1 airfoil dipart2

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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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2

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:

6

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

8

( )

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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Bending mode 41Hz

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Torsion mode 47 Hz

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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



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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



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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



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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



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Pole placement Experiment

both poles assigned to increase damping and frequencyspacing

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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



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Pole placement Experiment

frequency separation

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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



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Flutter Margin


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

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