Wang X. Vehicle Noise and Vibration Refinement
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Mid- and high-frequency problems in vehicle noise 143
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quantifi ed or modifi ed by changing mass, stiffness and damping or changing
geometry, adding isolators, foams, fi bres, etc.; and the receiver can be
quantifi ed or optimized by design for subjective response, sound quality,
reduction in levels, etc.
In order to predict noise and vibration, classic low-frequency approaches
are applied to establish system dynamic properties or system dynamic equa-
tions in a large number of degrees of freedom by use of fi nite element
dynamic analysis or an experimental modal analysis method, then to calcu-
late the system responses from excitation and the dynamic properties or
equations. For a typical saloon car, there are 3 × 10
6
structural modes and
1 × 10
6
acoustic modes at frequencies less than 10 kHz. Higher-order modes
are extremely sensitive to uncertainties in boundary conditions, material
properties and physical properties. It takes a long computational time to
solve the dynamic equations in the large number of degrees of freedom in
the high frequencies, and the results of the solutions are uncertain due to
the uncertainties caused by the material, manufacturing and assembly
process variations as shown in Fig. 7.1. This is because subsystem properties
and boundary conditions are not known precisely; short-wavelength
responses in higher-order modes are very sensitive to small uncertainties.
Therefore, traditional deterministic analysis methods are not appropriate
for problem solutions at high frequencies due to the expense and amount
of detail required. Alternative approaches are needed for high-frequency
noise and vibration prediction.
Statistical energy analysis (SEA) was originally developed in the 1960s
as a method of predicting the high-frequency response of dynamic struc-
tures [1]. Statistical energy analysis is a fi eld of study in which subsystems
are statistically described in order to simplify the analysis of complicated
structural–acoustic problems; the ‘S’ indicates systems drawn from a popu-
lation or ensemble, ‘E’ represents the fact that energy is the primary
response quantity of interest, and ‘A’ illustrates that SEA is an analysis
frame instead of one method. The power injection method and experimen-
tal techniques in comparison with analytical calculations such as imped-
ance/modal density calculations were developed in the 1970s. Commercial
codes such as VAPEPs, SEAM, AutoSEA, etc., appeared in the 1980s.
Leppington et al. [2] proposed radiation effi ciency formulations and k-space
approaches; more generic coupling loss factor calculations based on line
wave impedances were developed in the 1990s. Langley and co-workers
[3–6] generalized the wave approach and developed the hybrid FEA–SEA
method; based on the wave approach, variance estimation of energy vari-
ables was extended to generic subsystems in the 2000s [7].
In SEA, the whole structure and acoustic cavity system are considered
as a network of subsystems coupled through joints. A subsystem is defi ned
as a fi nite region with a resonant behaviour, involving a number of modes
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144 Vehicle noise and vibration refi nement
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of the same type. Each subsystem’s response is represented by a space and
frequency band averaged energy level. Based on the principle of conserva-
tion of energy, a band limited power balance matrix equation for the con-
nected subsystems can be derived and easily resolved. Because of the
reduced number of degrees of freedom of the models, the SEA calculations
are usually very fast (from few minutes to few hours), allowing a fast
analysis process in the design phase. The method predicts the ‘mean’ energy
level in each of the subsystems, in the sense that the system is considered
to have random properties. The method would, for example, predict the
mean interior noise level for a fl eet of cars manufactured on the same pro-
duction line. A variance prediction method for built-up structures was
developed by Langley and Cotoni [7] in which prediction of the statistics
of the response of a highly random system without knowledge of the nature
of the underlying physical uncertainties became possible. The method
Frequency (Hz)
(a)
(b)
050
60
40
20
0
–20
–40
–60
100 200 300 400 500450350150
FRF (dB re 1 Pa/N)
60
40
20
0
–20
–40
–60
FRF (dB re 1 Pa/N)
250
Frequency (Hz)
0 50 100 200 300 400 500450350150 250
7.1 Uncertainty of measured frequency response functions (FRF);
sampled data range: (a) one vehicle, repeated 12 times; (b) 98
nominally identical vehicles. The excitation and response points are
identical for the FRF measurement.
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Mid- and high-frequency problems in vehicle noise 145
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allows for the long-established SEA method to be extended to variance
prediction. Both the mean energy level and its variance give estimations of
subsystem responses. However, this method is only applicable in mid- to
high-frequency ranges.
The objectives of this chapter are to provide an overview of the modern
predictive SEA method, to highlight the problems associated with model-
ling noise and vibration at mid- and high frequencies, to summarize the
derivation of the SEA equations and list the main assumptions, and to
illustrate the concepts and methods used to derive the parameters of an
SEA model.
7.2 Modal approach
In order to predict vibration energy transmission, as mentioned above, a
deterministic approach is possible in principle but not practical, and an
alternative approach is to adopt a statistical description of response and look
at time and spatial averages of vibration energy. The modal approach is
based on the assumption that two oscillator results apply to coupled multi-
modal systems, that is, the coupling power proportionality of the two oscil-
lators applies to the multi-modal system. In order to illustrate the modal
approach, modal resonators and their properties are fi rst illustrated.
7.2.1 Modal resonators and their distribution
If the generalized displacement of the system is denoted by y:
ρÿ + c
γ
y
+ Λy = p (7.1)
where p is the distributed excitation, ρ is the mass density, c
γ
is a viscous
resistance coeffi cient and Λ is a linear operator consisting of differentiations
with respect to space.
In the case of a fl at plate, for example, ρ is the mass per unit area of the
plate and Λ = β∇
4
, where β is the bending rigidity. The use of simple viscous
damping is a valuable simplifi cation and does not affect the utility of our
results as long as the damping is fairly small. Where the system is bounded,
with well-defi ned boundary conditions, the solution is frequently sought by
expansion in eigenfunctions ψ
n
which are solutions to the equation:
1
2
ρ
ψωψ
Λ
nnn
=
(7.2)
where the functions ψ
n
satisfy the same boundary conditions that y does,
and the quantities ω
n
2
are determined by the boundary conditions and the
operator ρ
−1
Λ. The response and the excitation are then expanded in the
functions
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yytx
nn
n
=
(
)
(
)
∑
ψ
(7.3)
p
M
ft x
nn
n
ρ
ψ
=
(
)
(
)
∑
1
(7.4)
Multiplying Equation 7.2 by ψ
m
, and integrating over the region occupied
by the structure and subtracting from this the same equation with the
indices reversed, we get:
ψψψψ ωω ψρψ
ψρ ψ
ψ
mn nm n m m n
mn
xxx
xx nm
ΛΛ−
[]
=−
()
()
()
=≠
∫∫
∫
dd
dfor
22
0
nn
xxM nm
2
ρ
()
==
⎫
⎬
⎪
⎪
⎭
⎪
⎪
∫
dfor
(7.5)
Normalization of the amplitude of eigenfunctions gives:
1
M
x
mn mn mn
ψρψ ψψ δ
ρ
d ≡=
∫
,
(7.6)
ρ
ρ
ωψ
ρ
ψ
y
c
yy
M
f
n
y
nnnn
n
nn
n
++
⎡
⎣
⎢
⎤
⎦
⎥
=
∑∑
2
(7.7)
if c
y
= ρΔ where Δ is a constant. Multiplying Equation 7.7 by ψ
m
(x) and
integrating it over the system domain gives:
my y y ft
nn n nn n
++
[]
=
(
)
Δ
ω
2
(7.8)
where modal mass is m
n
= M, modal stiffness is ω
n
2
m
n
and mechanical resis-
tance is m
n
Δ = c
n
.
For example, the two-dimensional simply supported isotropic and homo-
geneous rectangular fl at plate of dimensions L
1
, L
2
(see Fig. 7.2) has mode
shapes for bending motion that are:
Supported edges
0
y
n
1
,n
2
= 2 sin
L
1
L
1
L
2
x
1
n
1
px
1
x
2
sin
L
2
n
2
px
2
7.2 Mode shapes for bending motion of a 2D simply supported
isotropic and homogeneous rectangular fl at plate of dimensions L
1
, L
2
.
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ψ
nn
nx
L
nx
L
12
2
11
1
22
2
,
sin sin=
ππ
(7.9)
The resonance frequencies are given by:
ω
nn
ll
n
L
n
L
RC k k RC
12
2
1
1
2
2
2
2
2
22
1
2
2
2
2
22
,
=
⎛
⎝
⎞
⎠
+
⎛
⎝
⎞
⎠
⎡
⎣
⎢
⎤
⎦
⎥
=+
()
=
ππ
kkRC
l
422
(7.10)
where n
1
and n
2
are integers, R is the radius of gyration of the plate cross-
section,
CE
l
=
ρ
is the longitudinal wave speed in the plate material, E
is Young’s modulus and ρ is the material density.
Equation 7.10 suggests a very convenient ordering of the modes, which
is shown in Fig. 7.3, where every ‘wave number lattice’ corresponds to a
mode. The distance from the origin to that point will determine the value
of the resonance frequency of the mode ω
n
1
,n
2
.
Each lattice corresponds to an area ΔA
k
= π
2
/A
p
where A
p
= L
1
L
2
is the
area of the plate. As the wave number increases from k to k + Δk a new
area
1
2
πkkΔ
will include
1
2
πkk A
k
ΔΔ
new modes, therefore the average
number of modes per unit increment of wave number,
nk
kk
Ak
k
A
kk
(
)
==
ππΔ
ΔΔ Δ22
(7.11)
which is termed the modal density in the wave number. The modal density
in the radian frequency n(ω) is derived from the relation:
n(ω) Δω = n(k) Δk (7.12)
n
k
A
kk
CA
kk
ω
ω
(
)
==
ππ
22Δ
Δ
ΔΔ
g
(7.13)
where C
g
= Δω/Δk is the group velocity for waves in a system that has a
phase velocity C
p
= ω/k.
p/L
1
Δk
k
p/L
2
k
2
k
1
7.3 A convenient ordering of modes.
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For a fl at plate, the group velocity C
g
is twice the phase velocity which is
given by
C
k
RC
lp
==
ω
ω
(7.14)
The modal density in cycles per second (Hz) is:
nf n
f
A
C
A
RC
A
hC
ll
(
)
=
(
)
===
ω
ω
ω
d
d
p
p
pp
2
42
3
2
22
π
π
(7.15)
where
Rh= 23
, the radius of gyration for a homogeneous plate of thick-
ness h. In this case the modal density is independent of frequency.
The kinetic energy of vibration is:
1
2
1
2
1
2
2
2
ddx
y
t
xytyt x x
Myt
m
mn
nmn
n
ρρψψ
∂
∂
(
)
=
′
(
)
′
(
)
(
)
(
)
=
′
(
)
[
∫
∑
∫
,
]]
∑
n
(7.16)
Since the kinetic and potential energies of each resonator are equal at reso-
nance, 〈y
n
′
2
〉 = ω
n
2
〈y
n
2
〉, the total energies of the modes simply add to form the
total system energy.
For the response of a system to a point force excitation, a point force of
amplitude ƒ
0
is applied at a location x
s
on the structures modal amplitude
from Equation 7.4:
f t ft x x x x ft x
mm m
()
=
() ( )
−
()
=
() ( )
∫
00
ψδ ψ
ss
d
(7.17)
If the excitation is given by f
0
(t) = F
0
e
iωt
, Equation 7.8 becomes:
MyFx
nnnn
t
ωηωωω ψ
ω
22
0
+−
()
=
(
)
ie
s
i
(7.18)
The response is expressed as:
yxt
F
M
xx
t
nn
nn
,
(
)
=
(
)
(
)
−+
∑
0
22
e
i
i
s
ω
ψψ
ωω ηωω
(7.19)
The driving point mobility function is written:
Y
vx
fM
x
YY
n
nn
n
=
()
=
()
−+
≡
[]
+
[]
∑
ss
i
i
i
,
Re Im
ωω ψ
ωω ηωω
0
2
22
(7.20)
Re Yaxg
n
n
n
[]
=
(
)
(
)
∑
s
ω
(7.21)
where
axM
nn n
=
ψηω
2
()/
s
and g
n
= (ξ
2
+ 1)
−1
which is defi ned below.
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Im Yaxb
n
n
n
[]
=
(
)
(
)
∑
s
ω
(7.22)
where b
n
= ξ(ξ
2
+ 1)
−1
assuming the damping η < 0.3 is small enough so that
the individual modal mobility a
n
(g
n
+ ib
n
) is fairly sharp in frequency where:
a
x
M
g
n
n
n
n
=
(
)
=
+
ψ
ηω ξ
2
2
1
1
s
,
(7.23)
ξ
ω
ηω
ω
ηω
ξ
ω
ηω
=− ≈
n
n
n
,d
d
(7.24)
Yagb
nn n
=+
(
)
∑
i
(7.25)
g
nn
n
ω
ωξ
ω
=
+
−∞
∞
∫
1
2
1
1
2
Δ
d
(7.26)
Assuming ω is fi xed and the ω
n
are random variables, ω
n
tend to ω. Δω is
the interval of resonance frequency, and if
Δ
ω
π
ηω
2
,
g
n
n
ω
ηω
ωξ
ξ
ηω
ω
=
+
=
−∞
∞
∫
2
1
12
2
ΔΔ
d
π
(7.27)
which is simply the ratio of the modal bandwidth to the averaging band-
width. In the interval Δω the number of modes that can contribute to the
average is n(ω) Δω.
Re
,
Y
n
M
n
M
n
x
n
[]
=
(
)
=
(
)
ω
ωω
ηω
ηω
ω
ψ
ω
s
Δ
Δ
ππ
22
2
(7.28)
For a fl at plate, n(ω) = A
p
/4πRC
l
and M = ρ
s
A
p
, where ρ
s
is the surface
density and A
p
is the surface area,
〈Re[Y]〉 = (8ρ
s
RC
l
)
−1
(7.29)
If the input force has a fl at spectrum of S
f
over the band Δω, the power fed
into any one mode can be found from the dissipation ηω
0
M〈
y
2
〉:
P
n
M
S
Yft
n
x
=
(
)
=
[]
(
)
π
π
ωω
ω
22
2
f
s
Δ
Re
,
(7.30)
7.3 Energy sharing between two oscillators
As shown in Fig. 7.4, two linear resonators can be coupled by spring, mass
and gyroscopic elements. The kinetic and potential energies are given by
KE
c
=++ +
(
)
1
2
1
2
1
8
11
2
22
2
12
2
my my m y y
(7.31)
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PE
c
=++ −
(
)
1
2
1
2
1
2
11
2
22
2
21
2
ky ky k y y
(7.32)
The equation of motion in the absence of velocity-dependent forces is
given by:
d
d
KE PE
ty y
f
ii
i
∂
∂
−
∂
∂
=
(7.33)
Substituting Equations 7.31 and 7.32 into Equation 7.33 gives:
mmycykkyfkygymy
1111111222
1
4
1
4
+
(
)
+++
(
)
=+ + −
cccc
(7.34)
mmycykkyfkygymy
2222222111
1
4
1
4
+
(
)
+++
(
)
=+ + −
cccc
(7.35)
where the gyroscopic coupling forece g
y
is included.
If we let y
2
=
y
2
= ÿ
2
= 0 in Equation 7.34 and let y
1
=
y
1
= ÿ
1
= 0 in Equation
7.35, oscillators 1 and 2 become clamped and are called clamped systems 1
and 2. That is, we set the response variables of the other system to be zero
in order to obtain the decoupled system.
Equations 7.34 and 7.35 can be rewritten as:
yyy yyRyF
1111
2
12221
1
+++ +−
(
)
=Δ
ω
λ
μν
(7.36)
yyy yyRyF
2222
2
21112
+++ +−
(
)
=Δ
ωλμν
(7.37)
y
2
y
1
f
1
(t) f
2
(t)
k
1
k
2
m
2
m
1
m
c
k
c
g
c
1
c
2
7.4 Coupled linear resonators.
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where
Δ
i
i
i
i
i
i
c
m
m
kk
m
m
=
+
()
=
+
+
()
c
c
c
44
2
,
ω
μν
=
(
)
+
(
)
+
(
)
=
+
()
+
()
−
m
m
m
m
mg
m
m
m
m
cc c
cc
44 4
44
1
1
2
2
1
2
1
1
2
2
1
2
,
R
k
m
m
m
m
m
m
m
m
F
f
m
m
i
i
i
=
+
()
+
()
=+
(
)
+
(
)
=
+
−
c
cc
cc
1
1
2
2
1
2
1
1
2
2
1
2
44
44
,,
λ
cc
4
()
Considering both ƒ
1
and ƒ
2
as independent white noise sources, the power
fl ows between the resonators are calculated. The power supplied by the
source P
1
= 〈f
1
y
1
〉:
fy m m yy c y k k yy k yy
gyy
11 1 11 1 1
2
11121
21
1
4
=+
()
+++
()
−
−
ccc
++
1
4
21
myy
c
fy c y m yy k yy g yy
11 1 1
2
21 21 21
1
4
=+ − −
cc
(7.38)
fy c y k yy m yy g yy
22 2 2
2
12 12 12
1
4
=− + +
cc
(7.39)
Using the fact that:
d
dt
yy yy yy
12 12 12
0
(
)
=+=
d
dt
yy yy yy
12 12 12
0
(
)
=+=
summation of Equations 7.38 and 7.39 gives:
fy fy c y c y
11 22 1 1
2
22
2
+=+
From Equation 7.38 or 7.39, the power fl ow from resonator 1 to resonator
2 gives:
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P k yy g yy m yy
12 21 21 21
1
4
=− − +
cc
If Δ
1
= ω
1
η
1
, Δ
2
= ω
2
η
2
and η
i
is the loss factor of each uncoupled resonator,
therefore when clamping system 2, y
2
=
y
2
= ÿ
2
= 0.
ε
ωω
ωη
1
11 111
1
11
2
1
4
2
1
4
1
4
b
ff
S
mm
S
mm
mm
(
)
≡
(
)
+
()
=
(
)
+
()
=+
(
)
ππ
Δ
cc
c
yy
1
2
(7.40)
which is the uncoupled or blocked energy of resonator 1 according to
Equation 2.1.36 in reference [1]. Similarly,
ε
ω
2
22
22
2
2
2
1
4
1
4
b
f
S
mm
mmy
(
)
≡
(
)
+
()
=+
(
)
π
Δ
c
c
(7.41)
is the energy of resonator 2 when y
1
=
y
1
= ÿ
1
= 0.
The power fl ow from resonator 1 to resonator 2 is:
PA
bb
12
12
=−
()
(
)
(
)
εε
(7.42)
which states:
1. The power fl ow is dominated by the resonator interaction between
the two resonators. The quantity A is very large when ω
1
and ω
2
are within a resonance bandwidth of each other, but very small
otherwise.
2. The power fl ow is directly proportional to the difference in decoupled
energy of the resonators.
3. The quantity A is positive defi nite; the average power fl ow is from the
resonator of greater energy to the resonator of less energy.
4. A is also symmetric in the system parameters so that an equal difference
of resonator energies in either direction (1 → 2 or 2 → 1) will result in
an equal power fl ow; that is, the power fl ow is reciprocal.
The power fl ow is proportional to the difference in the actual kinetic energy,
or potential energy, or total energy. The power fl ow can be calculated either
from input power to the system or from measured vibration. The calcula-
tion can be based either on uncoupled systems or on the actual vibration
energy of the coupled systems.
If the average energy of resonator 1 is ε
1
, and the average energy of reso-
nator 2 is ε
2
, the power fl ow between them is:
〈P
12
〉 = B(ε
1
− ε
2
) (7.43)
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