Identification of Nonlinear Mechanical Systems: State of the Art and Recent Trends

- Acceleration (m/s 2 ) Identification of Nonlinear Mechanical Systems: State of the Art and Recent Trends 5 Force level: 67 N rms Gaëtan Kerschen Space Structures and Systems Laboratory Aerospace and Mechanical Eng. Dept. University of Liège 4 3 2 1 0-1 -2-3 -4-5 -4-3 -2-1 0 1 2 3 4 Relative displacement (m) x 10-4 Colleagues and Collaborators: J.P. Noël, J. Schoukens, K. Worden, B. Peeters

Why Is NSI Important in Mechanical Engineering? Contact nonlinearities 300% error between predictions and measurements 2

Why Is NSI Important in Mechanical Engineering? Introduction of two poles around one nonlinear mode FRF F-16 6.5 Frequency (Hz) 7.5 Linear tools and methods may fail 3

Outline A brief review of sources of nonlinearity in mechanics. State-of-the-art: seven families of NSI methods in mechanical engineering. Future trends, including identification for design. 4

3 Main Assumptions for Linear Mechanical Systems Mx(t) ሷ + Cx(t) ሶ + Kx(t) = f(t) Linear elasticity nonlinear materials Small displ. and rotations geometric nonlinearity nonlinear boundary conditions Viscous damping nonlinear damping mechanisms 5

Source 1: Material Nonlinearities Stress Hyperelastic material (e.g., rubber) Strain Stress Shape memory alloy Strain 6

Anti-Vibration Mounts of a Helicopter Cockpit Rubber Decrease in stiffness Decrease in damping A. Carrella, IJMS 2012 7

Source 2: Displacement-Related Nonlinearities Decrease in stiffness Increase in stiffness l 0 l 1 x ሷ θ + ω 0 2 sin θ = 0 F = 2k l 1 l 0 x l 1 = 2kx 1 l 0 x 2 + l 0 2 sin θ = θ θ3 6 + l 0 x 2 + l 0 2 = 1 x2 2l 0 2 + 3x4 8l 0 4 + O(x6 ) 8

A Widely-Used Benchmark in Mech. Engineering Beam @ ULg Increase in stiffness Displ. (m) Frequency (Hz) 9

The SmallSat Spacecraft Mechanical stops SmallSat spacecraft (EADS Astrium) NL isolation device for reaction wheels Elastomer plots Goals Micro-vibration mitigation Large amplitude limitation Solutions Elastomer plots Mechanical stops 10

Contact Can Generate Complex Dynamics Accel. 100 excitation Dangerous nonlinear resonance 1 g Acc. (m/s 2 ) 0 0.1 g EADS Astrium satellite -100 5 20 30 70 Sweep frequency (Hz) 11

Source 3: Damping Nonlinearities Nonlinear damping, a pleonasm! Extremely complex. Present in virtually all interfaces between components (e.g., bolted joints). Key parameter, because it dictates the response amplitude. Bouc-Wen benchmark. 12

Sliding Connection in the F-16 Aircraft SLIDING 13

Impact of the Connection on the F-16 Dynamics -20 Decrease in stiffness Increase in damping FRF -40 Low High -60-80 2 4 6 8 10 Frequency (Hz) 14

Outline A brief review of sources of nonlinearity in mechanics. State-of-the-art: seven families of NSI methods in mechanical engineering. Future trends, including identification for design. 15

A Three-Step Process Yes or No? 1. Detection What? Where? How? f nl? 3 ( x, x ) x,sin x, x 2. Characterization How much?? 3 3 3 f nl ( x, x) 0.1x,1.2 x, 3x 3. Parameter estimation More information but increasingly difficult 16

Three Main Differences MECHANICS The quantification of the impact of nonlinearity is not often performed. EE/CONTROL Best-linear approximation (Schoukens et al.). White-box approach commonplace. The black-box approach seems to be very popular. It is only very recently that uncertainty is accounted for (e.g. Beck, Worden et al.). 1965 (!): Astrom and Bohlin introduced the maximum likelihood framework. A reputable engineer should never deliver a model without a statement about its error margins (M. Gevers, IEEE Control Systems Magazine, 2006) 17

White-Box Approach Commonplace in Mechanics Often, reasonably accurate low-dimensional models can be obtained from first principles. 18

But Not Always: Individualistic Nature of Nonlinearities Elastic NL (grey-box) Damping NL (black-box or further analysis) 19

Case Study: The F-16 Aircraft (With VUB & Siemens) Right missile Connection with the wing 20

Apply Your Favorite Modal Analysis Software Introduction of two poles around one nonlinear mode FRF F-16 6.5 Frequency (Hz) 7.5 21

Measured Time Series Nonlinearity can often be seen in raw acceleration signals. Acc. (m/s^2) Time (s) 7.2 Hz 22

Amplit ude (db) Measured Frequency Response Functions -15-20 FRF amplitude, sensor [4] DP_RIGHT:-Z 7.2 Hz Low level High level -25 Ampl. (db) -30-35 -40-45 -50-55 4 6 8 10 12 14 Frequency (Hz) At this stage, we should be convinced about the presence of nonlinearity. 23

Time-Frequency Analysis (Wavelet Transform) Objective: know more about the nonlinear distortions. Frequency (Hz) Time (s) 24

Restoring Force Surface Method (Masri & Caughey) Objective: visualize nonlinearities. Nonlinear connection instrumented on both sides. 25

Restoring Force Surface Method 26

A Three-Step Process Yes or No? 1. Detection What? Where? How? f nl? 3 ( x, x ) x,sin x, x 2. Characterization How much?? 3 3 3 f nl ( x, x) 0.1x,1.2 x, 3x 3. Parameter estimation More information but increasingly difficult 27

Classification in Seven Families 1. By-passing nonlinearity: linearization 2. Time-domain methods 3. Frequency-domain methods 4. Time-frequency analysis 5. Black-box modeling 6. Modal methods 7. Finite element model updating G. Kerschen, K. Worden, A.F. Vakakis, J.C. Golinval, Past, Present and Future of Nonlinear System Identification in Structural Dynamics, MSSP, 2006 28

Time- and Frequency-Domain Methods Often manipulation of equations of motion giving rise to a least-squares estimation problem (restoring force surface, conditioned reverse path, generalizations of subspace identification methods) The Volterra and higher-order FRFs theories are popular within our community, but have never found application on realistic structures. 29

Time-Frequency Analysis Hilbert transform and its generalization to multicomponent signals (empirical mode decomposition). Wavelet transform. Displacement (m) Instantaneous frequency (Hz) Time (s) Time (s) 30

Black-box Modeling Interesting when there is no a priori knowledge about the nonlinearity. But A priori information and physics-based models should not be superseded by any blind methodology. Overfitting may be an issue. Characterized by many parameters; difficult to optimize. 31

Modal Methods Different approaches exist in the literature: linearised modes intuitive, but do not account for NL phenomena. data-based modes straightforward, but limited theoretical background. nonlinear normal modes rigorous, and do fully account for NL phenomena. 32

Finite Element Model Updating FE model Feature extraction??? Experimental data Feature extraction??? Correlation Model updating Poor Good Reliable model Parameter selection O.F. min. 33

Where Do We Stand? 1970s 1980s 1990s-2000 Today First contributions Focus on 1DOF: Hilbert, Volterra Focus on MDOF: NARMAX, frequency-domain ID, finite element model updating Large-scale structures with localized nonlinearities: uncertainty quantification, extension of linear algorithms (nonlinear subspace ID) 34

Identification for Design Computer-aided modelling (FEM, ) MEASURE IDENTIFY MODEL UNDERSTD UNCOVER DESIGN F-16 aircraft SmallSat spacecraft 35

The SmallSat Spacecraft Mechanical stops SmallSat spacecraft (EADS Astrium) NL isolation device for reaction wheels Elastomer plots Goals Micro-vibration mitigation Large amplitude limitation Solutions Elastomer plots Mechanical stops 36

Troubleshooting Clearly Needed Acceleration (m/s²) 100 50? MEASURE IDENTIFY 0 MODEL -50 1 g base 0.1 g base -100 5 10 20 30 40 50 60 70 Sweep frequency (Hz) UNDERSTD UNCOVER DESIGN 37

24 Accels Close to the Suspected Nonlinear Device MEASURE IDENTIFY MODEL UNDERSTD UNCOVER DESIGN 38

Wavelet Transform: Nonsmooth Nonlinearity Instantaneous frequency (Hz) MEASURE 100 80 IDENTIFY 60 MODEL 40 20 5 5 7 9 11 13 Sweep frequency (Hz) UNDERSTD UNCOVER DESIGN 39

Time Series: Clearance Identification Relative displacement ( ) MEASURE 2 1 Jump IDENTIFY 0 MODEL -1 Discontinuity UNDERSTD UNCOVER -2 5 7 7.5 9 9.4 11 13 Sweep frequency (Hz) DESIGN 40

Acceleration Surface: Nonlinearity Visualization 0.6 g MEASURE -Acc. IDENTIFY 1 g Rel. displ. MODEL -Acc. UNDERSTD UNCOVER DESIGN Rel. displ. 41

Restoring force (N) Experimental Model of the Nonlinearity 800 400 Fitted model Experimental data MEASURE IDENTIFY 0 MODEL -400 UNDERSTD UNCOVER -800-2 -1 0 1 1.6 2 Relative displacement DESIGN 42

Finite Element Modeling Linear main structure = fairly easy to model numerically. MEASURE IDENTIFY MODEL UNDERSTD UNCOVER Nonlinear component = difficult to model numerically. DESIGN 43

Remember Acceleration (m/s²) 100 50? MEASURE IDENTIFY 0 MODEL -50 1 g base 0.1 g base -100 5 10 20 30 40 50 60 70 Sweep frequency (Hz) UNDERSTD UNCOVER DESIGN 44

Nonlinear Modal Analysis Objective: investigate the nonlinear resonances. MEASURE IDENTIFY MODEL UNDERSTD UNCOVER DESIGN 45

Nonlinear Mode #6 at Low Energy Level MEASURE IDENTIFY Frequency (Hz) MODEL UNDERSTD UNCOVER Energy (J) DESIGN 46

Nonlinear Mode #6 at Higher Energy Level MEASURE IDENTIFY Frequency (Hz) MODEL UNDERSTD UNCOVER Energy (J) DESIGN 47

Understand: 6 th Nonlinear Mode of the Satellite 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) 48

Troubleshooting: Problem Solved! 31.3 100 excitation 0 Frequency (Hz) 30-100 5 20 30 70 28.5 10 0 10 5 Energy (J) 2:1 interaction with NNM12 49

Uncovering Quasiperiodicity using Bifurcations Inertia wheel displacement ( ) MEASURE 2 1 IDENTIFY 0 MODEL -1 Sine-sweep at 168 N -2 20 25 30 35 40 Sweep frequency (Hz) UNDERSTD UNCOVER DESIGN 50

Eliminating Quasiperiodicity by Modifying Damping Inertia wheel amplitude ( ) MEASURE 1.32 Nominal value IDENTIFY 1.30 Annihilation of the NS bifurcations MODEL 1.28 1.26 60 65 70 75 80 Elastomer damping coefficient (Ns/m) 85 UNDERSTD UNCOVER DESIGN 51

Confirmation of the Elimination of QP Inertia wheel displacement ( ) MEASURE 2 1 Nominal design Improved design IDENTIFY 0 MODEL -1 UNDERSTD UNCOVER -2 20 25 30 35 40 Sweep frequency (Hz) 45 DESIGN 52

Concluding Remarks This is the uninformed/biased view of a mechanical engineer EE/CONTROL Very solid theoretical framework (MLE, prediction-error ID, subspace ID) MECHANICS Toolbox philosophy with ad hoc methods Essentially black box Often white box Explain the data Explain the system Identification for control Identification for design Importance of features (modes and FRFs) 53

Further Points for Discussion Choice of excitation signal (sine sweep vs. periodic random). Noise analysis and uncertainty bounds. Friction is challenging mechanical engineers. 54

- Acceleration (m/s 2 ) Thank you for your attention! 5 Force level: 67 N rms Gaëtan Kerschen Space Structures and Systems Laboratory Aerospace and Mechanical Eng. Dept. University of Liège 4 3 2 1 0-1 -2-3 -4-5 -4-3 -2-1 0 1 2 3 4 Relative displacement (m) x 10-4 Colleagues and Collaborators: J.P. Noël, J. Schoukens, K. Worden, B. Peeters