Wang X. Vehicle Noise and Vibration Refinement
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Random signal processing and spectrum analysis 113
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Its contribution to the response at time t is dependent upon the elapsed
time (t − a), or
fa aht a
(
)
−
(
)
Δ
(5.85)
where h(t − a) is the response to a unit impulse started at t = a. Because the
system we are considering is linear the principle of superposition holds.
By combining all such contributions the response to the arbitrary excita-
tion f(t) is represented by the integral:
xt f aht a a
t
(
)
=
(
)
−
(
)
∫
0
d
(5.86)
which is shown in Fig. 5.12. This integral is called the convolution integral.
If τ = t − a then
dd
τ
=− a
(5.87)
and
xt f t h f t h
t
t
(
)
=− −
(
)
(
)
=−
(
)
(
)
∫∫
τττ τττ
0
0
dd
(5.88)
For a linear system:
xt f t h
(
)
=−
(
)
(
)
−∞
∞
∫
τττ
d
(5.89)
where h(τ) = 0 when τ < 0. h(τ) is the response of the linear system to the
unit impulse delta function input. This relationship is presented in Fig. 5.13.
For a constant-parameter linear system a necessary and suffi cient condi-
tion for a linear system to be stable is:
h
ττ
(
)
<∞
−∞
∞
∫
d
(5.90)
For a real system to be physically realizable the system cannot respond
before the input occurs:
h
ττ
(
)
=<00for
(5.91)
x(t)
Linear system h(t)
y(t) = 兰h(t) x(t – t)dt
∞
−∞
5.13 Convolution integral relation for ideal constant-parameter linear
system.
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114 Vehicle noise and vibration refi nement
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For constant-parameter systems
ht h,
ττ
(
)
=
(
)
(5.92)
For any linear system, random input data with a theoretical Gaussian
probability density function describing its amplitude properties will produce
random output data that will also have a theoretical Gaussian probability
density function. For a non-linear system, if the random input is a Gaussian
process, the random output is a non-Gaussian process.
5.8 Relationship between complex frequency
response and impulsive response
If the excitation function F(ω) in the form of the unit impulse as shown in
Fig. 5.14:
ft t xt ht
(
)
=
(
)
(
)
=
(
)
δ
,
(5.93)
is
F ftt tt
tt
ωδ
ωω
(
)
=
(
)
=
(
)
=
−∞
∞
−
−∞
∞
−
∫∫
ed ed
ii
1
(5.94)
then the following relationship develops:
XFHH
ωωω ω
(
)
=
(
)
(
)
=
(
)
(5.95)
From the Fourier transform relationship:
xt X H ht
tt
(
)
=
(
)
=
(
)
=
(
)
−∞
∞
−∞
∞
∫∫
1
2
1
2ππ
ωω ωω
ωω
ed ed
ii
(5.96)
a
t
F
F(t)
5.14 Excitation force function.
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Random signal processing and spectrum analysis 115
© Woodhead Publishing Limited, 2010
which is
ht H
t
(
)
=
(
)
−∞
∞
∫
1
2π
ωω
ω
ed
i
(5.97)
The inverse Fourier transform relationship of Equation 5.97 gives:
Hhtt
t
ω
ω
(
)
=
(
)
−∞
∞
−
∫
ed
i
(5.98)
5.9 Frequency response functions
Fourier transforms of both sides of Equation 5.89 give the familiar relation
shown in Fig. 5.15 which describes constant-parameter linear systems in the
frequency domain, namely:
Xf HfFf
(
)
=
(
)
(
)
(5.99)
where X(f ), F(f ) and H(f ) are Fourier transforms of x(t), F(t) and h(τ)
respectively, as defi ned by:
Xf xt t
ft
()
=
()
−∞
∞
−
∫
ed
i2π
(5.100)
Ff Ft t
ft
()
=
()
−∞
∞
−
∫
ed
i2π
(5.101)
Hf h
f
()
=
()
−∞
∞
−
∫
WW
W
ed
i2π
(5.102)
A suffi cient condition for H(f ) to exist is that this linear system is stable.
The lower limit of Equation 5.102 becomes zero for physically realizable
systems where h(τ) = 0 for τ < 0.
The complex-valued quantity H(f ) is called the frequency response func-
tion of the system. It has the following properties:
Excitation F(t)
Transformed
excitation
F(w)
Transformed
response
X(w)
Frequency
response
H(w)
Response x(t)
Impulse response h(t)
5.15 Block diagram.
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Hf Hf H f H f
f
(
)
=
(
)
=
(
)
−
(
)
−
ei
i
RI
φ
()
(5.103)
Hf Hf f
R
(
)
=
(
)
(
)
cos
φ
(5.104)
Hf Hf f
I
(
)
=
(
)
(
)
sin
φ
(5.105)
Hf H f H f
(
)
=
(
)
+
(
)
[]
RI
22
1
2
(5.106)
φ
f
Hf
Hf
(
)
=
(
)
(
)
⎡
⎣
⎢
⎤
⎦
⎥
−
tan
1
I
R
(5.107)
The frequency response function has been widely applied in vehicle noise
and vibration measurement and analysis for determination of natural fre-
quency. It is one of the key frequency domain functions applied in modal
analysis.
5.10 Bibliography
Bendat, J.S. (1990), Non-linear System Analysis and Identifi cation from Random
Data, John Wiley & Sons.
Bendat, J.S. and Piersol, A.G. (1985), Random data – Analysis and Measurement
Procedures, John Wiley & Sons.
Newland, D.E. (1993), An Introduction to Random Vibrations, Spectral and Wavelet
Analysis, John Wiley & Sons.
Randall, R.B. (1987), Frequency Analysis, Brüel & Kjær.
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© Woodhead Publishing Limited, 2010
117
6
Theory and application of modal analysis in
vehicle noise and vibration refi nement
M. KRONAST, Ford-Werke GmbH, Germany
Abstract: Structural and acoustic resonances can amplify the excitation
acting on an operating vehicle to levels that degrade the noise and
vibration performance severely. Since one cannot avoid resonances
altogether, it is very important during vehicle design to have tools that
allow them to be determined and modifi ed. Modal analysis is a
mathematical tool that enables engineers to determine characteristic
values that describe the resonances and then to build an analytical
model from this information. This chapter outlines the theory of
vibration, shows how the modal model is derived and discusses the usage
of this method in vehicle noise and vibration refi nement.
Key words: experimental modal analysis, noise, vibration and harshness
(NVH), vehicle sound design, automotive NVH refi nement, structural
and acoustic resonance.
6.1 Introduction
Every structure or cavity will show a tactile or acoustic response when
excited by forces that act on it. Instead of looking at the excitation forces
and structural responses in the time domain, it is usual and often more
informative to describe these in the frequency domain. If so described, then
we will typically see that the amplitudes of both will vary in an uncorrelated
way with changing frequency, i.e. a force amplitude decrease going from a
lower to a higher frequency may result in either a response amplitude
increase, an equal value or even a decrease (see also Fig. 6.1). This behavior
is caused by acoustic and structural resonances that will amplify the input
excitation at frequencies near the resonance frequency, but that can also
attenuate the excitation at frequencies away from the resonance peaks.
These resonances can amplify response levels so much that a structure can
be destroyed. But even if the structure does not fail, high tactile or acoustic
responses are generated that can degrade the noise and vibration perfor-
mance of a vehicle severely. It is therefore very important to take this into
consideration during vehicle noise and vibration development.
Modal analysis is a mathematical tool that is used to determine the
characteristic values of the resonances. Based on measurements taken on
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118 Vehicle noise and vibration refi nement
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an existing test object, or using a model of the physical structure or enclo-
sure, the relevant dynamic characteristics can be extracted. With this
knowledge the development engineer can plan to avoid having resonances
at frequencies with high excitation, or having several resonances very close
in frequency to each other, and can therefore reduce the risk of failure or
excessively high structural vibration or sound pressure levels. With advances
in performance of the test equipment and computer capabilities, modal
analysis has become a standard tool in vehicle noise, vibration and harsh-
ness (NVH) development throughout the industry.
6.2 Application of modal analysis in
vehicle development
For the occupants of a vehicle the operating sound and vibration responses
are directly perceivable, contributing to the overall impression of vehicle
refi nement. It is the development task of the sound and vibration engineer
to design the vehicle hardware in such a way that customer expectations,
legal requirements and vehicle manufacturing targets with respect to the
sound and vibration characteristics are met. This requires a detailed under-
standing of the complex generation and transfer of sound and vibration to
the occupants of the vehicle cabin (Kronast and Hildebrandt, 2000).
A fi rst step in reducing complexity can be attained if the vehicle response
is separated into the contributing excitation forces on the one side, and the
structural and acoustic properties of the vehicle on the other side. The
structural and acoustic properties are then characterized by the sound and
vibration transfer functions, which contain all relevant information on the
structure’s dynamic properties, so these are often also used for comparing
vehicles and setting targets. Transfer functions derived by testing a struc-
ture can be broken down further through the application of modal analysis.
The breakdown into the modal characteristics delivers valuable informa-
Excitation spectrum
System FRF
Response spectrum
Force
Frequency
H
(f)
Response
=x
Frequency
Frequency
6.1 Distortion of the system response to a linear force input by the
transfer characteristics (courtesy of Ford of Europe, Germany).
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Theory and application of modal analysis in vehicle noise 119
© Woodhead Publishing Limited, 2010
tion for the sound and vibration development engineer, yielding a tool that
can be used to assess, change and predict a structure’s dynamic properties.
In the next sections we will discuss the outlined process.
6.2.1 Spatial, modal and response model
There are several ways to gain an understanding of the root cause of a
distinct response. The one with the least amount of analysis work uses
sound pressure or vibration data obtained at several locations on the oper-
ating vehicle together with a geometrical shape, allowing the generation of
‘operating defl ection shapes’ (ODS) (Johansen and Madsen, 1992). These
animated patterns of motion are processed and made visible using a com-
puter. They show areas of high and low response and thus help in identify-
ing the root cause of poor vehicle sound or vibration performance. Near a
dominating resonance the ODS can look very similar to an animation of
modal analysis eigenvectors, and indeed, as shown in Fig. 6.2, there is a
direct relation between a mode shape and an operating defl ection shape
(de Siqueira and Nogueira, 2001). Often this type of ODS investigation
already allows one to draw fi rst conclusions with respect to an NVH concern,
but remember that both force excitation and the structural properties are
mixed into the result and cannot be separated from each other at this point.
When the density of resonances is high but there is no dominating reso-
nance, the ODS will show a complex pattern of motion since several reso-
nances will then be contributing to a defl ection pattern at one frequency.
At this point it clearly becomes diffi cult to identify the root cause for an
NVH concern.
When the result of an ODS analysis does not yield suffi ciently clear
information, then modal analysis might be the right tool for the task. Modal
analysis employs structural and/or acoustic response data derived through
a test with artifi cial excitation, or a computer model of the object’s mass,
System characteristics
Dynamic forces
Imbalance
Combustion
Shock
Aerodynamic
Pressure
System responses
Operating deflection shapes
Natural frequency
Damping factor
Mode shape vector
Sound
Vibration
Stress
Modal analysis
6.2 Contribution of excitation and modal characteristics to a system’s
forced response (courtesy of Ford of Europe, Germany).
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stiffness and damping distribution properties. If the object under investiga-
tion fulfi lls the prerequisites for applying this method, which we will discuss
later, then we are able to determine the modes of a structure in a chosen
frequency band. Each mode of a structure is characterized by its natural
frequency, the modal damping value and the mode shape vectors. Figure
6.3 shows the principle and details for analytical and experimental modal
analysis. Avitabile (1998–2008) describes the basics of this method, while
Allemang (1992) and He and Fu (2001) present more details.
6.2.2 Application to vehicle NVH development
Having determined the modal parameters as described above, we now have
the possibility of formulating a mathematical model of the dynamic pro-
perties of the object under investigation (Inman, 2000). This then allows
us to calculate the frequency response function, i.e. acceleration/force or
pressure/force, between all defi ned structure points although, in the case of
experimental modal analysis, we may not have explicitly measured them.
Multiplication of the vibration or noise transfer functions by the respective
excitation forces acting on the vehicle and summation of all these contribu-
tions then yields the absolute response, or forced response, at the defi ned
response location, as shown in Fig. 6.4. Vice versa, looking at the vehicle
development process where absolute vehicle noise and vibration levels are
targeted at important response locations, one is now, at least in theory, able
to break this down into modal contributions. At the model level the con-
tribution of each mode to the response at a certain frequency can be ranked
against the other mode contributions, sensitivity investigations can be
performed and the fi ndings can be used to modify the modal model. The
Acoustic/structural
model
Modes
Response
properties
Analytical approach (FE)
Experimental approach
Physical properties:
Mass
Stiffness
Damping
Modal model:
Natural frequency
Modal damping
Mode shape vectors
Frequency response
function
6.3 Analytical and experimental approach to modal analysis (courtesy
of Ford of Europe, Germany).
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Theory and application of modal analysis in vehicle noise 121
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modifi ed model then allows a prediction of how the responses will change
if the structural or acoustic dynamic properties of the investigated
object are changed. This relation yields a powerful tool for vehicle NVH
development.
Often during vehicle development one is not interested in using the
modal model to perform calculations. The pure knowledge of a structure’s
mode frequencies and mode shapes already gives the experienced develop-
ment engineer suffi cient information, enabling him or her to set up a plan
for the distribution of these over frequency. This concept is often called
modal alignment, although it is really a method to avoid having several
modes strongly coupling at one frequency or having a strong excitation
and relevant modes near one common frequency. Figure 6.5 shows typical
H
1
H
2
H
3
X
seat
F
2 engine
X
2 engine
X
1 engine
F
1 engine
F
3 road
X
3 road
=
+
+
6.4 Multiple excitation and transfer path contributions to the structural
response of a vehicle (courtesy of Ford of Europe, Germany).
Rigid P/T
modes
Suspension
modes
Full vehicle body
modes
Steering
column
modes
5 101520 30 403525
Frequency (Hz)
1
2
3
4
6.5 Typical resonance frequencies of important vehicle systems.
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122 Vehicle noise and vibration refi nement
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frequency bands for major passenger vehicle modes. Other usages of the
modal properties include the possibility to compare the full vehicle, subsys-
tems or components with competitor dynamic performance, to correlate
models to test derived results, or to set targets for system and component
development.
6.3 Theory of modal analysis
As shown in Fig. 6.3, determination of the structural dynamic properties
can be divided into three steps. The fi rst step is building a physical model,
also called a spatial model, which means defi ning the mass, stiffness
and damping distribution for a structure or cavity. The second step is the
extraction of the mode frequencies, damping and shapes from this
model, which then yields the modal model. Finally, the forced response is
derived by applying excitation forces to the modal model (Thomson
and Dahleh, 1997). Vice versa, in test-based modal analysis the natural
response of a structure to an excitation force is measured. A transfer func-
tion is calculated from this data by dividing the acoustic or vibration
response by the excitation force. Modal parameters are then derived from
these transfer functions (Ewins, 2001; Heylen et al., 1997). In principle it is
further possible to derive the spatial model from the test-based modal
model, but this is rarely done. In the next sections we will look at the math-
ematical formulation of the spatial model, see how the modal model is
derived from this spatial model and discuss some of its important
characteristics.
6.3.1 The physical model
The dynamics of real structures must be represented by a multiple degree
of freedom (mdof) model. But a multiple degree of freedom system can be
represented by a linear superposition of a number of single degree of
freedom (sdof) models. For simplicity we will therefore start our investiga-
tions by looking at the characteristics of a single degree of freedom system.
Figure 6.6 presents the basic model.
m
k
c
x(t)
6.6 Free vibration model of a single degree of freedom system.
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