Expression de l’intensité de structure

L’intensité de structure

Jean-Claude Pascal LAUM, ENSIM

Plan

Expression de l’intensité dans les poutres

Formulation approchée

La mesure

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

( )^{t = −}

∑

i

j

j ij

i σ &

( ) ^{= −}

∑ {

}

=

j

j ij i

i i t w

I Re σ &

2 1

( )

{

}

ij t σ e ^ω

σ = Re wj( )t = Re

{ }

4

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

Définition de l’intensité des ondes de flexion

( )^x

{

}

I = − Re ^&∗ + θ^&∗

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 x EI w

x Q

x x EI w

x M

x x x w

∂

− ∂

=

∂

= ∂

∂

= ∂ θ

( ) ( ) ( ) ( ) ( )









∂

∂

∂

− ∂

∂

= ∂

∗

∗

x x w x

x x w

x w x w x EI

I & &

&

&

2 2 3

3

2ω Im

ω j w

w = &

Utilisation des différences finies pour exprimer les dérivées spatiales

Intensité approchée par différence finie

( ) ( )

1 2

3 4

3 3 2

1 2

3 4

2 2

2 3

1 2

3 3

2 2

x

w w

w w

x w x

w w

w w

x w

x w w

x w w

w w

∆

− +

≈ −

∂

∂

∆

+

−

≈ −

∂

∂

∆

≈ −

∂ + ∂

≈

&

&

&

&

&

&

&

&

&

&

&

&

&

&

&

&

x

∆x

1 2 3 4

( ) {

}

≈ ∆ ₃ Im 4 _{2 3 1 3 2 4}

4 w w w w w w

x

I EI & & & & & &

ω ω

j w

w& = && ^{⇒ I ≈ EI Im}

{

}

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

= ^{w j}ω ⇒

w& &&

( )

w = ₁ ⁻ + ₂ ⁻ + ₃ + ₄

4

EI

k ρ A

ω

=

x k w x

w w x k

w

∂

− ∂

∂ ≈

− ∂

∂ ≈

∂ & &

&

&

2 3

3 2

2 2

{

}

∗

≈ ∆

→

 







∂

≈ 2 Im ∂ Im w₁ w₂

x A I EI

x w w A

EI

I & & &

& ρ

ρ

{

}

≈ ₂ ∆ Im w₁ w₂ x

A

I EI && &&

ω ρ

∆x 1 2

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

MEASURED ENERGETIC QUANTITIES

Force distribution Divergence of the structural intensity Potential energy density

Kinetic energy density

Structural intensity

{

}

=

⋅

∇ B ₄ vv

2ω Im I

( )

(

)

j y B

x

F ,  ∇⁴ , − _B⁴ ,



 



= 

ω

4 v2

h T = ρ

( )

























∂

∂

− ∂









∂

∂

∂

− ∂

−

∇

=

∗ 2 2

2 2 2 2 2

2

2 2 1 Re

4 x y

v y

v x

v v

V B ν

ω

( ) ( )







 − ∇×∇× ∇

−

∇

∇

−

∇

∇

= B v v^∗ v v^∗ v v^∗

2 Im 1

2

2

2 υ

I ω

MEASURED ENERGETIC QUANTITIES :

ADVANCED METHODS OF WAVENUMBER PROCESSING Use of SFT for calculation of spatial derivatives

( )

) , ( ) (

) , (

y x n

y m

x TF

n m

n m

k k V jk

jk y

x

y x

v − −

∂

∂

∂ ⁺

a

is the Spatial Fourier Transform of )

, (k_x k_y

V ^v(^x,y)

y x

SFT

kx

ky

(x y)

v , V(k_x,k_y)

SFT on truncated signals amplifies the components of high wavenumbers, bringing large contributions of the high wavenumber components to the

The derivatives of vibrating velocity are easily calculated by

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).

-0.20 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

0.5 1 1.5 2 2.5 3 3.5 4 4.5

5x 10^-5

mA plit ud e

S igna l

dis ta nce (m)

Mirror s igna l + Origina l s igna l Origina l s igna l

MEASURED ENERGETIC QUANTITIES :

ADVANCED METHODS OF WAVENUMBER PROCESSING

EXPERIMENTAL CONFIGURATION

930 mm

850 mm

20 mm

clamped edge 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

ASSEMBLY PLATE ANALYSIS : STRUCTURAL INTENSITY

Wavenumber processing was used

Data integrated over a frequency band of 525 to 1000 Hz

A I

I

I = _φ + _A = −∇φ + ∇×

Standard structural intensity

Irrotational structural intensity

F orce s F =668 Hz

Divergence 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

ASSEMBLY PLATE ANALYSIS :

INJECTED OR DISSIPATED POWER AND FORCE DISTRIBUTION

ASSEMBLY PLATE ANALYSIS :

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

EQUIVALENT 1D MODEL :

AVERAGED POWER FLOW ON PLATE

x

y 0

forc e

Plate 1

Plate 2 join

( )

∫ ( )

Ly x

x x I x y dy

0

φ ,

( )

∫

( )

y y I x y dx

0

φ ,

Total power flow in x-

direction Total power flow in y-

direction

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

EQUIVALENT 1D MODEL :

ONE-DIMENTIONAL BEAM-LIKE MODEL

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

EQUIVALENT 1D MODEL :

ONE-DIMENTIONAL BEAM-LIKE MODEL

+

a2 +

a1

plate 1 junction plate 2

−

a2

−

a3

−

a1



 



 −

=₂¹ ₁^{+ 2} ₁⁻²

1 a a

P

2 2 3 3 1

− −

= a

P



 



 −

=₂¹ ₂^{+ 2} ₂^{− 2}

2 a a

P

plate 1 junction plate 2

P2

P3

P1

Expression of the conservation of flow

“entering” in the junction

3 0

2

1 + P + P = P

2

2 2 1

3

e t e

c

P _{g d} +

−

≈

An approximation of the dissipated power



 





















=

















+ +

−

−

−

2 1

3 2 1

a a t

r t

t t r

a a a

d d

Far-field scattering matrix

cg the group velocity for the flexural waves

td the dissipation coefficient

2 1, e

e the densities of total energy on both sides of the junction the reflection and transmission coefficients

r, t

EQUIVALENT 1D MODEL :

ONE-DIMENTIONAL BEAM-LIKE MODEL

USE ENERGY CONSERVATION LAW TO COMPUTE JOINT DISSIPATION

From the exact conservation law of energy

(

)

(

)

⋅

∇ I (^{x, y})

I

(^{x y})

W_dis ,

) , (x y W

is the measured structural intensity

is represented by a simplified model for dissipation proposed by Nefske & Sung and Bouthier & Bernhard :

(x y) e(x y)

W_dis , ≈ηω ,

the local average of the total energy density

is the injected or dissipated power by external elements (forces, joint, …)

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

( )

( )

(

)

dx x d

x

x +ηω = δ −

φ

the density of the total energy integrated in y direction (in J/m). The differential form of the conservation law is then written by

( )^x

E_x

( )

[

]

0 0

0

, ,

, y dy I x y W x y dy W x x x

x I

y y

y L

dis L

y L

x + + = −

∂

∂

∫ ∫

(^{x y})^{dy E} ( )^x

e _x

Ly

ω η ω

η ∫ =

0

,

USE ENERGY CONSERVATION LAW

EQUATION OF 1-D ENERGY CONSERVATION

For a point force in (^x₁^,y₁)

( )^{x Wx} (^{x x} )^{dx E}_x^x

x δ ηω

φ = ∫ − −

0

1

( ) ^E ( )^{x W} (^{x x}₁)

dx x d

x

x +ηω = δ −

φ

x x

(

)

Wδ −

W 0

x1 L_x

( )

φ

0 ^x₁ ^L_x

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 φ_x( )0 = 0, φ_x( )L_x =W −ηωE_xL_x = 0

The loss factor can be estimated

by: ∫ ∫ ( )

= −

= Lx Lx

x x

x

xL e x y dxdy

E

W _{max min}

ω ,

φ φ

η ω

USE ENERGY CONSERVATION LAW EXEMPLE WITH ONE FORCE

0

x

x 0

plate 2 plate 1

( ) (^{x x x}₀)

E

L_{y x} −

−ξ ω δ

( )^x₀

E L_yω _x ξ

(^{x x}₁)

Wδ −

W

x1

x0 L_x

Lx

USE ENERGY CONSERVATION LAW ONE FORCE AND JOINT

x

y 0

force

Plate 1

Plate 2 joint

( ) ^E ( )^{x W} (^{x x}₁) ^{L E} ( ) (^{x x x}₀)

dx x d

x y x

x +ηω = δ − −ξ ω δ −

φ

( ) ( ) _{( )}

0 0

2 0

1 joint

2E x L E x

x L E

W ωξ _y ^x + ^x =−ξω _{y x}

−

=

+

−

y d

gt L

c ω ξ ≡ ² Use the following differential equation

Dissipated power by joint

uppe r part

lowe r part x01

x02

joint

Position of the joint identified by the measured forces

( )

( ) e(x y)dxdy

L x e L

dy dx y x L e

e x

y x x

L

x L

y x

L x

y

∫

∫

∫ ∫

+

−

>= −

<

>=

<

ε ε

0 0

1 , 1 ,

0 0 2

0 0

0 1

The average density of energy in each of the two plates:

The loss factor

x x

x x

L ω E

φ η =φ ^{max− min}

The linear loss factor of density of dissipation

( )

( _{1 2})^/2

0 2

0 min 1

max

x x

y

x x x

x x

E E

L

x L E x E +

− +

−

= φ −φ ωη

ξ

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

USE ENERGY CONSERVATION LAW CHARACTERIZATION OF JOINT

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

Introduction (

Introduction (con’tcon’t))

The finite-difference-approximation method

Disadvantage

Too high sensitivity to phase differences between sensors due to the use of the finite difference technique

The finite-difference-approximation method

Disadvantage

Too high sensitivity to phase differences between sensors due to the use of the finite difference technique

Methods to compute wavenumbers

Introduction (

Introduction (con’tcon’t))

Methods to compute wavenumbers

Use of Fourier Transform (SFT)

Disadvantage

The use of the direct Fourier Transform results significant errors in the computations because of truncated signals.

Use of Fourier Transform (SFT)

Disadvantage

The use of the direct Fourier Transform results significant errors in the computations because of truncated signals.

Introduction (

Introduction (con’tcon’t))

Methods to compute wavenumbers

Spatial correlation approach

y x jk

jktx ty

e e^{− −}

ty tx k k ,

EESTIMATION OF FLEXURAL WAVENUMBER IN STIMATION OF FLEXURAL WAVENUMBER IN TWO-TWO-DIMENSIONAL STRUCTURESDIMENSIONAL STRUCTURES

First step

Use non dissipative energy equation of plate to derive the effective flexural wavenumbers

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

i i

B v F

k j v

D δ r r

ω

4 4

Fi

D kB

v

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

Derive flexural wavenumbers from energy concept

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

{ }

2 Im

4 =

∇ vv^∗ D

ω

( ) ^{( ) ( )}

( ) ( )

∑

∑

− +

=

−

=

−

∇ ^{∗ ∗}

i

i i

i i

i i

i B

jQ W

v F v

k j vv

D

r r

r r r

δ

ω ² δ

2 1 4 4

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

⋅

∇ I_s

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

Derive flexural wavenumbers from energy concept

{ }

Re ∇⁴v v^∗ − k_B² v² =

Estimators

of effective wavenumber of flexural waves:

{ }

2

Re 4











 ∇

=

∗

v

v v

γa

{ }

2

Re 4















 ∇

=

∗

v v v γb

The brackets < > denote the spatial average over the points, that is, outside the mechanical excitation zones.

CONSIDERATION OF DISSIPATIVE TERMS CONSIDERATION OF DISSIPATIVE TERMS

EFFECTIVE LOSS FACTOR EFFECTIVE LOSS FACTOR

Second step

Introduice dissipative terms in plate equation to obtain an

estimator of loss factor

CONSIDERATION OF DISSIPATIVE TERMS CONSIDERATION OF DISSIPATIVE TERMS

EFFECTIVE LOSS FACTOR EFFECTIVE LOSS FACTOR

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

i i

a F

p v

h v

j D ω ρ δ r r

ω

2

1 4

( jη)

D D = 1+

η pa

CONSIDERATION OF DISSIPATIVE TERMS CONSIDERATION OF DISSIPATIVE TERMS

EFFECTIVE LOSS FACTOR EFFECTIVE LOSS FACTOR

Assumptions

non external mechanical forces

no local damping or absorptions

{ }

{ }

T vv

v

v η η

η = +

∇

∇

= −

* 4

* 4

Re Im

Estimator of the total loss factor

•

• the loss factor due to acoustic radiations

is radiation efficiency coefficient η

ηa

h c v

h I_n

a ωρ

σ ρ

η ωρ ⁰

2 2

2

=

= σ

Maximum Magnitude order at critical frequency

for brass plate

•

•

10 4

4 .

9 × ⁻

a <

η

3

4 5 10

10

7× ⁻ <η_T < × ⁻

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

{ } ^1/4

2

Re 4











 ∇

=

∗

v v v γ a

{ } ^1/4

2

Re 4















 ∇

=

∗

v v v γ b

{ }

{

}

* 4

Re Im

v v

v v

T ∇

∇

= − η

Local W

Local Wavenumberavenumber EstimatorEstimator

Average

Averagedd WavenumberWavenumber EstimatorEstimator

LossLoss Factor Factor EstimatorEstimator

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.

4v

v ∇

Develop computation methods

METHOD OF COMPUTATION OF ESTIMATORS METHOD 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 absorbing

zones

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 ESTIMATORS METHOD OF COMPUTATION OF ESTIMATORS

Pre-processing and Spatial Fourier Transform (SFT)

{ }

2

Re 4















 ∇

=

∗

v

v v γ b

4v

∇

(

)

4

y x y

x SFT

K K V K

K

v +

∇ a

To reduce the distorsions caused by truncated signal, Pre-processing such as mirror method is applied before performing SFT.

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 proportional to exteral power flow due to forces acting on the brass plate

( f = 1500 Hz)

{

}

Im

Damping zone Excitation zone Hotpots

Use of Histogram of

distributions of the values of estimator over the plate

negative ones and ‘very large’ ones

METHOD OF COMPUTATION OF ESTIMATORS

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

Method to remove excitation or damping zones from computations

4

γa

4

γa

The points corresponding to those values are the Zones of

Zones of unwanted unwanted

Map of estimator

Trace map of estimator

Trace the points in the excitation or damping zones (circles in cyan color)

METHOD OF COMPUTATION OF ESTIMATORS

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

Method to remove excitation or damping zones from computations

4

γa

It is shown that the excluding points in excitation or damping zones can be determined by the methods proposed here.