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Flutter control using vibration test data: theory, rig design
and preliminary results
E. Papatheou1, X. Wei1, S. Jiffri1, M. Prandina1, M. Ghandchi
Tehrani2, S. Bode1, K. Vikram Singh3, J. E. Mottershead1, J. E.
Cooper4 1University of Liverpool, Centre for Engineering Dynamics,
Liverpool L69 3GH, United Kingdom. e-mail:
[email protected] 2Southampton University, Institute for
Sound and Vibration Research, United Kingdom. 3Miami University,
Department of Mechanical and Manufacturing Engineering, Oxford, OH.
4University of Bristol, Department of Aerospace Engineering, United
Kingdom.
Abstract The problem of flutter suppression in aeroelasticity
may be treated using eigenvalue assignment. The conventional
approach to active vibration control requires not only the
structural mass, damping and stiffness terms, but also aeroelastic
damping and stiffness contributions. This requirement becomes
unnecessary when using the receptance method which depends only
upon vibration measurements. In the cases considered in this paper
it requires wind-tunnel ‘wind on’ measurement of frequency response
functions. The set-up includes a two degree-of-freedom pitch-plunge
aerofoil, which allows for adjustment of the open-loop eigenvalues
and extrapolation of the closed-loop eigenvalues with wind speed
enabling the prediction of flutter velocities. One of the
advantages of using the receptance method is that in principle the
controller can be continuously corrected using in-flight
measurements with consequent improvements to aircraft
manoeuvrability and possibly survivability in the event of damage
to the aircraft.
The paper describes the first experimental study of flutter
suppression carried out in the Liverpool low-speed wind-tunnel
including physical tests and simulated application of the
receptance method using test data. Actuation of a flap is achieved
by two piezo-stacks in a ‘V’ configuration and the vibration
response is measured using ICP accelerometers mounted externally to
the wind-tunnel. The purpose of the research is to demonstrate the
delayed onset of flutter by increasing the flight envelope of
stable air speeds. Preliminary experimental results are
presented.
1 Introduction
Flutter of an aircraft, or its components, is a dynamic
instability associated with aeroelastic systems. It involves
interaction and coupling of modes (typically wing bending/torsion,
wing torsion/control surface or wing/engine) due to the surrounding
aerodynamic forces such that energy is extracted from the airstream
leading to negatively damped modes and unstable oscillations. For a
given Mach number, at some critical speed (flutter speed) the
system eigenvalues exhibit instability leading to sustained
oscillations and eventually fatigue or failure. In flutter analysis
the eigenvalues (natural frequencies and damping) of an aeroelastic
system are computed for varying speeds, altitudes and Mach number.
In aeroelastic control, the problem is to suppress flutter or
extend the flutter boundary by assigning stable poles using
feedback control forces, preferably supplied by available control
surfaces; this problem may
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be treated using eigenvalue assignment. In addition to the usual
structural mass, damping and stiffness terms the conventional
vibration control approach based on the state-space formulation
requires the matrices of aeroelastic damping and stiffness,
obtainable by double-lattice subsonic lifting surface theory (DLM)
available in commercial codes such as MSC-NASTRAN or ASTROS.
The problem of flutter control has been the subject of research
investigation for decades. In the 1980s and 1990s mainly analytical
and computational procedures were developed. Ueda and Dowel [1] and
Lee and Tron [2] studied the effect of aerodynamic and structural
nonlinearities respectively. Lee [3] developed an iterative
procedure for flutter analysis in large-scale systems; Tang and
Dowell [4] considered flutter instability and forced response of a
non-rotating helicopter blade; and Lee, Jiang and Wong [5] analysed
the flutter dynamics of an aerofoil with a cubic restoring force.
Towards the end of the 1990s the first experimental studies of
binary flutter in aerofoils appeared and studies of this kind have
continued up to the present day. Block and Strganac [6] used
full-state feedback with an optimal observer to stabilise linear
and nonlinear systems beyond the open-loop flutter speed;
Platanitis and Strganac [7] applied feedback linearisation to a
nonlinear wing section using control surfaces at both the leading
and trailing edges and Trickey et al. [8] made an experimental
study of limit cycle oscillations in a 3 degree of freedom
aerofoil.
The receptance method of eigenvalue assignment was developed by
Ram and Mottershead [9]. One of the advantages of the approach is
that the controller is formulated entirely on measurements rather
than on the conventional matrix theory, typically in state space.
In practice, the methodology requires the in-flight measurement of
frequency response functions (FRFs) so that there is no requirement
to know or to evaluate the structural mass, damping and stiffness
matrices (denoted A, D, E in this paper) or the aerodynamic
matrices (denoted B, C). Also there is no requirement for model
reduction or for the use of an observer to estimate the unmeasured
state variables. In principle the receptance-based controller can
be continuously corrected using in-flight measurements with
consequent improvements to aircraft manoeuvrability and possibly
survivability in the case of damage to the aircraft. The receptance
method has been developed for (i) partial pole placement (so that
selected poles are assigned while other chosen poles remain
unchanged) [10] and (ii) the assignment is carried out robustly
with respect to noise on the measured receptances [11]. It was
applied for vibration control of an AgustaWestland W30 helicopeter
airframe using electro-hydraulic actuators originally designed to
work as active vibration isolators separating the airframe and the
raft-mounted engine and gearbox [12].
This paper describes the first stages of an experimental study
of flutter suppression carried out in the Liverpool low-speed
wind-tunnel. The test set-up allows for the adjustment of open-loop
eigenvalues at different wind speeds and the extrapolation of
closed-loop eigenvalues with wind speed enables the prediction of
flutter velocities. The practical application of the receptance
method is described and simulated results, based on test data, are
presented.
2 Theory
The governing equation of an aeroelastic system may be cast in
matrix form [13],
( ) ( ) pfqECqDBqA +=++++ 2VV ρρ &&& (1) where, A,
B, C, D, E are the structural inertia, aerodynamic damping,
aerodynamic stiffness, structural damping and structural stiffness
matrices respectively, ( )tq is the vector of generalised
coordinates, ( )tf is the control force and ( )tp represents an
external disturbance such as measurement noise or turbulence. In
aeroelasticity, both circulatory and noncirculatory forces
generated by the wake, for a chosen Mach number and reduced
frequency, are expressed as additional contributions to the system
matrices. In equation (1) these terms appear as matrices B and C
which, in general, are frequency dependent. Often these
wake-induced forces are combined together in the form of the
aeroelastic influence coefficient (AIC) matrix at a set of discrete
frequencies. Finite element codes such as MSC-NASTRAN or ASTROS
provide the necessary modelling routines.
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For the open-loop homogenous system and by separation of
variables, ( ) ( )tt jn
jjj λα exp
2
1∑==
φq , the
eigenvalue equation of the jth mode may be expressed as,
( ) ( )( ) 0φECDBA =++++ jjj VV 22 ρλρλ (2)
where jα is the jth modal coordinate. The complex eigenvalues,
in equation (2), may be written in terms of
the j th damping and natural frequency, the real and imaginary
parts of the following expression. When the matrices, A, B, C, D, E
are strictly real the eigenvalues (and eigenvectors) appear in
complex conjugate pairs
2
, 1i jjjjjnj ζωωζλ −±−=+ , .,,2,1 nj K= (3)
The real part of the eigenvalues defines the stability of the
system and hence, when the real part of the eigenvalues jλ in
equation (3) is positive, the system is said to be unstable
(resulting in flutter). For all other values the aeroelastic system
is either stable or marginally stable.
The receptance FRF matrix may be expressed in theory as the
inverse of the aeroelastic dynamic stiffness matrix,
( ) ( ) ( )( )122 ii
−++++−= ECDBAH VV ρρωωω (4)
In practice, when applying the receptance method, ( )ωiH is
estimated from measured vibration data using well established
techniques. Curve fitting to the estimated ( )ωiH , for example by
the LMS PolyMAX routine [13], allows the determination of ( )sH by
substituting s for iω in the curve-fitted approximation [10]. In
this paper we assume that the matrix of receptances can be
determined from ‘wind on’ measurements carried out in the low-speed
wind tunnel. It will be demonstrated that any input and output
signals may be used in aeroelastic eigenvalue assignment, in which
case the dynamics of the actuators and sensors (including the
effects of time delay) may be included in the measurement,
rendering unnecessary the need for mathematical models to
approximate the behaviour of actuators and sensors.
The receptance method depends upon an elegant result from the
linear algebra, namely the Sherman-
Morrison formula, which produces a modified inverse matrix, ( )
11 −− += TuvZZ , when a known rank-1 modification, Tuv , and
original inverse matrix, 1−Z , are available such that,
( ) ( ) ( ) ( ) ( )( ) ( ) ( )sss
ssssss
T
T
uZv
ZvuZZZ
1
1111
1)( −
−−−−
+−= (5)
In single-input control, the control force is typically given
by,
( ) ( )tut bf = (6) and
( ) qgqf &&&TTtu += (7)
for velocity and acceleration feedback.
It is easily demonstrated that ( )Tss gfb 2+ represents a rank-1
modification to the open-loop dynamic stiffness matrix of the
aeroelastic system. Then by combining equations (5) and (7) it is
found that,
( ) ( ) ( ) ( )( ) ( )bHgfHgfbH
HHsss
ssssss
T
T
2
2
1)(
++
+−= (8)
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where ( ) ( )ss 1−= ZH and ( ) ( ) ( )ssss TT vugfb =+ 2 The
poles of the closed-loop system are given by the zeros of the
denominator of equation (11). Thus for
the assignment of the complex-conjugate pair of poles *1 jj µµ
=+ the control gains f, g must be chosen to satisfy the
equations,
( ) ( )( ) ( ) nkkjjTjj
jT
jj ,,2,1,121
1
12
11
2
K=−=
=++
=++
+++ 0bHgf
0bHgf
µµµ
µµµ (9)
Re-arranging and combining equations (9) into a single matrix
expression leads to,
−
−−
=
1
1
1
Mf
gG (10)
with
=
Tnn
Tnn
TT
TT
22222
22222
11121
rr
rr
rr
G
µµ
µµµµ
MM (11)
where,
( ) ( )( ) ( ) nkkjjjj
jjj,,2,1,12
111K=−=
==
+++ bHr
bHr
µµµµ
(12)
which allows the determination of f and g by inversion of the
matrix G. Ram and Mottershead [9] showed that (a) G is invertible
when the system is controllable and the poles ( )n221 µµµ L are
distinct, and (b) f and g are real when G is invertible and the set
( )n221 µµµ L are closed under conjugation.
When G is a square matrix, there is a unique solution for ( )TTT
fg and when the system (11) is under-determined (fewer poles to be
assigned than the number of gain terms - f, g) a minimum norm
solution is available for the minimization of control effort.
Alternatively, in the latter case, the gains may be chosen that
assign the chosen eigenvalues, while at the same time minimizing
the sensitivity of the assigned poles to inaccuracy and noise in
the measured receptances. A robust pole-placement approach to noise
on the measured receptances was described by Tehrani et al.
[11].
3 The Experimental Rig
The wind tunnel experiment consists of a working section
containing a NACA0018 aerofoil (chord = 0.35 m, span = 1.2m),
supported by adjustable vertical and torsional leaf springs. The
aerofoil can be modelled as a 2D system with pitch and plunge
degrees of freedom as illustrated in Figure 1.
The design allows the adjustment of stiffnesses kh and kθ (the
vertical and torsional springs) as well as the possibility of
adding external masses me to vary the location of the centre of
mass C, the mass of the wing mw and its moment of inertia IC. The
maximum air speed for the wind tunnel used is around 20 m/s. The
aim of the design is to explore regions close to the flutter speed
of the system.
The vertical spring arrangement is shown in Figure 2. By varying
the clamp location in the direction shown in the figure, it is
possible to vary the stiffness of the vertical springs, one on each
side of the wind tunnel, which support the wing (attached to the
shaft on the left of the figure). The variation of the vertical
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stiffness versus the beams length was determined analytically
and is shown in Figure 3 and may be varied from about 200 to 23000
N/m.
The adjustable torsional spring is shown in Figure 4. By moving
the device in the direction indicated by the arrows it is possible
to increase or reduce the torsional stiffness in the range shown in
Figure 5, approximately from 10 to 320 Nm/rad.
Figure 1: Pitch and plunge 2D model
Figure 2: Adjustable vertical spring
Using these ranges of stiffness, it is possible to vary the
flutter speed of the aeroelastic system approximately between 10
and 70 m/s. The open working section (with sides removed and
separated from the wind tunnel) is shown in Figure 6. A torsional
bar is used in order to maintain the same vertical displacement on
the two sides of the test section. The external mass of the system
was calculated to be around 6.5 Kg.
Plunge motion
Adjustable clamp location
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Figure 3: Vertical stiffness range
Figure 4: Adjustable torsional spring
Active vibration control described in [15, 16] is achieved by
means of a ‘V’-stack piezoelectric actuator shown in Figure 7
acting on the control surface of the wing, allowing a flap
deflection of about ±7°. The actuator consists of two piezo-stacks
(Noliac SCMAP09-H80-A01) in a ‘V’ formation. The flap is actuated
when one arm of the ‘V’ is made to extend while the other retracts
by an equal amount – caused by applying equal voltages to the two
piezo stacks but with 180° phase difference. Khron Hite wideband
power amplifiers, model 7500, were used. The ‘V’ stack actuator
[15] is known to behave as a pure gain provided that its natural
frequency is well above the frequencies of the assigned poles. The
receptance method was applied with voltage input to the stacks and
acceleration output, measured using four Kistler ICP accelerometers
mounted externally to the wind-tunnel on small rigid beams (two
accelerometers on
Pitch motion
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each side). Thus both the pitch and plunge degrees of freedom
were measureable. The term receptance is used unconventionally in
this case as a generalised input-output FRF.
Figure 5: Torsional stiffness range
Figure 6: Open Working Section
4 Preliminary Results
A modal test of the experimental rig was carried out using
hammer excitation. Figure 8 shows a measured FRF in the frequency
band 0-102.4 Hz with resolution 0.1 Hz. The first two modes are the
two designed pitch and plunge modes of the system, at the currently
used setup - pitch at 3.9 Hz and plunge at 6.7 Hz.
Torsion Bar
Torsional Stiffness
Aerofoil
Vertical Stiffness
Flap
SELF-EXCITED VIBRATIONS 3053
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The first bending and torsion modes were found at around 40 Hz
(well away from the modes to be controlled) and other modes,
between 20 and 40 Hz were found to be modes of the supporting
structure. Work is presently going on to stiffen these latter modes
to take them further away from the controlled pitch and plunge
modes. All the experimental results were obtained using an LMS
SCADAS III data acquisition system running on a DELL desktop
PC.
(a) (b)
Figure 7: (a) ‘V’ Stack Actuator; (b) Actuator In-Situ
Figure 8: Preliminary FRF of the whole experimental rig under
impact hammer excitation
In order to realise active vibration control, as described in
[15, 16], FRFs were recorded with the ‘V’ stack actuator acting as
an input to the system while the acceleration was recorded through
the externally mounted ICP accelerometers at different wind speeds.
Figures 9-12 show the FRFs obtained from stepped sine tests in the
frequency band 0.5 – 12.5 Hz with resolution 0.05 Hz using just one
of the four accelerometers. The wind speeds used were 6, 7, 9 and
9.5 m/s. The effects of A/D and D/A conversion and of the filters
implemented using dSPACE were included in the measured FRFs. A
low-pass Butterworth filter with a cut-off frequency at 10 Hz was
included.
The FRFs were found to be noisier than those obtained by regular
vibration test procedures, probably because of the nature of the
excitation, aerodynamically by the flap moving in the airflow, and
possibly due to the piezo stacks which often produced audible
noise. As a first step in the eigenvalue assignment, a single input
– single output (SISO) approach was used with the response from one
of the four accelerometers. The open loop FRFs recorded are shown
in Figures 9-12 along with curve-fitted
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approximations obtained using Structural Dynamics toolbox [17]
in MATLAB with a pole/residue model. It should be noted that
although the FRFs were all of the accelerance type, the actual
sensitivity of the transducers was not used, so Figures 9-12 simply
display voltage/voltage transfer functions.
The curve fitted FRFs were then used in order to calculate the
feedback gains f and g. Velocity feedback was applied by
integration of the acceleration signal in dSPACE. The pole of the
second, plunge, mode at 6.7 Hz was assigned a damping increase from
1 % to 2 %. Therefore the assigned closed-loop pole became -0.9 ±
41.63i from the open-loop pole which was -0.48 ± 41.63i. The
actuator gains were calculated from the curve fitted FRF measured
with wind speed at 7 m/s and were found to be: g = 77.103 and f =
-1756.7.
The same gains were subsequently applied for all the remaining
cases of 6, 9 and 9.5 m/s. The simulated closed-loop FRFs were
calculated by using the Sherman-Morrison formula in equation (8)
with curve fitted data from the experimental measurements. Figures
13-16 show the open- and simulated closed-loop FRFs when the
damping of second mode of the system was increased for all the wind
speed cases. The simulated response of the closed-loop system is
shown using a linear scale (Figures 13-16) in order to clarify the
effect of the damping increase in the plunge mode of the system. It
was found that although the measured FRFs were noisy, the method
was adequately robust and worked successfully for all cases. The
damping of the plunge mode was always increased, but the effect was
not exactly the same for all wind speeds, something which was
expected since the actuator gains were calculated at a specific
speed of 7 m/s and subsequently implemented at the other
speeds.
Figure 9: Measured and curve fitted FRF with wind speed at 6
m/s, stepped sine test
5 Discussion and Conclusions
This paper describes the first results from a
flutter-suppression study carried out in the Liverpool low-speed
wind-tunnel using the receptance method. The theory, along with the
rig design is described and experimental and simulated results are
presented. Acceleration and velocity feedback was used together
with test measurements to obtain the first simulated closed-loop
results. Acceleration was measured though transducers and velocity
was derived by integration. FRFs were measured at different wind
speeds and pole assignment by an increase in the damping of the
plunge mode was performed on the curve fitted data originating from
a test with a wind speed at 7 m/s. The actuator gains g and f were
then used for other
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wind speeds, higher and lower, and it was found that the
simulated closed-loop system behaved precisely as expected.
Overall, this paper has shown that FRFs can be measured with the
‘V’ stack actuator acting as an input to the system and although
there was some noise in the measured data, the receptance method
worked well in simulation providing confidence for full
experimental closed-loop implementation in real time for flutter
suppression and the extension of flutter boundaries.
Figure 10: Measured and curve fitted FRF with wind speed at 7
m/s, stepped sine test
Figure 11: Measured and curve fitted FRF with wind speed at 9
m/s, stepped sine test
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Figure 12: Measured and curve fitted FRF with wind speed at 9.5
m/s, stepped sine test
Figure 13: Pole placement (damping increase) of the second mode,
closed-loop is simulated, at wind
speed of 7 m/s
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Figure 14: Pole placement (damping increase) of the second mode
at wind speed of 6 m/s when g,f gains
were calculated from the curve fitted FRF with wind speed at 7
m/s
Figure 15: Pole placement (damping increase) of the second mode
at wind speed of 9 m/s when g,f gains
were calculated from the curve fitted FRF with wind speed at 7
m/s
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Figure 16: Pole placement (damping increase) of the second mode
at wind speed of 9.5 m/s when g,f
gains were calculated from the curve fitted FRF with wind speed
at 7 m/s
Acknowledgement
The authors gratefully acknowledge the support of the US Air
Force, EOARD Grant FA8655-10-1-3054 and EPSRC Grant
EP/J004987/1.
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