Transcript Document 7634320

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
Lack of knowledge
Lack of confidence arising from either the
computational aeroelastic method or the
fidelity of modelling assumptions
reducible by further information
Aleatory (irreducible)
Variability in structural parameters arising from
the accumulation of manufacturing tolerances
or environmental erosion
Uncertainty in joints
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
B
Aerodynamic damping matrix, a function of Mach number, and reduced
frequency, k
D
k
c
2V
modal aerodynamic stiffness matrix, a function of Mach number, and
reduced frequency, k
=reduced frequency
Flutter and sensitivity analysis
This equation may be written as:
0
q  

  
1
2
q  M K   V D

 q 
     0  p  Sp  0.
1
 M C   c V B   q 
I

By assuming p  p h e t
S ph   ph

 
eigenvalue
    i 
transient decay rate coefficient/ aerodynamic damping.
Flutter and sensitivity analysis


  C   c V B / k q  K   V 2 D q  0
Mq
  
  
 1 
  
  
 2

S  f  .


.


.







  m 
‘’Flutter sensitivity computes the rates of
changes in the transient decay rate coefficient
wrt changes in the chosen parameters.
is
defined in connection with the complex
eigevanlue
     i 
The solution is semi-analytic in nature with
either forward differences (default) or central
differences (PARAM,CDIF,YES)’’
Propagation methods: Interval analysis
Determine:
 ,    min  , max  
i
i
i
i
Subject to:
Sθ,  i   i I u i   0;
θθθ

:Lower bound

:Upper bound
•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

2
2










M
θ

θ



c
V
B
/
k
θ

C

θ



V
E

K
θ



 uθ   0

4
2






 i   i (θ )   i
j 1 θ j
m
mi1   i θ  
mir 
 2 i
θi  θi   j 1k1
θ j θ k
m m
θ j θ j

θ j θ j
θ k  θk
θ
j
 θ j θ k  θk   ...

1
trace Gi θ covθ, θ 
2
r  1! trace G θ covθ, θ 
r!
T
r 2
g  i θ  covθ , θ G  i θ  covθ , θ g  i θ  
i
2
2




dp i 
a i
a i







p


p


exp
d


i
i
i
 b b b  2
d i
b0  b1 i  b2 i2
 0 1 i 2 i



r
Pearson’s theory
Numerical example:
Goland wing without structural damping
Thicknesses of skins
Area of spars cap
Thicknesses of spars Thicknesses of ribs
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:
Goland wing without structural damping
First Aeroelastic mode mean+maximum
First Normal
& 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
1
2
3
4
Damping Coefficient
3.403772×10-2
1.345800×10-2
4.506277×10-2
4.539254×10-2
Frequency
1.966897
4.046777
9.653923
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!