Formulation of a complete structural uncertainty model for robust flutter prediction Brian Danowsky Staff Engineer, Research Systems Technology, Inc., Hawthorne, CA firstname.lastname@example.org (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 Application of Unsteady Aerodynamics 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 Application of the Unsteady Aerodynamics n Aerodynamic Forces Vector of panel forces *Aerodynamic forces calculated in different coordinates than structure Vector of non-dimensional pressure coefficients Application of the Unsteady 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 n Unsteady Aerodynamic Rational Function 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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