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Evangelos Papatheou, John E Mottershead and
Jonathan Cooper
Centre for Engineering Dynamics
University of Liverpool
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Active Flutter Suppression by theMethod of Receptances:
Experimental Results
DiPaRT Loads and Aeroelastics Workshop
Bristol, 13th
of December 2012
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Outline
Background & receptance method
Experimental rig
Testing & simulation results
Closed-loop experiments
Future work
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Flutter suppression in aeroelasticity
Flutter is a self-feeding, unstable vibration, potentially
destructive
In general, flutter will be caused by the coupling of the
modes of the system and/or at least one mode will have
zero damping
Flutter suppression may be treated with eigenvalue
assignment by increasing damping or the frequencyspacing between modes
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1. Background
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Flutter Margin
4. Closed-loop experiments
Zimmerman N.H. and Weissenburger J.T. (1964)
Quadratic fit flutterspeed prediction
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Receptance method
Active vibration control by pole assignment
Known as Receptance Method, can be used for anyinput-output transfer function
Based on vibration measurements, not on physicsbased models (M,K,C matrices)
No need for model order reduction or observers
Single-input method which can be extended to multi-
input
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1. Background
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Single input control force:
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xgxfTT
tu = &)(
)()( tut bf =
(t)r(t))()()( fKCM +=++ txtxtx&&&
General system with feedback
For velocity and displacementFeedback typically
Input: r(t)
1. Background
(1)
(2)
Receptance method
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( ) ( ) ( )sssss rxfgbKCM =++++ T2
Closed-loop system transfer function
( )( )1T2
)()( ++++= fgbKCM
rx sss
s
s
1. Background
Combining eq. (1) & (2) and Laplace transform:
(3)
(4)
only modification from open loop
rank 1 modification[ ] 12)(
)( ++= KCM
r
xss
s
s
Open loop receptance
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Sherman-Morrison Formula
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( )
11
and
+=T
uvZZZ
-1 ( )( ) ( ) ( ) ( )
( ) ( ) ( )sss
ssssss
T
T
uZv
ZvuZZZ
1
1111
1)(
+
=
( )
( ) ( ) ( )
( ) ( )bHgf
HgfbH
HH ss
sssss
T
T
++
+=
1)(
Determine the inverse of a matrix after a rank-1
modification if the original inverse matrix and themodification are known:
1. Background
)(TT
s gfb +
(4)
(5)rank 1 modification
Closed-loopreceptance
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1. Background
Pole placement by receptance method
H(s) can be measured through H(j) and bycurve fitting like rational fraction polynomials
e.g. PolyMAX
( )( ) ( ) ( )
( ) ( )bHgf
HgfbHHH
ss
sssss
T
T
++
+=
1)(In practice
bH ),(s { }n221 L
n
Rgn
Rf
( ) 1)( =+ bHfgk
T
k
Given: , and a complex set
closed under conjugation
such thatFind:
g and freal if system is controllable
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Experimental rig
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2 DOF system pitch / heave
NACA 0018 airfoil Chord 0.35 m
Span 1.2 m
2. Experimental rig
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Torsion Bar
Torsional Stiffness
Vertical Stiffness
Flap
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Experimental rig
V-stack piezo actuators
2. Experimental rig
Aerofoil
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Pitch mode 3.9 Hz
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Modal analysis3. Testing & simulation results
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Heave mode 6.7 Hz
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Modal analysis3. Testing & simulation results
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Bending mode 41Hz
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Modal analysis3. Testing & simulation results
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Torsion mode 47 Hz
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Modal analysis3. Testing & simulation results
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3. Testing & simulation results
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Single input single output feedback
Acceleration velocity (derived) feedback Curve fitting of open-loop FRFs
Assign the second mode (heave) by increasing the
damping by 1 %
Use the measured data to calculate the feedback
gains g and f
Sherman Morrison formula for closed-loop system
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Receptance method - pole assignment
3. Testing & simulation results
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Curve fitting
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Fit pole-residue model (rational fraction polynomial) to obtain H(j)
Pole assigned from -0.48 41.63j to -0.9 41.63j g = 77 f= -1756
3. Testing & simulation results
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Pole assignment - simulation
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Damping increased in the plunge mode from 1 %
to 2 %, pitch mode is also affected
3. Testing & simulation results
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Single input single output feedback (first approach)
dSPACE for real-time control Curve fitting of open-loop dSPACE FRFs get g and f
Velocity (laser) displacement (laser)
SISO approach
SIMO (two outputs - two sensors)
Displacement and velocity (derivative)
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Real time control - pole assignment
4. Closed-loop experiments
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Pole placement Experiment
both poles assigned to increase damping and frequencyspacing
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4. Closed-loop experiments
Heave placed from 6.83 Hz to 7 Hz and 0.5 % damping increase
Pitch from 3.89 Hz to 3.5 Hz and damping 1.5 % increase
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Damping increase in heave mode4. Closed-loop experiments
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Damping increase in heave mode
4. Closed-loop experiments
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Pole placement Experiment
frequency separation
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4. Closed-loop experiments
Heave placed at same frequency and 0.5 % damping increase
Pitch from 5.65 Hz to 6.04 Hz and damping 0.5 % increase
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Frequency damping trends root locus
4. Closed-loop experiments
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Flutter Margin
4. Closed-loop experiments
Predicted flutterspeed increases from17 m/s to 20 m/s
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Conclusions & future work
It is possible to assign damping and frequency spacing inmodes of a 2-DOF airfoil with the receptance method
By assigning poles in one wind speed and using the samegains g, f we can increase the flutter speed by 15 %
FM assignment
Introduction of non-linearity on the structure with theultimate goal of non-linear active control