also nearly equal, about “25% of βcand
βs.” (The statement in quotes is qualified next.)
The wake states represent wake velocity, and they are of aerodynamic
origin; whereas the rotor and body states represent rigid-body rotations,
and they are of structural origin. Thus, the wake states α1
2and β1
2do not
seem to permit a sensible comparison with the structural states such as θ
and βc. That is, the preceding observation that the wake states are “25%
of βcand βs” does not provide a basis to rule out that this mode is not
a wake mode. In fact, this mode may be a wake mode since it is not
observed when dynamic inflow is not included. Therefore, it is expedient
to study the corresponding mode without wake; this is taken up next.
Figure 12 shows the eigenvector of a mode with a frequency of
0.19 rev (13.0885, 1/s), which is fairly close to the uncoupled RFM
frequency of 0.25 rev (Table 2). Moreover, the eigenvector shows that
βcand βsare nearly equal and that they represent the largest compo-
nent. Accordingly, this mode is identified as RFM (without wake). A
comparison of eigenvalues given in Figs. 11 and 12 shows that there is
an appreciable decrease in damping and frequency, damping by a factor
of 4.5 and frequency by 2.2, with inclusion of wake. As demonstrated
earlier for RFM of configuration 1, here as well, an additional metric of
phase angles between βcand βsand between α1
2and β1
2is used. As seen
from Fig. 13, these two sets of phase angles remain close to 90◦; with
0 c s 0 c s 1
0
2
1
2
1
States
0
1
2
3
4
5
Magnitude
10−3−23.58138+148.0024i
Fig. 14. Modal content of progressive flap mode for configuration 4
at 650 rpm.
c
s
2
1
2
1
Fig. 15. Progressive flap mode phase angles for configuration 4 at
650 rpm.
the additional information of βc≈βsand α1
2≈β1
2from Fig. 11, this
mode is identified as RFM. Because of wake, it appears as though RFM
(without wake, Fig. 12) is morphed into the mode shown in Fig. 11. This
shows the inadequacy of the two widely used metrics of mode identifi-
cation: modal frequency and the relative participation of the states in the
eigenvector and the necessity of using phase information to identify the
mode correctly.
Figures 14 and 15 present the eigenvector of a mode for an eigenvalue
of (−23.58138 +148.0024i,1/s) and also the phase diagram: phase
angles between βcand βsand also between α1
2and β1
2. This eigenvalue
is nearly equal to that of PFM of configuration 1 in Fig. 4; as expected,
the eigenvector and the phase angles are virtually identical to those of
PFM of configuration 1. Accordingly, this mode is identified as PFM of
configuration 4.
Recall that both RFM and PFM exhibit three characteristics. As seen
from the eigenvectors in Figs. 11 and 14, βc≈βsand α1
2≈β1
2. The third
characteristic refers to the phase angles: Figures 13 and 15 show that the
phase angles between βcand βsand also between α1
2and β1
2are close to
90◦. Because of these three characteristics, variation of dynamic inflow
magnitude |λ(r, ψ )|exhibits axi-symmetry; that is, for a prescribed radial
position r, RFM and PFM see inflow whose magnitude is invariant of
042010-7
S. S. NAIR JOURNAL OF THE AMERICAN HELICOPTER SOCIETY
Inflow magnitude
x
y
0
1
0.02
1
0.04
−3.6633+5.8681i
0
0.06
0
−1−1
0.01
0.02
0.03
0.04
0.05
Fig. 16. Magnitude of inflow for regressive flap mode for configura-
tion 4 at 650 rpm ([−1 1] correspond to rotor disk).
0 c s 0 c s 1
0
2
1
2
1
States
0
0.005
0.01
0.015
0.02
0.025
Magnitude
−33.8911+2.2086i
Fig. 17. Modal content of wake mode for configuration 4 at 650 rpm.
the azimuthal location. Figure 16 shows this axisymmetric variation of
inflow magnitude with respect to rand ψ, for RFM. (For PFM, |λ(r, ψ )|
is similar to that for RFM and it is omitted. Also see Fig. 6, which shows
the corresponding inflow variation for RFM of configuration 1.)
The next set of two figures refers to an eigenvalue of (−33.8911 +
2.2086i,1/s); Fig. 17 shows the eigenvector, and Fig. 18 shows the
phase angles between βcand βsand also between α1
2and β1
2. This mode
is characterized by very high damping and very low frequency. As an
illustration, in comparison to RFM (Fig. 11), the damping is 9.26 times
higher and the frequency is 2.66 times lower. Moreover, βsrepresents
the largest eigenvector component, and βcthe next largest; thus, this
mode may be a possible candidate for RFM. Given the appreciable
participation of α1
2and β1
2, this mode may well be a wake mode. Now
the phase diagram in Fig. 18 shows that this mode is not RFM, for which
the phase angles between βcand βsand between α1
2and β1
2remain
close to 90◦; by comparison, the corresponding angles are 68◦and 118◦.
Accordingly, this mode is identified as the wake mode.
Eigenvector, frequency, and phase angle with variation
Thus far, the modes were identified for a representative rotational
speed =650 rpm. Moreover, the modal content, modal frequency,
c
s
2
1
2
1
Fig. 18. Wake mode phase angles for configuration 4 at 650 rpm.
400 500 600 700 800 900
Rotor rpm
0
1
2
3
4
5
Frequency (Hz)
Pitch mode
RFM
Wake mode
Experimental data (Ref. 1)
Fig. 19. Variation of frequency of three modes with rotor speed and
experimental results (configuration 1).
and phase—the phase angles between βcand βsand also between α1
2and
β1
2—were used as mode identification metrics. Since these metrics could
change with , their sensitivity to is presented in the next set of six
figures for 400 ≤rpm ≤1000, covering the soft-in-plane region of the
database. (The sensitivity of the eigenvector to has been investigated
earlier by Rosen and Issar (Ref. 5); hence it is not discussed here).
Figures 19–21 address configuration 1 and Figs. 22–24, configuration
4; therein, the focus is on regressive flap, pitch, and wake modes, for
which the literature (Refs. 1, 3–5) shows considerable variation in mode
identification. Moreover, Figs. 19 and 22, showing the frequency versus
, merit special mention in that Fig. 19 includes a comparison of how
predictions of pitch mode frequencies correlate with the measurements
for configuration 1 and Fig. 22 includes similar correlations for both pitch
mode and RFM for configuration 4 and these correlations also provide
the setting to study how the predicted frequency of a specific mode has
been identified with different modes in the literature.
Figure 19 shows the modal frequency versus . As expected, the fre-
quencies at =650 rpm, agree with the corresponding values: 2.13 Hz
for the pitch mode (Fig. 2), 0.524 Hz for RFM (Fig. 3), and 0.19 Hz for
the wake mode (Fig. 7). More importantly, the frequencies remain fairly
constant over the entire sweep (400 ≤rpm ≤1000), and measure-
ments are available only for the pitch mode and they agree remarkably
well with the predictions.
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INFLUENCE OF DYNAMIC INFLOW STATES ON COUPLED ROTOR FUSELAGE MODES 2017
400 500 600 700 800 900
Rotor rpm
/2
3/4
5 /4
3/2
7/4
2
Phase difference between c and s
Pitch mode
RFM
Wake mode
Fig. 20. Variation of phase difference between flap cosine and sine
states of three modes with rotor rpm for configuration 1.
400 500 600 700 800 900
Rotor rpm
0
/4
/2
3/4
5 /4
3/2
7/4
2
Phase difference between 2
1 and 2
1
Pitch mode
RFM
Wake mode
Fig. 21. Variation of phase difference between wake cosine and sine
states of three modes with rotor rpm for configuration 1.
400 500 600 700 800 900
Rotor rpm
0
1
2
3
4
5
Frequency (Hz)
RFM
Pitch mode
Wake mode
Experimental data (Ref. 1)
Experimental data (Ref. 1)
Fig. 22. Variation of frequency of three modes with rotor speed and
experimental results (configuration 4).
400 500 600 700 800 900
Rotor rpm
0
/4
/2
3/4
5 /4
3/2
7/4
2
Phase difference between c and s
RFM
Pitch mode
Wake mode
Fig. 23. Variation of phase difference between flap cosine and sine
states of three modes with rotor rpm for configuration 4.
400 500 600 700 800 900
Rotor rpm
5 /4
3/2
7/4
2
Phase difference between 2
1 and 2
1
RFM
Pitch mode
Wake mode
Fig. 24. Variation of phase difference between wake cosine and sine
states of three modes with rotor rpm for configuration 4.
Figures 20 and 21 show phase angles between βcand βsand also
between α1
2and β1
2, respectively. That for RFM, both sets of phase
angles remain fairly close to −90◦(2π−0.5π) over the entire sweep
is noteworthy. By comparison, the corresponding phase angles for the
pitch mode remains close to −81◦(2π−0.45π) between βcand βs
(Fig. 20) and close to −52◦(2π−0.29π) between α1
2and β1
2(Fig. 21).
The phase angles in these two figures for the wake mode show different
trends compared to the pitch and regressive flap modes. Observe that the
“discontinuity” in these figures for the wake mode is due to the real part
of the ratio of the eigenvector components of the two wake states (phase
of which is plotted) is changing sign and passing through zero.
Figure 22 of configuration 4 is the counterpart of Fig. 19 of configu-
ration 1; here as well, the frequencies of pitch and regressive flap modes
remain essentially constant for 400 ≤rpm ≤1000. From this figure at
=650 rpm and also from cross-reference to Figs. 10, 11, and 17, it is
seen that the frequencies are close to 2.35 Hz for the pitch mode, 0.93 Hz
for RFM, and 0.35 Hz for the wake mode. It is also noteworthy that the
frequency measurements of pitch mode and RFM agree well with the
predictions. However, the modes identified with these predictions show
considerable variation: The mode referred to as the pitch mode here is
identified as RFM in Ref. 3 and the mode referred to as RFM here is iden-
tified as the wake mode in Ref. 1. Figures 23 and 24 show, respectively,
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S. S. NAIR JOURNAL OF THE AMERICAN HELICOPTER SOCIETY
0
−5
0.005
−2.90724+15.0206i
15
−4
−3
0.01
−2
States
−110
n
0125
345
Magnitude
2
4
6
8
10
12
14
16
18
Higher
harmonic
wake states
Fig. 25. Fourier coefficients for periodic eigenvector for pitch mode
(configuration 4, M=3, =650 rpm).
0
0.005
−5
−3.8653+6.0849i
15
−4
Magnitude
−3
0.01
−2
States
−110
n
0125
345
2
4
6
8
10
12
14
16
18
Higher
harmonic
wake states
Fig. 26. Fourier coefficients for periodic eigenvector for regressive
flap mode (configuration 4, M=3, =650 rpm).
the phase angles between βcand βsand between α1
2and β1
2. As was the
case in Figs. 20 and 21, here as well, the phase angles for the regressive
flap and pitch modes remain essentially constant for the entire sweep.
(In Fig. 23, the “discontinuity” in phase difference for the wake mode is
due to the equivalence between 0 and 2π.)
Finally, it is worth revisiting Figs. 20 and 21 of configuration 1 and
then study Figs. 23 and 24 of configuration 4, and study both sets of
phase angles of RFM for 400 ≤rpm ≤1000. This study shows that
for RFM, both phase angles—between βcand βsand also between α1
2
and β1
2—remain close to −90◦for the entire sweep. (This finding
applies to PFM as well; see Table 2 for 650 rpm.)
This finding that for RFM the phase angles between βcand βsand
between α1
2and β1
2remain close to 90◦has been fruitfully exploited in
two earlier cases. The first case refers to identifying RFM in Fig. 3 for
configuration 1 and in Fig. 11 for configuration 4. The second case refers
to identifying the wake modes for both configurations; see Fig. 7 for
configuration 1 and also Fig. 17 for configuration 4. Both Figs. 7 and
17 show that βcand βsrepresent (numerically) the most dominant states
in the eigenvector and that βcand βsdo not seem to permit a sensible
comparison with α1
2and β1
2. Therefore, the corresponding phase angle
information typified by Figs. 8 and 18 was used as a basis for identifying
the modes in Figs. 7 and 17 as wake modes. Stated otherwise, this phase
information served as a basis to determine that the modes of Figs. 7 and
17 are not RFM.
10
y
Inflow magnitude
x
0
1
2
1
−3
4
−3.8653+6.0849i
0
6
0
−1−1
1
2
3
4
5
10−3
Fig. 27. Magnitude of inflow for regressive flap mode for configura-
tion 4 at 650 rpm (M=3, [−1 1] correspond to rotor disk).
The last set of three figures, Figs. 25–27 for configuration 4, are
noteworthy in that they are based on a third harmonic inflow distribution,
M=3. (The body states couple only with the first harmonic states, and
the case for M=2 does not introduce any additional first harmonic
states to those with M=1. Hence inflow modeling for M=2 is not
considered.) As noted earlier, the case for M=3 leads to periodic
coefficient systems, and the modal treatment is based on the Fourier
series expansion of periodic eigenvectors. (Given the low-thrust and
ground-contact conditions, the effects of periodic coefficients are not
significant. Nevertheless, the results are based on a rigorous Floquet
approach.)
Figures 25 and 26 show the Fourier coefficients of the periodic eigen-
vector (also see Eq. (3)), for the pitch and regressive flap modes. The
contribution of the higher order inflow states such as α1
4,β1
4,α
2
3,andβ2
3to
the periodic eigenvector is virtually negligible. In other words, dynamic
inflow modeling with M=3 does not provide any more additional in-
formation than what is obtained with M=1. (Therefore, the results
for M=3 for configuration 1 are omitted.) Finally, Fig. 27 shows the
inflow distribution |λ(r, ψ )|for RFM; here as well, the axial symmetry
is clearly seen. It may be noted that for M=3, the inflow radial varia-
tion is no more linear (due to using third-order polynomial radial shape
functions); this is well depicted by the curved surface of the inverted
cone.
Conclusions
This paper has presented a detailed account of identifying the modes
for two configurations, one with unmatched flap and lag stiffnesses and
another with matched flap and lag stiffnesses. The rotor–body system
represents a three-bladed rotor on gimbal support that permits roll and
pitch motions, and the blades execute rigid flap and lag motions. The
Peters–He dynamic inflow model is used, and the rotor–body–inflow
system is linearized about an equilibrium position.
The mode identification for these two configurations leads to the
following conclusions:
1a) The phase differences between multiblade flap cosine and sine
states βcand βsas well as between first harmonic wake states α1
2and β1
2
are crucial to distinguish RFM from the inflow mode.
1b) This phase information, along with the conventional mode iden-
tification metrics of relative participation of different states in the eigen-
vector and modal frequency, provides a means of identifying the modes
of the rotor–body–inflow system unambiguously.
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INFLUENCE OF DYNAMIC INFLOW STATES ON COUPLED ROTOR FUSELAGE MODES 2017
2) In the eigenvector analysis, the magnitudes of the inflow states of
aerodynamic origin such as α1
2and β1
2do not seem to permit a sensible
comparison with the magnitudes of the structural states such as βc,βs,and
θ. Therefore, the mode identification, which is exclusively based on the
most dominant component of the eigenvector, could lead to precluding
the inflow mode from the set of modes of the rotor–body–inflow system.
References
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