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Formulation of a complete structural uncertainty model for

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Formulation of a complete structural uncertainty model for Powered By Docstoc
					                           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?

				
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posted:7/13/2014
language:English
pages:38