Diapositive 1 · Title: Diapositive 1 Author: Maxime Peeters Created Date: 4/29/2016 10:21:54 PM

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

-4 -3 -2 -1 0 1 2 3 4

x 10-4

-5

-4

-3

-2

-1

0

1

2

3

4

5Force level: 67 N rms

Relative displacement (m)

- A

ccele

ration (

m/s

2)

Colleagues and Collaborators:

J.P. Noël, J. Schoukens, K. Worden, B. Peeters

2

Why Is NSI Important in Mechanical Engineering?

Contact nonlinearities

300% error between predictions and measurements

3

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

4

OutlineOutline

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.

5

3 Main Assumptions for Linear Mechanical Systems

M ሷx(t) + C ሶx(t) + Kx(t) = f(t)

Linear elasticity → nonlinear materials

Viscous damping → nonlinear damping mechanisms

Small displ. and rotations → nonlinear boundary conditions

→ geometric nonlinearity

6

Stress

Strain

Stress

Strain

Hyperelastic material

(e.g., rubber)

Shape memory alloy

Source 1: Material Nonlinearities

7

Anti-Vibration Mounts of a Helicopter Cockpit

Decrease in stiffness

Decrease in damping

A. Carrella, IJMS 2012

Rubber

8

ሷ𝜃 + 𝜔02 sin 𝜃 = 0

sin 𝜃 = 𝜃 −𝜃3

6+…

𝑙0

𝑙1𝑥

𝐹 = 2𝑘 𝑙1 − 𝑙0𝑥

𝑙1= 2𝑘𝑥 1 −

𝑙0

𝑥2 + 𝑙02

𝑙0

𝑥2 + 𝑙02

= 1 −𝑥2

2𝑙02 +

3𝑥4

8𝑙04 + 𝑂(𝑥6)

Increase in stiffnessDecrease in stiffness

Source 2: Displacement-Related Nonlinearities

http://en.wikipedia.org/wiki/File:SimplePendulum01.JPG

9

Beam @ ULg

Increase in stiffness

Frequency (Hz)

Displ.

(m)

A Widely-Used Benchmark in Mech. Engineering

10

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

11

Contact Can Generate Complex Dynamics

5 20 30 70-100

0

100

Accel.

Sweep frequency (Hz)

Acc.

(m/s2)

0.1 g

1 gexcitation

EADS Astrium satellite

Dangerous nonlinear

resonance

12

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

13

SLIDING

Sliding Connection in the F-16 Aircraft

14

42 6

-40

-60

-808 10

-20

Low

HighFRF

Frequency (Hz)

Decrease in stiffness

Increase in damping

Impact of the Connection on the F-16 Dynamics

15

OutlineOutline

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.

16

A Three-Step Process

More information but

increasingly difficult

1. Detection

2. Characterization

3. Parameter

estimation

Yes or No ?

What ? Where ? How ?

How much ?

xxxxxfnl ,sin,),( 3

?

333 3,2.1,1.0),( xxxxxfnl ?

17

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)

18

White-Box Approach Commonplace in Mechanics

Often, reasonably accurate low-dimensional models can

be obtained from first principles.

19

But Not Always: Individualistic Nature of Nonlinearities

Elastic NL

(grey-box)

Damping NL

(black-box or further analysis)

20

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

Right missile Connection with the wing

21

Apply Your Favorite Modal Analysis Software

FRF

Frequency (Hz)6.5 7.5

F-16

Introduction of two poles

around one nonlinear mode

22

Measured Time Series

Nonlinearity can often be seen in raw acceleration signals.

7.2 Hz

Time (s)

Acc.

(m/s^2)

23

Measured Frequency Response Functions

4 6 8 10 12 14-55

-50

-45

-40

-35

-30

-25

-20

-15

Frequency (Hz)

Am

plitu

de

(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

of nonlinearity.

7.2 Hz

24

Time-Frequency Analysis (Wavelet Transform)

Objective: know more about the nonlinear distortions.

Frequency

(Hz)

Time (s)

25

Restoring Force Surface Method (Masri & Caughey)

Nonlinear

connection

instrumented

on both sides.

Objective: visualize nonlinearities.

26

Restoring Force Surface Method

27

A Three-Step Process

More information but

increasingly difficult

1. Detection

2. Characterization

3. Parameter

estimation

Yes or No ?

What ? Where ? How ?

How much ?

xxxxxfnl ,sin,),( 3

?

333 3,2.1,1.0),( xxxxxfnl ?

28

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

29

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.

30

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)

31

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.

32

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:

33

Finite Element Model Updating

FE model

O.F. min.

Correlation

Poor

???

Feature extraction

Experimental

data

???

Feature extraction

Parameter

selection

Model

updating Good Reliable model

34

Where Do We Stand ?

First contributions

1980s1970s 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)

35

MEASURE MODELIDENTIFYUNDERSTD

UNCOVERDESIGN

Computer-aided

modelling (FEM, …)

Identification for Design

F-16 aircraft

SmallSat

spacecraft

36

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

37

50

100

-50

-100

0

Acceleration (m/s²)

1 g base0.1 g base


30205 10 5040 60 70

?

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

Troubleshooting Clearly Needed

38

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

24 Accels Close to the Suspected Nonlinear Device

MEASURE

39

Wavelet Transform: Nonsmooth Nonlinearity

60

100

20

5

40

Instantaneous frequency (Hz)


975 11 13

80

MEASURE

MODEL

UNDERSTD

UNCOVER

DESIGN

IDENTIFY

40

Time Series: Clearance Identification

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

1

2

-1

-2

0

Relative displacement ( – )


975 7.5 119.4 13

Jump

Discontinuity

41

0.6 g

1 g

-Acc.

-Acc.

Rel. displ.

Rel. displ.

Acceleration Surface: Nonlinearity Visualization

MEASURE

MODEL

UNDERSTD

UNCOVER

DESIGN

IDENTIFY

42

Experimental Model of the Nonlinearity

MEASURE

MODEL

UNDERSTD

UNCOVER

DESIGN

IDENTIFY

Relative displacement

0-1 1-2 2

Resto

ring f

orc

e (

N)

-800

-400

400

800

0

1.6

Fitted model

Experimental data

43

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

Linear main structure =

fairly easy to model

numerically.

Nonlinear component =

difficult to model numerically.

Finite Element Modeling

44

Remember

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

50

100

-50

-100

0

Acceleration (m/s²)

1 g base0.1 g base


30205 10 5040 60 70

?

45

Objective: investigate the nonlinear resonances.

Nonlinear Modal Analysis

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

46

Nonlinear Mode #6 at Low Energy Level

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

Energy (J)

Frequency

(Hz)

47

Nonlinear Mode #6 at Higher Energy Level

Energy (J)

Frequency

(Hz)

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

48

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 !

49

Troubleshooting: Problem Solved!

excitation

100

105

28.5

30

31.3

Energy (J)

Frequency

(Hz)

2:1 interaction

with NNM12

5 20 30 70-100

0

100

50

Uncovering Quasiperiodicity using Bifurcations

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

0

2

-2

-1

Inertia wheel displacement ( – )


302520 35 40

1

Sine-sweep at 168 N

51

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)

706560 75 80

1.30

85

Nominal value

Annihilation of the

NS bifurcations

52

Confirmation of the Elimination of QP

MEASURE

MODEL

IDENTIFY

UNDERSTD

UNCOVER

DESIGN

0

2

-2

-1

Inertia wheel displacement ( – )


302520 35 40

1

Nominal design

Improved design

45

53

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

54

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

-4 -3 -2 -1 0 1 2 3 4

x 10-4

-5

-4

-3

-2

-1

0

1

2

3

4

5Force level: 67 N rms

Relative displacement (m)

- A

ccele

ration (

m/s

2)

Colleagues and Collaborators:

J.P. Noël, J. Schoukens, K. Worden, B. Peeters

Diapositive 1 · Title: Diapositive 1 Author: Maxime Peeters Created Date: 4/29/2016 10:21:54 PM

Documents

Transcript of Diapositive 1 · Title: Diapositive 1 Author: Maxime Peeters Created Date: 4/29/2016 10:21:54 PM