Formulation of a complete structural uncertainty model for robust flutter prediction Brian Danowsky Staff Engineer, Research Systems Technology, Inc., Hawthorne, CA (310) ex. 28 SAE Aerospace Control and Guidance Systems Committee Meeting #99

Copyright © Brian Danowsky, All rights reserved Acknowledgement Iowa State University Dr. Frank R. Chavez NASA Dryden Flight Research Center Marty Brenner NASA GSRP Program

Copyright © Brian Danowsky, 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, 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, All rights reserved Purpose of Research Flutter problem can be very sensitive to structural parameter uncertainty.

Copyright © Brian Danowsky, All rights reserved Wing Structural Model Governing Equation of Unforced Motion for Wing Modal Analysis: mode shapes and frequencies

Copyright © Brian Danowsky, All rights reserved Wing Structural Model

Copyright © Brian Danowsky, 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, All rights reserved Application of the Unsteady Aerodynamics Aerodynamic force: Pressure Coefficient c P = 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, 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, All rights reserved Complete Aeroelastic Wing Model Coordinate Transformation  Aerodynamic force calculations in a different domain than structural Modal Domain Approximation  Significantly reduce the dimension of the mass and stiffness matrices h = H  Matrix of retained mode shapes

Copyright © Brian Danowsky, All rights reserved Forced Aeroelastic Equation of Motion: Flutter prediction can now be done: v-g method Not suitable to be cast as a 1 st order state equation  AIC is not real rational in reduced frequency ( k ) Complete Aeroelastic Wing Model

Copyright © Brian Danowsky, 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, All rights reserved Complete Aeroelastic Wing Model 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, All rights reserved Complete Aeroelastic Wing Model State Space Representation  State Vector  First Order System Only a function of velocity for a fixed constant Mach number

Copyright © Brian Danowsky, All rights reserved Nominal Flutter Point Results V-g Flutter Point (no AIC or density approximation) Flutter Point calculated using stability of A NOM

Copyright © Brian Danowsky, All rights reserved Nominal Flutter Point

Copyright © Brian Danowsky, 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, 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, All rights reserved Robust Stability The Small Gain Theorem

Copyright © Brian Danowsky, 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, 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, All rights reserved Application of the Allowable Variation in the Freestream Velocity Velocity representation Applied to Aeroelastic Equation of motion

Copyright © Brian Danowsky, All rights reserved Application of the Allowable Variation in the Freestream Velocity Formulate M-  model with polynomial dependant uncertainty defined  Standard method to separate polynomial dependant uncertainty (Lind, Boukarim)  Introduce new feedback signals

Copyright © Brian Danowsky, All rights reserved Nominal Flutter Margin Only  V variation is considered

Copyright © Brian Danowsky, 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, 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, All rights reserved Application of Parametric Uncertainty in the Wing Structural Properties Apply J to Aeroelastic Equation of motion: Note: 2nd order  J 2 terms are neglected

Copyright © Brian Danowsky, All rights reserved Application of Parametric Uncertainty in the Wing Structural Properties Formulate M-  model   V =  VI   J =  JI

Copyright © Brian Danowsky, All rights reserved Robust Flutter Margin Determination Uncertainty operator, , a function of 2 parameters (  V,  J ) Calculation of  is necessary

Copyright © Brian Danowsky, All rights reserved Robust Flutter Margin Determination Formulate frequency dependant model 1/s s = js = j

Copyright © Brian Danowsky, All rights reserved Robust Flutter Margin Results

Copyright © Brian Danowsky, All rights reserved Robust Flutter Margin Results 30% uncertainty in  *

Copyright © Brian Danowsky, All rights reserved Robust Flutter Margin Results 30% uncertainty in E *

Copyright © Brian Danowsky, 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, 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, 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, All rights reserved Questions?