Formulation of a
complete structural
uncertainty model for
robust flutter prediction
Brian Danowsky
Staff Engineer, Research
Systems Technology, Inc., Hawthorne, CA
bdanowsky@systemstech.com
(310) 679-2281 ex. 28
SAE Aerospace Control and Guidance Systems Committee Meeting #99
Acknowledgement
Iowa State University
Dr. Frank R. Chavez
 NASA Dryden Flight Research Center
Marty Brenner
NASA GSRP Program
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Copyright © Brian Danowsky, 2004. All rights reserved
Outline
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Introduction to the Flutter Problem
Purpose of Research
Wing Structural Model
Application of Unsteady Aerodynamics
Complete Aeroelastic Wing Model
Review of Robust Stability Theory
Application of the Allowable Variation in the
Freestream Velocity
Application of Parametric Uncertainty in the Wing
Structural Properties
Conclusions and Discussion
Copyright © Brian Danowsky, 2004. All rights reserved
Introduction to The Flutter Problem
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Coupling between Aerodynamic Forces and
Structural Dynamic Inertial Forces
Can lead to instability and possible structural
failure.
Flight testing is still an integral part in estimating
the onset of flutter.
Current flutter prediction methods only account
for variation in flutter frequency alone, and do
not account for variation in structural mode
shape.
VIDEO
Copyright © Brian Danowsky, 2004. All rights reserved
Purpose of Research
Flutter problem can be very sensitive to structural
parameter uncertainty.
Flutter Points: Mach Number vs. Altitude
37000
36000
35000
34000
33000
32000
31000
30000
Altitude, ft.
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29000
28000
27000
26000
25000
24000
23000
22000
21000
20000
19000
18000
0.3
0.4
0.5
0.6
0.7
0.8
Mach Number
Copyright © Brian Danowsky, 2004. All rights reserved
0.9
Wing Structural Model
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Governing Equation of Unforced Motion
for Wing
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Modal Analysis: mode shapes and
frequencies
Copyright © Brian Danowsky, 2004. All rights reserved
Wing Structural Model
Copyright © Brian Danowsky, 2004. All rights reserved
Application of the Unsteady
Aerodynamics
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Aerodynamic Forces
Vector of panel forces
*Aerodynamic forces calculated in
different coordinates than structure
Vector of non-dimensional
pressure coefficients
Copyright © Brian Danowsky, 2004. All rights reserved
Application of the Unsteady
Aerodynamics
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Aerodynamic force: Pressure Coefficient
cP = vector of panel pressure
coefficients
w = vector of panel local
downwash velocities
Determined from the unsteady
doublet lattice method
AIC(k,Mach) = Aerodynamic
Influence Coefficient matrix
(complex)
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
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Since the structural model and the aerodynamic
model have been established the complete
model can be constructed
Representation of the Aeroelastic Wing
Dynamics as a First Order State Equation
 Needed
to Apply Robust Stability (m analysis)
 The dynamic state matrix will be a function of one
variable (U)
 Tailored for subsequent control law design, if desired
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
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Coordinate Transformation
 Aerodynamic
force calculations in a different
domain than structural
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Modal Domain Approximation
 Significantly
reduce the dimension of the
mass and stiffness matrices
h = Hh
Matrix of retained mode shapes
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
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Forced Aeroelastic Equation of Motion:
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Flutter prediction can now be done: v-g method
Not suitable to be cast as a 1st order state
equation
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 AIC
is not real rational in reduced frequency (k)
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
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Unsteady Aerodynamic Rational Function
Approximation (RFA)
If s = jw, then p = jk
With constant Mach number,
approximate as:
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
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Atmospheric Density Approximation
 Direct
relationship between atmospheric
density and freestream velocity
 Coefficients are a function of Mach number
 Based on the 1976 standard atmosphere
model
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
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State Space Representation
 State
 First
Vector
Order System
Only a function of
velocity for a fixed
constant Mach
number
Copyright © Brian Danowsky, 2004. All rights reserved
Nominal Flutter Point Results
V-g Flutter Point
(no AIC or density approximation)
Flutter Point calculated using
stability of ANOM
Copyright © Brian Danowsky, 2004. All rights reserved
Nominal Flutter Point
Copyright © Brian Danowsky, 2004. All rights reserved
Model with Uncertainty
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The flutter problem can be sensitive to
uncertainties in structural properties
A model accounting for uncertainty in structural
properties is desired
An allowable variation to velocity must be
accounted for to determine robust flutter
boundaries due to uncertainty in structural
properties
Robust flutter margins are found using Robust
Stability Theory (m analysis)
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Stability
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The Small Gain Theorem- a closed-loop
feedback system of stable operators is
internally stable if the loop gain of those
operators is stable and bounded by unity
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Stability
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The Small Gain Theorem
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Stability
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m: The Structured Singular Value
- With a known uncertainty structure a less
conservative measure of robust stability can
be implemented
stable if and only if
Copyright © Brian Danowsky, 2004. All rights reserved
Application of the Allowable Variation in
the Freestream Velocity
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Allowable variation to velocity must be accounted for to
determine robust flutter boundaries due to uncertainty in
structural properties.
System can be formulated with a stable nominal
operator, M, and a variation operator, D.
M - constant nominal operator representing the wing
dynamics at a stable velocity
D – variation operator representing the allowable
variation to the nominal velocity
Nominal flutter point can be determined using this M-D
framework which will match that found previously.
Copyright © Brian Danowsky, 2004. All rights reserved
Application of the Allowable Variation in
the Freestream Velocity
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Velocity representation
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Applied to Aeroelastic Equation of motion
Copyright © Brian Danowsky, 2004. All rights reserved
Application of the Allowable Variation in
the Freestream Velocity
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Formulate M-D model with polynomial
dependant uncertainty defined
 Standard
method to separate polynomial
dependant uncertainty (Lind, Boukarim)
 Introduce new feedback signals
Copyright © Brian Danowsky, 2004. All rights reserved
Nominal Flutter Margin
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Only dV variation is considered
Copyright © Brian Danowsky, 2004. All rights reserved
Application of Parametric Uncertainty in
the Wing Structural Properties
Must expand M-D model to account for
uncertainty in structural parameters
 Account for uncertainty in structural mode
shape and frequency
 Uncertain elements are plate structural
properties:
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Copyright © Brian Danowsky, 2004. All rights reserved
Application of Parametric Uncertainty in
the Wing Structural Properties
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Define uncertainty in any modulus (elasticity or
density)
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Structural mode shapes and frequencies are
dependant on this:
derivatives calculated analytically (Friswell)
Copyright © Brian Danowsky, 2004. All rights reserved
Application of Parametric Uncertainty in
the Wing Structural Properties
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Apply J to Aeroelastic Equation of motion:
Note: 2nd order dJ2
terms are neglected
Copyright © Brian Danowsky, 2004. All rights reserved
Application of Parametric Uncertainty in
the Wing Structural Properties
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Formulate M-D model
DdV = dVI
DdJ = dJI
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Flutter Margin
Determination
Uncertainty operator, D, a function of 2
parameters (dV, dJ)
 Calculation of m is necessary
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Copyright © Brian Danowsky, 2004. All rights reserved
Robust Flutter Margin
Determination
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Formulate frequency dependant model
1/s
s = jw
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Flutter Margin Results
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Flutter Margin Results
30% uncertainty in
G*
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Flutter Margin Results
30% uncertainty in
E*
Copyright © Brian Danowsky, 2004. All rights reserved
Conclusions and Discussion
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Complete Model
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State Space Model
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Direct mode shape and frequency dependence on structural
parameters
Analytical derivatives avoiding computational inaccuracies
Aerodynamic RFA
Flutter point instability matches V-g method
Well-Suited for Subsequent Control Law Design if Desired
Method can be easily applied to a much more complex
problem (i.e. entire aircraft)
Copyright © Brian Danowsky, 2004. All rights reserved
Major Contributions of this Work
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Inclusion of Mode Shape Uncertainty
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Dependence of Mode Shape and Frequency
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Traditionally only frequency uncertainty is considered
The uncertainty in both the structural mode shape and mode
frequency are dependant on a real parameter (E*,G*)
The individual mode shapes and frequencies are not
independent of one another
Complete M-D model with Uncertainty
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Well suited for subsequent control law design taking structural
parameter uncertainty into account (Robust Control)
Copyright © Brian Danowsky, 2004. All rights reserved
Areas of Future Investigation
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Abnormal flutter point
 Instability
reached with a decrease in velocity
 Abnormality due to Mach number dependence
 Wing created that would flutter at reasonable altitude
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Limited range of valid velocities
 Due
to Mach number dependence and standard
atmosphere
Copyright © Brian Danowsky, 2004. All rights reserved
Questions?
Copyright © Brian Danowsky, 2004. All rights reserved