Methods for Propagating Structural Uncertainty to Linear by elfphabet5

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									Methods for Propagating Structural
 Uncertainty to Linear Aeroelastic
        Stability Analysis

           February 2009
Contents:
• Introduction
• Flutter and sensitivity analysis
• Propagation methods
            - Interval analysis
            - Fuzzy method
            - Perturbation procedure
• Numerical case studies
   - Goland wing without structural damping
   - Goland wing with structural damping
   - Generic fighter
  Introduction
           Epistemic                               Aleatory (irreducible)
           Lack of knowledge                 Variability in structural parameters arising from
                                             the accumulation of manufacturing tolerances
Lack of confidence arising from either the                or environmental erosion
computational aeroelastic method or the
    fidelity of modelling assumptions                     Uncertainty in joints

    reducible by further information                    atmospheric uncertainty
Introduction
  Structural uncertainty
Flutter and sensitivity analysis
 General form for N DoF system:

                                     
Mq  C   c V B / k q  K   V 2 D q  0
                                                
 M       Mass matrix

 K  Stiffness matrix
C        Structural damping matrix
          Aerodynamic damping matrix, a function of Mach number, and reduced
B        frequency, k

D        modal aerodynamic stiffness matrix, a function of Mach number, and
          reduced frequency, k

     c
k          =reduced frequency
     2V
Flutter and sensitivity analysis
   This equation may be written as:


q  
                0                                I        q 
                                                           0  p  Sp  0.
                                                                        
q   M K   V D                   M C   c V B   q 
         1                               1
                 2




 By assuming p  p h e t

  S ph   ph

        eigenvalue        i 

         transient decay rate coefficient/ aerodynamic damping.
Flutter and sensitivity analysis
                           
 Mq  C   c V B / k q  K   V 2 D q  0
                                           
            
                  ‘’Flutter sensitivity computes the rates of
           1        changes in the transient decay rate coefficient
                  wrt changes in the chosen parameters.       is
                  defined in connection with the complex
           2        eigevanlue
S  f  .     
                          i 
          .     
          .         The solution is semi-analytic in nature with
                    either forward differences (default) or central
                  differences (PARAM,CDIF,YES)’’
            m 
                
Propagation methods: Interval analysis

                                                 
   Determine:

  ,   min  , max  
    i    i                i          i
                                                         :Lower bound

  Subject to:                                           :Upper bound
  Sθ,  i   i I u i   0;   θθθ   •Select uncertain structural parameters from
                                            sensitivity analysis and define their intervals.

                                            •Identify the unstable mode from deterministic
                                            analysis and carry out optimisation to find the
                                            maximum and minimum values of real parts of
                                            eigenvalues close to the deterministic flutter
                                            speed.

                                            •Check     for  unstable-mode        switching     for
                                            parameter change at low flutter speeds. If
                                            switching occurs, go to step 2; if not, go to step 4.

                                            •Fit curves to both the maximum and minimum
                                            real parts of the eigenvalues and find the
                                            minimum and maximum flutter speeds as in
                                            Figure 1.
Propagation methods: Fuzzy method




     α-level strategy, with 4 α-levels, for a function of two triangular fuzzy parameters
  [Moens, D. and Vandepitte, D., A fuzzy finite element procedure for the calculation of uncertain
                       frequency response functions of damped structures:
            Part 1 – procedure. Journal of Sound and Vibration 2005; 288(3):431–62.].
Propagation methods: Fuzzy method
Propagation methods:
Perturbation procedure using the theory of quadratic forms
   The uncertain flutter equation:
                 1                              1                
   Mθ2 θ     c V B / k θ  C  θ     V 2 E  K θ uθ  0
                 4                              2                


                                                              2 i
                                           θi  θi   1k1                        θ        θ j θ k  θk   ...
                     m                                  m m
    i   i (θ )   i
                    j 1 θ j                           j   θ j θ k
                                                                                          j
                                                                          θ j θ j
                                θ j θ j
                                                                          θ k  θk




                      2
                                           
      mi1   i θ   trace Gi θ covθ, θ 
                      1
                                                                

      mir 
              r!
                                                       
                 g i θ  covθ, θ G  i θ  covθ, θ g i θ  
                          T                    r 2                   r  1! trace G θ covθ, θ                          
                                                                                                                                 r
                                                                                      i
              2                                                          2

       dp i         a i                                     a i          
                                     p i   p i   exp 
                                                              b b b  2 d i 
                                                                                                                        Pearson’s theory
        d i      b0  b1 i  b2 i2                         0 1 i 2 i         
Numerical example:
Goland wing without structural damping




              Thicknesses of skins   Thicknesses of spars Thicknesses of ribs




  Area of spars cap          Area of ribs cap            Area of posts
Numerical example:
Goland wing without structural damping
  Sensitivity analysis
Numerical example:
Goland wing without structural damping
  Interval analysis
Numerical example:
Goland wing without structural damping
  Interval analysis
Numerical example:
Goland wing without structural damping

                          Probabilistic methods
Numerical example:
Goland wing without structural damping
    Numerical example:                                 First Normal

              wing without structural damping
    Goland Aeroelastic mode mean+maximum
        First
                                                       & Aeroelastic
                                                       mode




First Normal
& Aeroelastic
mode




Second Normal
& Aeroelastic
mode




                                                       Second Normal
                                                       & Aeroelastic

                Second Aeroelastic mode mean+maximum   mode
Numerical example:
Goland wing without structural damping
Numerical example:
Goland wing without structural damping
Numerical example:
Goland wing with structural damping




                   Modal damping coefficients achieved by Complex Eigenvalue Solution.

                  Mode Number             Damping Coefficient             Frequency
                       1                    3.403772×10-2                 1.966897
                       2                    1.345800×10-2                 4.046777
                       3                    4.506277×10-2                 9.653923
                       4                    4.539254×10-2                 13.44795
Numerical example:
Goland wing with structural damping
Numerical example:
Generic fighter




                       Mode 1    Mode 2     Mode 3    Mode 4       Mode 5


    Updated FE model   3.74 h1   5.91 α+θ   8.12 γ   11.00 h2+ α   11.51 θαT
         GVT           4.07 h1   5.35 α+θ   8.12 γ    12.25 h2
Numerical example: Generic fighter


                   Mode 1, first bending (h1) ,symmetric, 3.74Hz.




                   Mode 2, torsion+pitch (α+θ), symmetric, 5.91 Hz.




          Aeroelastic modes at velocity 350 m/s, (a): mode 1, 4.106Hz, (b): mode 2,
Numerical example: Generic fighter
Numerical example: Generic fighter




         Rotational spring coefficient: [0.7-1.3]×2000 kN m/rad,
        Young modulus of the root: [0.9-1.1] ×1.573×1011 N/m2
        Young modulus of the pylon: [0.9-1.1] ×9.67×1010 N/m2
           Mass density of the root: [0.9-1.1] ×5680 kg/m3,
           Mass density of the pylon: [0.6-1.1] ×3780 kg/m3,
            Mass density of the tip: [0.9-1.1] ×3780 kg/m3.
Conclusion
• Different forward propagation methods, interval, fuzzy and perturbation,
  were applied to linear aeroelastic analysis of a variety of wing models.

• MCS was used for verification purposes and structural-parameter
  uncertainties were assumed.

• Sensitivity analysis was used to select parameters for randomisation that
  had a significant effect on flutter speed.

•   Interval analysis was found to be an efficient method which produces
    enough information about uncertain aeroelastic system responses.

•   Nonlinear behaviour was observed in tails of the eigenvalue real-part pdfs
    of the flutter mode.

•   Second order perturbation and fuzzy methods were found to be capable
    of representing this nonlinear behaviour to an acceptable degree.
Thank you!

								
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