Document Sample

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 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?

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 0 |

posted: | 7/13/2014 |

language: | English |

pages: | 38 |

OTHER DOCS BY hcj

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.