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en ca on an quan ca on ostructural nonlinearities from measured

vibration response

.

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The Problem

-6-6-6-6

]

]

]

]

2222

4444

tanceFRF

[m/

tanceF

RF

[m/

tanceF

RF

[m/

tanceFRF

[m/ 1 N1 N1 N1 N

5 N5 N5 N5 N

10 N10 N10 N10 N

45454545 50505050 55555555 60606060 65656565 70707070 75757575 80808080 858585850000Rece

Rece

Rece

Rece

Frequency [Hz]

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The goal

taking Experimental Modal Analysis (EMA) from the

current state of ideal linear world to a higher level in

which nonlinearities are not ignored but, at least at a

,

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SOURCESOURCE StrategyStrategy

RECEIVERRECEIVER TRANSMISSIONTRANSMISSION

PATHPATH 4 research projects ongoing

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ENGINEERING DESIGN MEANS

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ENGINEERING DESIGN: the HOLY GRAIL

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My research area(s)

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In the presence of a nonlinearity

IGNORE IT!!(Nelson principle)

NONLINEARITY

WHAT DO WEDO??

I AM SURE WECAN GET

COOL! LETSDOCK HERE!

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Linear system excited with stepped sine 0.1, 0.4 and 10N: FRF

Li t (F 0 1N) d l l i

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Linear system (F = 0.1N): modal analysis

Li t (F 10N) d l l i

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Linear system (F =10N): modal analysis

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Nonlinear System (QD + CS): F = 0.1, 0.4,1N - FRF

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Nonlinear System (QD + CS): F = 0.1N modal analysis

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Nonlinear System (QD + CS): F = 10N modal analysis

N li S (QD) F 1N d l l i

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Nonlinear System (QD): F = 1N modal analysis

Approach to NLMT

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NonLinearSpatial model Modal model Response model

Modal Testing

Approach to NLMT

I AM SUREWE CAN GETAROUND IT!

NONLINEARITY

structure

1 .51 .51 .51 .5

2222

2 .52 .52 .52 .5x 1 0x 1 0x 1 0x 1 0

- 6- 6- 6- 6

eFRF

[m/N]

eFRF

[m/N]

eFRF

[m/N]

eFRF

[m/N]

1 N1 N1 N1 N

5 N5 N5 N5 N1 0 N1 0 N1 0 N1 0 N

, ,r r r i

4 54 54 54 5 5 05 05 05 0 5 55 55 55 5 6 06 06 06 0 6 56 56 56 5 7 07 07 07 0 7 57 57 57 5 8 08 08 08 0 8 58 58 58 50000

0 .50 .50 .50 .5

1111

Recep

tanc

Recep

tanc

Recep

tanc

Recep

tanc

F r e q u e n c y [ H z ]F r e q u e n c y [ H z ]F r e q u e n c y [ H z ]F r e q u e n c y [ H z ]

How to do NLMT? A review of the literature

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How to do NLMT? A review of the literature

The major references for the study so far are:

1. Past, present and future G. Kerschen et al, MSSP, 20,

505-592, 2006

2. Identificationwith Linearity Plots, D. Goge, Nonlinear

Mechanics, 40, 27-48, 2005

3. Textbook: Nonlinearity in Structural Dynamics, K. Worden,

G. Tomlinson, 2001

4. PhD: Identification of R. Lin, 1990, Imperial College

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From Kerschens review paper

Time Domain Methods: the signal analysed is acquired directly by the

sensor, i.e. no need of data processing

Time Domain Methods

r qu n y m n : y m p n n m .

This is usually preferred as it gives more visual information and physical

insight than a time history.

Modal Methods: these attempt to extend the concept of Modal Model to anonlinear system by extracting the modal parameters from measured data.

The basic concept is the Nonlinear Normal Mode (NNM). Because is highly

mathematical and theoretical further work needs to be done before it is

requency oma n e o s

Modal Methods

applied to engineering problems.

Restoring Force Surface (RFS)

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Conceptually very simple. The equation of motion of a system is

( , ) ( )mx f x x y t + =

By knowing the excitation1 ,the response and the mass m( )y t ( )x tit is possible to study the surface which

contains information on the stiffness and damping of the system( , ) ( )f x x y t mx=

Because the response is measured at discrete times, there is need of

fitting/interpolating the surface . Several mathematical methods

(e.g. Cebyshev polynomials) but not very physical

( , )f x x

, ,x x and the mass need to be known. Time consuming and laborious

1 the excitation has to contain multiple frequencies, e.g. pseudo-

random, and it needs to give uniform coverage of the phase plane


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.

kf p x p x p x p

p e p

= + + +

= + =

3 2

1 2 3 4

1 27 006 1 33910.5

1

1.5

x 104

. .p p= =3 4

-0.2 -0.15 -0.1-0.05 0 0.05

0.1 0.15-10

0

10-2

-1.5

-1

-0.5

0

displ

vel

force

0

0.5

1

1.5x 10

4

e

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15-2

-1.5

-1

-0.5

forc

Harmonic Balance Inverse Method

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Harmonic Balance Inverse Method

With the Harmonic Balance to a first order expansion, theresponse is assumed to be at the same frequency of the

excitation signal.

nverse e o s a me o ase on einversion of the Receptance FRF. However, this method

is based on the assumption that the mode to be

analysed is real.

Lin (1990) extended the previous work to account also

for modal complexity

My research stems from these previous techniques andaims at defining a new approach to modal testing which

does not ignore nonlinearities

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COde for Nonlinear CharacterisationCOde for Nonlinear Characterisation

rom m asure esponse ovibratiOn

rom m asure esponse ovibratiOn

CONCERTO the core principles

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CONCERTO the core principles

1) The mode analysed dominates theresponse

)sin()()( 0 tFxXkxXcxm e=++

2) The stiffness and damping are unknown functionof the amplitude of vibration, i.e. the e.o.m. is

a g ven amp u ei

e sys em s near se

as the damping and stiffness become coefficients

)sin(0 tFxkxcxm eii =++

Linearisation (e g hardening cubic stiffness)

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force

Linearisation(e.g. hardening cubic stiffness)

displacementX3

-X3

X2

-X2 -X1

X1

K K K<

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Numerical Simul. Stepped sine excitation 0.01Hz

F0= 0.2 N

31 3 0 cos( )emx cx k x k x F t + + + =

-30-30-30-30

1

6 3

3

6000 N/m

7 10 N/m

k

k

=

=

m = . g

c = 0.8 Ns/m

amplitud

e

-60-60-60-60

-50-50-50-50

-40-40-40-40

|H||H||H||H|

isplacem

ent

5555 10101010 15151515

-80-80-80-80

-70-70-70-70

Frequency [Hz]Frequency [Hz]Frequency [Hz]Frequency [Hz]

SDOF analysis points around resonance

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-30-30-30-30

points above and below resonance are defined at thesame amplitude of vibration

y p

-50-50-50-50

-45-45-45-45

-40-40-40-40

-35-35-35-35

|H||H||H||H|

displacement

9999 9.59.59.59.5 10101010 10.510.510.510.5 11111111-65-65-65-65

-60-60-60-60

----


SDOF analysis points around resonance

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At a given amplitude, correspond 2 points of

the measured FRF which are:

r rA i B+= =-40-40-40-40

-35-35-35-35

----

nt

y p

2 2 2

1

2 2 22 2 2

2

( )

r r r

r r

r r r

iA i B

H R i Ii

++= = +

+

Which can be used to find the 4 unknowns, , ,r r r r A B 9999 9.59.59.59.5 10101010 10.510.510.510.5 11111111-65-65-65-65

-60-60-60-60

-55-55-55-55

-50-50-50-50

-45-45-45-45


displacem

( ) ( )2 1 2 2 1 1 2 1 2 2 1 12

2 1 2 1

2 2r

R R R R I I I I

R R I I

+

+

=

( )( ) ( )( )

( ) ( )

2 1 2 2 1 1 2 1 2 2 1 1

2

2 1

2 2 2

1

2

2

2

2r

r

I I R R R R I I

R R I I

+

+

=

System Identification using the Inverse Method

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1 2

2

1H H k m j c

k m j c

= = + +

1 2

1

Re( )

Im( )

H k m

H c

=

=

STIFFNESS

y g

|H

|

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

A B


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1 2Re( )H k m = The slope of the AB lineis the mass of the

y g

RE

(H

-1)

A

be interpreted as themodal mass (MDOF)

2

BLinear(ised) system


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Once the mass (or modal mass),m, has been determined, by

knowing how the natural frequency and the loss factor change with

y g

,

function of the amplitude, that is:

2( ) ( )rk X X m=

c X X X m=

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Code ValidationCode Validation

Numerical SimulationNumerical Simulation

Library of common nonlinearities(from Goges paper)

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Examples of numerical simulations

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m = 1.5Kg c = 0.8 Ns/m

0( , ) cos( )emx f x x F t + =

Quadratic

Damping

Cubic

stiffness,hardening

Cubic

stiffness,

softening

ValuesStiffness Forceamp ng

ForceNonlinearity

cx3

1 3k x k x+ 1 6 33

6000 N/m

4 10 N/m

k

k

=

=

1

6 33

6000 N/m

7 10 N/m

k

k

=

=

3

1 3k x k x+cx2

cx c x x+

1

2

2

6000 N/m

8 Ns/m

k

c

=

=1k x

Pre-loaded

piecewise

linear stiffness

1

2 1 2

3 1 2 2 3

,

( ) ,( ) ( ) ,

x x

k x k k b b x g k x k k b k k g x g

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