# 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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