Wang X. Vehicle Noise and Vibration Refinement
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Mid- and high-frequency problems in vehicle noise 153
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where:
B
R
=
++ +
()
[]
++
()
+
()
+
μω ω ω ω
γμ ω ω
2
12
4
21
4
12 12
2
21
2
2
12
2
21
2
2
ΔΔΔΔΔΔ
ΔΔ RR
2
12
2
1
2
2
2
2
1212
2
21
2
1
ΔΔ
ΔΔΔ Δ
+
(
)
−
()
−
()
++
(
)
+
()
⎡
⎣
⎤
⎦
μωω ω ω
(7.44)
If resonator 2 has no direct excitation, S
f
2
= 0, then the power dissipated by
the resonator must equal the power transferred from 1 to 2.
〈P
2,dissipated
〉 = 〈P
1,2
〉; Δ
2
ε
2
= B(ε
1
− ε
2
)
or
ε
ε
2
12
=
+
B
BΔ
(7.45)
The largest value of ε
2
is ε
1
, which occurs when the coupling (determined
by B) is strong compared to the damping Δ
2
.
When the two resonators are identical and have stiffness coupling only,
then Equation 7.44 becomes:
B
RR
==
2
42
2
22
2
2
Δ
ΔΔ
ωω
(7.46)
ε
ε
ω
ω
2
1
222
222
2
12
=
+
R
R
Δ
Δ
(7.47)
Additional points can now be added in regard to the power fl ow from reso-
nator 1 to 2:
5. The power fl ow is proportional to the actual vibration energy difference
of the systems, the constant of proportionality being B.
6. The parameter B is positive defi nite and symmetric in system parame-
ters; the system is reciprocal and power fl ows from the more energetic
resonator to the less energetic one.
7. If only one resonator is directly excited, the greatest possible value of
energy for the second resonator is that of the fi rst resonator.
7.4 Energy exchange in multi-degree-
of-freedom systems
As shown in Fig. 7.5, the equations for the blocked or decoupled subsystems
are:
y
c
y
yP
i
i
b
r
i
i
b
i
i
b
i
i
i
i
(
)
(
)
(
)
+
⎛
⎝
⎜
⎞
⎠
⎟
+==
ρρρ
Λ
12, ,...
(7.48)
Λ
ii
i
ii
i
ψ
ρ
ωψ
α
αα
==
2
12, ,...
(7.49)
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with the orthogonal property and normalizations given by
〈ψ
iα
ψ
iβ
〉
ρ
i
= δ
α,β
(7.50)
The boundary conditions satisfi ed for ψ
iα
include the clamped condition.
Suppose that the spectral densities of the modal excitations F
iα
(t) = ∫p
i
ψ
iα
dx
i
are fl at over a fi nite range of frequency Δω and that within this band
there are N
i
= n
i
Δω modes of each subsystem. The modes for these subsys-
tems are illustrated in Fig. 7.6(a). Each mode group represents a model of
the subsystem.
In the application of SEA, this model has some very particular properties
as illustrated below.
1. Each mode α of subsystem i is assumed to have a natural frequency ω
iα
that is uniformly probable over a frequency interval Δω. This means that
each subsystem is a member of a population of systems that are gener-
ally similar physically, but differ enough to have randomly distributed
parameters. The assumption is based on the fact that normally identical
structures or acoustic spaces will have uncertainties in their modal
parameters, particularly at higher frequencies.
y
1
y
1
y
2
y
2
= 0
y
1
= 0
p
1
p
2
p
1
p
2
11
1
2
2
2
2
(a) (b)
(b)
y
2
(b)
‘Clamped’
(c)
7.5 Blocked or decoupled subsystems.
Subsystem 1
1
2
a
N
1
1
2
s
N
2
Subsystem 2 Subsystem 1 Subsystem 2
Total
Power Π
12
Π
as
(a) Blocked (b) Connected
1
2
a
N
1
1
2
s
N
2
7.6 Modes for subsystems: (a) individual modal groups with
subsystems blocked; (b) interactions of mode pairs with subsystems
connected at the junction.
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Mid- and high-frequency problems in vehicle noise 155
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2. We assume that every mode in a subsystem is equally energetic and that
the modal amplitudes
yt
y
m
x
i
ii i
i
α
α
ρψ
(
)
=
∫
d
are incoherent; that is,
yy y
ii i
αβ αβ α
δ
=
,
2
(7.51)
3. This assumption requires that we correctly select mode groups, which
is an important guide for appropriate SEA modelling. This also implies
that the modal excitation functions F
i
are drawn from random popula-
tions of functions that have certain similarities (such as equal frequency
and wave number spectra) but are individually incoherent.
4. The damping of each mode in a subsystem is the same.
‘Unlock’ the system and consider the new equations of motion, which are:
y
c
yy
pxyy xyyk
i
r
i
iii
iijijj
j
ij i j j i
i
++=
+
()
+−
()
()
+
ρ
μγ
Λ
,,1
jjj
i
y
ijij
ρ
≠=, , , ,...12
(7.52)
One now expands these two equations in the eigenfunctions ψ
1α
(x
1
) and
ψ
2σ
(x
2
) to obtain:
my y y F y y Ry
11
2
α α αα α ασσ ασσ ασσ
σ
ωμγ
++
[]
=+ + +
[]
∑
Δ
(7.53)
my y y F y y Ry
22
2
σ σ σσ σ σαα σαα σαα
α
ωμγ
++
[]
=+ + +
[]
∑
Δ
(7.54)
where α is for subsystem 1, and σ is for subsystem 2, and Δ
i
= c
r
i
/ρ
i
; the mass
of the subsystem is m
i
. The coupling parameters are [1]:
μμ ψψ
ασ α σ
=
(
)
[]
(
)
(
)
[]
∫
12 1 2 1 1 2 1 1
junction
detc.xxx x xx x,,
(7.55)
μμ ψψ
σα σ α
=
()
[]
()
[]
()
∫
21 1 2 2 1 2 2 2
junction
detc.xx x xx x x,,
(7.56)
where the integrations are taken along the junction between the subsystems
and, therefore, over the same range of x
1
, x
2
in both integrals. Conservative
coupling requirements are met by μ
ασ
= μ
σα
= μ, γ
ασ
= γ
σα
= γ, R
ασ
= R
σα
= R,
or μ
21
= μ
12
= μ, γ
12
= γ
21
= γ, R
12
= R
21
= R.
The coupled systems may now be represented as shown in Fig. 7.6(b)
with the interaction lines showing the energy fl ows that result from the
coupling. Suppose we study the energy fl ow between mode α of subsystem
1 and mode σ of subsystem 2. Modes α and σ have energies ε
α
and ε
σ
. The
modal energies of the subsystem 1 modes are all equal; that is ε
α
= ε
1
=
constant and ε
σ
= ε
2
= constant.
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156 Vehicle noise and vibration refi nement
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The inter-modal power fl ow:
PB
a
ασ ασ
ωω
σ
εε
=−
(
)
12
(7.57)
B
ασ
ωω
ασ
ασ
ω
λ
=
π
2
12
ΔΔ
Δ
(7.58)
where
λ
μω γ μ ω
≡
++
()
+
[]
22 2 2 2
12
2 RR
ΔΔ
(7.59)
when μ
2
<< 1, Δ/ω << 1 and λ << 1.
The total power fl ow from all N
1
modes of subsystem 1 to mode σ of
subsystem 2 is:
〈P
1,σ
〉 = 〈B
ασ
〉N
1
(ε
1
− ε
2
) (7.60)
Finally the total power fl ow from subsystem 1 to subsystem 2 is found by
summing over the N
2
modes of subsystem 2:
PBNN
RR
12 1 2 1 2
22 2 2 2
12
12
2
2
=−
(
)
=
++
()
+
−
(
)
ασ
εε
ω
μω γ μ ω
δω δω
εε
π
Δ
(7.61)
when δω
i
= Δω/N
i
is the average frequency separation between modes.
If the total energy of subsystems 1 and 2 is defi ned as E
1(total)
and E
2(total)
:
εε
1
1
2
2
2
==
(
)
(
)
E
N
E
N
1 total total
,
(7.62)
PBNN
E
N
E
N
E
N
N
12 1 2
1
1
2
2
12 1
1
=−
⎡
⎣
⎢
⎤
⎦
⎥
=−
(
)
(
)
(
)
ασ
ωη
total total
total
22
2
E
total
(
)
⎡
⎣
⎢
⎤
⎦
⎥
(7.63)
where η
12
= 〈B
ασ
〉N
2
/ω.
If we defi ne η
21
= N
1
η
12
/N
2
then:
〈P
12
〉 = ω (η
12
E
1(total)
− η
21
E
2(total)
) (7.64)
η
12
and η
21
are called the coupling loss factors for subsystems 1 and 2.
ωη
12
E
1(total)
represents the power lost by subsystem 1 due to coupling, just
as the quantity ω
1
η
1
E
1(total)
represents the power lost by subsystem 1 due to
damping. If subsystem 1 is connected to subsystem 2 with energy E
2(total)
,
subsystem 1 will receive an amount of power ωη
21
E
2(total)
from subsystem 2.
In the basic relationship N
1
η
12
= N
2
η
21
, replacing the modal count N
i
with
modal density time, the frequency bandwidth n
i
Δω gives
n
1
η
12
= n
2
η
21
(7.65)
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Mid- and high-frequency problems in vehicle noise 157
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The system shown in Fig. 7.6(b) is simply represented in Fig. 7.7. The power
fl ow equations for subsystems 1 and 2 are:
〈P
1,in
〉 = 〈P
1,dissipated
〉 + 〈P
12
〉
= ω [η
1
E
1(total)
+ η
12
E
1(total)
− η
21
E
2(total)
] (7.66)
〈P
2,in
〉 = 〈P
2,dissipated
〉 + 〈P
21
〉
= ω [η
2
E
2(total)
+ η
21
E
2(total)
− η
12
E
1(total)
] (7.67)
Considering the case where only one of the systems is directly excited by
one external source, that is, setting <P
2,in
> = 0:
E
E
N
N
2
1
2
1
21
221
total
total
(
)
(
)
=
+
η
ηη
(7.68)
A solution of Equations 7.66 and 7.67 gives:
E
PP
D
1
1 2 21 2 21
total
in in
(
)
=
+
(
)
+
,,
ηη η
ω
(7.69)
E
PP
D
2
2 1 12 1 12
total
in in
(
)
=
+
(
)
+
,,
ηη η
ω
(7.70)
where
D = (η
1
+ η
12
)(η
2
+ η
21
) − η
12
η
21
(7.71)
The dissipated power in subsystem 1 can be evaluated by
〈P
1,dissipated
〉 = 2πfη
1
E
1
(7.72)
where E
1
is the total dynamic energy of the subsystem modes at frequency
f (Hz) and η
1
is the damping loss factor.
As shown in Fig. 7.7, the net transmitted power can be represented in
several different forms, for example
〈P
12
〉 = 2π Δf β
12
(ε
1
− ε
2
) (7.73)
N
1
N
2
E
1,tot
E
2,tot
Π
12
Π
1,in
Π
1,diss
Π
2,in
Π
2,diss
7.7 An energy transfer and storage model.
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where ε
1
and ε
2
are the average modal energies (energies per mode) of the
two subsystems in a frequency band Δf, and β
12
is a coupling factor that
depends only on the physical properties of the two coupled subsystems:
ε
δ
==
E
N
E
f
fΔ
(7.74)
where N is the mode number and
δ
f
is the average frequency spacing
between the modes of the subsystem in the frequency bandwidth Δf.
〈P
12
〉 = 2πf(η
12
E
1
− η
21
E
2
) (7.75)
〈P
1,in
〉 = 〈P
1,dissipated
〉 + 〈P
12
〉 = 2πf(η
1
+ η
12
)E
1
− 2πfη
21
E
2
(7.76)
〈P
2,in
〉 = 〈P
2,dissipated
〉 + 〈P
21
〉 = 2πf(η
2
+ η
21
)E
2
− 2πfη
12
E
1
(7.77)
ε
ε
η
ηη
2
1
21
221
1
0=
+
=
()
P
,in
(7.78)
If N
1
= 1:
PEfEf
12 12 1 2 2
2=−
()
π
βδ
Δ
(7.79)
PEfEf
E
n
E
n
f
12 12 1 1 2 2 12
1
1
2
2
2
Δ
=−
()
=
(
)
−
(
)
⎛
⎝
⎜
⎞
⎠
⎟
π
βδ δ β
ωω
(7.80)
For a complex system with N subsystems, the SEA equations can be written
as
ω
ηη η η
ηηη
nnn
nn
i
i
NN
i
i
111 1
1
21 2 1
12 1 2 22 2
2
+
⎛
⎝
⎜
⎞
⎠
⎟
−−
−+
⎛
≠
≠
∑
∑
......
⎝⎝
⎜
⎞
⎠
⎟
−+
......
.
.
.
.
.
.
...... ......
.
.
.
...... ......
ηηη
11NNNN
nn
NNi
iN
N
N
E
n
E
n
≠
∑
⎛
⎝
⎜
⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎧
⎨
⎪
1
1
.
.
.
.
.
.
⎪⎪
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎪
⎪
⎪
⎪
⎪
⎪
=
⎧
⎨
⎪
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎪
⎫
⎬
⎪
⎪
⎪
⎪
⎭
⎪
P
P
N
1
.
.
.
.
.
.
⎪⎪
⎪
⎪
(7.81)
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If each subsystem has n degrees of freedom, the total of n × N dynamic
equations are reduced to the total of N SEA equations, which speeds up
the computation.
In summary, modes of a system become localized to various subsystems
where local modes of the subsystems of a system are statistically described;
local modal properties/wave properties are assumed to be maximally dis-
ordered in a state of maximum entropy. Simple expressions for ensemble
average energy fl ow between coupled subsystems exist, since vibration
energy fl ow between coupled subsystems is proportional to difference in
modal energies, that is, average energy per mode. The SEA power balance
equations which govern response of a system in a given frequency band can
be derived from the principle of energy conservation.
The above SEA modal description is good for a qualitative introduction
to SEA theory since the SEA equations derived from a modal approach
are based on several assumptions, in particular for the extension of the two
oscillator results to multimodal systems. The exact SEA equations as per
Equations 7.66 and 7.67 can also be derived from a wave approach [8]. This
is discussed in the next section.
7.5 Wave approach to statistical energy
analysis (SEA)
In order to illustrate the wave approach, room acoustics are fi rst reviewed.
A typical room of 100 m
3
has 1 × 10
7
acoustic modes at frequencies of
less than 10 kHz. If the connection between two such rooms is a rigid
piston, the component of fi eld associated with the radiation from such a
piston into unbounded space/subsystem is called a direct fi eld. The differ-
ence between the actual fi eld and the direct fi eld is called the reverberant
fi eld. The term ‘diffuse’ is used to describe a special set of statistics that are
obtained when averaging over a large enough ensemble of reverberant
fi elds, where the average can be taken over a set of nominally identical
subsystems (or sometimes across a frequency band) and the statistics rep-
resent a state of maximum disorder or maximum entropy, in other words
to get an equal partition of energy and incoherence of individual modes/
waves.
As shown in Fig. 7.8, the loading on the connecting piston can be repre-
sented by the direct fi eld impedance of the fl uid in the cavity subsystem and
diffuse reverberant loading (incident power or blocked force proportional
to the energy of the reverberant fi eld, αE
i
). Since the two reverberant fi elds
are incoherent, the responses can be considered separately (Fig. 7.8(b) and
(c)) and then superimposed. The power transmission from room cavity 1
to room cavity 2 is given by
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PP
EcA
V
12 1
11
1
4
==
ττ
inc
(7.82)
where the reverberant fi eld αE
2
is not included, as shown in Fig. 7.9;
P
inc
is incident power, E is cavity energy, c is speed of sound, A is area
of the connecting piston, V is cavity volume and τ is transmission
coeffi cient.
The power transmission from room cavity 2 to room cavity 1 is given
by
PP
EcA
V
21
22
2
4
==
ττ
inc2
(7.83)
where the reverberant fi eld αE
1
is excluded, as shown in Fig. 7.10. The
coupling power between room cavities 1 and 2 is given by
aE
2
aE
1
Cavity 2Cavity 1
Direct field
(a)
(b) (c)
Reverberant field
Reverberation
response
Sound envelope
7.8 Subsystem loading on connection.
7.9 Power transmission from room cavity 1 to room cavity 2; with
reverberant fi elds incoherent, the responses are calculated separately
and then superimposed.
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PP P A
Ec
V
Ec
V
12 2
11
1
22
2
44
=−= −
⎛
⎝
⎞
⎠
ττ τ
inc1 inc
(7.84)
which is proportional to the difference in energy density group velocity.
From the above room acoustics discussion:
• The response of a subsystem can be described in terms of direct and
reverberant fi elds.
• Averaging fi elds across an ensemble gives a diffuse reverberant fi eld
with a state of maximum entropy.
• Two diffuse reverberant fi elds become incoherent.
• Loading on the connection is related to the direct fi eld impedance and
diffuse reverberant fi eld loading.
• The diffuse reverberant fi eld loading is proportional to the energy
density and group velocity of the subsystem.
• Net coupling power is proportional to the difference between local
incident powers.
In the wave approach to SEA, a system is divided into a series of sub-
structures or subsystems that support wave propagation such as beams,
plates, shells, acoustic ducts, acoustic cavities, etc. where each substructure
or subsystem contains a number of wave types including bending, exten-
sional, shear waves, etc. Each wave type is represented by a separate SEA
subsystem which can receive, store, dissipate and transmit energy. From a
wave viewpoint, a subsystem is a collection of propagating waves, while
from a modal viewpoint it is considered as a collection of resonant modes.
A connection is defi ned as a geometric region which allows energy to fl ow
in or out of a subsystem. A direct fi eld is a component of the response
associated with direct fi eld radiation from connections, and is deterministic.
A reverberant fi eld is a component of the response associated with refl ec-
tions from boundaries of the subsystem and blocked connections, and is
statistical. Each SEA subsystem is represented in terms of superposition of
a direct fi eld and a reverberant fi eld as shown in Fig. 7.11.
7.10 Power transmission from room cavity 2 to room cavity 1; with
reverberant fi elds incoherent, the responses are calculated separately
and then superimposed.
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162 Vehicle noise and vibration refi nement
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The SEA wave approach is based on the following assumptions as shown
in Fig. 7.11:
• Neglect coherence between direct fi elds of different connections to the
same subsystem as shown in Fig. 7.11(a).
• There is signifi cant uncertainty regarding the properties of each subsys-
tem so that the reverberant fi elds are diffuse when viewed across the
ensemble as shown in Fig. 7.11(b).
The fi rst assumption is not valid for problems in which direct fi eld transmis-
sion between connections is a domain path, for example heavily damped
subsystems or subsystems that are small compared with a wavelength.
Separate corrections are needed in such cases. The second assumption
implies that the SEA prediction gives the ensemble average response and
leads to incoherence between direct and reverberant fi elds when averaged
across the ensemble.
In order to apply the SEA approach to engineering problems, the pro-
cedures below have to be followed.
7.6 Procedures of the statistical energy
analysis approach
In most cases, the following three steps in the application of SEA have to
be followed:
1. Defi ne the system model.
2. Evaluate the model parameters.
3. Evaluate the response variables.
(a) Direct field
Component of response
associated with direct
field radiation from
connections –
deterministic
Component of response
associated with reflections
from boundaries of subsystem
and blocked connections –
statistical
Reverberant field(b)
+
7.11 Each SEA subsystem is represented by superposition of (a) a
direct and (b) a reverberant fi eld.
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