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
IntroductionEpistemic Aleatory (irreducible)
Lack of knowledge
Lack of confidence arising from either the computational aeroelastic method or the
fidelity of modelling assumptions
reducible by further information
Variability in structural parameters arising from the accumulation of manufacturing
tolerances or environmental erosion
Uncertainty in joints
atmospheric uncertainty
IntroductionStructural uncertainty
Flutter and sensitivity analysis
0DKBCM qVqkVcq 2/
K
D
C
General form for N DoF system:
B Aerodynamic damping matrix, a function of Mach number, and reduced frequency, k
modal aerodynamic stiffness matrix, a function of Mach number, and reduced frequency, k
V
ck
2
=reduced frequency
M Mass matrix
Stiffness matrix
Structural damping matrix
Flutter and sensitivity analysis
eigenvalue i
transient decay rate coefficient/ aerodynamic damping.
.0
1210Spp
BCMDKM
I0
q
q
VcVq
q
This equation may be written as:
the
pp assumingBy
hh ppS
Flutter and sensitivity analysis
m
f
.
.
.2
1
S
‘’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)’’
0qDKqBCqM 2/ VkVc
Propagation methods: Interval analysis
iiii max,min,
θθθ0uIθS ;, iii
Determine:
Subject to:
•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.
:Lower bound
:Upper bound
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
0θuθKEθCθBθθM
22
2
1/
4
1VkVc
The uncertain flutter equation:
...θθθθθθ
θθθ
)(
θθ
θθ1 1
2
1θθ
kkjj
m
j
m
k kj
iii
m
j j
iii
kk
jjjj
θ
θθθθ ,covtrace2
11
iGm ii
rrTri iiii
rrm θθθGθgθθθGθθθg ,covtrace
2
!1,cov,cov
2
! 2
i
ii
iii
ii
i
i
i dbbb
app
bbb
a
d
dp
2210
2210
exp Pearson’s theory
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
First Normal
& Aeroelastic
mode
Numerical example: Goland wing without structural damping
Second Normal
& Aeroelastic
mode
First Aeroelastic mode mean+maximum
Second Aeroelastic mode mean+maximum
First Normal
& Aeroelastic
mode
Second Normal
& Aeroelastic
mode
Numerical example: Goland wing without structural damping
Numerical example: Goland wing without structural damping
Numerical example: Goland wing with structural damping
Mode Number Damping Coefficient Frequency1 3.403772×10-2 1.9668972 1.345800×10-2 4.0467773 4.506277×10-2 9.6539234 4.539254×10-2 13.44795
Modal damping coefficients achieved by Complex Eigenvalue Solution.
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
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/m2Young 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.
Numerical example: Generic fighter
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!