nonlinear vibrations of aerospace structures€¦ · a second test case: an f-16 fighter aircraft...
TRANSCRIPT
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Nonlinear Vibrations of
Aerospace Structures
Jean-Philippe Noël | [email protected] 22 October 2018
Detecting Nonlinearity
Lecture
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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)
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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)
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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, …).
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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!
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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
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… 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
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NSI in the Design Cycle of Eng. Structures = Troubleshooting!
MEASURE MODELIDENTIFYUNDERSTD
UNCOVERDESIGN
Computer-aided
modelling (FEM, …)
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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.
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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.
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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.
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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)
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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.
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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)
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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
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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).
?
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Why Is Nonlinear System Identification Difficult?
Challenge #5: True system may be (is!) outside the model class.
Selected model class
Tested system
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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.
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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).
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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.
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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.
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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, …
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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.
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24
A First Test Case: the SmallSat Spacecraft from Airbus DS
NL WEMS device
Challenges:
► Nonsmooth nonlinearities
► High damping
► Gravity-induced asymmetry
► Viscoelastic components
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WEMS Device: Piecewise-linear Behaviour
Elastomer plots
Isolation and
protection device.
Mechanical stops
Experimental
stiffness curve.
Displacement
Resto
ring f
orc
e
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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)
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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
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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)
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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
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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
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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
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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
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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
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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
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A Close-up Reveals the Appearance of Strong Harmonics
Sweep frequency (Hz)
8.4
Relative displacement ( – )
-2
2
0
1
-1
8.5
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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.
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A Second Test Case: an F-16 Fighter Aircraft
F16 aircraft,
Saffraanberg,
Belgium.
Joints between
substructures are
generic sources of NLs.
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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
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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
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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
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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
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42Frequency (Hz)
2
Amplitude (dB)
-20
20
-80
5 1510
-60
F-16 Measurement at Low Excitation Level
-40
0
Output Even
Noise Odd
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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
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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)
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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)
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46
Not affected AFFECTED
AFFECTED
Nonlinearity Location Can Be Inferred from Mode Shapes
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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
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48
Low level
High level
Nonlinearity Location Is Clearly Revealed
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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
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50
The Source of Nonlinearity … See Next Lecture
– Acceleration (m/s²)
Damping
nonlinearity!
-3
3
0
Velocity (mm/s)
-20-40 0 20 40
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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.
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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.
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Nonlinear Vibrations of
Aerospace Structures
Jean-Philippe Noël | [email protected] 22 October 2018
Detecting Nonlinearity
Lecture