Nonlinear Normal Modes: Theoretical Curiosity or Practical Concept?

Transcription

1 Nonlinear Normal Modes: Theoretical Curiosity or Practical Concept? Gaëtan Kerschen Space Structures and Systems Lab Structural Dynamics Research Group University of Liège

2 Linear Modes: A Key Concept Clear physical meaning Structural deformation at resonance Synchronous vibration of the structure Important mathematical properties Orthogonality Modal superposition Invariance

3 Linear Modal Analysis Is Mature Airbus A380 Envisat But structures may be nonlinear!

4 Objective of this Presentation 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?

5 Specific Efforts in Our Research Most contributions in the literature deal with systems with very low-dimensionality (typically 2-DOF systems): Progress toward more realistic, large-scale structures Most contributions in the literature use analytic methods limited to weak nonlinearity: Develop computational methods which can tackle strongly nonlinear systems. Nonlinear dynamics is complicated and generally not well understood: Focus on developments that can be understood and exploited by the practising engineer.

6 Theoretical Curiosity or Practical Concept? 1. Nonlinear normal mode? 2. Theoretical modal analysis 3. Experimental modal analysis

7 Theoretical Curiosity or Practical Concept? 1. Nonlinear normal mode? Definition Frequency-energy dependence 2. Theoretical modal analysis 3. Experimental modal analysis

8 Historical Perspective: Lyapunov For n-dof conservative systems with no internal resonances, there exist at least n different families of periodic solutions around the equilibrium point of the system. These n families define n NNMs that can be regarded as nonlinear extensions of the n LNMs of the underlying linear system.

9 Historical Perspective 1960s: First constructive methods (Rosenberg) 1970s: Asymptotic methods (Rand, Manevitch) 1980s:? 1990s: New impetus (Vakakis, Shaw and Pierre) 2000s: Computational methods (Cochelin, Laxalde, Thouverez)

10 Undamped NNM Definition: Rosenberg An NNM is a vibration in unison of the system (i.e., a synchronous oscillation). Displacement Time Important remark: not limited to conservative systems!

11 Damped NNM Definition: Shaw and Pierre An NNM is a two-dimensional invariant manifold in phase space.

12 Extension of Rosenberg s Definition An NNM is merely a periodic motion of a nonlinear conservative system Displacement (m) Displacement (m) Time (s) Time (s)

13

14 NNMs Are Frequency-Energy Dependent Time series Modal curves Increasing energy (in-phase NNM)

15 Appropriate Graphical Depiction of NNMs NNM Nonlinear frequencies (backbone curves) NNM

16 LNMs & NNMs: Clear Conceptual Relation This frequency-energy plot gives a clear picture of the action of nonlinearity on the dynamics. It can also be understood by the practising engineer.

17 LNMs & NNMs: Clear Conceptual Relation 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 Invariance YES YES YES NO NO YES

18 Some Fundamental Differences Frequency-energy dependence LNMs NO NNMs YES Stability YES YES NO Number DOFs = number modes YES NO Modal interactions NO YES

19 Why Normal? NNMs are not orthogonal to each other, as LNMs are. They are still referred to as normal modes, because they are normal to the surface of maximum potential energy (bounding ellipsoid). A.F. Vakakis, MSSP 11, 1997

20 Theoretical Curiosity or Practical Concept? 1. Nonlinear normal mode? 2. Theoretical modal analysis Proposed algorithm Demonstration in Matlab 3. Experimental modal analysis

21 Theoretical Modal Analysis

22 How To Compute an NNM? NNM: periodic motion of a nonlinear conservative system Solve a 2-point boundary value problem Newmark

23 Shooting Algorithm: 2-Point BVP Governing equations in state space Periodicity condition (2-point BVP) Numerical solution through iterations Initial guess Corrections

24 Shooting Algorithm: Newton-Raphson 2n x 2n Monodromy matrix n x 1

25 Computational Burden Reduction Jacobian matrix (shooting) Finite differences (perturb the ICs and integrate the nonlinear equations of motion) COMPUTATIONALLY INTENSIVE Sensitivity analysis VERY APPEALING ALTERNATIVE

26 Jacobian Matrix through Linear ODEs Governing equations in state space Differentiation of the governing equations

27 How To Account for Energy Dependence? Predictor step tangent to the branch Corrector steps to the predictor step

28 Algorithm: Shooting and Continuation

29 Numerical Demonstration in Matlab 2DOF system with a single cubic nonlinearity Cyclic assembly of 30 substructures ( 30 cubic nonlinearities, 120 state-space variables)

30 Modal Interactions in a 2DOF Nonlinear System This is neither abstract art nor a new alphabet

31 Localization in the Bladed Disk Frequency (rad/s) x Mode (1,14) Energy

32 Theoretical Curiosity or Practical Concept? 1. Nonlinear normal mode? 2. Theoretical modal analysis 3. Experimental modal analysis Nonlinear force appropriation Experimental demonstration

33 Experimental Modal Analysis

34 The Linear Case Phase separation: All modes excited at once. Random or sine sweep excitations. Time (e.g., stochastic subspace identification) or frequencydomain methods (e.g., polyreference least-squares). Phase resonance: The structure is excited in one of its normal modes. The modes are identified one by one. Harmonic excitation at the resonance frequency with a specific amplitude and phase distribution (at several input locations).

35 Phase Resonance or Phase Separation? Clear physical meaning: Structural deformation at resonance Synchronous vibration (but not always) Phase resonance: OK Mathematical properties: No modal superposition Invariance Phase separation: KO Frequency-energy dependence: OK

36 Nonlinear Phase Quadrature Criterion A nonlinear structure vibrates according to one of its NNMs if the degrees of freedom have a phase lag of 90º with respect to the excitation (for all harmonics!) NNM = 90º Nonlinear MMIF

37 Two-Step Methodology Step 1: isolate an NNM motion using harmonic excitation Step 2: turn off the excitation and induce single-nnm free decay Extract the backbone and the modal curves

38 ECL Benchmark: Geometric Nonlinearity Experimental set-up: 7 accelerometers along the main beam. One displacement sensor (laser vibrometer) at the beam end. One electrodynamic exciter. One force transducer.

39 Moving Toward the First Mode Stepped sine excitation

40 Phase Quadrature Is Obtained Stepped sine excitation OK!

41 Phase Quadrature Is Obtained OK! Nonlinear MMIF

42 The First Mode Vibrates in Isolation

43 Frequency-Energy Dependence of the Mode

44 Modal shapes at different energy levels Frequencies (wavelet transform; Argoul et al.)

45 Validation of the Methodology Independently, a reliable numerical model of the structure was identified using the conditioned reverse path method Experimental Theoretical

46 Validation of the Methodology

47 Isolation of the Second NNM OK! Nonlinear MMIF

48 The Second NNM is Much Less Nonlinear 39 Hz 30 Hz 144 Hz 143 Hz 500 Acc #7 Acc #7 0 Acc #4 Mode 1 Mode Acc #3

49 Theoretical Curiosity or Practical Concept? 1. Nonlinear normal mode? 2. Theoretical modal analysis 3. Experimental modal analysis

50 Conclusion: We can extract NNMs both from finite element models and from experimental data

51 Structure Linear modes Nonlinear modes Important modes for the end result? Classical phase separation ( LNM with SSI) Proposed nonlinear phase resonance ( NNM)

52 State-of-the-art for linear aircrafts: combining linear phase separation with linear phase resonance nonlinear Next step

53 Technical Details and Open Questions Many technical details (stable/unstable NNMs, modal interactions, bifurcations) were omitted in this lecture and are available in a series of journal publications. Open questions and challenges: Multi-sine and multi-point excitation (coupled NNMs). Imperfect force appropriation. Complex modes. Nonlinear damping.

54 Thanks for your attention! Gaëtan Kerschen Space Structures and Systems Lab Structural Dynamics Research Group University of Liège