Statistical Energy Analysis A. Le Bot

Tribology and system dynamics laboratory CNRS - Ecole centrale de Lyon, FRANCE

Contents: 1 Introduction 5 Modal approach of statistical energy analysis 2 Modal density 6 Coupling loss factors 3 Damping 7 Energy balance 4 Randon functions 8 Injected power

9 Entropy in statistical energy analysis

Reference: Foundation of statistical energy analysis in vibroacoustics, A. Le Bot, Oxford University Press, may 2015.

Up2HF, summer school, Celya, 1-3 july 2015

automobile < 1000 Hz

aircraft < 100 Hz

ship < 10 Hz

Launcher, building < 1 Hz

Frequency limits by FEM

When the frequency increases,

Huge number of dof

The modal density increases (!)

High sensitivity to data

Frequency limitation of FEM

Inefficiency of modal analysis

Poor predictibility

ωΔΔ= Nn

FRF

Hz BF MF HF

FRF

Hz

LF HF

Introduction

Statistical energy analysis is a statistical theory of sound and vibration when the number of modes is large and vibration is sufficiently disorganized.

uncoherent vibration coherent vibration

laminar flow turbulent flow

modes <-> molecules diffuse field <-> thermal equilibrium

Dimension 1 Resonance = integer number of half-wavelengths wavenumber dispersion relationship modal density group speed

Each finite structure (or bounded room) has a sequence of natural frequencies which tends to infinite. Ex. string: and for

(!)

n(!) =L

⇡cg

length

#modes wavenumber

group speed

1D modal density

length

Modal density

!i = i⇡

Li = 1, 2, ...

⇡L = N

n(!) =dN

d!=

L

⇡

d

d!

cg =d!

d

i(x) = sin⇣i⇡

x

L

⌘

Dimension 2 Resonance = integer number of half-wavelengths in each direction wavenumber vector number of modes below : is approximated by the area under the ellipse modal density group speed phase speed

surface

integer

phase speed

2D modal density

length integer width

!

n(!) =S!

2⇡cpcg

group speed

N(!) = Card

⇢(p, q) 2 N2,

⇣p⇡a

⌘2+

⇣q⇡b

⌘2 2(!)

�

N(!)

n(!) =dN

d!=

ab

2⇡d

d!

cg =d!

d cp =!

N(!) =⇡

4

a

⇡

b

⇡=

ab2

4⇡

= (x

,y

)

1 =⇣p⇡

a

⌘2+⇣q⇡

b

⌘2

(p, q) 2 N2y

⇡b = q

x

⇡a = p

Dimension 3 Resonance = integer number of half-wavelengths in each direction wavenumber vector number of modes below : is the volume enclosed by the ellipsoid surface modal density group speed phase speed

surface

integer

phase speed

3D modal density

length integer width

!

group speed

N(!)

cg =d!

dcp =

!

= (x

,y

,z

) y

⇡b = q

x

⇡a = p (p, q, r) 2 N3

integer height

z

⇡c = r

1 =⇣p⇡

a

⌘2+

⇣q⇡b

⌘2+

⇣r⇡c

⌘2

n(!) =V !2

2⇡2c2pcg

N(!) = Card

⇢(p, q, r) 2 N3,

⇣p⇡a

⌘2+⇣q⇡

b

⌘2+⇣r⇡

c

⌘2 2(!)

�

N(!) =1

8⇥ 4

3⇡a

⇡

b

⇡

c

⇡=

abc3

6⇡2

n(!) =dN

d!=

abc

2⇡22 d

d!

Damping At high frequencies, the vibrational level is controlled by damping -  viscous damping coefficient -  modal damping ratio -  damping loss factor -  half-power bandwidth

Half-power bandwidth half-power bandwidth damping loss factor This a low frequency method

Reverberation time Tr=Time for a decrease of 60 dB of vibrational level (or SPL) Time decrease of energy

Power balance Steady state condition Measurement of with an impedance head This is a high frequency method Measurement of with accelerometers or vibrometer

⇣⌘�

⇣ =c

2m!0!0 =

rk

m

⌘ = 2⇣ � = ⌘!0

c

FRF

Hz

-3 dB �

�⌘ =

�

!0

E(t) = E0 exp (�⌘!t) 10 log10 E(Tr)� 10 log10 E0 = �60

Tr =2.2⇥ 2⇡

⌘!Tr = 0.16

V

↵S

In structures In rooms

(Sabine’s formula)

Pinj = ⌘!EPinj

E

Random functions A random function is a map where is a random variable

< . > probabilitic expectation Auto-correlation – power spectral density

At

t 7! x(t) x(t)

Auto-correlation

hx(t)x(t+ ⌧)i = 1

2⇡

Z 1

�1S

xx

(!) exp (i!t)d!

Power spectral density (PSD)

time time delay

hx2i = 1

2⇡

Z 1

�1S

xx

(!)d!

⌧ = 0

Stationarity A random function is stationary if and do not depend on t. White noise Uncorrelation and are said uncorrelated if or Response to a linear system

frequency response function

PSD of output

0

0

Time

Signal

00

TimeCorrelation 1

Frequency

PSD

⇢S

xx

(!) = S0

hx(t)x(t+ ⌧i = S02⇡ �(t)

x(t) y(t) hx(t)y(t+ ⌧)i = 0 Sxy

(!) = 0

H(!) =X(!)

F (!)F (!) exp (i!t) X(!) exp (i!t)

hx(t)i hx(t)x(t+ ⌧i

Sxx

(!) = |H|2(!)Sff

(!) Sfx

(!) = i!H(!)Sff

(!)Sxx

(!) = !2|H|2(!)Sff

(!)

The goal of statistical energy analysis is to provide an analysis of vibrating structures in terms of energy and power. : vibrational energy in subsystem i. : kinetic energy in subsystem i. : elastic energy in subsystem i. How is the vibrational energy shared between the elastic and kinetic forms? : power injected in subsystem i by external forces : power dissipated in subsystem i. How to compute the injected power from spectrum of forces? Is the dissipated power related to vibrational energy? : power exchanged between subsystem i and j. Is there a relation between and , ?

Modal approach of statistical energy analysis

Flexible structure excited by random forces

Random forces

Exchange of energy

Dissipation of energy

1

2

Ki

Vi

Ei

Ei = Vi +Ki

Pinj,i

Pdiss,i

Pij

Pij Ei Ej

Governing equation Frequency response function Power balance

kinetic energy elastic energy dissipated power injected power vibrational energy

Single resonator Assumptions

• linear mechanical oscillator

•  stationary random force f(t)

•  white noise force of spectrum S0 ⇣ =c

2m!0

!0 =

rk

m

mx+ cx+ kx = f(t) H(!) =1

m!20

h1 + 2i !

!0� !2

!20

i

d

dt

1

2mx

2 +1

2kx

2

�+ cx

2 = fx

i) Equality of kinetic and elastic energies ii) Injected power iii) Mean power balance

hV i = 1

2khx2i = k

4⇡

Z 1

�1Sxx

(!)d! =kS0

4⇡

Z 1

�1|H|2(!)d! =

kS0

8⇣m2!30

hKi = 1

2mhx2i = m

4⇡

Z 1

�1Sxx

(!)d! =mS0

4⇡

Z 1

�1!2|H|2(!)d! =

mS0

8⇣m2!0

hPinj

i = hfxi = 1

2⇡

Z 1

�1Sfx

(!)d! =S0

2⇡

Z 1

�1i!H(!)d! =

S0

2m

hdEdt

i=hmxxi+ khxxi = m

2⇡

Z 1

�1Sxx

(!)d! +k

2⇡

Z 1

�1Sxx

(!)d!=mS0

2⇡

Z 1

�1i!3|H|2(!)d! +

kS0

2⇡

Z 1

�1i!|H|2(!)d!=0

hPdissi =c

mhEi

hPinji =c

mhEi

hV i = hKi

hPinji =S0

2m

hdEdt

i+ hPdissi = hPinji

hPdissi = chx2i = 2c

m

⇥ 1

2mhx2i = 2c

m

hKi = c

m

hEiodd function

Governing equation Frequency response matrix Power balance kinetic energy elastic energy dissipated power exchanged power injected power The elastic energy of the coupling has been shared between adjacent oscillators

Pair of resonators Assumptions

•  linear mechanical oscillators

•  stationary random forces

•  white noises force of spectrum

•  uncorrelated forces

•  conservative coupling (inertial, gyroscopic and elastic)

Lyon and Sharton 1968

✓m1 00 m2

◆✓x1

x2

◆+

✓c1 00 c2

◆✓x1

x2

◆+

✓k1 �K

�K k2

◆✓x1

x2

◆=

✓f1

f2

◆

M C K

H(!) =⇥�!2M+ i!C+K

⇤�1

d

dt

1

2mix

2i +

1

2kix

2i �

1

2Kx1x2

�+ cix

2i +

1

2K(xixj � xixj) = fixi

i)  Equality of kinetic and elastic energies It exists a unique way to share the coupling energy between oscillators to enforce the equality for each oscillator. ii) Injected power The injected power does not depend on the presence of a coupling iii) Mean power balance Since each product is a linear combination of the power spectrums Si and

So, by expanding After calculating all the integrals by the residue theorem

hVii = hKii

hPinj,ii =Si

2mi

hx(p)i x

(q)j i

hPiji =�A1 A2

�✓ S1

S2

◆

✓hE1ihE2i

◆=

✓E11 E12

E21 E22

◆✓S1

S2

◆ hPiji =�A1 A2

�✓ E11 E12

E21 E22

◆�1 ✓ hE1ihE2i

◆

hP12i = B1hEi �B2hE2i

B = B1 = B2 =K2(�1 +�2)

m1m2 [(!21 � !2

2)2 + (�1 +�2)(�1!2

2 +�2!21)]

hP12i = B(hEi � hE2i)« blocked » natural frequency half-power bandwidth

�i = ci/mi

!i =p

ki/mi

Governing equation Perturbation technique Frequency response matrix i) Equality of kinetic and elastic energies ii) Injected power iii) Mean power balance The coupling factor B is a second order term

Set of resonators Assumptions

•  white noise forces of spectrum Si for i=1,…,n

•  uncorrelated forces

•  conservative coupling

•  light coupling

Newland 1966

H(!) =⇥�!2M+ i!C+K

⇤�1

MX+CX+KX = F(t)K =

✓k1 OO kn

◆+ ✏

✓0 ⇥⇥ 0

◆

small parameter

hVii = hKii

hPinj,ii =Si

2mi

xi(t) = xi0(t) + ✏xi1(t) + ✏

2xi2(t) + o(✏2)

B = B1 = B2 =K2(�1 +�2)

m1m2 [(!21 � !2

2)2 + (�1 +�2)(�1!2

2 +�2!21)]

hP12i = B(hEi � hE2i) + o(✏2)

B = ✏2B0

✏ / K

Subsystems of resonators Assumptions

•  conservative and light coupling

•  subsystem = { uncoupled resonators + same mi, ci }

•  « rain-on-the-roof » forces =

white noise forces + uncorellated forces + PSD constant in subsystems

subsystem i, j mode α, β

Canonical problem By previous results But at order 0 ( ), Coupling power proportionality

Ei =X

↵

Ei↵

Pij =X

↵,�

Pi↵,j�

hPiji =X

↵,�

✏

2B

0i↵,j�(hEi↵i � hEj�i) + o(✏2)

✏ = 0 hEi↵i =Si

2ci+ o(1) hEi↵i =

hEiiNi

+ o(1)

hPiji =

0

@X

↵,�

Bi↵,j�

1

A✓hEiiNi

� hEjiNj

◆+ o(✏2)

Equipartition of energy PSD constant

damping constant

8<

:

mixi↵ + cixi↵ + ki↵xi↵ = fi↵ +P

j 6=i

P� ki↵,j�xj�

Ei↵ = 12mix

2i↵ + 1

2ki↵x2i↵ �

Pj 6=i

P�

12ki↵,j�xi↵xj�

Pi↵,j� = 12ki↵,j�(xi↵xj� � xi↵xj�)

Random resonators The coupling factor is a very complicated function that depends on « blocked » natural frequencies and half-power bandwidth of all resonators. In practice, its effective computation is costly (N~106) and requires the knowledge of all details of the subsystems. This is exactly what we want to avoid in statistical energy analysis To « forget » the exact position of eigenfrequencies, we need to approximate the discrete sum by an integral. How to choose the pdf?

Empirical cumulative distribution function

X

↵,�

Bi↵,j�

1

NiNj

X

↵,�

f(!↵,!�) =

Z Zf(!,!0)pi(!)pj(!

0)d!d!01

N

X

↵

f(!↵) =

Zf(!)p(!)d!

F!i(!) = Card {↵/!i↵ !}

pi(!) =d

d!F!i(!) =

ni(!)

Ni

modal density

# of modes in Δω

pdf

Δω Hz

�i << �!

Uniform probability density function

X

↵,�

Bi↵,j� =⇡K2NiNj

2mimj!20�!

X

↵,�

Bi↵,j� =K2

mimj

Z

�!

Z

�!

(�i +�j)d!id!j⇥(!i + !j)2 +

14 (�i +�j)2

⇤ ⇥(!i � !j)2 +

14 (�i +�j)2

⇤

Assumptions

•  light damping

Coupling power proportionality Reciprocity

hPiji = !0⌘ijNi

✓hEiiNi

� hEjiNi

◆⌘ijNi = ⌘jiNj

coupling loss factor

half-power bandwidth frequency bandwidth

smooth term fluctuating term

Ensemble of similar systems

number of members in the ensemble

number of modes in each member

1

NiNj

X

↵,�

� Z

�!

Z

�!�d!d!0

System with a large number of modes and

SEA applies to a unique system Population of similar systems with random variations (Gibb’s ensemble)

SEA applies to the average system

Ni >> 1 Nj >> 1

Ne ⇥Nm >> 1

Meaning of the approximation process

Coupled beams

coupling force

Assumptions

•  non-resonant modes are neglected

Governing equation Blocked modes

mi@

2ui

@t

2+ ci

@ui

@t

+ EiIi@

4ui

@x

4= fi(x, t) +K

@uj

@x

�

0(x)

external force

Lotz & Crandall 1971

EiIid

4

dx

4 i↵(x) = mi!

2i↵ i↵(x)-

-  modes are orthogonal -  they form a complete set ui(x, t) =

X

↵

Ui↵(t) i↵(x)

i↵(x)

Reduction to the canonical problem Subsituting into the governing equation leads to the following set of equations on modal amplitudes This is exactly the canonical problem Coupling power proportionality

ui(x, t) =X

↵

Ui↵(t) i↵(x)Ui↵(t)

8><

>:

miUi↵ + ciUi↵ +mi!2i↵Ui↵ = Fi↵ +

P� K

0i↵(0)

0j�(0)Uj�

Ei↵ = 12miU2

i↵ + 12mi!2

i↵U2i↵ � 1

2

P� K

0i↵(0)

0j�(0)Ui↵Uj�

Pi↵,j� = 12K

0i↵(0)

0j�(0)(Ui↵Uj� � Ui↵Uj�)

hPiji = !0⌘ijni

✓hEiini

� hEjini

◆⌘ij =

K2

Li(miEiIi)1/2m1/4j (EjIj)3/4!3/2

length mass per unit length bending stiffness coupling loss factor modal density

Wave approach Specific intensity: Transmission efficiency : Hence The problem reduces to the computation of the mean transmission efficiency

Coupling loss factors

power transmitted from 1 to 2 vibrational energy 1

2

reflected transmitted

incident

θ Area A1

Length L

Assumptions

•  diffuse field (homogeneous + isotropic)

P1!2 = !⌘12E1

I1 = cg1Ei

2⇡AiT (✓) =

Ptransmitted

Pincident

P1!2 =

Z

L

ZI1 cos ✓T (✓)d✓dL

P1!2 =

E1

2⇡A1cg1L

Z ⇡/2

�⇡/2T (✓) cos ✓d✓

⌘12 =

Lcg1⇡!A1

Z ⇡/2

0T (✓) cos ✓d✓ ⌘12 =

Scg14⇡!V1

Z 2⇡

0

Z ⇡/2

0T (✓,') cos ✓ sin ✓d✓d'

coupling length

Dimension 2

plate area

Dimension 3

coupling surface room volume

Identification procedure System =⎨ n subsystems⎬ The subsystems are alternatively excited by a source of unit power. :energy in subsystem j for a unit power in subsystem i Repeating the experiment for other source subsystems gives n2 equations. The coupling loss factors are obtained by solving these equations. Remark : reciprocity of CLF has not been used. It exists many variants of this procedure

Eij

0

BBBBBB@

0...

Pi = 1...0

1

CCCCCCA=

0

B@⌘11 . . . ⌘1n...

...⌘n1 . . . ⌘nn

1

CA

0

BBBB@

E1......

En

1

CCCCAn equations

imposed measured unknown

For each subsystem, the energy balance reads Loss by damping Loss by coupling Hence In a matrix form

Energy balance

exchanged power injected power dissipation

Pinj,i = Pdiss,i +X

j 6=i

Pij

Pdiss,i = ⌘i!Ei

Pij = !⌘ijEi � !⌘jiEj

Pi = ⌘i!Ei +X

j 6=i

(!⌘ijEi � !⌘jiEj)

!

0

B@

Pj ⌘1j �⌘ji

. . .�⌘ji

Pj ⌘nj

1

CA

0

B@E1...

En

1

CA =

0

B@P1...Pn

1

CA

A statistical estimation of the mean conductance ( real part of the mobility) gives,

Injected power

square mean force

modal density

mass of subsystem

Random force

F

mass M

Injected power

force velocity

mobility

P = hfvi = 1

2⇡

Z 1

�1Sfv(!)d!

P =1

2⇡

Z 1

�1Sff (!)Y (!)d!

P = hf2i 1

�!

Z

�!< [Y (!)] d!

P = hf2i ⇥ ⇡n(!)

2M

injected power

Clausius’ postulate Heat cannot be transferred from cold body to hot body without converting heat into work.

Clausius (1822-1888)

Q T2>T1 T1

€

dS =δQT

High modal energy

€

dS = N dEE

Low modal energy

Analogy : Heat

Coupling power proportionality

€

Pij =ωηijNiEiNi

−E j

N j

%

& '

(

) *

Entropy in statistical analysis

Heat spontaneously flows from hot body to cold body

Vibrational energy flows from high modal energy to low modal energy

Vibrational power

Vibrational energy Temperature T Modal energy E/N

Clausius entropy : Vibrational entropy Vibrational entropy

Variation of energy

# of modes

R.H. Lyon

Entropy balance

3E

2E

1E

Dissipation of entropy

Production of entropy

Production of entropy by mixing

Statistics at large scale

mean free path micrometer-cm m

statistical physics Quantum physics Continuous physics Complex physics

2nd statistics

Molecular dynamics

Quantum physics

Turbulence theory

Granular material

SEA

Kinetic theory of gases

Solid state physics

Fluid mechanics

Point material

Vibroacoustics

loss of information

thermodynamical entropy vibrational entropy

Vibrational entropy is a measure of missing information between FEM and SEA.