Nonlinear Normal Modes: Theoretical Curiosity or Practical Concept ?

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



⇒ Definition

⇒ Frequency-energy dependence



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


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




⇒ Proposed algorithm

⇒ Demonstration in Matlab


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



⇒ Nonlinear force appropriation

⇒ Experimental demonstration



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:


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





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

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Nonlinear Normal Modes: Theoretical Curiosity or Practical Concept ?