tutorial on nonlinear modal analysis of mechanical systems · tutorial on nonlinear modal analysis...

Tutorial on Nonlinear Modal Analysis of

Mechanical Systems

Gaëtan Kerschen

Aerospace and Mechanical Eng. Dept.

University of Liège, Belgium

[email protected]

2

Belgium ?

3

University of Liège ?

Founded in 1817 (Belgium was not existing yet !)

All disciplines covered

17000 undergraduate students

2000 graduate students

500 faculty members.

4

Aerospace and Mechanical Engineering Dept.

Aerospace

Materials

Energetics

Biomechanics

Computer-Aided Eng.

Facts and figures:

25 Professors

85 researchers

15 staff members

5 Masters, among which the Master

of Aerospace Engineering

5

Space Structures & Systems Laboratory (S3L)

Nonlinear dynamics

Spacecraft structures

Orbital mechanics

Nanosatellites

Facts and figures:

1 Professor

5 Postdoc, 6 PhD students, 1 TA

Collaborations: UIUC, Torino, VUB, LMS

Financial support: ERC, ESA, FNRS

BELSPO

System ID & modal analysis

Constructive utilization of NL

Bifurcation analysis & management

6

Linear ?

A linear system has an output that is directly proportional

to input.

Superposition principle: cornerstone of linear theory

A X

B Y

A+B X+Y LIN

LIN

LIN

7

Nonlinear ?

A nonlinear system is a system that is not linear.

Most physical systems are inherently nonlinear in nature !

(Navier-Stokes, planetary motion, rigid body rotation, etc.)

A X

B Y

A+B ??? NL

NL

NL

8

Why Care About Nonlinearity in Structural Dynamics?

1. Nonlinear dynamics is fun !

2. Nonlinear dynamics is complex.

3. There are industrial needs.

1. Meet escalating performance.

2. Nonlinearities are pervasive.

3. Nonlinearity may reveal damage.

9

Two Examples in Aerospace Engineering

SmallSat spacecraft

Collaborators:

J.B. Vergniaud @ EADS-Astrium,

B. Peeters @ LMS,

A. Newerla @ ESA.

Morane-Saulnier MS760

Collaborators:

C. Stephan @ ONERA,

P. Lubrina @ ONERA.

10

Outline of the Presentation

Sources of nonlinearity

(with real-life examples)

Fundamental differences between

linear & nonlinear dynamics

What about nonlinear modes ?

11

5 main assumptions for

Small vibrations around

an equilibrium

Small displ. and rotations

Mechanical vibrations

→ nonlinear boundary conditions

→ geometric nonlinearity

→ inherently nonlinear forces in

multiphysics applications

Mx (t) + Cx (t) + Kx(t) = f(t)

Linear elasticity → nonlinear materials

(hyperelastic, plasticity)

Linear Systems in Structural Dynamics

Viscous damping → nonlinear damping mechanisms

12

Nonlinear Materials: Elastomers, Foams, Rubber

Impact of elastomeric engine mounts

on A-400 M engine roll mode

J.R. Ahlquist et al., IMAC 2010

Airbus Military

Decrease in

stiffness

13

Geometric NL: Large Displacements/Rotations

Thin beam activated in large

displacements

Force

Rel. displ.

Increase in

stiffness

Cantilever beam with a thin beam

14

Combined Nonlinearity: Nonlinear BCs + Friction

Bolted connections between wing

tip and external fuel tank. Paris aircraft @ ONERA (France)

Force

Rel. displ.

Decrease in

stiffness

5 60-4

0

FRF (dB)

Frequency (Hz)

Low excitation level

High excitation level

15







Superposition principle

Linear Nonlinear

Uniqueness of the solutions

x2 x2

Invariance of FRFs and of

modal parameters

NO ! By definition …

NO ! Freq.-energy dependence

NO ! Steady state response

depends on transient

response

Bifurcations, quasi-periodicity

and chaos.

+

17

Superposition Principle (Linear Case)

0 0.03 0.05 0.10

0.3

0.7

Force F (N) with ω=1.1 rad/s

𝑥 + 0.05𝑥 + 𝑥 = 𝐹𝑠𝑖𝑛𝜔𝑡

Displ. x

(m)

18

No Superposition Principle (Nonlinear Case)

0 0.03 0.05 0.10

0.3

0.7

Force F (N) with ω=1.1 rad/s

𝑥 + 0.05𝑥 + 𝑥 + 𝑥3 = 𝐹𝑠𝑖𝑛𝜔𝑡

𝑥 + 0.05𝑥 + 𝑥 = 𝐹𝑠𝑖𝑛𝜔𝑡

Discontinuous

behavior, potentially

dangerous in practice Displ. x

(m)

19

0 100

0.1

1.5

Time (s)

𝑥 + 0.05𝑥 + 𝑥 + 𝑥3 = 0.15 𝑠𝑖𝑛 1.6𝑡

𝑥 0 = 0, 𝑥 0 = 0 𝑥 0 = 0, 𝑥 0 = 2

x15

Displ. x

(m)

Non-Uniqueness (Sensitivity to Initial Conditions)

20

350 400

-0.8

-0.1

0.1

0.8

Marked Sensitivity to Forcing (→ Jump)

Time (s)

𝑥 + 0.05𝑥 + 𝑥 + 𝑥3 = 0.05 𝑠𝑖𝑛 1.22𝑡

x8

𝑥 + 0.05𝑥 + 𝑥 + 𝑥3 = 0.05 𝑠𝑖𝑛 1.24𝑡

Displ. x

(m)

21

How to Compute the “FRFs” of a Linear System ?

𝑥 + 𝑥 = 𝐴𝑠𝑖𝑛 𝜔𝑡

𝑥(𝑡) = 𝐵𝑠𝑖𝑛 𝜔𝑡

−𝜔2𝐵 + 𝐵 = 𝐴

𝐵 =𝐴

1 − 𝜔2

Confirmation of the

superposition principle

22

How to Compute the “FRFs” of a Nonlinear System ?

𝑥 + 𝑥 + 𝑥3 = 𝐴𝑠𝑖𝑛 𝜔𝑡

𝑥(𝑡) = 𝐵𝑠𝑖𝑛 𝜔𝑡

−𝜔2𝐵𝑠𝑖𝑛 𝜔𝑡 + 𝐵𝑠𝑖𝑛 𝜔𝑡 +𝐵3𝑠𝑖𝑛3𝜔𝑡 = 𝐴𝑠𝑖𝑛 𝜔𝑡

𝑠𝑖𝑛3𝜔𝑡 = (3𝑠𝑖𝑛 𝜔𝑡 − 𝑠𝑖𝑛 3𝜔𝑡)/4

Nonlinear relation

between B and A

𝑥 𝑡 = 𝐵𝑠𝑖𝑛 𝜔𝑡 + 𝐶𝑠𝑖𝑛 3𝜔𝑡

Solution: infinite series of

harmonics

OPTION 1: EXACT OPTION 2: APPROXIMATION

−𝜔2𝐵𝑠𝑖𝑛 𝜔𝑡 + 𝐵𝑠𝑖𝑛 𝜔𝑡 + 3

4𝐵3𝑠𝑖𝑛 𝜔𝑡 = 𝐴𝑠𝑖𝑛 𝜔𝑡

Solve a third order

polynomial

23

Key Feature of Nonlinear Systems: Bifurcations

Pulsation ω (rad/s)

𝑥 + 0.05𝑥 + 𝑥 + 𝑥3 = 𝐹𝑠𝑖𝑛𝜔𝑡

0.6 1 1.1 1.5 2.20

1

1.8

3 solutions 1 solution 1 solution

Displ. x

(m)

24

FRF Peak Skewness Perturbs Linear Modal Analysis

Introduction of two stable poles around 1 nonlinear mode.

B. Peeters et al., IMAC 2011

Polymax applied to F-16

25

0.6 1 1.1 1.5 2.20

1

1.8

F=0.01 N

F=0.02 N

F=0.05 N

F=0.1 N

F=0.15 N

Displ. x

(m)

Pulsation ω (rad/s)

𝑥 + 0.05𝑥 + 𝑥 + 𝑥3 = 𝐹𝑠𝑖𝑛𝜔𝑡

3. Sensitivity to forcing

(F=0.05N,

ω=1.22/1.24)

2. Sensitivity to

initial conditions

(F=0.15N, ω=1.6)

1. No superp.

(various F, ω=1.1)

Explanation of Previous Results

26







27

Objectives of This Third Part

Can we extend modal analysis to nonlinear systems ?

3. How do we extract NNMs from experimental data ?

1. What do we mean by a nonlinear normal mode (NNM) ?

2. How do we compute NNMs from computational models ?

28

Normal Modes of Conservative Systems

Linear normal mode (LNM):

synchronous periodic motion.

Nonlinear normal mode (NNM):

synchronous periodic motion.

Rosenberg

(1960s)

0)()( tt qKqM

NDOF system: existence of at

least N families of periodic

solutions around the equilibrium.

Lyapunov

(1907)

0)()()( ttt NL qfqKqM

29

Fundamental Difference Between LNMs and NNMs

LNMs

𝑞 1 + 2𝑞1 − 𝑞2 = 0

𝑞 2 + 2𝑞2 − 𝑞1 = 0

𝐴 = 𝐵, 𝜔1= 1 rad/s

𝐴 = −𝐵, 𝜔2= 3 rad/s

𝑞1,2 = 𝐴, 𝐵𝑐𝑜𝑠𝜔𝑡

NNMs are frequency-

amplitude dependent !

𝑞 1 + 2𝑞1 − 𝑞2 + 0.5𝑞13 = 0

𝑞 2 + 2𝑞2 − 𝑞1 = 0

𝜔1 ∈ 1, 2 rad/s

𝜔2 ∈ 3,+ ∞ rad/s

𝐴 = ±8 𝜔2 − 2 𝜔2 − 1

3 𝜔2 − 2

𝐵 =𝐴

2 − 𝜔2

𝑞1,2 ≅ 𝐴, 𝐵𝑐𝑜𝑠𝜔𝑡

0.5 1 2 3-1.5

-0.5

0.5

1

4Initial conditions on displacements

Blue mass

Ora

ng

e m

ass

1 1

1 1

0.5

(cubic)

31

It Is Necessary to Extend the Definition of an NNM

LNM: synchronous

periodic motion.

NNM: synchronous

periodic motion.

Rosenberg

(1960s)

Modal

interactions

NNM: periodic motion

(nonnecessarily synchronous).

0)()()( ttt NL qfqKqM 0)()( tt qKqM

32

A Frequency-Energy Plot Is a Convenient Depiction

10-5

103

0

0.7

2DOF linear

Energy (J)

Frequency

(Hz)

33

A Frequency-Energy Plot Is a Convenient Depiction

10-5

103

0

0.7

Energy (J)

Out-of-phase

NNM

In-phase NNM

0.28

0.16

Frequency

(Hz)

2DOF nonlinear

34

LNMs and NNMs Have a Clear Conceptual Relation

10-5

103

0

0.7

2DOF linear

Energy

Freq.

10-5

103

0

0.7

2DOF nonlinear

Freq.

Energy

Frequency-energy dependence

Harmonics: modal interactions

Bifurcations: #NNMs > #DOFs

Bifurcations: stable/unstable modes

But …

35

Practical Significance for Damped-Forced Responses

10

8

Ampl.

(m)

Freq. (rad/s)

1 1.30

2.5

Ampl.

(m)

Freq. (rad/s)

Locus of LNMs

for various

energies

Locus of NNMs

for various

energies

Linear: resonances occur

in neighborhoods of LNMs.

Nonlinear: resonances occur

in neighborhoods of NNMs.

36

In Summary

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

37

Practical Computation of NNMs

10-5

103

0

0.7

Frequency-energy plot Complex finite

element model

38

Shooting and Pseudo-Arclength Continuation

Numerical integration

(Newmark)

t=0 t=T

z0, T zT, T

Newton-Raphson

10-5

103

0

0.7

Freq.

Energy

STEP 1: compute an

isolated periodic solution

Prediction tangent

to the branch

Correction to the

prediction

zp0

T

Energy

Freq.

10-5

103

0

0.7

STEP 2: compute the

complete branch ┴

39

Shooting

Periodicity condition

(2-point BVP)

Numerical solution through iterations:

2n x 1 2n x 2n Monodromy matrix

40

Numerical Demonstration in Matlab

41

In-Phase and Out-of-Phase NNMs Connected

102

104

0.208

0.23

Energy (J)

Frequency

(Hz)

3:1 modal interaction

Out-of-phase

NNM

In-phase

NNM

42

Neither Abstract Art Nor a New Alphabet

43

MS760 Aircraft (ONERA)

Front connection

Rear connection

Bolted connections

between external fuel

tank and wing tip

44

Finite Element and Reduced-Order Modeling

Finite element model (2D shells

and beams, 85000 DOFs)

Reduced model accurate in

[0-100] Hz, 548 DOFs

Condensation of the linear

components of the model

Craig-Bampton technique

8 remaining nodes

+

500 internal modes

45

A Close Look at Two Modes

Wing bending

Wing torsion (symmetric)

46

Wing Bending Mode not Affected by Nonlinearity

MAC = 1.00 MAC = 0.99

Frequency

(Hz)

Energy (J)

47

Wing Torsion Mode Is Affected by Nonlinearity

9:1

MAC = 1.00

MAC = 0.98

31.1

30.4

10-4 104

Frequency

(Hz)

Energy (J)

3:1

5:1

Decrease of the natural

frequency

Internal resonances

Mode shape slightly

affected

48

6th NNM of the Spacecraft

100

105

28.5

30

31.3

Energy (J)

Frequency

(Hz)

9:1

3:1

26:1

2:1 interaction

with NNM12

49

6th NNM: From a Local Mode to a Global Mode

NNM6: local deformation at

low energy level

NNM6: global deformation at

a higher energy level

(on the modal interaction branch)

The top floor vibrates

two times faster !

5 30 60 90-5.9

0

5.9

Excitation frequency (Hz)

Acc.

(m/s2)

90 60 5

3.3

5.4

Sine sweep

Accelerometer

at top floor

20N

80N

? 80N: large top floor motion

20N: no top floor motion

x4 x4

x4 x4

x4

30

51

Experimental Confirmation

Accel.

Base excitation

(sine sweep)

5 10 20 30 40 50 60 70-100

-50

0

50

100

Excitation frequency (Hz)

Acc.

(m/s2)

0.1 g

1 g

No mode of the top floor in

this frequency range

?

52

Experimental Identification of NNMs ?

Nonlinear phase separation: NO !

► If several modes are excited (arbitrary forcing and/or ICs),

extraction of individual NNMs is not possible generally, because

modal superposition is no longer valid in the nonlinear case.

► Because the NNMs attract the dynamical flow, in some cases,

the motion may end up along one dominant NNM.

53

Experimental Identification of NNMs ?

If the motion is initiated on one specific NNM, the

remaining NNMs remain quiescent for all time.

Nonlinear phase resonance: YES !

► Excite a single NNM

► Exploit invariance principle:

► Identify the resulting modal parameters

54

Two-Step Methodology

NNM free decay

Inducing single-NNM

free decay

Turn off the

excitation ON OFF

NNM force appropriation

Isolating an NNM motion using

harmonic excitation

Extract the

NNM

55

Necessary Ingredients

1. Appropriate excitation to isolate one NNM

2. Modal indicator (are we on the NNM of interest ?)

3. Invariance principle

4. Wavelet transform and NNM extraction

Extraction of NNMs and their oscillation

frequencies directly from experimental data

56

Nonlinear Phase Lag Quadrature Criterion

A nonlinear structure vibrates according to an NNM if the

degrees of freedom have a phase lag of 90° with respect to

the excitation (for all harmonics !).

Nonlinear mode

indicator function (MIF)

57

Invariance Principle

If the motion is initiated on one specific NNM,

the remaining NNMs remain quiescent for all time.

Turn off the

shaker

58

Wavelet Transform & NNM Extraction

Modal curves of NNMs:

► Extracted directly from the time series

Oscillation frequencies of NNMs:

► Extracted using the wavelet transform

Reconstructed FEP from experimental data:

► Compute energy and eliminate time

Frequency vs. time

Frequency vs. energy

59

Experimental Demonstration

Experimental set-up:

► 7 accelerometers along the main beam.

► One displacement sensor (laser vibrometer) at the beam end.

► One electrodynamic exciter.

► One force transducer.

Beam with a geometrical nonlinearity;

benchmark of the European COST Action F3.

60

Moving Toward the First Mode

Stepped sine

excitation

61

Moving Toward the First Mode

Stepped sine

excitation

OK !

62

Modal Indicator

OK ! Nonlinear mode

indicator function (MIF)

63

The First Mode Vibrates in Isolation

64

Frequency-Energy Dependence

Modal shapes at

different energy levels

Frequencies

(wavelet transform)

66

Validation of the Methodology

Modal curves Modal shapes

Acc

#7

Acc #3

Theoretical Experimental

Acc

Position along the beam

67

Validation of the Methodology

68

Concluding Remarks

Virtually all engineering structures are nonlinear …

but they may vibrate in linear regimes of motion.

Nonlinearity is not “plug and play”:

Code for NNM computation available online.

► Highly individualistic nature of nonlinear systems.

► Even weakly nonlinear systems can exhibit complex dynamics.

► No universal method; toolbox philosophy.

Thank you for your attention.

Gaetan Kerschen

Aerospace and Mechanical Eng. Dept.

University of Liège, Belgium

[email protected]

tutorial on nonlinear modal analysis of mechanical systems · tutorial on nonlinear modal analysis...

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