Wang X. Vehicle Noise and Vibration Refinement

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Theory and application of modal analysis in vehicle noise 123

The governing equation for the free but damped response of this type of

system is:



(t) + c



(t) + k x(t) = 0 (6.1)

Here the ﬁ rst term containing a lumped mass represents the inertia forces,

the second term represents a velocity-proportional viscous damping force,

and the third term is the force generated by a linear spring with negligible

mass. The solution is assumed to be:

x(t) = xe

(6.2)

Substituting Equation 6.2 and its derivatives into Equation 6.1 yields

Equation 6.3:

0++=

(6.3)

This is a quadratic equation whose solutions are:

1=− ± − =− ± −

ωζ ω ζ

(6.4)

where

= km

represents the undamped natural frequency of the system

and ζ = c/2mω

the damping ratio. Both are important factors that charac-

terize system dynamic properties, oscillation only being possible for a

damping ratio below 1. Introducing the above solutions into Equation 6.2

yields the equation of oscillation:

x(t) = xe

−ω

ζt

iω

(6.5)

where

ωζ

=−1

(6.6)

is the damped natural frequency of the system.

This result implies that a mode for a damped system consists of two parts,

an imaginary part representing the oscillation with a frequency of ω

and a

real part characterizing the decay rate of ω

ζ. Figure 6.7 illustrates this

behavior.

6.3.2 Forced harmonic vibration

When an excitation force acts on a structure the system is forced to vibrate

at those frequencies contained in the excitation spectrum. Although there

are many types of excitation it makes sense to consider pure harmonic

excitation ﬁ rst. It is often encountered in engineering systems and the ana-

lytical handling of the resulting equations is fairly easy. We will therefore

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124 Vehicle noise and vibration reﬁ nement

now look at the properties of a viscously damped sdof system undergoing

harmonic excitation:



(t) + c



(t) + kx(t) = F(t) (6.7)

If we assume that x(t) = xe

iωt

and f(t) = fe

iωt

then we obtain the complex

equation:

(−mω

+ iωc + k)xe

iωt

= fe

iωt

(6.8)

Rearranging and substituting ω

into Equation 6.8 yields the reception form

of the system transfer function. This function contains both amplitude and

phase information:

fm c

ωω ω

()

−

(

)

(6.9)

Figure 6.8 shows a plot of the transfer function on a logarithmic scale.

One can distinguish three areas in which one of the inﬂ uencing factors m,

c or k dominates the system response characteristics. For frequencies far

below resonance, i.e. ω << ω

, the transfer function becomes h

(ω→0)

= 1/k.

So far below resonance the frequency response function is dominated by

the stiffness in the oscillating system. For ω >> ω

the transfer function

reduces to h

(ω→∞)

= 1/−mω

. The response is dominated by the mass of the

oscillating system. Finally, for ω = ω

the transfer function can be written

as h

(ω = ω

)

= 1/iωc. Near resonance the system is dominated by its damping.

These three relations are very important in designing structures.

Depending on the relation between the exciting frequency and the system

characteristics, one must carefully choose to change either the mass, the

stiffness or the damping in a system to best cope with the effects of

resonance.

(t)

Decay ≈ e

–w

z t

Oscillation ≈ sinw

6.7 Real and imaginary solution for the mode of a damped vibration

system.

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Theory and application of modal analysis in vehicle noise 125

6.3.3 Multiple degree of freedom systems

and modal properties

In reality the dynamic characteristics of most structures or cavities cannot

be described by a sdof system, but must be modeled as a mdof system. The

equations of a mdof system are similar to that derived in the previous sec-

tions for the sdof system, but now matrices and vectors are used to repre-

sent the numerous contributions to the response. Omitting damping, the

equation for the free vibration of a general mdof system is:

[M] ⋅ {



(t)} + [K] ⋅ {x} = 0 (6.10)

where [M] is the matrix of mass elements, [K] the matrix of stiffness ele-

ments and {x} the vector of displacement versus time. Substituting again an

exponential solution of the form {x(t)} = {x}e

iωt

and {



(t)} = ω

{x}e

iωt

into

Equation 6.10 yields:

([K] − ω

[M]){x}e

iωt

= 0 (6.11)

The only non-trivial solution for Equation 6.11 is:

det

[K] − ω

[M]

= 0

This means that we can extract a number of ω

values, r = 1, . . . , N,

that in this case represent the system’s undamped natural frequencies.

Substituting these back into Equation 6.11 yields corresponding displace-

ment vectors {x

} called the mode shape vectors. While natural frequencies

are unique values with a ﬁ xed quantity, the mode shape vectors are not.

The mode shape vectors are arbitrarily scaled; the amplitude of the result-

ing mode shape is therefore not unique and ﬁ xed. But since the scaling

0.01

0.10

1.00

10.00

100.00

10.01.00.1

w /w

H(w)

k = 1 N/m

k = 10 N/m

k = 0.1 N/m

m = 1 kg

m = 10 kg

m = 0.1 kg

6.8 Single degree of freedom system compliance transfer function

with stiffness (horizontal) and mass (slanted) asymptotes (courtesy of

Ford of Europe, Germany).

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126 Vehicle noise and vibration reﬁ nement

factor acts on all mode shape vectors in the same way, the mode shape itself

is not changed. The following numerical example demonstrates these rela-

tionships in more detail.

Although real models may have a very high count of modes, the princi-

ples are the same as in very small models. For simplicity we will therefore

ﬁ rst study an example of an undamped 2 dof system and derive the modal

properties from this system. Finally we will look at some important rela-

tions of the modal model.

Numerical example

The system shown in Fig. 6.9 can be seen as a very simple model of a vehicle

suspension. The wheel is represented by a lumped mass m with the tire

acting as a spring with stiffness 10k. The attached body is represented by

the relevant lumped mass 4m and is attached to the wheel by a spring with

stiffness k. This is a 2 dof system and we expect it to exhibit two modes.

The equations of motion for the free response of this model are:



+ 10kx

+ k(x

− x

) = 0 (6.12)



− k(x

− x

) = 0 (6.13)

Substituting an exponential solution as in the previous section to Equations

6.12 and 6.13 yields:

(10k + k − ω

m)x

− kx

= 0 (6.14)

−kx

+ (k − ω

/4m)x

= 0 (6.15)

6.9 Free vibration model of an undamped 2 dof system.

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Theory and application of modal analysis in vehicle noise 127

Equations 6.14 and 6.15 are satisﬁ ed for any x if the following determinant

is zero:

kmk

kkm

−

(

)

−

−−

(

)

(6.16)

This yields the condition that

ωω

0−

()

(6.17)

If α is substituted for ω

then Equation 6.17 is transformed into a quadratic

equation for α whose solution is:

=±

()

−

()

(6.18)

We now have the two modal frequencies of our 2 dof system:

αωα

11 0232 0 2268== ==., .

(6.19)

Assuming k = 80 kN/m and m = 120 kg, we obtain f

= 13.6 Hz and f

2 Hz. These are typical modal frequencies for light commercial vehicle

suspension systems.

With the knowledge of the modal frequencies we can now determine the

ratio of the displacement vectors x. From Equations 6.14 and 6.15 we obtain

the two expressions

−

(6.20)

Introducing the two modal frequencies from Equation 6.19 into either of

these two equations leads to:

0 0232

43 1

10 7732

0 093

−

=− = =

(6.21)

This result means that at the higher mode frequency ω

, the displacement

of both masses is opposed in phase, while at the lower mode frequency of

both masses move in phase. At this stage it is important to realize that

the displacement vectors do not possess a ﬁ xed value but are arbitrarily

scaled and only the ratios of the displacement vectors within one mode are

ﬁ xed. We therefore choose to set the vectors x

1,1

and x

1,2

to 1 and scale the

other vector so that the displacement ratio is retained. One thus obtains

vectors that are called normal mode shapes and often designated by Φ

i(x)

In our example the two normal mode shape vectors are

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128 Vehicle noise and vibration reﬁ nement

ΦΦ

0 0232

10 8

(

)

(

)

−

{}

(6.22)

6.3.4 Properties of the eigenvalues

The modal model possesses some important properties. These are known

as the orthogonal properties of the eigenvectors. It can be shown that the

eigenvectors are orthogonal to the mass and stiffness matrices. If ω

≠ ω

i and j being two different modes, then the following relationships exist:

{x}

⋅ [M] ⋅ {x}

= 0 (6.23)

{x}

⋅ [K] ⋅ {x}

= 0 (6.24)

Substituting the normal mode shape vectors and masses from our above

example into Equation 6.23 demonstrates that our modal model is

orthogonal.

1 0 0232

10 8

1 0 0232

43 2

0−

(

)

⋅

⎡

⎣

⎢

⎤

⎦

⎥

⋅

{}

=−

(

)

⋅

{}

=−=.

The same result can be obtained when using Equation 6.24. Finally, if i = j

then Equations 6.23 and 6.24 yield:

{x}′

⋅ [M] ⋅ {x}

= [m

], {x}′

⋅ [K] ⋅ {x}

= [k

]

The elements on the right-hand side of both equations are called the modal

or generalized mass and stiffness. Often the mode shape vectors are scaled

with the generalized mass.

6.4 Methods for performing modal analysis

As shown in Fig. 6.3 there are two methods for performing modal analysis.

When hardware is available, a test-based modal analysis is possible. In the

case of benchmarking of competitor products experimental modal analysis

is the only tool available to extract modal information. If a model of the

mass, stiffness and damping distribution within a structure is available, then

the modal model can be derived by digital calculation. In both cases the

modal model can then be used further to perform sensitivity analysis, modi-

ﬁ cation and transfer function prediction. We will now discuss in more detail

the application of the two methods in vehicle development.

6.4.1 Test-based modal analysis

The application of test-based modal analysis in road vehicle development

became more widespread with the introduction of computers and high-

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Theory and application of modal analysis in vehicle noise 129

performance data acquisition equipment in the 1980s. Although equipping

a test facility with the necessary software and hardware for multi-channel

transfer function measurements and modal analysis is very expensive, in

the order of $250,000, test-based modal analysis has become a standard tool

throughout the automotive industry. In the vehicle development process it

is most often used for benchmarking competitor products, and obtaining a

modal status of prototype and production vehicles and their components,

e.g. for correlation with FE models or for noise and vibration problem

solving. There are some further niche applications, among others determin-

ing the center of gravity/inertia characteristics of a structure or determining

the equivalent point static stiffness from a single measured transfer

function.

An experimental modal analysis can be divided roughly into four stages:

ﬁ rst, setup of the test object; second, acquisition of the measurement data;

third, test data analysis and determination of the modal model; and fourth,

application of the modal model for prediction purposes. With hardware

unavailability early in the vehicle development process and the evolution

of ﬁ nite element modal analysis, prediction from test-based modal models

has become somewhat less important. In the next section we will therefore

focus on the ﬁ rst three stages of experimental modal analysis.

6.4.2 Test object setup

At the beginning of an experimental modal investigation the response loca-

tions for the test object need to be deﬁ ned. The response locations must be

chosen so that the mode shapes in the frequency band of interest can clearly

be identiﬁ ed during analysis. This means that test objects with high com-

plexity, i.e. structures built up from many components, afford a higher

number of response locations than simple structures. An investigation of

the global modes of an automobile body structure up to 100 Hz may require

only 40 response locations, while the analysis of a complete vehicle up to

the same frequency may require 100 and more. The frequency band of

interest is another factor when planning the response locations. With

increasing frequency the distance between the individual response locations

must be reduced, since wavelengths on the structure reduce and higher-

order mode shapes are encountered. Once the response locations are

deﬁ ned they are marked on the structure and the geometric positions mea-

sured. With this data a grid model is created in the analysis software which

afterwards allows one to view the animated mode shapes.

After the deﬁ nition of the response geometry the test object is set up in

the laboratory. There are two possibilities for supporting the test object.

The simplest is a suspension which resembles free–free boundary condi-

tions. This means that the test object’s possibility of moving freely in space

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130 Vehicle noise and vibration reﬁ nement

is restricted as little as possible. In this case the structure is often suspended

by soft elastic cords or supported on very soft springs, e.g. air-ﬁ lled tubes.

This separates the six rigid body modes by ideally a factor of 10 in frequency

from the lowest frequency ﬂ exible test object mode, thus avoiding interac-

tion between rigid and ﬂ exible modes. Figure 6.10 shows a vehicle body

structure setup to determine the required transfer functions.

When subsystems or components are to be tested, their modal charac-

teristics are often very different from the case when they are attached to

the vehicle structure. If a test of these on the vehicle in normal built-in

conditions is not possible, the setup becomes more difﬁ cult. The subsystem

or component must then be attached to a rig for which the attachment

points exhibit similar impedance characteristics as the vehicle attachments,

or at least the rig modes must be well outside the frequency band of inter-

est. This is important in order to avoid interference from resonances that

are characteristic of the rig structure but have nothing to do with the modes

of the test object under investigation.

When the test object is set up for data acquisition the excitation locations

must be chosen. The most important requirement here is that all modes in

the frequency band of interest are excited. For simple structures where one

has a good idea of what modes can be expected, it is sufﬁ cient to use only

one excitation location. For more complex structures or with less knowl-

Response

sensors

Shaker

Air cushion

suspension

6.10 Uniﬁ ed vehicle sheet metal body setup on air mounts for the

measurement of vibration transfer functions. An electrodynamic

shaker is attached to the right rear rail for vertical excitation. Small

magnets are attached to the thin panels for mass compensation and

3-D sensors measure the vibration response (courtesy of Ford of

Europe, Germany).

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Theory and application of modal analysis in vehicle noise 131

edge it is often better to apply excitation at two or more locations, in order

to reduce the risk of having the excitation near nodal points and therefore

not exciting some of the modes. For investigations on the sheet metal body

of a car, excitation may be applied at the front and rear load-carrying rails

to excite the structure in the vertical and lateral directions. On a complete

car additional excitation locations can be used in the middle of the tunnel

and on the powertrain.

Two types of excitation mechanisms are commonly used (Brown et al.,

1977). The ﬁ rst is a hammer that has a force transducer attached to the tip

with which the test object is impacted. This type of excitation is of a tran-

sient type and avoids adding an additional system to the test object. When

roving excitation measurements are to be made, a hammer reduces the time

required to change positions. On the other hand it is more difﬁ cult to

choose and control a speciﬁ c excitation force spectrum as closely as with

an electrodynamic shaker.

For vehicle structures the electrodynamic shaker is more often used to

obtain the necessary frequency response function data. It must be attached

carefully to the structure, avoiding force inputs in other directions than the

intended, avoiding resonances of the attachment structure, often a rod or

wire, and avoiding grounding which can induce unwanted reactions, espe-

cially when more than one shaker is used in the setup. On the other hand,

since the exciter stays attached to the structure during the entire measure-

ment, it is now possible to measure the driving point with a common

element for force and acceleration, called an impedance head. This setup

comes very close to the ideal one-point impedance measurement. Also, the

amplitude, frequency band and type of excitation can now be controlled

very well. A practical disadvantage is of course that the equipment is much

more expensive than a hammer, and that changing the excitation location

requires more time.

6.4.3 Pre-investigations and data acquisition

The excitation point, also called the driving point, is a very important part

of a modal test. It is therefore understandable that some prerequisites that

govern modal analysis are checked here before data acquisition. At the

driving point where by deﬁ nition the excitation force and acceleration

response act exactly at the same location, the imaginary part of the mea-

sured point impedance must exhibit peaks at the resonance frequencies that

are all either positive or negative. Since modal analysis is a method based

on the linear behavior of a structure, this also needs to be checked. The

input force is varied and the resulting measured transfer functions should

be checked for changes of resonance amplitudes or frequency shifts. For a

multiple excitation setup it is important that the individual excitation

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132 Vehicle noise and vibration reﬁ nement

spectra are not coherent. If this is the case then the reciprocity of the trans-

fer functions between the driving points can also be checked. A typical

measurement equipment setup for the determination of structural and

acoustic transfer functions is shown in Figure 6.11.

When the number of measurement locations exceeds the available data

acquisition channels on the measurement equipment, the measurements

must be taken in several groups over a certain period in time. This is espe-

cially true for data acquisition on complex test objects such as a full vehicle

where a high number of response locations are required. Here every effort

must be made to ensure the integrity of the measurement data, otherwise

subsequent analysis of the test data will become difﬁ cult or impossible.

Complex test objects can change their dynamic characteristics over time,

therefore it is essential to track any such changes. This can best be done

either by comparing transfer functions of response locations that are not

changed during the entire test, or by tracking the driving point frequency

response functions for the entire test. Another concern arises from the

small but sometimes not negligible mass loading through the accelerome-

ters themselves. This can be avoided by adding dummy masses of equal

weight as the sensors to locations where no accelerometers are positioned

during a group measurement, and where a not negligible sensitivity to a

mass change is presumed.

With a well-prepared test object setup and good results from the pre-

investigation, data acquisition is a straightforward task. If the dynamic

characteristics of the structure under investigation are susceptible to chang-

ing environmental conditions, e.g. effect of temperature on certain materi-

als, then these conditions must be controlled. It is in any case best to try to

get all required data in as short a time as possible. Figure 6.12 graphically

illustrates the process.

Sensor

Filtering,

amplification and

A/D conversion

Data processing

and

storage

Excitation

Signal generation

and

D/A conversion

Control

Amplifier

6.11 Excitation and data acquisition equipment layout for the

measurement of transfer functions.

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