Nonlinear Vibrations of

Aerospace Structures

Jean-Philippe Noël | jp.noel@uliege.be 22 October 2018

Detecting Nonlinearity

Lecture

2

System Identification in Structural Dynamics?

Construction of an accurate mathematical model of the dynamics

of a real structure based on input and output measurements.

𝑦 = ℱ(𝑢)ℱ is a dynamic input-

output mapping

Input

𝑢(𝑡)

Output

𝑦(𝑡)

3

System Identification: 3 Basis Ingredients and User-choices

Data + Model + Distance

𝑦 = ℱ(𝑢)ℱ is a dynamic input-

output mapping

Input

𝑢(𝑡)

Output

𝑦(𝑡)

4

Linear System Identification (= Modal Analysis) Is Mature

AIRBUS A350XWBGovers et al., ISMA 2014.

Theoretical modal

analysis for design

Experimental modal

analysis for certification

(eigenvalue solver in all

commercial FE packages).

(stochastic subspace

identification, PolyMAX, …).

5

But Every Modern Mechanical System Is Nonlinear …

Force amplitude (N)

0 5 10 15 20 25 30

0.58

0.55

0.56

0.57

5

2

3

4

Natural

freq. (Hz)

Damping

ratio (%)

> 5 % shift

> 180 % shift

A350 ground vibration testing, Spring 2013,

Suspected nonlinearity sources:

clearance in actuators, wing slenderness,

engine mounts, fuselage composite, …

“Linearity” plot!

6

An example from SONACA: the Piccolo tube.

Stress underestimated by more than to 200 % led to cracks in a weld.

… and FE Models Fail to Capture Most Nonlinearities

Forcing frequency (Hz)

1e-1

100 1000

Stress Power Spectral

Density (daN/mm²/Hz)

1e-3

1e-5

1e-7

Experimental test

Nonlinear FEM in

Samcef Mecano

Linear FEM

10

7

… and Commercial Testing Software Fail to Identify Them

FRF (dB)

Frequency (Hz)

6.6

F16 FighterPeeters et al., IMAC 2011.

7

2 poles around

1 nonlinear mode

6.8 7.2

8

NSI in the Design Cycle of Eng. Structures = Troubleshooting!

MEASURE MODELIDENTIFYUNDERSTD

UNCOVERDESIGN

Computer-aided

modelling (FEM, …)

9

Where Do We Stand?

1980s 1990s 2000s Today

First analyses of

SDOF systems.

MDOF systems and

continuous structures with

localised nonlinearities.

Real-life structures

with strong nonlinearities,

nonlinear modal analysis,

numerical model updating.

10

Outline of this Lecture

What are the challenges in nonlinear system identification?

How to build a nonlinear experimental model?

Nonlinearity detection in sine conditions: SmallSat spacecraft.

Nonlinearity detection in broadband conditions: F-16 aircraft.

11

Outline of this Lecture

What are the challenges in nonlinear system identification?

How to build a nonlinear experimental model?

Nonlinearity detection in sine conditions: SmallSat spacecraft.

Nonlinearity detection in broadband conditions: F-16 aircraft.

12

Why Is Nonlinear System Identification Difficult?

Challenge #1: Sensitivity to the type of input signal.

Distorted resonance

under random excitation.

Distorted resonance

under sine excitation.

Time (s) Frequency (Hz)

Amplitude (m) Amplitude (dB)

13

Why Is Nonlinear System Identification Difficult?

Challenge #2: Specific and demanding test campaigns.

Increase the sampling frequency.

Record throughput time histories.

Measure the force signal.

Instrument potential nonlinearities with sensors on both sides.

Consider multiple levels of excitations/types of excitations.

Study multiple sets of initial conditions, and reverse the sweep.

14

Why Is Nonlinear System Identification Difficult?

Challenge #3: Inapplicability of linear concepts.

FRFs are not invariant.Modal properties are not invariant.

Energy (J) Frequency (Hz)

Frequency (Hz) Amplitude (dB)

15

Why Is Nonlinear System Identification Difficult?

Challenge #4a: Individualistic nature of structural nonlinearities.

Disp.

Disp.

Vel.

Disp.Re

st.

fo

rce

Re

st.

fo

rce

Re

st.

fo

rce

Re

st.

fo

rce

16

Why Is Nonlinear System Identification Difficult?

Challenge #4b: Limited amount of prior knowledge.

𝑀 ሷ𝑞 + 𝐶𝑣 ሶ𝑞 + 𝐾 𝑞 + 𝑔(𝑞, ሶ𝑞) = 𝑝

Determining nonlinear functional forms is an integral part of

nonlinear system identification (and arguably the most difficult).

?

17

Why Is Nonlinear System Identification Difficult?

Challenge #5: True system may be (is!) outside the model class.

Selected model class

Tested system

18

Outline of this Lecture

What are the challenges in nonlinear system identification?

How to build a nonlinear experimental model?

Nonlinearity detection in sine conditions: SmallSat spacecraft.

Nonlinearity detection in broadband conditions: F-16 aircraft.

19

Three-Step NL System ID Process in Structural Dynamics

Do I observe nonlinear effects? Yes.

Should I build a nonlinear model? Yes.

3

2

1

Detection

Characterisation

Estimation

Information

vs. complexity

Where is the nonlinearity located? At the joint.

What is the underlying physics? Dry friction.

How to model its effects? 𝑓𝑛𝑙 𝑞, ሶ𝑞 = 𝑐 𝑠𝑖𝑔𝑛 ሶ𝑞 .

Model parameters?

How uncertain are they?

𝑐 = 5.47.

𝑐 = 𝒩(5.47,1).

20

Nonlinearity Detection

Evidence nonlinear behaviour in experimental data.

This is not particularly challenging:

This is crucial:

A wide variety of techniques exists in the literature.

Sine and broadband excitations can be addressed.

A single excitation level is generally sufficient.

Detection supports an important decision as building a nonlinear model

demands more time, capabilities and efforts than a linear model.

21

Nonlinearity Characterisation

Infer a suitable nonlinearity model from experimental data.

This is challenging:

This is crucial:

Prior knowledge is most often very limited.

Physical mechanisms resulting in nonlinearity are extremely diverse.

Nonlinearity may translate into a plethora of dynamic phenomena.

The success of the parameter estimation step is conditional upon an

accurate characterisation of all observed nonlinearities.

22

Nonlinear Parameter Estimation

Fit a mathematical model to experimental data.

This is challenging:

This is crucial:

If characterisation was not successfully completed.

If nonlinearities were not instrumented.

If the input signals were not measured.

Quantitative models are required to carry out response prediction,

numerical model updating and validation, design optimisation, …

23

Outline of this Lecture

What are the challenges in nonlinear system identification?

How to build a nonlinear experimental model?

Nonlinearity detection in sine conditions: SmallSat spacecraft.

Nonlinearity detection in broadband conditions: F-16 aircraft.

24

A First Test Case: the SmallSat Spacecraft from Airbus DS

NL WEMS device

Challenges:

► Nonsmooth nonlinearities

► High damping

► Gravity-induced asymmetry

► Viscoelastic components

25

WEMS Device: Piecewise-linear Behaviour

Elastomer plots

Isolation and

protection device.

Mechanical stops

Experimental

stiffness curve.

Displacement

Resto

ring f

orc

e

26

A “Good” Excitation: the Sine-sweep Signal

Linear sine sweep:

𝑢(𝑡) = 𝐴 𝑠𝑖𝑛 2𝜋𝑓0 𝑡 − 𝑡0 + 2𝜋𝑟

2𝑡 − 𝑡0

2 + 𝜑0

Time (s) Time (s)

Force (N) Forcing frequency (Hz)

27

A “Good” Excitation: the Sine-sweep Signal

Linear sine sweep:

Fully deterministic signal, so easy to interpret data visually.

Strongly activates nonlinearities, as energy is concentrated.

Very useful in nonlinearity characterisation.

Logarithmic sweep may be used to focus on low frequencies.

𝑢(𝑡) = 𝐴 𝑠𝑖𝑛 2𝜋𝑓0 𝑡 − 𝑡0 + 2𝜋𝑟

2𝑡 − 𝑡0

2 + 𝜑0

28

A “Good” Excitation: the Sine-sweep Signal

Log sine sweep:

𝑢(𝑡) = 𝐴 𝑠𝑖𝑛2𝜋 𝑓0

log 2 𝑟2𝑟 𝑡−𝑡0 − 1 + 𝜑0

Time (s) Time (s)

Force (N) Forcing frequency (Hz)

29

Envelope-based Analysis of a Raw Time History

Sweep frequency (Hz)

5 9

Relative displacement ( – )

-213

2

0

1

-1

7 11

Axial sine sweep at 1 g,

positive rate of 4 oct/min

7.5 9.4

30

Skewness and Nonsmoothness Result in a Jump

Sweep frequency (Hz)

5 9

Relative displacement ( – )

-213

2

0

1

-1

7 117.5 9.4

Axial sine sweep at 1 g,

positive rate of 4 oct/min

31

Discontinuity in Slope due to a Clearance in the System

Sweep frequency (Hz)

5 9

Relative displacement ( – )

-213

2

0

1

-1

7 11

Axial sine sweep at 1 g,

positive rate of 4 oct/min

7.5 9.4

32

Asymmetry owing to Gravity-induced Prestress

Sweep frequency (Hz)

5 9

Relative displacement ( – )

-213

2

0

1

-1

7 11

Axial sine sweep at 1 g,

positive rate of 4 oct/min

7.5 9.4

33

Breakdown of the Principle of Superposition

Sweep frequency (Hz)

5 9

Relative displacement ( – )

-2

2

0

1

-1

7

1 g/ up

0.6 g/ up

0.4 g/ up

0.1 g/ up

34

Existence of Nonlinear Hysteresis

Sweep frequency (Hz)

5 9

Relative displacement ( – )

-2

2

0

1

-1

7

1 g/ up

1 g/ down

0.4 g/ up

0.4 g/ down

35

A Close-up Reveals the Appearance of Strong Harmonics

Sweep frequency (Hz)

8.4

Relative displacement ( – )

-2

2

0

1

-1

8.5

36

Outline of this Lecture

What are the challenges in nonlinear system identification?

How to build a nonlinear experimental model?

Nonlinearity detection in sine conditions: SmallSat spacecraft.

Nonlinearity detection in broadband conditions: F-16 aircraft.

37

A Second Test Case: an F-16 Fighter Aircraft

F16 aircraft,

Saffraanberg,

Belgium.

Joints between

substructures are

generic sources of NLs.

38

Another “Good” Excitation: the Multisine Signal

Multisine with uniformly-distributed random phases:

𝑢(𝑡) = 𝑁−1/2𝑘𝑈𝑘 𝑒𝑥𝑝 𝑗2𝜋 𝑘

𝑓𝑠𝑁𝑡 + 𝑗 𝜑𝑘

Frequency (Hz)

Amplitude (dB)

Frequency (Hz)

Phase (rad)

3.14

-3.14

39

Multisine with uniformly-distributed random phases:

Random appearance in TD, but deterministic amplitudes in FD.

All frequencies excited throughout the test.

Periodicity allows to remove transients and characterise noise.

Gaussian signal for a sufficiently large number of frequencies.

Very useful in nonlinear parameter estimation.

Another “Good” Excitation: the Multisine Signal

𝑢(𝑡) = 𝑁−1/2𝑘𝑈𝑘 𝑒𝑥𝑝 𝑗2𝜋 𝑘

𝑓𝑠𝑁𝑡 + 𝑗 𝜑𝑘

40Frequency (Hz)

2 4

Amplitude (dB)

-40

-20

-806 108

-60

Superposed FRFs do not Match at Two Excitation Levels

12.4 N RMS

73.6 N RMS

41

𝑓/𝑓0

NL Detection using a Carefully-selected Input Spectrum

𝑓/𝑓0

𝑖𝑛𝑝𝑢𝑡

𝑓/𝑓0

𝑜𝑢𝑡𝑝𝑢𝑡

=

1 1198 1072 40 63 5

1 1198 1072 40 63 5

𝑛𝑜𝑖𝑠𝑒

𝑓/𝑓0

𝑙𝑖𝑛𝑒𝑎𝑟

1 1198 1072 40 63 5 𝑓/𝑓0

𝑜𝑑𝑑

1 1198 1072 40 63 5

𝑓/𝑓01 1198 1072 40 63 5

𝑒𝑣𝑒𝑛

1 1198 1072 40 63 5

42Frequency (Hz)

2

Amplitude (dB)

-20

20

-80

5 1510

-60

F-16 Measurement at Low Excitation Level

-40

0

Output Even

Noise Odd

43Frequency (Hz)

2

Amplitude (dB)

-20

20

-80

5 1510

-60

Output Even

Noise Odd

Odd and Even NLs Are Clearly Detected at High Level

-40

0

44

7.2 Hz (mode 3)

5.2 Hz (mode 1)9.2 Hz

(mode 4)

Sine-sweep Time Series at Low Level (on the Right Wing)

45

9.1 Hz (mode 4)

7 Hz

(mode 3)

5 Hz (mode 1)

6.3 Hz (mode 2)

Sine-sweep Time Series at High Level (on the Right Wing)

46

Not affected AFFECTED

AFFECTED

Nonlinearity Location Can Be Inferred from Mode Shapes

47

Low level

High level

Check for a departure from a

planar motion in phase space

7.3 Hz

mode

Map for Nonlinearity Detection and Location

48

Low level

High level

Nonlinearity Location Is Clearly Revealed

49

The Source of Nonlinearity … See Next Lecture

Displacement (mm)

-0.4 0

– Acceleration (m/s²)

-50.4

5

0

2.5

-2.5

Stiffness

nonlinearity!

-0.2 0.2

50

The Source of Nonlinearity … See Next Lecture

– Acceleration (m/s²)

Damping

nonlinearity!

-3

3

0

Velocity (mm/s)

-20-40 0 20 40

51

Concluding Remarks

Nonlinear system identification is a three-step process.

Nonlinearity detection is a fairly easy task, but supports an

important decision.

Sine and broadband data are addressed using specific techniques

in a toolbox philosophy.

52

Further Reading

J.P. Noël, G. Kerschen, Nonlinear system identification in structural

dynamics: 10 more years of progress, Mechanical Systems and Signal

Processing, 2016.

J. Schoukens, M. Vaes, R. Pintelon, Linear system identification in a

nonlinear setting, IEEE Control System Magazine, 2016.

Nonlinear Vibrations of

Aerospace Structures

Jean-Philippe Noël | jp.noel@uliege.be 22 October 2018

Detecting Nonlinearity

Lecture