Identification of Nonlinear Mechanical Systems: State of the Art and Recent Trends Gaëtan Kerschen Space Structures and Systems Laboratory Aerospace and Mechanical Eng. Dept. University of Liège 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 7.5 Frequency (Hz) Linear tools and methods may fail 3
Outline 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 M ሷ x(t) + C ሶ x(t) + Kx(t) = f(t) → nonlinear materials Linear elasticity → geometric nonlinearity Small displ. and rotations → nonlinear boundary conditions → nonlinear damping mechanisms Viscous damping 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 Decrease in stiffness Decrease in damping Rubber A. Carrella, IJMS 2012 7
ሷ Source 2: Displacement-Related Nonlinearities Decrease in stiffness Increase in stiffness 𝑚 0 𝑚 1 𝑦 𝑦 𝑚 0 𝜄 + 𝜕 0 2 sin 𝜄 = 0 𝐺 = 2𝑙 𝑚 1 − 𝑚 0 = 2𝑙𝑦 1 − 𝑚 1 𝑦 2 + 𝑚 0 2 𝜄 3 = 1 − 𝑦 2 2 + 3𝑦 4 𝑚 0 sin 𝜄 = 𝜄 − 6 +… 4 + 𝑃(𝑦 6 ) 2𝑚 0 8𝑚 0 𝑦 2 + 𝑚 0 2 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 NL isolation device for Elastomer plots (EADS Astrium) reaction wheels Goals Solutions Micro-vibration Elastomer plots mitigation Large amplitude Mechanical stops limitation 10
Contact Can Generate Complex Dynamics 100 Dangerous nonlinear resonance Accel. 1 g excitation 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 Decrease in stiffness Increase in damping -20 Low -40 High FRF -60 -80 2 4 6 8 10 Frequency (Hz) 14
Outline 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 ? 2. Characterization ? 3 f nl ( x , x ) x , sin x , x How much ? 3. Parameter ? estimation 3 3 3 f nl ( x , x ) 0 . 1 x , 1 . 2 x , 3 x More information but increasingly difficult 16
Three Main Differences MECHANICS EE/CONTROL The quantification of the impact of Best-linear approximation nonlinearity is not often performed. (Schoukens et al.). White-box approach commonplace. The black-box approach seems to be very popular. It is only very recently that 1965 (!): Astrom and Bohlin uncertainty is accounted for introduced the maximum (e.g. Beck, Worden et al.). 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 Damping NL (grey-box) (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 7.5 Frequency (Hz) 21
Measured Time Series Nonlinearity can often be seen in raw acceleration signals. Acc. (m/s^2) 7.2 Hz Time (s) 22
Measured Frequency Response Functions FRF amplitude, sensor [4] DP_RIGHT:-Z -15 7.2 Hz Low level -20 High level -25 Amplitude (dB) -30 Ampl. -35 (dB) -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 ? 2. Characterization ? 3 f nl ( x , x ) x , sin x , x How much ? 3. Parameter ? estimation 3 3 3 f nl ( x , x ) 0 . 1 x , 1 . 2 x , 3 x 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. Instantaneous frequency (Hz) Displacement (m) 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 Experimental FE model data Feature extraction Feature extraction ??? ??? Correlation Model Poor Good Reliable model updating Parameter selection O.F. min. 33
Where Do We Stand ? Today 1970s 1980s 1990s-2000 First contributions Focus on 1DOF: Focus on MDOF: Large-scale structures with Hilbert, Volterra NARMAX, localized nonlinearities: frequency-domain uncertainty quantification, ID, finite element extension of linear algorithms model updating (nonlinear subspace ID) 34
Identification for Design Computer-aided modelling (FEM, …) UNDERSTD MEASURE IDENTIFY DESIGN MODEL UNCOVER F-16 aircraft SmallSat spacecraft 35
The SmallSat Spacecraft Mechanical stops SmallSat spacecraft NL isolation device for Elastomer plots (EADS Astrium) reaction wheels Goals Solutions Micro-vibration Elastomer plots mitigation Large amplitude Mechanical stops limitation 36
Troubleshooting Clearly Needed Acceleration (m/s²) MEASURE 100 ? IDENTIFY 50 0 MODEL -50 UNDERSTD 1 g base UNCOVER 0.1 g base -100 5 10 20 30 40 50 60 70 Sweep frequency (Hz) 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 IDENTIFY 80 60 MODEL 40 UNDERSTD 20 UNCOVER 5 5 7 9 11 13 Sweep frequency (Hz) DESIGN 39
Time Series: Clearance Identification Relative displacement ( – ) MEASURE 2 Jump IDENTIFY 1 0 MODEL -1 UNDERSTD Discontinuity 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 Rel. displ. MODEL 1 g UNDERSTD -Acc. UNCOVER DESIGN Rel. displ. 41
Experimental Model of the Nonlinearity 800 MEASURE Fitted model Experimental data 400 IDENTIFY Restoring force (N) 0 MODEL -400 UNDERSTD UNCOVER -800 -2 0 1 2 -1 1.6 DESIGN Relative displacement 42
Finite Element Modeling MEASURE Linear main structure = fairly easy to model numerically. IDENTIFY MODEL UNDERSTD UNCOVER DESIGN Nonlinear component = difficult to model numerically. 43
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