L’intensité de structureperso.univ-lemans.fr/~jcpascal/Cours/ENSIM3A... · plate 2 P2 P3 P1...

1

ENSIM 3A

L’intensité de structure

Jean-Claude Pascal

LAUM, ENSIM

2

ENSIM 3A

Plan

� Expression de l’intensité dans les poutres

� Formulation approchée

� La mesure

3

ENSIM 3A

Expression de l’intensité de structure

Expression de l’intensité de structure

L’intensité structurale ou vibratoire correspond à la densité de flux de

puissance [en W/m2] transporté par les ondes vibratoires

densité de force (tenseur des contraintes) x vecteur vitesse

L’intensité moyenne dans le temps

avec les grandeurs complexes et

( ) ( ) ( ) 3,2,1, =−= ∑ jitwttij

jiji&σ

( ) { }∑ ∗−==

j

jijii wtiI &σRe2

1

( ) { }tj

ijij etωσσ Re= ( ) { }

jj wtw &Re=

4

ENSIM 3A

L’intensité structurale des ondes de flexion dans une poutre se réduit à

[W]

Définition de l’intensité des ondes de flexion

Expression de l’intensité des ondes de flexion

( ) ( ) ( ) ( ) ( ){ }xxMxwxQxI∗∗ +−= θ&&Re

2

1

La théorie d’Euler–Bernouilli permet d’exprimer toutes les quantités à

partir du déplacement

� déplacement angulaire

� moment de flexion

� effort tranchant

avec

( ) ( )

( ) ( )

( ) ( )3

3

2

2

x

xwEIxQ

x

xwEIxM

x

xwx

∂

∂−=

∂

∂=

∂

∂=θ

( ) ( ) ( ) ( ) ( )

∂

∂

∂

∂−

∂

∂=

∗∗

x

xw

x

xwxw

x

xwEIxI

&&&

&

2

2

3

3

Im2ω

ωjww &=

5

ENSIM 3A

Utilisation des différences finies pour exprimer les dérivées spatiales

Intensité approchée par différence finie

( ) ( )3

1234

3

3

2

1234

2

2

2312

33

2

2

x

wwww

x

w

x

wwww

x

wx

ww

x

wwww

∆

−+−≈

∂

∂

∆

+−−≈

∂

∂∆

−≈

∂

∂+≈

&&&&&&&&&&

&&&&&&

x

x∆

1 2 3 4

( ){ }∗∗∗ −−

∆≈ 4231323

4Im4

wwwwwwx

EII &&&&&&

ω

ωjww &&& =( )

{ }∗∗∗ −−∆

≈⇒ 423132334Im

4wwwwww

x

EII &&&&&&&&&&&&

ω


6

ENSIM 3A

Expression générale du déplacement

avec le nombre d’onde de flexion

En champ lointain quand il n’y a pas d’ondes évanescentes

Approximation de champ lointain

⇒= ωjww &&&


( ) kxjkxkxjkxeAeAeAeAxw 4321 +++= −−

4

EI

Ak

ρω=

x

wk

x

wwk

x

w

∂

∂−≈

∂

∂−≈

∂

∂ &&&

& 2

3

32

2

2

{ }∗∗

∆≈→

∂

∂≈ 21ImIm2 ww

x

AEII

x

wwAEII &&

&&

ρρ

{ }∗

∆≈ 212

Im wwx

AEII &&&&

ω

ρx∆

1 2

7

ENSIM 3A

CHARACTERISATION OF A DISSIPATIVE ASSEMBLY

BY STRUCTURAL INTENSITY

A. How to calculate energetic quantities from laser

vibrometer measurements

B. Analysis of assembly plate using energetic

quantities

C. Transformation of 2D model to 1D junction model

D. Use energy conservation low to compute joint

dissipation

8

ENSIM 3A

MEASURED ENERGETIC QUANTITIES

Force distribution

Divergence of the

structural intensity

Potential energy

density

Kinetic energy

density

Structural intensity

{ }∗∇=⋅∇ vvB 4Im

2ωI

( ) ( ) ( )( )yxvkyxvj

ByxF B ,,, 44 −∇

=

ω

4

2v

hT ρ=

( )

∂∂

∂−

∂

∂

∂

∂−−∇=

∗ 22

2

2

2

222

2Re12

4 yx

v

y

v

x

vv

BV ν

ω

( ) ( )

∇×∇×∇−

−∇∇−∇∇= ∗∗∗vvvvvv

B

2

1Im

2

22 υ

ωI

9

ENSIM 3A

MEASURED ENERGETIC QUANTITIES :

ADVANCED METHODS OF WAVENUMBER PROCESSING

Use of SFT for calculation of spatial derivatives

( )),()()(

,yx

ny

mx

TF

nm

nm

kkVjkjkyx

yxv−−

∂∂

∂ +

a

is the Spatial Fourier Transform of ),( yx kkV ( )yxv ,

y

x

SFT

xkyk

( )yxv , ),( yx kkV

SFT on truncated signals amplifies the components of high wavenumbers, bringing large contributions of the high wavenumber components to the results, especially in the case of the high-order derivatives of the velocity.

The derivatives of vibrating velocity are easily calculated by

10

ENSIM 3A

Mirror methods used to reduce errors caused by operation of SFT

The idea of the mirror method is to build a continuous and

periodic signal (the resulting signal) from the signal to be

processed by SFT (the original signal).

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

-5

Amplitude

Signa l

dis ta nce (m)

Mirror s ignal + Origina l s ignalOrigina l s ignal

MEASURED ENERGETIC QUANTITIES :

ADVANCED METHODS OF WAVENUMBER PROCESSING

11

ENSIM 3A

EXPERIMENTAL CONFIGURATION

93

0

mm

850 mm

20 m

m

clampededge

free edge

x

y

� A scanning vibrometer use a OFV

300 optical head

� Two galvo-driven mirrors direct the

laser beam horizontally and vertically

� 32x32 measurement points

� Test assembly consists of two steel

plates of thickness 1 mm

� The two opposite edges are clamped.

The two other edges are free

� A normal point force is acting on the plate

12

ENSIM 3A

ASSEMBLY PLATE ANALYSIS :

STRUCTURAL INTENSITY

� Wavenumber processing was used

� Data integrated over a frequency band of 525 to 1000 Hz

AIII A ×∇+∇−=+= φφ

Standard structural

intensity

Irrotational structural

intensity

13

ENSIM 3A

Force s F=668 HzDivergence of the structural intensity Force distribution

At frequency 668 Hz. these two quantities show that the dissipation produced by the joint is maximum at the positions where the joint is constrained by the tightening of the bolts


INJECTED OR DISSIPATED POWER AND FORCE DISTRIBUTION

14

ENSIM 3A


FORCE DISTRIBUTIONS AVERAGED IN FREQUENCY BAND

[549,600] Hz [601,649] Hz [650,699] Hz [700,750] Hz

[751,800] Hz [801,850] Hz [851,900] Hz [901,950] Hz

The dominating zones of dissipation correspond to the points of maximum constraints introduced by the bolts ensuring the contact of the two parts of the plate on the joint. However this behaviour of the

joint will depend largely on the frequency

15

ENSIM 3A

EQUIVALENT 1D MODEL :

AVERAGED POWER FLOW ON PLATE

x

y0

forc

e

Plate

1

Plate 2join

t( ) ( )∫=Ly

xx dyyxIx

0

,φ( ) ( )∫=

Lx

yy dxyxIy0

,φ

Total power flow in x-

direction Total power flow in y-direction

16

ENSIM 3A

The evolution of the

averaging power flow over

one direction of a plate

reveals the similar

behaviour to that of a one-

dimensional system like a

beam in the other direction

of the plate

F

Beam


ONE-DIMENTIONAL BEAM-LIKE MODEL

17

ENSIM 3A

� The basic idea is thus to represent the joint like a node of

junction of one-dimensional elements

� The third branch comprising only an outgoing wave is used to

express the power dissipated by the joint. Thus the power of each

branch entering in the junction element can respectively be

written by



+2a

+1a

plate 1 junction plate 2

−2a

−3a

−1a

−= −+ 2

1

2

121

1 aaP

2

321

3−−= aP

−= −+ 2

2

2

221

2 aaP

18

ENSIM 3A

plate 1 junction

plate 2

2P

3P

1P

Expression of the

conservation of flow

“entering” in the junction

0321 =++ PPP

2

2123

eetcP dg

+−≈

An approximation of the dissipated

power

=

+

+

−

−

−

2

1

3

2

1

a

a

t

r

t

t

t

r

a

a

a

dd

Far-field scattering matrix

gc the group velocity for the flexural waves

dt the dissipation coefficient

21, ee the densities of total energy on both sides of the junction

the reflection and transmission coefficientstr,



19

ENSIM 3A

USE ENERGY CONSERVATION LAW

TO COMPUTE JOINT DISSIPATION

From the exact conservation law of energy

( ) ( ) ),(,, yxWyxWyx dis =+⋅∇ I

( )yx,I

( )yxWdis ,

),( yxW

is the measured structural intensity

is represented by a simplified model for dissipation

proposed by Nefske & Sung and Bouthier & Bernhard :

( ) ( )yxeyxWdis ,, ωη≈

the local average of the total

energy density

is the injected or dissipated power by external

elements (forces, joint, …)

20

ENSIM 3A

Integration over direction y leads to a system with one dimension in x

( ) ( ) ( )1xxWxEdx

xdx

x −=+ δωηφ

the density of the total energy integrated in y direction (in

J/m). The differential form of the conservation law is then

written by

( )xEx

( ) ( )[ ] ( ) ( )1

0

0

0

,,, xxWdyyxWyxIdyyxIx

y

y

y L

dis

L

y

L

x −=++∂

∂∫∫ δ

( ) ( )xEdyyxe x

Ly

ωηωη =∫0

,


EQUATION OF 1-D ENERGY CONSERVATION

For a point force in ( )11, yx

21

ENSIM 3A

( ) ( ) xEdxxxWx x

x

x ηωδφ −−= ∫0

1

( ) ( ) ( )1xxWxE

dx

xdx

x −=+ δωηφ

x x

( )1xxW −δ

W 0

1xxL

( )xxφ

0 1x xL

Injected power in the plate The evolution of the averaged power

flow along the x dimension

� For an isolated plate system, the boundary conditions

are ( ) ,00 =xφ ( ) 0=−= xxxx LEWL ωηφ

� The loss factor can be estimated

by: ( )∫∫

−==

xx LL

xx

xxdxdyyxe

LE

W

00

minmax

,ω

φφ

ωη


EXEMPLE WITH ONE FORCE

22

ENSIM 3A

0

x

x

0

plate 2plate 1

( ) ( )0xxxEL xy −− δωξ

( )0xEL xyωξ

( )1xxW −δ

W1x

0x xL

xL


ONE FORCE AND JOINT

x

y0

forcePlate 1

Plate 2joint

( )( ) ( ) ( ) ( )01 xxxELxxWxE

dx

xdxyx

x −−−=+ δωξδωηφ

( ) ( ) ( )00201

joint2

xELxExE

LW xyxx

y ωξξω −=+

−=+−

ydg Ltc ωξ 2≡

Use the following differential equation

Dissipated power by joint

23

ENSIM 3A

uppe r part

lowe r part

x01

x02

joint

Position of the joint identified

by the measured forces

( )

( )( ) dydxyxe

LxLe

dydxyxeLx

e

xy

x

L

x

L

yx

L x

y

∫∫

∫ ∫

+

−

−>=<

>=<

ε

ε

0

0

,1

,1

002

0 001

The average density of energy in each of the two

plates:

The loss factor

xx

xx

LEω

φφη minmax

−=

The linear loss factor of density of dissipation

( ) ( )[ ]( ) 2/21

0201minmax

xxy

xxxxx

EEL

xLExE

+

−+−−=

ηωφφξ

At 668 Hz for the plate loss factor and for the linear loss factor of the joint are respectively 0.03 % and 5%


CHARACTERIZATION OF JOINT

24

ENSIM 3A

EFFECTIVE PARAMETER IDENTICAFATION OF 2D

STRUCTURES FROM MEASUREMENTS USING A

SCANNING LASER VIBROMETER

� Introduction

� Methods for evaluating parameters of structures

� Energy methods by using measuring data by a Scanning

Laser Vibrometer

� Estimation of flexural wavebumbers and loss

factor in 2-D structures

� Energy methods to obtain dispersion curve

� General techniques for computation

� Results of measurements from the Scanning Laser Vibrometer

� Introduction

� Methods for evaluating parameters of structures

� Energy methods by using measuring data by a Scanning

Laser Vibrometer

� Estimation of flexural wavebumbers and loss

factor in 2-D structures

� Energy methods to obtain dispersion curve

� General techniques for computation

� Results of measurements from the Scanning Laser Vibrometer

25

ENSIM 3A

Introduction (Introduction (con’tcon’t))

The finite-difference-approximation method

� Use three accelerometers to estimate the flexural

wavenumbers in one-dimensional structures such as

beams

� It is directly based on the wave equation associated

with the far-field approximation

Disadvantage

� Too high sensitivity to phase differences between

sensors due to the use of the finite difference technique

The finite-difference-approximation method

� Use three accelerometers to estimate the flexural

wavenumbers in one-dimensional structures such as

beams

� It is directly based on the wave equation associated

with the far-field approximation

Disadvantage

� Too high sensitivity to phase differences between

sensors due to the use of the finite difference technique

Methods to compute wavenumbers

26

ENSIM 3A



Use of Fourier Transform (SFT)

� Determine the maximum of wavenumber spectrum in beams, which

was then used to identify the value of natural flexural wavenumber

� To reduce the distortions brought by Spatial Fourier Transform

(SFT) a regressive method was proposed

Disadvantage

� The use of the direct Fourier Transform results significant errors in

the computations because of truncated signals.

Use of Fourier Transform (SFT)

� Determine the maximum of wavenumber spectrum in beams, which

was then used to identify the value of natural flexural wavenumber

� To reduce the distortions brought by Spatial Fourier Transform

(SFT) a regressive method was proposed

Disadvantage

� The use of the direct Fourier Transform results significant errors in

the computations because of truncated signals.

27

ENSIM 3A



Spatial correlation approach

� Correlation of the measurements with the wavefield

� The choice of that maximises the correlation gives the

best estimate of the flexural wavenumbers

� It is used for estimation of wavenumbers in 2D structures

Spatial correlation approach

� Correlation of the measurements with the wavefield

� The choice of that maximises the correlation gives the

best estimate of the flexural wavenumbers

� It is used for estimation of wavenumbers in 2D structures

yjkxjk tytx ee−−

tytx kk ,

28

ENSIM 3A

EESTIMATION OF FLEXURAL WAVENUMBER IN STIMATION OF FLEXURAL WAVENUMBER IN

TWOTWO--DIMENSIONAL STRUCTURESDIMENSIONAL STRUCTURES

First step

Use non dissipative energy equation of plate to derive the

effective flexural wavenumbers

29

ENSIM 3A

EESTIMATION OF FLEXURAL WAVENUMBER STIMATION OF FLEXURAL WAVENUMBER ((concon’’tt))

A thin isotropic plate

� excited by one or more mechanical forces

� neglecte the structural dissipation and the losses by radiation

The equation of Kirchhoff is expressed by

is the bending stiffness of plate,

natural flexural wavenumber in vacuum ,

flexural velocity

A thin isotropic plate

� excited by one or more mechanical forces

� neglecte the structural dissipation and the losses by radiation

The equation of Kirchhoff is expressed by

is the bending stiffness of plate,

natural flexural wavenumber in vacuum ,

flexural velocity

Derive flexural wavenumbers from energy concept

( ) ( )∑ −=−∇i

iiB Fvkvj

Drrδ

ω44

iF

D

Bk

v

30

ENSIM 3A



Multiplying the above equation by the complex conjugate of the velocity yields

{ } 0Im2

4 =∇ ∗vv

D

ω

( ) ( ) ( )

( ) ( )∑

∑

−+=

−=−∇ ∗∗

i

iii

i

iiiB

jQW

vFvkvvj

D

rr

rrr

δ

δω 2

1244

2

Consider a non-dossipation plate. In the zone where there are non-

excitation forces, no damping, no absorptions, we can obtain

two equations:

Leading the divergence of the structural intensity to be zero.

0=⋅∇ sI

31

ENSIM 3A



{ } 0Re224 =−∇ ∗ vkvv B

Estimators

of effective wavenumber of flexural waves:

{ }4/1

2

4Re

∇=

∗

v

vvaγ

{ }4/1

2

4Re

∇=

∗

v

vvbγ

The brackets < > denote the spatial average over the points,

that is, outside the mechanical excitation zones.

32

ENSIM 3A

CONSIDERATION OF DISSIPATIVE TERMSCONSIDERATION OF DISSIPATIVE TERMS

EFFECTIVE LOSS FACTOREFFECTIVE LOSS FACTOR

Second step

Introduice dissipative terms in plate equation to obtain an estimator of loss factor

33

ENSIM 3A



If the dissipations, losses due to structural dissipation and

losses by radiations, are taken into consideration, equation

of Kirchhoff are expressed by

� is the complex bending stiffness

� the structural loss factor

� is the acoustic radiation pressure on the two sides of the plate

( ) ( )∑ −+−=−∇i

iia FpvhvDj

rrδρωω

241

( )ηjDD += 1

η

ap

34

ENSIM 3A



Assumptions

� non external mechanical forces

� no local damping or absorptions

{ }{ } aT

vv

vvηηη +=

∇

∇−=

*4

*4

Re

Im

Estimator of the total loss factor

• the structural loss factor

• the loss factor due to acoustic

radiations

is radiation efficiency coefficient

η

aη

h

c

vh

Ina ρω

σρ

ωρη 0

22

2==

σ

Maximum Magnitude order

at critical frequency

for brass plate

•

•

4104.9

−×<aη

34105107

−− ×<<× Tη

35

ENSIM 3A

THREE EFFECTIVE ESTIMATORS FOR 2D STRUCTURESTHREE EFFECTIVE ESTIMATORS FOR 2D STRUCTURES

{ }4/1

2

4Re

∇=

∗

v

vvaγ

{ }4/1

2

4Re

∇=

∗

v

vvbγ

{ }{ }*4

*4

Re

Im

vv

vvT

∇

∇−=η

Local WLocal Wavenumberavenumber EstimatorEstimator

AverageAveragedd WWavenumberavenumber EstimatorEstimator

LLossoss FFactor actor EEstimatorstimator

� They are derived from energetic conception : they are independent of

the resolution in wavenumber domain

� They are function of and

� They are based on the assumption : there are no external mechanical

forces and no local damping.

v4∇v

Develop computation methods

36

ENSIM 3A

METHOD OF COMPUTATION OF ESTIMATORSMETHOD OF COMPUTATION OF ESTIMATORS

Third step

Find solutions to compute the double

Laplacian of vibrating velocity and to exclude the points in local excitation or absorbingzones

37

ENSIM 3A

To compute the double

Laplacian of the vibrating

velocity, the technique of

wavenumber processing

associated with the Spatial

Fourier Transform (SFT) is

employed.

METHOD OF COMPUTATION OF ESTIMATORSMETHOD OF COMPUTATION OF ESTIMATORS

Pre-processing and Spatial Fourier Transform (SFT)

{ }4/1

2

4Re

∇=

∗

v

vv

bγ

v4∇

( ) ),(2224yxyx

SFT

KKVKKv +∇ a

To reduce the distorsions

caused by truncated signal,

Pre-processing such as mirror

method is applied before

performing SFT.

38

ENSIM 3A

METHOD OF COMPUTATION OF ESTIMATORS METHOD OF COMPUTATION OF ESTIMATORS ((concon’’tt))

Method to remove excitation or damping zones from computations

An experimental example is used

to show how to exclude the data in

excitation or damping zones

� A brass plate with dimension

350 x 200 x 3 mm

� The plate is excited by a shaker

� Normal vibrating velocity was

measured by using Scanning

Laser vibrometer

Map of proportionalto exteral power flow due to forces acting on the brass plate

( f = 1500 Hz)

{ }∗∇ vv4

Im

Damping zone

Excitation zone

Hotpots

39

ENSIM 3A

Use of Histogram of

� The histogram shows the

distributions of the values of

estimator over the plate

� The unwanted values are

negative ones and ‘very large’ ones



4aγ

4aγ

The points corresponding to those values are the

zones of energy transfer due to external forces

Zones of Zones of

unwantedunwanted

valuesvalues

40

ENSIM 3A

Map of estimator

� Trace map of estimator

� Trace the points in the

excitation or damping zones

(circles in cyan color)



4aγ

It is shown that the excluding points in excitation or

damping zones can be determined by the methods

proposed here.

41

ENSIM 3A

How to determine values in the

excitation zones

� Construct a vector in the way of

sorting the values in

ascending order.

� Select a threshold value with

the help of the curve of the vector

� The values greater than the

threshold value are unwanted

values and are removed from the

computation, resulting in excluding

the excitation or damping zones.

How to determine values in the

excitation zones

� Construct a vector in the way of

sorting the values in

ascending order.

� Select a threshold value with

the help of the curve of the vector

� The values greater than the

threshold value are unwanted

values and are removed from the

computation, resulting in excluding

the excitation or damping zones.



{ }∗∇ vv4

Im

42

ENSIM 3A

� The values are averaged over the

points selected using the histogram of � The values are averaged over the

points selected using the histogram of


ExamplesHistogram of at frequency indicatedaγ

aγbγ � The is computed

and shown by yellow band� The is computed

and shown by yellow band

42// bhD γωρ =

Experimental values have good agreement with Table values

43

ENSIM 3A

EXPERIMENTAL RESULTS OF EFFECTIVE PARAMETERS

Scanning Laser Vibrometer

� A scanning vibrometer use a OFV

300 optical head

� Two galvo-driven mirrors direct the

laser beam horizontally and vertically

� 32x32 measurement points

Structure for testing

�Test assembly consists of two steel plates

of thickness 1 mm

� The two opposite edges are clamped.

The two other edges are free

� A normal point force is acting on the plate

x

93

0

mm

850 mm

20 m

m

clampededge

free edge

y

Divergence of the structural intensity

44

ENSIM 3A

EXPERIMENTAL RESULTS OF EFFECTIVE PARAMETERS

The curve in dashed line is the wavenumber computed by

( ) 41eB Dhk ρω=

with

( )ω

ωγ

ρω

ωω

ω

ω

dh

Db

e ∫−=

2

1

4

2

12

1

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