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Franco Mastroddi Daniele Dessi Luigi Morino
Davide Cantiani
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for Limit-Cycle
Oscillations
Bifurcation and Model Reduction Techniques for Large Multi-
Disciplinary Systems
University of Liverpool 26-27 June 2008
Istituto Nazionale per Studi ed
Esperienze di Architettura NavaleDipartimento di Ingegneria
Meccanica ed Industriale
Università “ROMA TRE”
Summary of presentation
Page 2Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
1. Normal Form (NF) approach for nonlinear (NL) (aeroelastic) systems experiencing bifurcations: highlights of theory for (aeroelastic) model reduction
2. NL analysis based on 3rd order NF theory
• Appl.#1: LCO’s control
• Appl.#2: Chaos control
3. NL analysis based on 5th order NF theory
• Appl.#3: Gust response of a nonlinear aeroelastic system
• Appl.#4: Freeplay modeling: direct num. integration v.s. NF approach
4. Concluding remarks
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 3
Reduced
Complexity
Our goal
Aeroelastic System
Topics at a glance
Fluid / Structure
+
Non - Linear
Complexity
Physical
Reduction
Modal Description
/ Finite State
Aerodynamics
1st Level
Mathematical
Reduction
Simpliflied
Non - Linear
Dynamics
2nd Level
Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 4
Considered aeroelastic applications and relevance of NF
Modal Model /
/ Finite State
Aerodynamics
Nonlinearity
Sources
Normal Form
Reduction
PREDICT
Prevision of safe conditions
• flutter speed limits
• admissible gust CONTROL
Phenomenon intensity
• LCO amplitude
• Chaos behaviour
Wing Panels in
Supers. Flow
Wing Typical Section
(plunge, pitch, flap) in
Subs. Unsteady Flow
Piston – theory for
aerodynamics
1. Nonlinear pitch
stiffness
2. Free-play in control
surface
3-order Analysis
Pitchfork Bifurcation
5-order Analysis
Knee Bifurcation
Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 5
Authors‟ papers related to the presentation
Some theoretical hightlights on
NORMAL FORM (NF) approach for bifurcation
-A tool for analysis of nonlinear (aeroelastic) systems
with polynomial nonlinearities
IFASD 2007
Stockholm 18-20 June 2007 A Reduced-Order Modeling for the
Aeroelastic Stability of a Launch VehiclePage 6
Ali H. Nayfeh, Method of Normal Forms, Wiley Series in
Nonlinear Science
Page 7Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Normal Form (I)
• Nonlinear Aeroelastic Problem reduced to :
• Taking a Taylor expansion of around an equilibrium point the
dynamical system can be rewritten as:
nonlinear-term vector of type (generic NL’s can be locally
reduced in this form) :
• Assuming analytically dependent on , one obtains
Similarily:
Control parameter (e.g., flight speed)
Page 8Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Normal Form (II)
• Setting the transformation (diagonalize the linear-ized part), one has ( diagonal matrix of eigenvalues ):
where
• The ordering parameter is introduced such that .
Motivation: scaling the contributions of any terms in each equations
on the base of the amplitude of the original state space variable .
z = ²u
• Balancing the nonlinear terms with the perturbation on the linear
terms (condition for a local bifurcation) implies
Page 9Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Normal Form (III)
• Recasting equations in the new state space variable
or
• The normal form method consists of simplifying the differential
problem through the “near identity” coordinate transformation
where have to be chosen so as to simplify the problem.
Suitable small positive parameter
defining the resonance conditions
Page 10Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
• Collecting terms of same order
where
Normal Form (IV)
• Substituting ( ):
• Choosing for the same functional dependence as
• The following expressions are obtained for coefficients of
(near)-Resonance
conditions
NEW nl term
OLD nl termDIFFERENCE
Page 11Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
(2) ( )y Λy g y
(2) ( )z y w y
1
ˆ :Nj j i
Given system
Pseudo-diagonal system
Near-Resonance condition Normal form
Nonlinear transformation
ˆ
ˆ
In the complex plane, the -points can be plotted for every nonlinear term (green for 3rd order terms,
blue for linear terms); by enlarging the circle of radius more nonlinear terms are taken into account
until a satysfying solution is reached.
The radius is obtained by a trial and error procedure.
ˆ
Normal Form (V) RELEVANT ISSUE #1:
opportunity of defining the near-identity transformation
Page 12Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
• Looking for a solution of the type:
Normal Form (V) RELEVANT ISSUE #2:
LCO solution analytically obtained (resonance
conditions)
• Third order system
• Consider the pair of complex conjugate equations given by the “manifold”
• The system becomes:
Around a complex
conjugated stable
Complex eigenvalueANALYTIC SOLUTION FOR LCO
Directly related to the linear part
Directly related to the lnoninear part
Page 13Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Normal Form (VI) (the other equations are “slaves”
• Consider the other eigenvalues
Stable Complex
eigenvalue
Stable Real
eigenvalue
• Final analytical Solutions
Page 14Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Normal Form (V) RELEVANT ISSUE #3:
othe stability of LCO can be discussed (Hopf theorem)
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 15
Fluid (linear)
Structure
(linear)
Elastic
Displacement
Unsteady
Load
Aeroelastic applications background: linear case
Initial
Values
Gust
Load
Flow Speed
Excitation mechanism
System parameter
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Page 16
• Hopf (pitchfork) bifurcation:
subrcritical-detrimental flutter
• Nonlinear structural stiffness arising from large displacement gradients
• Freeplay in the control surfaces
Fluid (linear)
Structure
(NONlinear)
Elastic
Displacement
Unsteady
Load
• Hopf (pitchfork) bifurcation:
supercritical-benign flutter
Flow Speed > Critical speed
Stable
Limit Cycles
Flow Speed < Critical speed
UNStable
Limit Cycles
Background: nonlinear case (1)
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 17
Unsteady potential
incompressible
unbounded 2D flow
Nonlinear structure with
torsional and bending
modes of vibration.
Uniform and constant flow horizontal velocity
+ vertical deterministic gust
Vibrating structure
with nonlinear (soft) torsional spring
hypothesis
Heave & PitchAerodynamic augmented states
DOFs
Physical model
Page 18Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Methematical model: A simple (very used ) aeroelastic model:
Typical Section With Control Surface
• A three degree of
freedom aeroelastic
typical section with a
trailing edge control
surface
NONLINEAR TERMS
APPLICATION #1 based on NF theory: control the
LCO amplitude and “taming” of explosive flutter
Page 19Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
• Nonlinear aeroelastic system with a nonlinear control
• Then the previous Hopf analysis can be repeated with a new
“closed-loop” nonlinear coefficient:
with
Nonlinear feedback
• Condition for stable limit cycle in closed loop conditions
• NB: it is always possible to choose such as to satisfy previous Eq.
it is possibile to control the LCO amplitude by a nonlinear feedback
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 20
APPLICATION #1: LCO nonlinear control
Different LCO amplitude with different values
for non-linear gain
Ref.: Morino, L., Mastroddi, F., “Limit-cycle oscillation control with
aplication to flutter,” The Aeronautical Journal, Nov. 1996.
Linear system unstable response
envelope
Unstable linear system
nonlinearly controlled: LCO envelope
Unsteable linear system with nonlinear
control Different LCO amplitude with
different values for non-linear gain
Page 21Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
• Periodic orbits– Harmonic Limit Cycle Oscillations
• Quasi Periodic orbits– Non-periodic Oscillations
• Strange Attractors– Chaos
APPLICATION #2 based on NF theory: periodic, quasi periodic, or chaotic solution by transforming
the original problem with different
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 22
APPLICATION #2 based on NF theory: periodic, quasi periodic, or chaotic solution by transforming
the original problem with different
“Classic” nonlinear Panel flutterSimply supported panel in a supersonic flow with dynamic pressure with
a buckling load R x and structural stabilizing nonlinearities
PDE system solved “a la Galerkin” assuming
for the solution
Page 23Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
APPLICATION #2 based on NF theory: periodic, quasi periodic, or chaotic solution by transformin
the original problem with different
Nonlinear Panel flutter
• The greater is the assumed radius , the less simply harmonic the solution is
• the NF is able to identify the NL terms which are responsible of chaotic solution Ref.: Morino, M., Mastroddi, F., Cutroni, M.,``Lie Transformation Method for Dynamical System
having Chaotic Behavior,“ Nonlinear Dinamics, Vol. 7, No.4, June 1995, pp. 403-428.
=0 =100 =500
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 24
Some theoretical hightlights on
(fifth order) NORMAL FORM (NF) approach
-A tool for nonlinear analysis of aeroelastic
system with polynomial nonlinearities
- …. For more complicated bifurcation …. Pre-
critical instabilities ….
Ali H. Nayfeh, Method of Normal Forms, Wiley Series in
Nonlinear Science
Ref: Dessi, D., Morino, L., Mastroddi, F., ``A Fifth-Order Multiple-Scale
Solution for Hopf Bifurcations," Computers and Structures, Vol. 82, 2004,
pp. 2723-2731.
Page 25Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Fifth-order NF Normal Form (very brief details)
• Considering the fourth order terms the following equations need to be simplified by the NF procedure:
• Again the NF-method consists of searching for a new state space coordinates through the “near-identity” transformation
• The transformed dynamical system will be in the form
• By using the coordinate transformation
Page 26Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Loss of Stability for Nonlinear Aeroelastic Systems
• Eigenvalue crossing of the imaginary axis (linear analysis)
• Pitchfork/Hopf bifurcation: supercritical (benign flutter) and subcritical (explosive flutter)
• Beyond the Hopf bifurcation:
“knee” bifurcation
(precritical LCO’s)
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 27
Fluid (linear)
Structure
(NONlinear)
Elastic
Displacement
Unsteady
Load
• Wind tunnel model with free-play
• Wind tunnel model with transonic effects• Hopf (knee) bifurcation:
“subrcritical to supercritical”
Turning point < Flow Speed < Flutter speed
UNStable & Stable
Limit Cycles
Flow Speed > Flutter speed
Stable
Limit Cycles
Aeroelastic
interpretation
Page 28Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Physical Sources of Structural Nonlinearities in the Fixed
Wing Model
• Approximation with Polinomial Nonlinearities
• 8 states variables (aerodynamic ones included)
• Nonlinear structural stiffness arising from large
displacement gradients (Lee, B.H.K., et al.,
1989)
• Freeplay in the control surfaces - bilinear stiffness
due to loosely connected structural components
(Conner et al., 1997)
Page 29Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Nonlinear Analysis: 3 order v.s. 5 order NF
using =03
3)( cM
The NF-method has to be extended to fifth-
order to capture at least the qualitative
behavior of the aeroelastic LCO in the case of
„knee‟ bifurcations.
No 3 order NF analysis is not able to describe
the special kind of bifurcation
Page 30Bifurc. and Model Reduction Tech.
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Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
3
3)( cM
• To improve the accuracy, more equations and terms have to be added.
Including the near-resonant terms, an extended (in the sense of Center-Manifold)
NF-method is implemented
Nonlinear Analysis: 5° order NL using >0
Page 31Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
A detail on this analysis:
3
3( )M c
• Third order and fifth order normal form (CM) provide a satisfactory approximation
0 5 10 15 20 25 30 35 40 45 50-0.015
-0.01
-0.005
0
0.005
0.01
0.015
t(s)
Tim
e H
isto
ry (
)
LCO, IC<aLC
3rd order approximation
5th order approximation
exact
0 5 10 15 20 25 30 35 40 45 50-0.03
-0.02
-0.01
0
0.01
0.02
0.03
t(s)
Tim
e H
isto
ry
LCO, IC>aLC
3rd order approximation
5th order approximation
exact
0 1 2 3 4 5 6 7
x 10-3
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
LC
O a
mplit
ude
bifurcation curve
exact
Normal 5th order
Normal 3rd order
exact
Normal 5th order
Normal 3rd order
exact
Normal 5th order
Normal 3rd order
20 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 21-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
t(s)
LCO, IC>aLC
3rd order approximation
5th order approximation
numerical
Page 32Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
As sub-product of this analysis:identified a parameter moving the system from
Hopf to knee-Bifurcation
Plunge Bifurcation Diagrams
for different elastic axis
positions
3
3)( cM
ha
Fisically: the elastic axis position
evealed to havea clear influence on the type
of bifurcation, subcritical knee-like bif. or supercritical pitchfork bif.
Page 33Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Dessi, D., Mastroddi, F. , “A nonlinear analysis of stability and gust response of
aeroelastic systems, Journal of Fluids and Structures, 24 (3), p.436-445, Apr 2008
NF METHODOLOGY/APPROACH
- The nonlinear analysis for gust-excited problem
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 34
CASE A:
• Initial conditions
• no gust excitation0
U;U
xx
gxfxAx
(0)
)()()(
THIS IS THE CASE PRESENTED BEFORE!
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 35
CASE B: APPLICATION #3
• No initial conditions
• “discrete”
(=time impulsive) gust excitation
0(0)
)()()(
x
gxfxAx U;U
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 36
In the case of the gust problem, the system dynamics is excited only by the “discrete” gust
profile with zero initial conditions
00
U;U
)(
)()()(
x
gxfxAx
0 200 400 600 800 1000 1200
-0.01
0
0.01
0.02
0.03
0.04
0.05
v
w0
CASE B: Gust excitation with no IC
It is assumed a discrete gust of the wave form
where:
• Gust intensity
• Gust gradient
G
0G Cos1
2
ww )(
0 100 200 300
-0.01
-0.005
0
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 37
Speed4.925 4.93 4.935
Gu
st
inte
nsity
0
0.02
0.04
-0.010
140 150 160 170 180 190
-0.0002
0
0.0002
0.0004
0.0006
• Fix the gust gradient G to a value
• Define a region in the plane of the
parameters and consider a matrix of
numerical simulations for each value of
the grid nodes
• Introduce the logarthmic damping
coefficient between consecutive peaks in
the amplitude modulated periodic solution
n1n
n
(max)
i1n
(max)
i1n
)(x)(xlog
Gust analysis
2D
gu
st-
pa
ram
ete
r s
pa
ce
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 38
• For each run, consider the logarithmic damping for , and obtain the
response surface
• The level curve can be determined by interpolation between the grid nodes
• It represents the set of parameters (gust intensity for a given flight speed) that
leads the state-vector to be very close to a stable or unstable periodic solution
(LCO)
Speed4.925
4.93
4.935
Gust intensity
0
0.01
0.02
0.03
0.04
-0.01
0
XY
Z
Undamped oscillations
Damped oscillations
Line of critical gust intensity
N
0N
G0NN ;w,UNn
Gust analysis
The level curve
does not depend
on the final time step
0N
Primary critical speed
Secondary critical speed
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 39
• The branch of the curve for which across it,
represents the critical gust intensity characterized by the property that
Speed
Gu
st
inte
nsity
4.926 4.928 4.93 4.932 4.934 4.936 4.9380
0.01
0.02
0.03
0.04
0.05
Lower basin of attraction of stable LCO
Upper basin of attraction of stable LCO
Basin of attraction of fixed point solution
• Thus, a basin of attraction for the solution has been identified in the space of
physical parameters.
•This analysis has to be repeated for several gust gradients G
G0NN ;w,U 0w/ 0N
)U(ww )c(
0
)c(
0
dampedissolutiontheww
undampedissolutiontheww)c(
00
)c(
00
Basin of attraction: critical gust intensity
… and what happens if the nonlinearities
are not polynomials?....
For example: How is important to correctly model
the discontinuity of a freeplay? (i.e., avoiding
polynomials)
Page 40Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
Page 41Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
APPLICATION #5: physically more realistic modeling
for Freeplay/Hysteresis Nonlinearity• DOMAIN SUB-DIVISION: Zone 1 &
Zone 3
• Zone 2 & Zone 4
-4 -3 -2 -1 0 1 2 3 4-15
-10
-5
0
5
10
15
zone 1 zone 3Preload
0
,
c
,
zone 2
displacement/rotation
Forc
e/M
om
ent
freeplay
-4 -3 -2 -1 0 1 2 3 4-15
-10
-5
0
5
10
15
displacement/rotation
Forc
e/M
om
ent
hysteresis
zone 1 zone 3Preload
0
,
c
,
zone 2
zone 4
Ampl.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 42
Freeplay Results: direct numerical integration
5 10 15 20 25 30-0.03
-0.02
-0.01
0
0.01
0.02
0.03Bifurcation graphic
flow speed (m/s)
LC
O a
mplit
ude
5 10 15 20 25 30-10
0
10
20
30
40
50
60
70
80
speed flow (m/s)
frequency (
Hz)
characteristic frequency
First frequency max PSD
Second Frequency max PSD
• Amplitude of LCOs increases
with flow speed and activated
between the two linear stability
limit
• Second Frequency
characteristic 3
times the first
(linear) Flutter
speed flap
disconnected
(linear) Flutter
speed flap
connected
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 43
How to apply Normal Form in this case? (1)
-5 -4 -3 -2 -1 0 1 2 3 4 5
x 10-3
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
Rotation/Displacement
Mom
ent/F
orce
Approximation
freeplay
cubic
0 100 200 300 400 500 600 700 800 900-0.01
-0.005
0
0.005
0.01cubic approximation Vs freeplay
t(s) cubic approximation
freeplay
700 700.1 700.2 700.3 700.4 700.5 700.6 700.7 700.8 700.9 701-0.01
-0.005
0
0.005
0.01
t(s)
ZOOM
• Freeplay must be approximated by
a polinomial nonlinearity in order to
perform the NF analysis
• The cubic approximations give
satisfactory results for the amplitude
and the frequency of LCOs. However
the transitory is quite different
• All the polynomial modeling for the
freeplay discontinuity works locally
well for LCO (around bifurcation point)
5 10 15 20 25 30-0.03
-0.02
-0.01
0
0.01
0.02
0.03
flow speed (m/s)
LCO
Am
plitu
de
Bifurcation graphic
cubic
freeplay
• …but …
Page 44Bifurc. and Model Reduction Tech.
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A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOs
43.2 43.25 43.3 43.35 43.4 43.45 43.5 43.55 43.6
-5
-4
-3
-2
-1
0
1
2
3
4
5x 10
-3
t(s)
Approximation via Normal Form
freeplay
cubic approximation
Normal Form 5th order
• all the equations must be used ( <4) to obtain a
correct approximation for LCO
How to apply Normal Form in this case? (2)
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Nonlinear Aeroelasticity for LCOsPage 45
CONCLUDING REMARKS 1/2 (from ourown experience
in nonlinear aeroelasticity)• Once a system can be mathematically modeled by
nonlinear first order differential equation with polynmial
nonlinearities (many aeroelastic systems can be modelled
so), NF approach may represent a local powerfull tool
o To obtain the analytic solution around a Hopf bif.
o To reduce the system size to a restricted set equations
(Center Manifold theorem)
o To study the LCO stability in precritical condition (with a
higher order analysis (basin of attraction defined with I.C.
or with suitable input)
o To find a nonlinear feedback to “tame” linear and
nonlinear oscillations
o To identify the nonlinear contributions responsible of
chaotic behavior
Bifurc. and Model Reduction Tech.
for Large Multi-Disc. Systems
Liverpool 26-27 June 2008
A Singular Perturbation Approach in
Nonlinear Aeroelasticity for LCOsPage 46
• If system nonlinearities are not in a polynomial form (it is
the case of the freeplay modeling)
o A polynomial approximation of the nonlinearity could
be used and the NF approach can efficiently capture the
nonlinear behavior of the system if near-resonace terms
are included
o An extension of the NF theory should be developed
(something is existing like Lie Transformation) in this
case
CONCLUDING REMARKS 2/2
Comment
The freeplays nonlinearities can be trivially identified but not so easily
analyzable by NF approach
Polynomial nonlinearity are (not so-trivially) analyzable by NF
approach but are not identifieable by actual measurements at all
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