# Nonlinear Vibrations of Aerospace Structures A Second Test Case: an F-16 Fighter Aircraft F16...

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Nonlinear Vibrations of

Aerospace Structures

Jean-Philippe NoΓ«l | 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.

π¦ = β±(π’) β± is a dynamic input-

output mapping

Input

π’(π‘)

Output

π¦(π‘)

3

System Identification: 3 Basis Ingredients and User-choices

Data + Model + Distance

π¦ = β±(π’) β± is a dynamic input-

output mapping

Input

π’(π‘)

Output

π¦(π‘)

4

Linear System Identification (= Modal Analysis) Is Mature

AIRBUS A350XWB Govers 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, β¦).

5

But Every Modern Mechanical System Is Nonlinear β¦

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, β¦

βLinearityβ plot!

6

An example from SONACA: the Piccolo tube.

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

β¦ and FE Models Fail to Capture Most Nonlinearities

Forcing frequency (Hz)

1e-1

100 1000

Stress Power Spectral

Density (daN/mmΒ²/Hz)

1e-3

1e-5

1e-7

Experimental test

Nonlinear FEM in

Samcef Mecano

Linear FEM

10

7

β¦ and Commercial Testing Software Fail to Identify Them

FRF (dB)

Frequency (Hz)

6.6

F16 Fighter Peeters 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 MODELIDENTIFY UNDERSTD

UNCOVER DESIGN

Computer-aided

modelling (FEM, β¦)

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.R e

s t.

f o

rc e

R e

s t.

f o

rc e

R e

s t.

f o

rc e

R e

s t.

f o

rc e

16

Why Is Nonlinear System Identification Difficult?

Challenge #4b: Limited amount of prior knowledge.

π α·π + πΆπ£ αΆπ + πΎ π + π(π, αΆπ) = π

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? πππ π, αΆπ = π π πππ αΆπ .

Model parameters?

How uncertain are they?

π = 5.47.

π = π©(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, β¦

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:

βΊ Nonsmooth nonlinearities

βΊ High damping

βΊ Gravity-induced asymmetry

βΊ Viscoelastic components

25

WEMS Device: Piecewise-linear Behaviour

Elastomer plots

Isolation and

protection device.

Mechanical stops

Experimental

stiffness curve.

Displacement

R e s to

ri n g f

o rc

e

26

A βGoodβ Excitation: the Sine-sweep Signal

Linear sine sweep:

π’(π‘) = π΄ π ππ 2ππ0 π‘ β π‘0 + 2π π

2 π‘ β π‘0

2 + π0

Time (s) Time (s)

Force (N) Forcing frequency (Hz)

27

A βGoodβ 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.

π’(π‘) = π΄ π ππ 2ππ0

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