Methods for Propagating Structural Uncertainty to Linear Aeroelastic Stability Analysis

1. Methods for Propagating Structural Uncertainty to Linear Aeroelastic Stability Analysis February 2009

2. 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

3. Introduction Epistemic 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

4. Introduction Structural uncertainty

5. Flutter and sensitivity analysis General form for N DoF system: Mass matrix Stiffness matrix Structural damping matrix 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

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

7. 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)’’

8. 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

9. 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.].

10. Propagation methods: Fuzzy method

11. Propagation methods: Perturbation procedure using the theory of quadratic forms The uncertain flutter equation: Pearson’s theory

12. 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

13. Numerical example: Goland wing without structural damping Sensitivity analysis

14. Numerical example: Goland wing without structural damping Interval analysis

15. Numerical example: Goland wing without structural damping Interval analysis

16. Numerical example: Goland wing without structural damping Probabilistic methods

17. Numerical example: Goland wing without structural damping

18. Numerical example: Goland wing without structural damping First Normal & Aeroelastic mode First Aeroelastic mode mean+maximum First Normal & Aeroelastic mode Second Normal & Aeroelastic mode Second Normal & Aeroelastic mode Second Aeroelastic mode mean+maximum

19. Numerical example: Goland wing without structural damping

20. Numerical example: Goland wing without structural damping

21. Numerical example: Goland wing with structural damping Modal damping coefficients achieved by Complex Eigenvalue Solution.

22. Numerical example: Goland wing with structural damping

23. Numerical example: Generic fighter

24. 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,

25. Numerical example: Generic fighter

26. 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.

27. 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.

28. Thank you!