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Formulation of a complete structural uncertainty model for robust flutter predictionBrian DanowskyStaff Engineer, ResearchSystems Technology, Inc., Hawthorne, CAbdanowsky@systemstech.com(310) 679-2281 ex. 28
SAE Aerospace Control and Guidance Systems Committee Meeting #99
Copyright © Brian Danowsky, 2004. All rights reserved
Acknowledgement
Iowa State University
Dr. Frank R. Chavez NASA Dryden Flight Research Center
Marty Brenner
NASA GSRP Program
Copyright © Brian Danowsky, 2004. All rights reserved
Outline 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
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
Flutter Points: Mach Number vs. Altitude
18000
19000
20000
21000
22000
23000
24000
25000
26000
27000
28000
29000
30000
31000
32000
33000
34000
35000
36000
37000
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Mach Number
Alt
itud
e, ft
.
Purpose of Research
Flutter problem can be very sensitive to structural parameter uncertainty.
Copyright © Brian Danowsky, 2004. All rights reserved
Wing Structural Model
Governing Equation of Unforced Motion for Wing
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 Aerodynamic Forces
Vector of panel forces
Vector of non-dimensional pressure coefficients
*Aerodynamic forces calculated in different coordinates than structure
Copyright © Brian Danowsky, 2004. All rights reserved
Application of the Unsteady Aerodynamics Aerodynamic force: Pressure Coefficient
cP = vector of panel pressure coefficients
w = vector of panel local downwash velocities
AIC(k,Mach) = Aerodynamic Influence Coefficient matrix (complex)
Determined from the unsteady doublet lattice method
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
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 ( 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
Coordinate TransformationAerodynamic force calculations in a different
domain than structural Modal Domain Approximation
Significantly reduce the dimension of the mass and stiffness matrices
h = HMatrix of retained mode shapes
Copyright © Brian Danowsky, 2004. All rights reserved
Forced Aeroelastic Equation of Motion:
Flutter prediction can now be done: v-g method Not suitable to be cast as a 1st order state
equation AIC is not real rational in reduced frequency (k)
Complete Aeroelastic Wing Model
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
Unsteady Aerodynamic Rational Function Approximation (RFA)
With constant Mach number,approximate as:
If s = j, then p = jk
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
Atmospheric Density Approximation
Direct relationship between atmospheric density and freestream velocity
Coefficients are a function of Mach numberBased on the 1976 standard atmosphere
model
Copyright © Brian Danowsky, 2004. All rights reserved
Complete Aeroelastic Wing Model
State Space RepresentationState Vector
First Order SystemOnly 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
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 ( analysis)
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Stability
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
The Small Gain Theorem
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Stability
: 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 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, .
M - constant nominal operator representing the wing dynamics at a stable velocity
– variation operator representing the allowable variation to the nominal velocity
Nominal flutter point can be determined using this M- framework which will match that found previously.
Copyright © Brian Danowsky, 2004. All rights reserved
Application of the Allowable Variation in the Freestream Velocity
Velocity representation
Applied to Aeroelastic Equation of motion
Copyright © Brian Danowsky, 2004. All rights reserved
Application of the Allowable Variation in the Freestream Velocity
Formulate M- model with polynomial dependant uncertainty definedStandard method to separate polynomial
dependant uncertainty (Lind, Boukarim) Introduce new feedback signals
Copyright © Brian Danowsky, 2004. All rights reserved
Nominal Flutter Margin
Only V variation is considered
Copyright © Brian Danowsky, 2004. All rights reserved
Application of Parametric Uncertainty in the Wing Structural Properties
Must expand M- model to account for uncertainty in structural parameters
Account for uncertainty in structural mode shape and frequency
Uncertain elements are plate structural properties:
Copyright © Brian Danowsky, 2004. All rights reserved
Application of Parametric Uncertainty in the Wing Structural Properties
Define uncertainty in any modulus (elasticity or density)
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
Apply J to Aeroelastic Equation of motion:
Note: 2nd order J2 terms are neglected
Copyright © Brian Danowsky, 2004. All rights reserved
Application of Parametric Uncertainty in the Wing Structural Properties
Formulate M- model V = VI
J = JI
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Flutter Margin Determination Uncertainty operator, , a function of 2
parameters (V, J) Calculation of is necessary
Copyright © Brian Danowsky, 2004. All rights reserved
Robust Flutter Margin Determination Formulate frequency dependant model
1/s
s = j
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
*
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
Complete Model Direct mode shape and frequency dependence on structural
parameters Analytical derivatives avoiding computational inaccuracies
State Space Model 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
Inclusion of Mode Shape Uncertainty Traditionally only frequency uncertainty is considered
Dependence of Mode Shape and Frequency The uncertainty in both the structural mode shape and mode
frequency are dependant on a real parameter (E*,*) The individual mode shapes and frequencies are not
independent of one another
Complete M- model with Uncertainty 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
Abnormal flutter point Instability reached with a decrease in velocity Abnormality due to Mach number dependence Wing created that would flutter at reasonable altitude
Limited range of valid velocities Due to Mach number dependence and standard
atmosphere
Copyright © Brian Danowsky, 2004. All rights reserved
Questions?
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