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 Aleatory (irreducible) Lack of knowledge Variability in structural parameters arising from the accumulation of manufacturing tolerances Lack of confidence arising from either the or environmental erosion computational aeroelastic method or the fidelity of modelling assumptions Uncertainty in joints reducible by further information 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 Aerodynamic damping matrix, a function of Mach number, and reduced B frequency, k D modal aerodynamic stiffness matrix, a function of Mach number, and reduced frequency, k c k =reduced frequency 2V Flutter and sensitivity analysis This equation may be written as: q 0 I q 0 p Sp 0. q M K V D M C c V B q 1 1 2 By assuming p p h e t S ph ph eigenvalue i transient decay rate coefficient/ aerodynamic damping. Flutter and sensitivity analysis Mq C c V B / k q K V 2 D q 0 ‘’Flutter sensitivity computes the rates of 1 changes in the transient decay rate coefficient wrt changes in the chosen parameters. is defined in connection with the complex 2 eigevanlue S f . i . . The solution is semi-analytic in nature with either forward differences (default) or central differences (PARAM,CDIF,YES)’’ m Propagation methods: Interval analysis Determine: , min , max i i i i :Lower bound Subject to: :Upper bound Sθ, i i I u i 0; θθθ •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 Mθ2 θ c V B / k θ C θ V 2 E K θ uθ 0 4 2 2 i θi θi 1k1 θ θ j θ k θk ... m m m i i (θ ) i j 1 θ j j θ j θ k j θ j θ j θ j θ j θ k θk 2 mi1 i θ trace Gi θ covθ, θ 1 mir r! g i θ covθ, θ G i θ covθ, θ g i θ T r 2 r 1! trace G θ covθ, θ r i 2 2 dp i a i a i p i p i exp b b b 2 d i Pearson’s theory d i b0 b1 i b2 i2 0 1 i 2 i 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 Numerical example: First Normal wing without structural damping Goland Aeroelastic mode mean+maximum First & 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 Damping Coefficient Frequency 1 3.403772×10-2 1.966897 2 1.345800×10-2 4.046777 3 4.506277×10-2 9.653923 4 4.539254×10-2 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!
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