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
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
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.