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 EpistemicAleatory (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

Introduction Structural uncertainty

Flutter and sensitivity analysis General form for N DoF system: 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 =reduced frequency Mass matrix Stiffness matrix Structural damping matrix

Flutter and sensitivity analysis eigenvalue transient decay rate coefficient/ aerodynamic damping. This equation may be written as:

Flutter and sensitivity analysis ‘’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 The solution is semi-analytic in nature with either forward differences (default) or central differences (PARAM,CDIF,YES)’’

Propagation methods: Interval analysis 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 The uncertain flutter equation: Pearson’s theory

Numerical example: Goland wing without structural damping Thicknesses of skinsThicknesses of sparsThicknesses 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 with structural damping Mode NumberDamping CoefficientFrequency × × × × Modal damping coefficients achieved by Complex Eigenvalue Solution.

Numerical example: Goland wing with structural damping

Numerical example: Generic fighter Mode 1Mode 2Mode 3Mode 4Mode 5 Updated FE model3.74 h α+θ8.12 γ11.00 h 2+ α11.51 θ αT GVT4.07 h α+θ8.12 γ12.25 h 2

Mode 1, first bending (h 1 ),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

Rotational spring coefficient: [ ]×2000 kN m/rad, Young modulus of the root: [ ] ×1.573×1011 N/m2 Young modulus of the pylon: [ ] ×9.67×1010 N/m2 Mass density of the root: [ ] ×5680 kg/m3, Mass density of the pylon: [ ] ×3780 kg/m3, Mass density of the tip: [ ] ×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.

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