Methods for Propagating Structural Uncertainty to Linear Aeroelastic Stability Analysis

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


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


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


Sensitivity analysis


Interval analysis


Interval analysis


Probabilistic methods


First Normal

& Aeroelastic

mode


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 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,



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.


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!

Documents

Methods for Propagating Structural Uncertainty to Linear Aeroelastic Stability Analysis