Identification of Nonlinear Mechanical Systems: State of the Art and - - PowerPoint PPT Presentation

Gaëtan Kerschen

Space Structures and Systems Laboratory Aerospace and Mechanical Eng. Dept. University of Liège

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

Colleagues and Collaborators: J.P. Noël, J. Schoukens, K. Worden, B. Peeters

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Why Is NSI Important in Mechanical Engineering?

Contact nonlinearities  300% error between predictions and measurements

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Why Is NSI Important in Mechanical Engineering?

FRF Frequency (Hz)

6.5 7.5 F-16

Introduction of two poles around one nonlinear mode

 Linear tools and methods may fail

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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.

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3 Main Assumptions for Linear Mechanical Systems

M ሷ x(t) + C ሶ x(t) + Kx(t) = f(t)

Linear elasticity

→ nonlinear materials

Viscous damping

→ nonlinear damping mechanisms

Small displ. and rotations

→ nonlinear boundary conditions → geometric nonlinearity

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Stress Strain Stress Strain

Hyperelastic material (e.g., rubber) Shape memory alloy

Source 1: Material Nonlinearities

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Anti-Vibration Mounts of a Helicopter Cockpit

Decrease in stiffness Decrease in damping

• A. Carrella, IJMS 2012

Rubber

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ሷ 𝜄 + 𝜕02 sin 𝜄 = 0 sin 𝜄 = 𝜄 −

𝜄3 6 +…

𝑚0 𝑚1 𝑦

𝐺 = 2𝑙 𝑚1 − 𝑚0 𝑦 𝑚1 = 2𝑙𝑦 1 − 𝑚0 𝑦2 + 𝑚0

2

𝑚0 𝑦2 + 𝑚0

2

= 1 − 𝑦2 2𝑚0

2 + 3𝑦4

8𝑚0

4 + 𝑃(𝑦6)

Increase in stiffness Decrease in stiffness

Source 2: Displacement-Related Nonlinearities

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Beam @ ULg

Increase in stiffness

Frequency (Hz) Displ. (m)

A Widely-Used Benchmark in Mech. Engineering

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

The SmallSat Spacecraft

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Contact Can Generate Complex Dynamics

5 20 30 70

• 100

100 Accel.

Sweep frequency (Hz) Acc. (m/s2) 0.1 g 1 g excitation EADS Astrium satellite

Dangerous nonlinear resonance

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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.

Source 3: Damping Nonlinearities

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SLIDING

Sliding Connection in the F-16 Aircraft

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4 2 6

• 40
• 60
• 80

8 10

• 20

Low High

FRF Frequency (Hz)

Decrease in stiffness Increase in damping

Impact of the Connection on the F-16 Dynamics

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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.

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A Three-Step Process

More information but increasingly difficult

• 1. Detection
• 2. Characterization
• 3. Parameter

estimation Yes or No ? What ? Where ? How ? How much ?

x x x x x fnl   , sin , ) , (

3



?

3 3 3

3 , 2 . 1 , 1 . ) , ( x x x x x fnl    ?

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Three Main Differences

The quantification of the impact of nonlinearity is not often performed. White-box approach commonplace. It is only very recently that uncertainty is accounted for (e.g. Beck, Worden et al.). Best-linear approximation (Schoukens et al.). The black-box approach seems to be very popular. 1965 (!): Astrom and Bohlin introduced the maximum likelihood framework. MECHANICS EE/CONTROL

A reputable engineer should never deliver a model without a statement about its error margins (M. Gevers, IEEE Control Systems Magazine, 2006)

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White-Box Approach Commonplace in Mechanics

Often, reasonably accurate low-dimensional models can be obtained from first principles.

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But Not Always: Individualistic Nature of Nonlinearities

Elastic NL (grey-box) Damping NL (black-box or further analysis)

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Case Study: The F-16 Aircraft (With VUB & Siemens)

Right missile Connection with the wing

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Apply Your Favorite Modal Analysis Software

FRF Frequency (Hz)

6.5 7.5 F-16

Introduction of two poles around one nonlinear mode

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Measured Time Series

Nonlinearity can often be seen in raw acceleration signals.

7.2 Hz Time (s) Acc. (m/s^2)

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Measured Frequency Response Functions

4 6 8 10 12 14

• 55
• 50
• 45
• 40
• 35
• 30
• 25
• 20
• 15

Amplitude (dB)

FRF amplitude, sensor [4] DP_RIGHT:-Z

Frequency (Hz) Ampl. (dB) Low level High level

At this stage, we should be convinced about the presence

• f nonlinearity.

7.2 Hz

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Time-Frequency Analysis (Wavelet Transform)

Objective: know more about the nonlinear distortions.

Frequency (Hz) Time (s)

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Restoring Force Surface Method (Masri & Caughey)

Nonlinear connection instrumented

• n both sides.

Objective: visualize nonlinearities.

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Restoring Force Surface Method

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A Three-Step Process

More information but increasingly difficult

• 1. Detection
• 2. Characterization
• 3. Parameter

estimation Yes or No ? What ? Where ? How ? How much ?

x x x x x fnl   , sin , ) , (

3



?

3 3 3

3 , 2 . 1 , 1 . ) , ( x x x x x fnl    ?

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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

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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.

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Time-Frequency Analysis

Hilbert transform and its generalization to multicomponent signals (empirical mode decomposition). Wavelet transform.

Instantaneous frequency (Hz) Time (s) Displacement (m) Time (s)

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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.

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Modal Methods

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

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Finite Element Model Updating

FE model O.F. min. Correlation Poor

???

Feature extraction

Experimental data

???

Feature extraction

Parameter selection Model updating Good Reliable model

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Where Do We Stand ?

First contributions

1980s 1970s 1990s-2000 Today

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)

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MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

Computer-aided modelling (FEM, …)

Identification for Design

F-16 aircraft SmallSat spacecraft

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

The SmallSat Spacecraft

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50 100

• 50
• 100

Acceleration (m/s²) 1 g base 0.1 g base Sweep frequency (Hz) 30 20 5 10 50 40 60 70

?

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

Troubleshooting Clearly Needed

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MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

24 Accels Close to the Suspected Nonlinear Device

MEASURE

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Wavelet Transform: Nonsmooth Nonlinearity

60 100 20 5 40 Instantaneous frequency (Hz) Sweep frequency (Hz) 9 7 5 11 13 80

MEASURE MODEL UNDERSTD UNCOVER DESIGN IDENTIFY

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Time Series: Clearance Identification

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

1 2

• 1
• 2

Relative displacement ( – ) Sweep frequency (Hz) 9 7 5 7.5 11 9.4 13

Jump Discontinuity

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0.6 g 1 g

• Acc.
• Acc.
• Rel. displ.
• Rel. displ.

Acceleration Surface: Nonlinearity Visualization

MEASURE MODEL UNDERSTD UNCOVER DESIGN IDENTIFY

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Experimental Model of the Nonlinearity

MEASURE MODEL UNDERSTD UNCOVER DESIGN IDENTIFY

Relative displacement

• 1

1

• 2

2

Restoring force (N)

• 800
• 400

400 800

1.6 Fitted model Experimental data

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MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

Linear main structure = fairly easy to model numerically. Nonlinear component = difficult to model numerically.

Finite Element Modeling

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Remember

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

50 100

• 50
• 100

Acceleration (m/s²) 1 g base 0.1 g base Sweep frequency (Hz) 30 20 5 10 50 40 60 70

?

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Objective: investigate the nonlinear resonances.

Nonlinear Modal Analysis

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

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Nonlinear Mode #6 at Low Energy Level

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

Energy (J) Frequency (Hz)

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Nonlinear Mode #6 at Higher Energy Level

Energy (J) Frequency (Hz)

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

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Understand: 6th Nonlinear Mode of the Satellite

NNM6: local deformation at low energy level NNM6: global deformation at a higher energy level (on the modal interaction branch)

The top floor vibrates two times faster !

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Troubleshooting: Problem Solved!

excitation

10 10

5

28.5 30 31.3 Energy (J) Frequency (Hz) 2:1 interaction with NNM12

5 20 30 70

• 100

100

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Uncovering Quasiperiodicity using Bifurcations

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

2

• 2
• 1

Inertia wheel displacement ( – ) Sweep frequency (Hz) 30 25 20 35 40 1

Sine-sweep at 168 N

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Eliminating Quasiperiodicity by Modifying Damping

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

1.28 1.32 1.26 Inertia wheel amplitude ( – ) Elastomer damping coefficient (Ns/m) 70 65 60 75 80 1.30 85 Nominal value Annihilation of the NS bifurcations

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Confirmation of the Elimination of QP

MEASURE MODEL IDENTIFY UNDERSTD UNCOVER DESIGN

2

• 2
• 1

Inertia wheel displacement ( – ) Sweep frequency (Hz) 30 25 20 35 40 1

Nominal design Improved design

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Concluding Remarks

Very solid theoretical framework (MLE, prediction-error ID, subspace ID) Essentially black box Explain the data Identification for control Toolbox philosophy with ad hoc methods Often white box Explain the system Identification for design Importance of features (modes and FRFs) EE/CONTROL MECHANICS

This is the uninformed/biased view of a mechanical engineer

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Further Points for Discussion

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

Thank you for your attention!

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