Tutorial on Nonlinear Modal Analysis of Mechanical Systems

Transcription

1 Tutorial on Nonlinear Modal Analysis of Mechanical Systems Gaëtan Kerschen Aerospace and Mechanical Eng. Dept. University of Liège, Belgium

2 Belgium? 2

3 University of Liège? Founded in 1817 (Belgium was not existing yet!) All disciplines covered undergraduate students 2000 graduate students 500 faculty members. 3

4 Aerospace and Mechanical Engineering Dept. Aerospace Materials Energetics Biomechanics Facts and figures: 25 Professors 85 researchers 15 staff members 5 Masters, among which the Master of Aerospace Engineering Computer-Aided Eng. 4

5 Space Structures & Systems Laboratory (S3L) Nonlinear dynamics System ID & modal analysis Constructive utilization of NL Bifurcation analysis & management Spacecraft structures Orbital mechanics Nanosatellites Facts and figures: 1 Professor 5 Postdoc, 6 PhD students, 1 TA Collaborations: UIUC, Torino, VUB, LMS Financial support: ERC, ESA, FNRS BELSPO 5

6 Linear? A linear system has an output that is directly proportional to input. Superposition principle: cornerstone of linear theory A B LIN LIN X Y A+B LIN X+Y 6

7 Nonlinear? A nonlinear system is a system that is not linear. A B NL NL X Y A+B??? NL Most physical systems are inherently nonlinear in nature! (Navier-Stokes, planetary motion, rigid body rotation, etc.) 7

8 Why Care About Nonlinearity in Structural Dynamics? 1. Nonlinear dynamics is fun! 1. Meet escalating performance. 2. Nonlinear dynamics is complex. 2. Nonlinearities are pervasive. 3. There are industrial needs. 3. Nonlinearity may reveal damage. 8

9 Two Examples in Aerospace Engineering Morane-Saulnier MS760 Collaborators: C. ONERA, P. ONERA. SmallSat spacecraft Collaborators: J.B. EADS-Astrium, B. LMS, A. ESA. 9

10 Outline of the Presentation Sources of nonlinearity (with real-life examples) Fundamental differences between linear & nonlinear dynamics What about nonlinear modes? 10

11 Linear Systems in Structural Dynamics 5 main assumptions for Small vibrations around an equilibrium Linear elasticity Viscous damping Small displ. and rotations Mechanical vibrations Mx (t) + Cx (t) + Kx(t) = f(t) nonlinear boundary conditions nonlinear materials (hyperelastic, plasticity) nonlinear damping mechanisms geometric nonlinearity inherently nonlinear forces in multiphysics applications 11

12 Nonlinear Materials: Elastomers, Foams, Rubber Decrease in stiffness J.R. Ahlquist et al., IMAC 2010 Airbus Military Impact of elastomeric engine mounts on A-400 M engine roll mode 12

13 Geometric NL: Large Displacements/Rotations Cantilever beam with a thin beam Increase in stiffness Force Rel. displ. Thin beam activated in large displacements 13

14 Combined Nonlinearity: Nonlinear BCs + Friction Paris ONERA (France) Decrease in stiffness 0 Bolted connections between wing tip and external fuel tank. Low excitation level High excitation level Force FRF (db) Rel. displ Frequency (Hz) 14

15 Outline of the Presentation Sources of nonlinearity (with real-life examples) Fundamental differences between linear & nonlinear dynamics What about nonlinear modes? 15

16 Linear Nonlinear Superposition principle NO! By definition x2 x2 Uniqueness of the solutions NO! Steady state response depends on transient response Invariance of FRFs and of modal parameters NO! Freq.-energy dependence + Bifurcations, quasi-periodicity and chaos.

17 Superposition Principle (Linear Case) 0.7 Displ. x (m) 0.3 x x + x = Fsinωt Force F (N) with ω=1.1 rad/s 17

18 No Superposition Principle (Nonlinear Case) 0.7 x x + x + x 3 = Fsinωt Displ. x (m) 0.3 Discontinuous behavior, potentially dangerous in practice x x + x = Fsinωt Force F (N) with ω=1.1 rad/s 18

19 Non-Uniqueness (Sensitivity to Initial Conditions) 1.5 x x + x + x 3 = 0.15 sin 1.6t x15 Displ. x (m) 0.1 x 0 = 0, x 0 = 0 x 0 = 0, x 0 = Time (s) 19

20 Marked Sensitivity to Forcing ( Jump) 0.8 x x + x + x 3 = 0.05 sin 1.22t x8 Displ. x (m) x x + x + x 3 = 0.05 sin 1.24t Time (s) 20

21 How to Compute the FRFs of a Linear System? x + x = Asin ωt x(t) = Bsin ωt ω 2 B + B = A B = A 1 ω 2 Confirmation of the superposition principle 21

22 How to Compute the FRFs of a Nonlinear System? x + x + x 3 = Asin ωt x(t) = Bsin ωt ω 2 Bsin ωt + Bsin ωt +B 3 sin 3 ωt = Asin ωt Nonlinear relation between B and A sin 3 ωt = (3sin ωt sin 3ωt)/4 x t OPTION 1: EXACT = Bsin ωt + Csin 3ωt OPTION 2: APPROXIMATION ω 2 Bsin ωt + Bsin ωt B3 sin ωt = Asin ωt Solution: infinite series of harmonics Solve a third order polynomial 22

23 Key Feature of Nonlinear Systems: Bifurcations x x + x + x 3 = Fsinωt solution 3 solutions 1 solution Displ. x (m) Pulsation ω (rad/s) 23

24 FRF Peak Skewness Perturbs Linear Modal Analysis Introduction of two stable poles around 1 nonlinear mode. B. Peeters et al., IMAC 2011 Polymax applied to F-16 24

25 Explanation of Previous Results x x + x + x 3 = Fsinωt 1.8 Displ. x (m) 1 1. No superp. (various F, ω=1.1) 3. Sensitivity to forcing (F=0.05N, ω=1.22/1.24) 2. Sensitivity to initial conditions (F=0.15N, ω=1.6) F=0.01 N F=0.02 N F=0.05 N F=0.1 N F=0.15 N Pulsation ω (rad/s) 25

26 Outline of the Presentation Sources of nonlinearity (with real-life examples) Fundamental differences between linear & nonlinear dynamics What about nonlinear modes? 26

27 Objectives of This Third Part Can we extend modal analysis to nonlinear systems? 1. What do we mean by a nonlinear normal mode (NNM)? 2. How do we compute NNMs from computational models? 3. How do we extract NNMs from experimental data? 27

28 Normal Modes of Conservative Systems Mq ( t) Kq( t) 0 ( t) 0 Mq ( t) Kq ( t) f NL q Linear normal mode (LNM): synchronous periodic motion. Lyapunov (1907) NDOF system: existence of at least N families of periodic solutions around the equilibrium. Rosenberg (1960s) Nonlinear normal mode (NNM): synchronous periodic motion. 28

29 Fundamental Difference Between LNMs and NNMs q 1 + 2q 1 q 2 = 0 q 2 + 2q 2 q 1 = 0 q 1,2 = A, Bcosωt q 1 + 2q 1 q q 1 3 = 0 q 2 + 2q 2 q 1 = 0 q 1,2 A, Bcosωt A = B, ω 1 = 1 rad/s A = ± 8 ω2 2 ω ω 2 2 A = B, ω 2 = 3 rad/s B = A 2 ω 2 LNMs ω 1 1, 2 rad/s ω 2 3,+ rad/s NNMs are frequencyamplitude dependent! 29

30 4 Initial conditions on displacements (cubic) Orange mass Blue mass

31 It Is Necessary to Extend the Definition of an NNM Mq ( t) Kq( t) 0 ( t) 0 Mq ( t) Kq ( t) f NL q LNM: synchronous periodic motion. Rosenberg (1960s) NNM: synchronous periodic motion. Modal interactions NNM: periodic motion (nonnecessarily synchronous). 31

32 A Frequency-Energy Plot Is a Convenient Depiction 0.7 2DOF linear Frequency (Hz) Energy (J) 32

33 A Frequency-Energy Plot Is a Convenient Depiction 0.7 2DOF nonlinear Frequency (Hz) Out-of-phase NNM 0.28 In-phase NNM Energy (J) 33

34 LNMs and NNMs Have a Clear Conceptual Relation 0.7 2DOF linear 0.7 2DOF nonlinear Freq. Freq Energy Energy But Frequency-energy dependence Harmonics: modal interactions Bifurcations: #NNMs > #DOFs Bifurcations: stable/unstable modes 34

35 Practical Significance for Damped-Forced Responses 8 Locus of LNMs for various energies 2.5 Locus of NNMs for various energies Ampl. (m) Ampl. (m) Freq. (rad/s) Freq. (rad/s) Linear: resonances occur in neighborhoods of LNMs. Nonlinear: resonances occur in neighborhoods of NNMs. 35

36 In Summary Clear physical meaning Structural deformation at resonance Synchronous vibration of the structure LNMs YES YES NNMs YES YES, BUT Important mathematical properties Orthogonality Modal superposition YES YES NO NO Invariance YES YES 36

37 Practical Computation of NNMs Complex finite element model Frequency-energy plot 37

38 Shooting and Pseudo-Arclength Continuation Freq. 0.7 STEP 1: compute an isolated periodic solution t=0 t=t z 0, T Numerical integration (Newmark) z T, T Newton-Raphson Energy 0.7 STEP 2: compute the complete branch z p0 Correction prediction to the Freq. Prediction tangent to the branch Energy 38

39 Shooting Periodicity condition (2-point BVP) Numerical solution through iterations: 2n x 2n Monodromy matrix 2n x 1 39

40 Numerical Demonstration in Matlab 40

41 In-Phase and Out-of-Phase NNMs Connected :1 modal interaction Out-of-phase NNM Frequency (Hz) In-phase NNM Energy (J) 41

42 Neither Abstract Art Nor a New Alphabet 42

43 MS760 Aircraft (ONERA) Bolted connections between external fuel tank and wing tip Front connection Rear connection 43

44 Finite Element and Reduced-Order Modeling Condensation of the linear components of the model Craig-Bampton technique 8 remaining nodes internal modes Finite element model (2D shells and beams, DOFs) Reduced model accurate in [0-100] Hz, 548 DOFs 44

45 A Close Look at Two Modes Wing bending Wing torsion (symmetric) 45

46 Wing Bending Mode not Affected by Nonlinearity Frequency (Hz) MAC = 1.00 MAC = 0.99 Energy (J) 46

47 Wing Torsion Mode Is Affected by Nonlinearity 31.1 Decrease of the natural frequency 3:1 Frequency (Hz) MAC = 1.00 Internal resonances 30.4 MAC = :1 9:1 Mode shape slightly affected Energy (J) 4 47

48 6 th NNM of the Spacecraft :1 3:1 Frequency (Hz) 30 26: Energy (J) 2:1 interaction with NNM12 48

49 6 th NNM: From a Local Mode to a Global Mode The top floor vibrates two times faster! NNM6: local deformation at low energy level NNM6: global deformation at a higher energy level (on the modal interaction branch) 49

50 ? 80N: large top floor motion 20N: no top floor motion 3.3 x4 x4 x4 x4 x4 Acc. 0 (m/s 2 ) 80N 20N Accelerometer at top floor Sine sweep Excitation frequency (Hz)

51 Experimental Confirmation No mode of the top floor in this frequency range 100? Accel. Acc. (m/s 2 ) g 0.1 g Base excitation (sine sweep) Excitation frequency (Hz) 51

52 Experimental Identification of NNMs? Nonlinear phase separation: NO! If several modes are excited (arbitrary forcing and/or ICs), extraction of individual NNMs is not possible generally, because modal superposition is no longer valid in the nonlinear case. Because the NNMs attract the dynamical flow, in some cases, the motion may end up along one dominant NNM. 52

53 Experimental Identification of NNMs? Nonlinear phase resonance: YES! Excite a single NNM Exploit invariance principle: If the motion is initiated on one specific NNM, the remaining NNMs remain quiescent for all time. Identify the resulting modal parameters 53

54 Two-Step Methodology NNM force appropriation Isolating an NNM motion using harmonic excitation NNM free decay Inducing single-nnm free decay ON Turn off the excitation OFF Extract the NNM 54

55 Necessary Ingredients 1. Appropriate excitation to isolate one NNM 2. Modal indicator (are we on the NNM of interest?) 3. Invariance principle 4. Wavelet transform and NNM extraction Extraction of NNMs and their oscillation frequencies directly from experimental data 55

56 Nonlinear Phase Lag Quadrature Criterion A nonlinear structure vibrates according to an NNM if the degrees of freedom have a phase lag of 90 with respect to the excitation (for all harmonics!). Nonlinear mode indicator function (MIF) 56

57 Invariance Principle If the motion is initiated on one specific NNM, the remaining NNMs remain quiescent for all time. Turn off the shaker 57

58 Wavelet Transform & NNM Extraction Modal curves of NNMs: Extracted directly from the time series Oscillation frequencies of NNMs: Extracted using the wavelet transform Frequency vs. time Reconstructed FEP from experimental data: Compute energy and eliminate time Frequency vs. energy 58

59 Experimental Demonstration Beam with a geometrical nonlinearity; benchmark of the European COST Action F3. Experimental set-up: 7 accelerometers along the main beam. One displacement sensor (laser vibrometer) at the beam end. One electrodynamic exciter. One force transducer. 59

60 Moving Toward the First Mode Stepped sine excitation 60

61 Moving Toward the First Mode Stepped sine excitation OK! 61

62 Modal Indicator Nonlinear mode indicator function (MIF) OK! 62

63 The First Mode Vibrates in Isolation 63

64 Frequency-Energy Dependence 64

65 Modal shapes at different energy levels Frequencies (wavelet transform)

66 Acc #7 Acc Validation of the Methodology Modal curves Modal shapes Acc #3 Position along the beam Theoretical Experimental 66

67 Validation of the Methodology 67

68 Concluding Remarks Virtually all engineering structures are nonlinear but they may vibrate in linear regimes of motion. Nonlinearity is not plug and play : Highly individualistic nature of nonlinear systems. Even weakly nonlinear systems can exhibit complex dynamics. No universal method; toolbox philosophy. Code for NNM computation available online. 68

69 Thank you for your attention. Gaetan Kerschen Aerospace and Mechanical Eng. Dept. University of Liège, Belgium