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1
Effect of rotor wake on aeromechanical instabilities in helicopters
Salini S Nair
Research Scholar
IIT Madras
Chennai, Tamil Nadu, India
snairsalini@gmail.com
Ranjith Mohan
Assistant Professor
IIT Madras
Chennai, Tamil Nadu, India
ranjith.m@iitm.ac.in
ABSTRACT
The influence of rotor wake and ground effect on aeromechanical instabilities, specifically ground resonance is
investigated. The analytical model considered has fuselage pitch, roll and blade flap, lag degrees of freedom. To
analyze the system stability at different rotor rotational speeds, modal damping and frequency is computed using
Floquet method. The Peter He dynamic inflow model is incorporated into the system equations to analyze the
influence of rotor wake on system stability. The effect of collective pitch on the system stability, with uniform
inflow as well as dynamic inflow, is considered. It has been observed that rotor wake influences the effect of
collective pitch on system stability. The influence of full ground effect as well as partial ground effect on the system
stability is investigated.
NOTATION
Damping matrix
Mass matrix
Influence coefficient matrix
Pressure
,
Normalized associated Legendre functions
Mass flow parameter
,
Velocity expansion coefficients of main rotor
,
Ground rotor velocity states
Distance from the ground, non-dimensional on
blade radius
,
Order of associated Legendre function
,
Degree of associated Legendre function
Induced velocity in the axial direction
,,,
Main rotor ellipsoidal coordinates
,,
Ground rotor ellipsoidal coordinates
,
Ground rotor pressure expansion coefficients
,
Main rotor pressure expansion coefficients
INTRODUCTION
The paper investigates the effect of rotor wake and
ground effect on aeromechanical instabilities, specifically
Presented at the 4th Asian/Australian Rotorcraft Forum, IISc,
India, November 16 - 18, 2015. Copyright © 2015 by the
Asian/Australian Rotorcraft Forum. All rights reserved.
ground resonance. The experimental model described in Ref.
1 is considered for the analysis. The degrees of freedom
considered in the analysis are flap, lag degrees of freedom of
the rotor and the pitch, roll degrees of freedom of the
fuselage.
Ground resonance is an aeromechanical instability that
can occur in helicopters when operating near or on the
ground. It happens mainly due to coupling between lead-lag
and fuselage degrees of freedom. Detailed experimental and
theoretical analysis on ground resonance is presented in
references 2, 3, 4, 5 and 6. These studies and our own results
have shown that it is the regressive lag mode of the blades,
which couple with the fuselage motion leading to instability.
Ground resonance is studied here by computing modal
damping and frequency associated with the system at a
particular rotational frequency (Ref.7,8). If the modal
damping is positive, the system is unstable at that particular
rotational frequency. This is also indicated by coalescence
between modal frequencies. However, frequency
coalescence is not a sufficient condition for ground
resonance. If sufficient damping in active or passive form is
present in the system, then ground resonance will not occur.
The first step in the analysis is the computation of the
equations of motion for the system. The equations of motion
for the coupled rotor-fuselage system are derived in
Mathematica using the Lagrangian energy method. The rotor
being in the rotating frame, the resultant equations are
periodic with period equal to the rotor rotation period.
2
Depending on whether the helicopter configuration has
isotropic or anisotropic rotor and fuselage, it may be
possible to convert the system equations to constant
coefficients using either rotating frame transformation (Ref.
7) or Mutiblade Coordinate Transformation (Ref. 9). In
general case, when both the rotor and fuselage are
anisotropic, the system equations cannot be converted to
constant coefficient form.
In case of periodic coefficient equations, the
conventional eigenvalue method cannot be used to compute
the modal damping and frequency. Hence, Floquet analysis
(Ref 10, 11) is required to determine the regions of
instability. Floquet analysis involves the computation of
Floquet Transition Matrix from which the modal damping
and frequency associated with the system can be computed.
In all our analysis, Floquet method has been used.
Finite state inflow model
In the initial analysis, the inflow through the rotor is
assumed constant and uniform. However, the inflow through
the rotor disc varies in the azimuthal and radial directions. In
the early studies, based on experimental and theoretical
analysis, several different models were proposed to account
for the variation of inflow over the rotor plane. This includes
several linear inflow models and Mangler and Squire inflow
model (Ref. 12). CFD-based approaches have also been
proposed in literature for the computation of inflow through
the rotor. The disadvantage of CFD-based approaches is that
the computation is complex, time consuming and
incorporation into the stability analysis model is difficult.
Hence, there was the requirement for a wake model that
is accurate and is in a form amenable to stability analysis.
Finite state models or more often called the dynamic inflow
models was proposed initially by Pitt and Peters (Ref. 13)
based on the potential theory equations. The model was
further improved by Peters and He in Ref. 14 and it was
found that the model predicted inflow accurately.
The equations were derived starting from the basic
potential theory equations for momentum and energy
conservation. In order to satisfy the appropriate boundary
conditions, pressure and velocity are represented by
equations 1 and 2 using Legendre functions and ellipsoidal
coordinates.
=
+
=+1
=0 (1)
=
+
=+1
=0 (2)
=
cos
=
cos
(3)
=
sin
=
sin
(4)
In equations 3 and 4, (,,
) represent the coordinates of
the rotor in the ellipsoidal coordinate system and
and
are the associated Legendre polynomials of first and
second kind respectively. The
,
terms represent the
velocity states. The
,
terms with +=
correspond to the pressure discontinuities and +=
correspond to the mass source terms. The pressure
discontinuity terms can be computed from the lift on the
rotor. In Peters He model (Ref 15), the mass source terms
were not considered.
By using the Galerkin approach and the potential theory
equations, a set of differential equations relating the velocity
states to the pressure potential was developed (Eqn. 5). Each
of the matrices in the equation has closed form expressions
as given in Ref. 14.
+ 1
=
/2 (5)
In the second part of the paper, dynamic inflow model
based on references 14 and 15 is incorporated in the system
equations and the effect of rotor wake on system stability is
analyzed using Floquet method and is compared with the
experimental results in Ref. 1. The effect of collective pitch
on rotor stability is analyzed in detail in Ref. 5. However,
the influence of rotor wake has not been considered. In this
paper, the effect of collective pitch variation on system
stability in the presence of rotor wake is presented.
Though Peter-He model shows excellent correlation
with the experimental solutions, it cannot be used to predict
the inflow outside the rotor disc. Hence, many modifications
were proposed to extend its applicability. The history of
evolution of finite state dynamic inflow model in its present
state is given in references 13, 16, 17 and 18. Using the
model proposed in Ref. 18, it is possible to compute the
inflow on or off, below or above the rotor plane.
The model developed in Ref. 16 is capable of including
mass source perturbations in the model. The model could be
used to predict the inflow anywhere on or above the rotor
disc. The general differential equation for the model is of the
form given in Eqn. 6. Here again all the matrices have closed
form representations as given in Ref. 16. In axial flow, the
equation is reduced to the form in Eqn. 7. Here +=
as well as odd terms were added to the system.
However, = terms were not considered, so the net mass
flux through the system was zero (Ref.19).
+ 1
=
(6)
+
=
(7)
Ground effect finite state model
In the third part of the paper, the influence of ground
effect on system stability is analyzed. The presence of
ground varies the inflow through the main rotor as the
ground surface acts like an obstruction that deflects the flow
(Ref. 20). The proximity of the ground constrains the rotor
wake thereby reducing the inflow through the rotor. Hence,
3
more thrust is produced with the same power (Ref 21). The
effect of ground on the rotor inflow has been experimentally
verified in literature using flow visualization techniques
(Ref. 22). It was observed that the presence of ground
actually creates an upwash on the rotor plane. Ground effect
is particularly important when the rotor is hovering above
buildings or near ship decks.
Ground effect can be modeled by using an image rotor
placed at a distance below the ground equal to the height of
the rotor above the ground (Ref. 22). The disadvantage with
image rotor method is that only planar static ground surface
effects could be analyzed. The dynamic motion of the
ground surface like in case of ship decks or partial ground
effect cannot be represented.
From the experimental results, empirical formulas were
developed to represent the influence of ground by comparing
the thrust generated or power required in OGE (Out of
Ground Effect) and IGE (In Ground Effect). The relating
factor was called the ground effect factor and several
formulas for the same were developed (Ref. 19, 22). Out of
the different formulas, Hayden's formula (Eqn. 8) is
considered the most accurate.
=1
0.9926+.03794 (2/)^2 (8)
Here h is the non-dimensional distance from the ground. The
influence of ground on inflow can be represented using the
generalized dynamic inflow model by dividing the mass
flow parameter in the model by the kG factor (Ref. 22).
However, again this method could be used only for planar
surfaces and full ground effect.
Later on, equating the pressure due to the main rotor
(Eqn. 9) on the ground to an equivalent pressure at the
ground was proposed (Ref. 22). The ground pressure (Eqn.
10) was represented as mass source perturbations and hence
+= terms alone are taken to represent the ground
pressure potential. By equating the two, the ground pressure
states and can be determined from the main rotor
pressure states
and
using appropriate coordinate
transformations (Eqn. 11). The [
] matrix is a function of
ellipsoidal coordinates of the rotor and varies with
orientation and shape of the ground surface.
=
+
=+1,+3...
=0 (9)
= +
=,+2
=0 (10)
=
(11)
=+ 1, + 3 ,=,+ 2
The OGE induced velocity at the rotor disc and the
ground interference velocities are represented by equations
12 and 13. Here
and
represent the main rotor and
ground interference velocity states respectively.
=
+
=+1
=0 (12)
=
+
=+1
=0 (13)
In Ref. 22, the unsteady part of the ground pressure
perturbation is neglected and using potential theory
equations, the ground interference velocity states are directly
computed from the ground pressure states. The model was
able to get fairly good results for complete, partial and
dynamic ground effect. In addition, the implementation
involved the computation of several integrals, which is time
consuming.
The representation of ground effect by a finite state
model was developed in Ref. 20 by incorporating a ground
rotor into the system. The ground rotor generates pressure at
the ground rotor disc equal to the pressure due to the main
rotor on the ground thereby satisfying the no penetration
condition. The ground rotor does not generate lift and is
assumed to have infinite number of blades. The model was
an extension of the model proposed in Ref. 16 (Eqn. 7) by
including the m=n terms in the pressure perturbations of the
ground rotor. Thus, another set of differential equations, one
representing the main rotor (Eqn. 14), and the other
representing the ground rotor (Eqn. 15) were developed. In
this model, the velocity at the main rotor plane due to the
ground rotor, which in effect accounts for the ground effect,
is determined from the states of Eqn. 15. Hence, the
incorporation into the stability analysis model is easy.
+
=
(14)
+
=
(15)
The model gave fairly good correlation with the
experimental results for complete and partial ground effect
and can be used to represent ground surface of any shape by
calculating appropriate [
] matrix. The model is
incorporated in our stability analysis model to see the
influence of ground on system stability. Partial ground effect
is also analyzed considering ground surfaces of different
dimensions. The model has been developed in the linear and
non-linear form (Ref. 19) but the results obtained using
either model was similar. In all our analysis, linear model is
used.
METHODOLOGY
The experimental model from Ref. 1 is used for
analysis. The model has a hingeless rotor with anisotropic
fuselage. The degrees of freedom considered in the analysis
are the blade lead lag and flap degrees of freedom and the
fuselage roll and pitch degrees of freedom.
4
Figure 1. Variation of RLM damping with rotor
rotational speed with uniform and dynamic inflow
compared with the experimental results for zero
collective pitch angle
The equations of motion are obtained from Lagrange
approach using Mathematica. The resulting equations are
periodic with frequency equal to the rotor rotational
frequency. In case either the rotor or the fuselage is
isotropic, it is possible to convert the set of equations from
the periodic coefficient form to constant coefficient form.
The analysis on constant coefficient systems is
computationally simpler. However, in real cases, both the
fuselage and the rotor will be anisotropic and hence the
system equations cannot be converted from their periodic
form. In our analysis, we retain the equations in the periodic
form and Floquet method is used for computing damping
and frequency.
Inflow model
Initially, the inflow through the rotor is assumed to be
uniform. In the second part of the analysis, inflow is
modeled based on the Peters-He dynamic inflow model (Ref.
15). The radial and azimuthal variation of dynamic inflow is
taken into account. The model damping is again computed at
different rotor rotational frequencies. Figure 1 compares the
model damping associated with the least stable mode
(identified as the Regressive Lag Mode) with and without
dynamic inflow and the experimental results from Ref. 1. It
is observed that good correlation exists between the results
obtained with dynamic inflow and the experimental results.
A small shift in resonance frequency is observed, but this
could be due to some parameter errors and other studies for
the same coupled rotor fuselage model also observed this
deviation.
Peter He model assumes the number of harmonics to
extend from zero to infinity. However as the number of
harmonics is increased, the modal damping settles to
constant values (Figure 2).
Figure 2. Variation of modal damping corresponding to
RLM mode at 760 rpm for zero collective pitch angle
with increasing number of harmonics
Figure 3. Variation of modal damping of RLM with
collective pitch at 760 rpm in the presence and absence of
wake
Hence, in further analysis, the maximum number of
harmonics is restricted to 5. The number of radial shape
functions is determined using the method proposed by Peters
and He (Ref. 15).
Effect of Collective pitch angle
The variation of modal damping with collective pitch
angle has already been studied in Ref. 5 but the effect of
rotor wake was not taken into consideration. Reference 5
showed that with increasing collective pitch angle the
stability of the system is reduced. Here we include dynamic
inflow into the system equations and it is observed the wake
has a stabilizing effect on the system dynamics at higher
collective pitch angles as seen in Figure 3. Here a linear
relation between angle of attack and lift coefficient is
assumed and stall effect is not considered.
5
Figure 4. Variation of average perturbed inflow through
the rotor at 760 rpm and zero collective pitch angle with
and without the inclusion of kG factor at = .
GROUND EFFECT
Ground effect represents the influence of the ground on
the inflow through the rotor and hence on the thrust. Though
Peter-He model gives good correlation with the experimental
results, it has to be modified further to represent the
influence of ground proximity. The mass flow parameter in
the Peter-He model is divided by the kG factor (Eqn. 8) to
include ground effect. This reduces the inflow through the
rotor since the presence of ground creates an upwash. In
results presented here for the perturbed inflow, small
perturbation is given to the flap degree of freedom of one of
the blades. The average is computed by integrating the
inflow over the radial and azimuthal direction. The
comparison of average perturbed inflow through the rotor at
a height of .5 times the rotor radius with and without the
inclusion of kG factor in the model is shown in Figure 4.
The finite state space model for ground effect (Ref. 20)
was incorporated in our system equations to analyze the
effect of ground on stability. This involves computation of
[
] matrix relating the ground rotor pressure states to the
main rotor pressure states (Eqn. 11). In axial flow analysis,
=
and using the orthogonality properties of Legendre
functions, the [
] matrix takes the form of Eqn. 16 in full
ground effect. The relations connecting (,) and (,) can
be obtained using appropriate coordinate transformation
matrices (Ref. 22).
=
, = 0, =
1
0 (16)
However though the inflow through the rotor was
modified with ground effect, as seen in Figure 5, the
influence on modal damping was found to be of very small
magnitude as seen in Figure 6. The analysis was performed
assuming the rotor is at a distance .5 times the rotor radius
from the ground.
Figure 5. Variation of average perturbed inflow through
the rotor at 760 rpm and zero collective pitch angle with
and without the inclusion of GE state space model at
= .
Figure 6. Variation of RLM mode damping with rotor
rotational frequency at zero collective pitch angle in the
presence and absence of ground effect at = .
As expected, it was observed that the influence of
ground decreases with increase in distance from the ground.
The variation of RLM modal damping with distance from
the ground is plotted in Figure 7. It is seen that the damping
values settle to a constant value at a distance of about 1 to
1.5 times the rotor radius.
It is observed that the state space model under predicts
the effect of ground compared to kG factor. Modifications to
the model like increasing the ground pressure due to the
main rotor by a correction factor (Ref. 19) and inclusion of
= terms in the velocity expansion (Ref. 17) for ground
has been proposed. However, even with the present
formulation it can be concluded that the effect of a planar
ground surface on the system, though small, is stabilizing.
6
Figure 7. Variation of modal damping of RLM at 760
rpm and zero collective pitch angle with distance from
the ground
Figure 8. Variation of RLM damping at 760 rpm and
zero collective pitch angle with sector area (blackened
region) ( indicates the angular span of sector)
PARTIAL GROUND EFFECT
Here the range of integration in the radial and azimuthal
direction depends on the shape of the ground. The pressure
equivalence relation is applied only over that area which is
under the influence of ground. The equation for computing
the ground rotor pressure sine terms () is given in Eqn.
17.The cosine terms () can be computed in similar
manner.
=1
1
0
1
0
= +1
=0
cos
+
sin
sin(r
)
, = 0
(17)
Figure 9. Variation of (ground interference
velocity) with time for different overlap areas.
Here,
0,
1,0,1form the limiting coordinates for the
region under ground effect (Ref. 22). As far as the modal
damping is concerned, it is observed that the partial ground
effect does not have much influence on the modal damping
results. The variation in modal damping with increasing
sector area of the main rotor under ground effect is plotted in
Figure 8. The angle indicates the angular span of the
sector. The variation of the ground rotor velocity with time
for various values of is plotted in Figure 9. It is observed
that the ground interference velocity is an upwash and
increases with increasing rotor area overlapping the ground.
CONCLUSIONS
The finite state wake model is incorporated into the
system dynamics equations to investigate the effect of wake
on rotor aeromechanical stability. The effect of collective
pitch, full ground effect and partial ground effect on system
stability with dynamic inflow model was investigated. It is
observed that dynamic inflow has a very significant effect on
system stability, also the analysis with varying collective
pitch indicate that the inflow dynamics is dependent on the
collective pitch angle. The presence of ground or any
obstruction varies the inflow through the rotor but the effect
on system stability as such is not significant.
ACKNOWLEDGMENTS
We acknowledge the support provided by HPCE
facility, IIT Madras.
7
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