Nonlinear Normal Modes of a Full-Scale Aircraft

Transcription

1 Nonlinear Normal Modes of a Full-Scale Aircraft M. Peeters Aerospace & Mechanical Engineering Dept. Structural Dynamics Research Group University of Liège, Belgium

2 Nonlinear Modal Analysis: Motivation? Most contributions in the literature deal with low-dimensional systems (typically 2 or 3 DOFs) Few computational methods for NNM calculation (e.g., Cochelin et al., Laxalde et al., Shaw et al., Touze et al.) A numerical computation of NNMs is targeted to progress toward a practical modal analysis of real-life structures.

3 The Long-Term Objective Numerical NNM Computation

4 Outline NNMs: What Are They? Proposed Algorithm for NNM Computation Application: Full-Scale Aircraft

5 Nonlinear Normal Mode (NNM) Discrete mechanical systems (finite element method), Conservative (undamped and unforced) systems. Extension of Rosenberg s definition (undamped NNM): An NNM motion is a (non-necessarily synchronous) periodic motion of the conservative nonlinear system.

6 Frequency-Energy Dependence of NNMs 2DOF system with a cubic stiffness: Three in-phase NNM motions of increasing energies: Time series Modal curves

7 Frequency-Energy Plot Modal curves

8 NNMs: Fundamental Properties Similar in spirit to linear normal modes, but Frequency-energy dependence Modal interactions Number of NNMs number of DOFs (bifurcations) Stable / unstable

9 Outline NNMs: What Are They? Proposed Algorithm for NNM Computation Application: Full-Scale Aircraft

10 Numerical Algorithm for NNM Computation NNM? A periodic motion of the nonlinear conservative system:

11 Shooting Method Governing equations Periodicity condition (2 point BVP) Numerical solution through iterations Initial guess Corrections

12 Shooting Method: Newton-Raphson

13 Shooting Method: Newton-Raphson 2n x 2n Monodromy matrix 2n x 1

14 Pseudo-Arclength Continuation Starting from the corresponding linear normal mode. Finding successive pair of initial conditions and period of NNM motions.

15 Pseudo-Arclength Continuation Starting from the corresponding linear normal mode. Finding successive pair of initial conditions and period of NNM motions. Predictor step tangent to the branch

16 Pseudo-Arclength Continuation Starting from the corresponding linear normal mode. Finding successive pair of initial conditions and period of NNM motions Predictor step tangent to the branch Corrector steps to the predictor step (shooting)

17 Pseudo-Arclength Continuation Starting from the corresponding linear normal mode. Finding successive pair of initial conditions and period of NNM motions. Predictor step tangent to the branch Corrector steps to the predictor step (shooting)

18 Numerical Features

19 Sensitivity Analysis Jacobian matrix (shooting) 2n x 2n?

20 Sensitivity Analysis Jacobian matrix (shooting) 2n x 2n Finite differences (perturb the ICs and integrate the nonlinear equations of motion) COMPUTATIONALLY INTENSIVE Sensitivity analysis VERY APPEALING ALTERNATIVE

21 Sensitivity Analysis Differentiation of the governing equations

22 Sensitivity Analysis Differentiation of the governing equations Linear ODEs!!!

23 Algorithm Implementation MATLAB environment using a GUI: Frequency-energyplot (FEP)

24 Numerical Example: Nonlinear 2DOF System 2DOF system with a cubic stiffness:

25 2DOF System 1 st NNM FEP of the first NNM (in-phase NNM): Efficiency of the NNM computation algorithm (even for higher energy).

26 2DOF System FEP Fundamental in-phase and out-of-phase NNMs:

27 2DOF System 3:1 Internal Resonance Smooth transition from the in-phase mode to the out-of-phase mode.

28 2DOF System Others internally resonant NNMs

29 2DOF System Stability Results Floquet multipliers

30 Outline NNMs: What Are They? Proposed Algorithm for NNM Computation Application: Full-Scale Aircraft

31 Full-Scale Aircraft Paris MS 670 Aircraft ONERA (Office National d Etudes et de Recherches Aéronautiques) Finite element model (Samcef FE software) 2D elements (shells and beams)

32 Nonlinear Wing-Tank Connection Bolted connections between external fuel tank and wing tip Front connection Rear connection

33 Experimental Characterization of Nonlinearity Instrumentation of the connections Swept sine testing Restoring force surface (RFS) results: Restoring force Stiffness curve Softening nonlinear behavior

34 The Reduced-Order Model is Accurate in [0-100] Hz Condensation of the linear components of the model Craig-Bampton technique 8 remaining nodes internal modes Full-scale finite element model ~ DOFs Reduced-order model 548 DOFs

35 Nonlinear Model Reduced-order linear model (548 DOFs) with: Negative cubic springs between the remaining nodes of each connection: Qualitative value estimated from RFS results:

36 Frequency (Hz) First Tail Bending Mode MAC = 1.00 MAC = 1.00 Energy (J)

37 Frequency (Hz) First Wing Bending Mode MAC = 1.00 MAC = 0.99 Energy (J)

38 First Wing Torsional Mode Frequency (Hz) 3:1 internal resonance MAC = :1 internal resonance 9:1 internal resonance MAC = 0.98 Energy (J)

39 Frequency (Hz) Second Wing Torsional Mode MAC = 1.00 MAC = :1 internal resonance Energy (J)

40 3:1 Internal Resonance Frequency (Hz) (a) (b) 3:1 internal resonance (c) First Wing Torsional Mode Energy (J) (a) (b) (c)

41 3:1 Internal Resonance (a) (b) (c) Tank tip Horizontal tail

42 Concluding Remarks Objective: practical nonlinear modal analysis tools for real life structures. Further research: 1. Distributed and nonsmooth nonlinearities 2. Bifurcation handling 3. Filtering of branches of high order internal resonances for systems with high modal density 4. And many open questions (including damping nonlinearities)

43 Thank you for your attention