# Formulation of a complete structural uncertainty model for by hcj

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```									                           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
n Iowa  State University
Dr. Frank R. Chavez
n NASA Dryden Flight Research Center
Marty Brenner
NASA GSRP Program
Outline
n   Introduction to the Flutter Problem
n   Purpose of Research
n   Wing Structural Model
n   Complete Aeroelastic Wing Model
n   Review of Robust Stability Theory
n   Application of the Allowable Variation in the
Freestream Velocity
n   Application of Parametric Uncertainty in the Wing
Structural Properties
n   Conclusions and Discussion
Introduction to The Flutter Problem
n   Coupling between Aerodynamic Forces and
Structural Dynamic Inertial Forces
n   Can lead to instability and possible structural
failure.
n   Flight testing is still an integral part in estimating
the onset of flutter.
n   Current flutter prediction methods only account
for variation in flutter frequency alone, and do
not account for variation in structural mode
shape.
VIDEO
Purpose of Research
n   Flutter problem can be very sensitive to structural
parameter uncertainty.
Wing Structural Model
n   Governing Equation of Unforced Motion
for Wing

n   Modal Analysis: mode shapes and
frequencies
Wing Structural Model
Aerodynamics
n   Aerodynamic Forces

Vector of panel forces

*Aerodynamic forces calculated in
different coordinates than structure               Vector of non-dimensional
pressure coefficients
Aerodynamics
n   Aerodynamic force: Pressure Coefficient

cP = vector of panel pressure
coefficients

w = vector of panel local
downwash velocities

AIC(k,Mach) = Aerodynamic
Determined from the unsteady   Influence Coefficient matrix
doublet lattice method         (complex)
Complete Aeroelastic Wing Model
n   Since the structural model and the aerodynamic
model have been established the complete
model can be constructed
n   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
Complete Aeroelastic Wing Model

n   Coordinate Transformation
¨ Aerodynamic  force calculations in a different
domain than structural
n   Modal Domain Approximation
reduce the dimension of the
¨ Significantly
mass and stiffness matrices
h = Hh
Matrix of retained mode shapes
Complete Aeroelastic Wing Model

n   Forced Aeroelastic Equation of Motion:

n   Flutter prediction can now be done: v-g method
n   Not suitable to be cast as a 1st order state
equation
¨ AIC   is not real rational in reduced frequency (k)
Complete Aeroelastic Wing Model

Approximation (RFA)        If s = jw, then p = jk

With constant Mach number,
approximate as:
Complete Aeroelastic Wing Model

n   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
Complete Aeroelastic Wing Model

n   State Space Representation
¨ State   Vector

Only a function of
velocity for a fixed
¨ First   Order System       constant Mach
number
Nominal Flutter Point Results
V-g Flutter Point           Flutter Point calculated using
(no AIC or density approximation)          stability of ANOM
Nominal Flutter Point
Model with Uncertainty
n   The flutter problem can be sensitive to
uncertainties in structural properties
n   A model accounting for uncertainty in structural
properties is desired
n   An allowable variation to velocity must be
accounted for to determine robust flutter
boundaries due to uncertainty in structural
properties
n   Robust flutter margins are found using Robust
Stability Theory (m analysis)
Robust Stability
n   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
Robust Stability
n   The Small Gain Theorem
Robust Stability
n   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
Application of the Allowable Variation in
the Freestream Velocity
n   Allowable variation to velocity must be accounted for to
determine robust flutter boundaries due to uncertainty in
structural properties.
n   System can be formulated with a stable nominal
operator, M, and a variation operator, D.
n   M - constant nominal operator representing the wing
dynamics at a stable velocity
n   D – variation operator representing the allowable
variation to the nominal velocity
n   Nominal flutter point can be determined using this M-D
framework which will match that found previously.
Application of the Allowable Variation in
the Freestream Velocity
n   Velocity representation

n   Applied to Aeroelastic Equation of motion
Application of the Allowable Variation in
the Freestream Velocity
n   Formulate M-D model with polynomial
dependant uncertainty defined
¨ Standard  method to separate polynomial
dependant uncertainty (Lind, Boukarim)
¨ Introduce new feedback signals
Nominal Flutter Margin
n   Only dV variation is considered
Application of Parametric Uncertainty in
the Wing Structural Properties
n Must expand M-D model to account for
uncertainty in structural parameters
n Account for uncertainty in structural mode
shape and frequency
n Uncertain elements are plate structural
properties:
Application of Parametric Uncertainty in
the Wing Structural Properties
n   Define uncertainty in any modulus (elasticity or
density)

n   Structural mode shapes and frequencies are
dependant on this:

derivatives calculated analytically (Friswell)
Application of Parametric Uncertainty in
the Wing Structural Properties
n   Apply J to Aeroelastic Equation of motion:

Note: 2nd order dJ2
terms are neglected
Application of Parametric Uncertainty in
the Wing Structural Properties
n   Formulate M-D model         DdV = dVI
DdJ = dJI
Robust Flutter Margin
Determination
n Uncertainty operator, D, a function of 2
parameters (dV, dJ)
n Calculation of m is necessary
Robust Flutter Margin
Determination
n   Formulate frequency dependant model

1/s

s = jw
Robust Flutter Margin Results
Robust Flutter Margin Results

30% uncertainty in
G*
Robust Flutter Margin Results

30% uncertainty in
E*
Conclusions and Discussion
n   Complete Model
¨   Direct mode shape and frequency dependence on structural
parameters
¨   Analytical derivatives avoiding computational inaccuracies
n   State Space Model
¨   Aerodynamic RFA
¨   Flutter point instability matches V-g method
¨   Well-Suited for Subsequent Control Law Design if Desired
n   Method can be easily applied to a much more complex
problem (i.e. entire aircraft)
Major Contributions of this Work
n   Inclusion of Mode Shape Uncertainty
¨   Traditionally only frequency uncertainty is considered
n   Dependence of Mode Shape and Frequency
¨   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
n   Complete M-D model with Uncertainty
¨   Well suited for subsequent control law design taking structural
parameter uncertainty into account (Robust Control)
Areas of Future Investigation
n   Abnormal flutter point
reached with a decrease in velocity
¨ Instability
¨ Abnormality due to Mach number dependence
¨ Wing created that would flutter at reasonable altitude

n   Limited range of valid velocities
¨ Due to Mach number dependence and standard
atmosphere
Questions?

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