nonlinear normal modes: theoretical curiosity or practical concept ?
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Nonlinear Normal Modes: Theoretical Curiosity or Practical Concept ?
Gaëtan Kerschen
Space Structures and Systems LabStructural Dynamics Research GroupUniversity of Liège
Linear Modes: A Key Concept
Clear physical meaning
Important mathematical properties
Orthogonality
Modal superposition
Invariance
Structural deformation at resonance
Synchronous vibration of the structure
Linear Modal Analysis Is Mature
Airbus A380
Envisat
But structures may be nonlinear !
Objective of this Presentation
Can we extend modal analysis to nonlinear systems ?
3. How do we extract NNMs from experimental data ?
3.
1. What do we mean by a nonlinear normal mode (NNM) ?
1.
2. How do we compute NNMs from computational models ?
2.
Specific Efforts in Our Research
Most contributions in the literature deal with systems with very low-dimensionality (typically 2-DOF systems):
⇒ Progress toward more realistic, large-scale structures
⇒ Develop computational methods which can tackle strongly nonlinear systems.
Most contributions in the literature use analytic methods limited to weak nonlinearity:
⇒ Focus on developments that can be understood and exploited by the practising engineer .
Nonlinear dynamics is complicated and generally not well understood:
Theoretical Curiosity or Practical Concept ?
1. Nonlinear normal mode ?
2. Theoretical modal analysis
3. Experimental modal analysis
Theoretical Curiosity or Practical Concept ?
1. Nonlinear normal mode ?
⇒ Definition
⇒ Frequency-energy dependence
2. Theoretical modal analysis
3. Experimental modal analysis
Historical Perspective: Lyapunov
For n-DOF conservative systems with no internal resonances, there exist at least n different families of periodic solutions around the equilibrium point of the system.
These n families define n NNMs that can be regarded as nonlinear extensions of the n LNMs of the underlying linear system.
Historical Perspective
1960s: First constructive methods (Rosenberg)
1970s: Asymptotic methods (Rand, Manevitch)
1980s: ?
1990s: New impetus (Vakakis, Shaw and Pierre)
2000s: Computational methods (Cochelin, Laxalde, Thouverez)
Undamped NNM Definition: Rosenberg
An NNM is a vibration in unison of the system (i.e., a synchronous oscillation).
Important remark: not limited to conservative systems !
Dis
plac
emen
t
Time
Damped NNM Definition: Shaw and Pierre
An NNM is a two-dimensional invariant manifold in phase space.
Extension of Rosenberg’s Definition
An NNM is merely a periodic motion of a nonlinear conservative system
Dis
plac
emen
t (m
)
Time (s)
Dis
plac
emen
t (m
)
Time (s)
NNMs Are Frequency-Energy Dependent
Time series
Modal curves
Increasing energy (in-phase NNM)
Appropriate Graphical Depiction of NNMs
Nonlinear frequencies (backbone
curves)
NNM
NNM
LNMs & NNMs: Clear Conceptual Relation
This frequency-energy plot gives a clear picture of the action of nonlinearity on the dynamics. It can also be understood by the practising engineer.
LNMs & NNMs: Clear Conceptual Relation
Clear physical meaning
Important mathematical properties
Orthogonality
Modal superposition
Invariance
Structural deformation at resonance
Synchronous vibration of the structure
YES
YES
YES
YES
YES
LNMs
YES, BUT…
NO
NO
YES
YES
NNMs
Some Fundamental Differences
Frequency-energy dependence
LNMs
NO YES
NNMs
YESStability YESNO
Number DOFs = number modes YES NO
YESModal interactions NO
Why Normal ?
NNMs are not orthogonal to each other, as LNMs are.
They are still referred to as normal modes, because they are normal to the surface of maximum potential energy (bounding ellipsoid).
A.F. Vakakis, MSSP 11, 1997
Theoretical Curiosity or Practical Concept ?
1. Nonlinear normal mode ?
2. Theoretical modal analysis
⇒ Proposed algorithm
⇒ Demonstration in Matlab
3. Experimental modal analysis
Theoretical Modal Analysis
Newmark
NNM: periodic motion of a nonlinear conservative system
⇒ Solve a 2-point boundary value problem
How To Compute an NNM ?
Governing equations in state space
Periodicity condition (2-point BVP)
Numerical solution through iterations
Initial guess Corrections
Shooting Algorithm: 2-Point BVP
n x 12n x 2n Monodromy matrix
Shooting Algorithm: Newton -Raphson
Finite differences(perturb the ICs and integrate the
nonlinear equations of motion)
COMPUTATIONALLYINTENSIVE
Sensitivity analysis
VERY APPEALING ALTERNATIVE
Jacobian matrix (shooting)
Computational Burden Reduction
Differentiation of the governing equations
Governing equations in state space
Jacobian Matrix through Linear ODEs
How To Account for Energy Dependence ?
Predictor step tangent to the branch
Corrector steps ⊥to the predictor step
Algorithm: Shooting and Continuation
Numerical Demonstration in Matlab
Cyclic assembly of 30 substructures (⇒ 30 cubic nonlinearities, 120 state-space variables)
2DOF system with a single cubic nonlinearity
Modal Interactions in a 2DOF Nonlinear System
This is neither abstract art nor a new alphabet…
10-6
10-4
10-2
100
1022
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Mode (1,14)
Energy
Fre
quen
cy (
rad/
s)
1 5 9 13 17 21 25 29-10
0
4
1 5 9 13 17 21 25 29-4
0
4x 10-4
Localization in the Bladed Disk
Theoretical Curiosity or Practical Concept ?
3. Experimental modal analysis
⇒ Nonlinear force appropriation
⇒ Experimental demonstration
1. Nonlinear normal mode ?
2. Theoretical modal analysis
Experimental Modal Analysis
Phase separation:
The Linear Case
All modes excited at once.
Random or sine sweep excitations.
Time (e.g., stochastic subspace identification) or frequency-domain methods (e.g., polyreference least-squares).
The structure is excited in one of its normal modes.
The modes are identified one by one.
Harmonic excitation at the resonance frequency with a specific amplitude and phase distribution (at several input locations).
Phase resonance:
Clear physical meaning:
Structural deformation at resonance
Synchronous vibration (but not always)
Mathematical properties:
No modal superposition
Invariance
Phase separation: KO
Phase resonance: OK
Frequency-energy dependence: OK
Phase Resonance or Phase Separation ?
Nonlinear Phase Quadrature Criterion
NNM = 90º
A nonlinear structure vibrates according to one of its NNMs if the degrees of freedom have a phase lag of 90ºwith respect to the excitation (for all harmonics ! )
Nonlinear MMIF
Two-Step Methodology
Step 1: isolate an NNM motion using harmonic excitation
Step 2: turn off the excitation and induce single-NNM free decay
Extract the backbone and the modal curves
ECL Benchmark: Geometric Nonlinearity
Experimental set-up:
7 accelerometers along the main beam.
One displacement sensor (laser vibrometer) at the beam end.
One electrodynamic exciter.
One force transducer.
Moving Toward the First Mode
Stepped sine excitation
Phase Quadrature Is Obtained
Stepped sine excitation
OK !
Phase Quadrature Is Obtained
OK !
Nonlinear MMIF
The First Mode Vibrates in Isolation
Frequency-Energy Dependence of the Mode
Modal shapes at different energy levels
Frequencies (wavelet transform;
Argoul et al.)
Validation of the Methodology
Independently, a reliable numerical model of the structure was identified using the conditioned reverse path method.
Experimental
Theoretical
-50 -40 -30 -20 -10 0 10 20 30 40 50-80
-60
-40
-20
0
20
40
60
80
Validation of the Methodology
Isolation of the Second NNM
Nonlinear MMIF
OK !
The Second NNM is Much Less Nonlinear
-400 -200 0 200 400-500
0
500
Mode 1
Acc #3Acc #4
Acc
#7
Acc
#7
39 Hz → 30 Hz 144 Hz → 143 Hz
Mode 2
Theoretical Curiosity or Practical Concept ?
1. Nonlinear normal mode ?
2. Theoretical modal analysis
3. Experimental modal analysis
Conclusion: We can extract NNMs both from finite element models and from experimental data
Structure
Classical phase separation (⇒ LNM with SSI)
0 100 200 300 400 500
Linear modes Nonlinear modes
Proposed nonlinear phase resonance (⇒ NNM)
Important modes for the end result ?
State-of-the-art for linear aircrafts: combining linear phase separation
with linear phase resonancenonlinear
Next step
Technical Details and Open Questions
Many technical details (stable/unstable NNMs, modal interactions, bifurcations) were omitted in this lecture and are available in a series of journal publications.
Multi-sine and multi-point excitation (coupled NNMs).
Imperfect force appropriation.
Complex modes.
Nonlinear damping.
Open questions and challenges:
Thanks for your attention !
Gaëtan Kerschen
Space Structures and Systems LabStructural Dynamics Research GroupUniversity of Liège