Wang X. Vehicle Noise and Vibration Refinement

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Theory and application of modal analysis in vehicle noise 133

6.4.4 Data analysis

Before commencing the modal analysis of the measured transfer function

data some sanity checks should be done. Transfer functions should show

the mass and stiffness line asymptote behaviors already discussed in Section

6.3.2 and shown in Fig 6.8. If these inspections are passed then one can gain

an initial overview of the investigated structure’s dynamic characteristics

by calculating the summation and mode indicator functions: see Fig. 6.13.

The summation function is calculated by summing the amplitudes of all

measured transfer functions, omitting the phase information, and then nor-

malizing with the number of measured data sets. The resonance peaks are

pronounced and one can get a good idea of frequencies that exhibit strong

modes and areas of high modal density. For complex test objects the sum-

mation functions for subsystems and components can also be calculated.

The overview is complemented by the mode indicator function that shows

a sharp drop to zero where resonances are detected.

The next step is that of extracting the modal frequencies, modal damping

and mode shape vectors from the matrix of transfer function data. This

process is often called curve ﬁ tting (Lembregts, 1989), which basically

means that analytical functions are ﬁ tted to the measured individual

response functions such that the difference between the measured response

function and the analytical expression is minimal. An example of this

process will be shown below for a very simple case study, but for more

complex systems a computer software package must be used. The modern

(Multiple) Force or

pressure excitation

Test structure

Acceleration and

pressure responses

Frequency response function matrix

. . . h

. .

. . . h

. .

. . . h

= Drive point FRF

Frequency (Hz)

0 1000 2000

1.2

0.5

0.0

m/s

6.12 Example of the data acquisition of transfer functions as required

for modal analysis (courtesy of Ford of Europe, Germany).

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134 Vehicle noise and vibration reﬁ nement

poly-reference algorithms implemented in these packages work in the time

or frequency domain and are capable of ﬁ tting multiple modes across mul-

tiple response functions for a number of references.

If we assume damping to be proportional to the stiffness, then the for-

mulation for the problem we are trying to solve is:

KCM

[]

−

[]

{}

⎛

⎝

⎜

⎞

⎠

⎟

−

ωω

(6.25)

For a single degree of freedom system the solution of this equation is

simple, but for large multiple degree of freedom systems this becomes very

inefﬁ cient since the inversion of large matrices need to be calculated.

Therefore a different formulation of this equation is used. The new formu-

lation can be derived by pre- and post-multiplying Equation 6.25 by the

eigenvector matrices [Φ]

and [Φ]. For the transfer function of a sdof system

this then results in:

ωω ω

()

−

(

)

(6.26)

1067

584

Mode

indicator

function

Summation

function

Frequency (Hz)

(a)

(b)

m/s

12008004000

1.0

0.5

0.0

1.0

m/s

0.5

0.0

20001600

6.13 (a) A mode indicator function dip towards zero indicates the

presence of a mode. (b) The summation function enhances resonance

peaks. Both give a ﬁ rst impression of mode strength and distribution

(courtesy of Ford of Europe, Germany).

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Theory and application of modal analysis in vehicle noise 135

To illustrate the process of extracting the modal data we will apply a simple

method that has been in use for a long time called the peak picking method:

see also Fig. 6.14. This method can be applied when the modes are clearly

separated from each other and the damping is light, so that contributions

from other modes are negligible. Very often this is not the case, but the

method is adequate to get a ﬁ rst estimate on the measured transfer function

data. We are now looking for a solution to Equation 6.26. The unknowns

in this equation are the modal frequency, the modal damping and the mode

shape.

First a resonance peak from the plot of the measured transfer function

is chosen and the corresponding resonance frequency is read from the

frequency axis. Then the half power points to both sides of the chosen reso-

nance peak are deﬁ ned by following the amplitude of the plotted frequency

response function downwards until it reaches an amplitude equal to the

resonance peak amplitude value divided by

. These points are called the

half power points. Again the corresponding frequencies are read from the

frequency axis. If the damping is light, i.e. below 3%, then we can calculate

the damping factor to be:

ωω

−

(6.27)

With

ζω

we have a solution for the damping. The only unknown left,

, can now be calculated directly from Equation 6.26. This factor is often

represented by an A and is called the modal constant or residue. When the

investigated system contains more than one mode, which will be the usual

- Circle fit

Fit one mode at a time:

- Peak picking

z =

– ω

For light damping:

2·ω

6.14 Example of the determination of the modal frequency and

damping by the simple peak picking method.

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136 Vehicle noise and vibration reﬁ nement

case for real structures, then the transfer function between two locations

will have several single degree of freedom contributions: see Fig. 6.15. The

governing equation then becomes

ir jr

ωω ω

()

−

(

)

∑

ΦΦ

(6.28)

where r denotes the respective mode number.

6.4.5 Predictive modal analysis

If minor structural changes are made to an existing product then very often

these can be cobbled and one has hardware available for an experimental

modal investigation. But at the beginning of a vehicle development process

for a fully new product, or when major structural changes have been made,

there is no hardware available that fully represents the ﬁ nal dynamic char-

acteristics. Since it is not possible to wait for the availability of hardware,

which is usually the case, it becomes inevitable that one ﬁ nds a different

means of deriving these. It is therefore usual to use predictive methods

based on design intent and using experience from earlier correlation work

done on predecessor structures to drive noise and vibration decisions early

in a vehicle development program. Here ﬁ nite element analysis, where

mass, stiffness and damping are distributed across a known geometry, and

methods that model the structure with lumped masses connected with

Investigated

Lower residual

mass contribution

Upper residual

stiffness contribution

Measured FRF

Mode contributions

–

Inertance ( )

m/s

6.15 Mode and residual contribution to a transfer function (courtesy of

Ford of Europe, Germany). The signs of neighboring modes govern

whether the transfer function exhibits a smooth or a sharp (anti-

resonance) transition between peaks.

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Theory and application of modal analysis in vehicle noise 137

springs and damping elements are most commonly used. Since these

methods are described in detail in other chapters of this book, we will

concentrate in this section on a comparison of experimental and predictive

CAE tools, outlining how to best utilize both in developing the required

reﬁ nement of vehicle noise and vibration.

The increase in performance and conﬁ dence for the predictive methods

has changed the distribution of modal analysis work between test-based

and predictive methods. Today CAE models are used to give design guid-

ance on increasingly complex models, i.e. comparison of dynamic proper-

ties such as the frequency response functions for complete vehicle models,

and this development is being backed and accelerated by the industry. But

the dynamic behavior of a full vehicle is still so complex that both methods

are often used in parallel.

When ﬁ rst CAD models of the future product become available, these

are taken and, e.g., ﬁ nite element models created. These models can then

be used for forced response or modal analysis calculations. The ﬁ rst step

here is to create a grid across the substructures, components and possible

cavities, the ﬁ nite element grid, on which the equations representing the

acoustic and structural motion are solved. The substructure and component

models are then joined to form the complete vehicle model. At this stage

the boundary and coupling conditions must be deﬁ ned, which is an impor-

tant and often difﬁ cult task.

For full vehicles the meshes have hundreds of thousands of elements,

compared to maybe 150 measurement points in experimental investiga-

tions. This high element count is necessary if the details of the structure are

to be represented properly, which for the test-based analysis is given by the

hardware, but they make the work of creating the model, the meshing, more

time-consuming and also prolong the time required for computation. This

is a serious concern since further reﬁ nement of the grids is sometimes con-

sidered necessary in order to even better represent important design details

of the vehicle. Therefore a balance between the required accuracy of the

model and the restrictions in time available for vehicle development is

required. The ﬁ ne distribution of elements leads to the FE model picking

up more modes than a test-based model. On the one hand this is an advan-

tage, since the resulting modal model is more ‘complete’ than the test-based

model; on the other hand the evaluation of the FE data is more difﬁ cult

and again time-consuming since many of the calculated modes are not

important for the noise and vibration characteristics of the vehicle as expe-

rienced by the passengers.

The predictive models are in general of linear type, the damping often

modeled as being proportional to the mass and stiffness matrices. In reality

the full vehicle exhibits non-linear behavior and this is a point of concern.

But this problem also affects the experimental modal analysis investigation,

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138 Vehicle noise and vibration reﬁ nement

since the applied excitation forces during a modal test are seldom equal to

the forces acting during operation. A typical example is the suspension

modes of the vehicle. The piston that moves in the damper strut during

on-road driving requires a minimal force to overcome the stick friction.

Modal tests seldom reach an excitation level as encountered in real vehicle

operating conditions, therefore modal values are not correct in this case.

So both approaches struggle with the non-linear behavior of the object

under investigation.

The assumption of damping being proportional is also not strictly appli-

cable but makes solution of the ﬁ nite element problem easier. When

damping is chosen to be proportional to the mass and stiffness matrices,

the equations can be solved for the undamped case ﬁ rst and the results

corrected for the effects of damping afterwards. Further simpliﬁ cations

often need to be made, i.e. one global damping value deﬁ ned over the entire

structure and frequency band of interest. Since the damping encountered

in vehicles is mostly at or below 3% critical damping, this assumption does

not affect the mode frequency or mode shapes very much, but it does have

a strong effect on amplitudes. But it is the amplitudes that engineers have

set targets for and are therefore interested in most. Here one must conclude

that the prediction of absolute values today is conﬁ ned to simple structures,

and noise and vibration predictions for full vehicles are best used in an A–B

comparison to give a design direction. Deﬁ ning a correct damping is a criti-

cal skill and it is here that an intensive exchange between both experimental

and predictive tools is a prerequisite for success.

For practical engineering it is not sufﬁ cient to describe only the dynamic

properties of a full vehicle or its systems in the form of a modal model, but

it is also important that in the case when requirements are not met, some

indication of what needs to be done is given. Since FE models possess a

ﬁ ne representation of a structure they can also deliver detailed structural

change proposals for the vehicle. The modal model extracted from test-

based analysis is restricted to structural optimization of general mass, stiff-

ness or damping changes between the rather coarse positions of the response

locations. This is a clear drawback in the development of a vehicle’s dynamic

properties.

In modal analysis, be it test-based or predictive, much depends on the

expertise of the engineer doing the evaluation. In the test-based analysis

process the evaluator decides how to analyze the data, determining the

modal frequency and damping by his or her own judgment and giving the

mode shapes a description. The FE engineer does not have to decide what

modal frequencies to include in the modal model, but the much higher

number of modes derived from the calculation must be evaluated, and also

mode shape descriptions must be formulated. The mode shape description

can then be used for correlation or sign-off against set mode shape targets.

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Theory and application of modal analysis in vehicle noise 139

But in any case, it is a demanding task to manually order two sets of mode

shapes for comparison, and one would wish to have a more automated tool

to do this. There are parameters available that allow a comparison of mode

shapes, be it from calculation or test. The modal scale factor, MSF, and

modal assurance criterion, MAC, allow the comparison of two vectors. The

MSF gives an estimate of the ratio between two vectors, the MAC a cor-

relation factor describing whether the vector is orthogonal or not. Some

investigations have been conducted towards the usage of these two values

to automate mode shape comparisons, but unfortunately with only limited

success. For full vehicle mode shapes it is still inevitable to visually inspect

and agree on an individual description before commencing with the task of

correlating two sets of mode shapes.

Last but not least is the fact that a model describes the design intent.

When a prototype or series hardware is ﬁ nally built from this design, then

it is certain that there will be deviations and in the case of a series produc-

tion additional spread within the individual vehicles. This must be kept in

mind when correlating models to experimental data, not expecting to ﬁ nd

identical properties but realizing that there will be a certain variation. Of

course it would be best to investigate a number of test objects to have an

idea of the variation and this has been done in single projects. Within the

constraints of usual vehicle development this is impossible since sufﬁ cient

hardware is often not available for testing, and since experimental modal

investigations are very resource and time demanding, so that the ﬁ nal result

will come too late for actual program application.

6.5 Limitations and trends

The limitations already discussed for test- and model-based modal analysis

deﬁ ne the future trend for the application of this method in vehicle design.

The use of model-based predictive methods will continue to increase, the

focus being on raising the conﬁ dence levels when working with complex

structures such as full vehicle models, and reducing the time required for

the investigation. This method can be applied early in the design process

without the need for expensive prototype hardware and facilities and it can

give direct design recommendations. The use of test-based modal analysis

has declined and might be decreased further, but one cannot avoid it

altogether. Correlation of complex structures and benchmarking of com-

petitor products require an experimental input.

To keep up with the growing expectations, experimental modal analysis

has been adapted to speciﬁ c test applications. Mass loading of a test struc-

ture through sensors can change the dynamic properties, especially in the

case of very light objects such as a sheet metal oil pan, quite a bit. The

application of a scanning laser doppler vibrometer helps to avoid this error

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140 Vehicle noise and vibration reﬁ nement

(Stanbridge and Ewins, 1999). The upper frequency limit of a modal test

depends on the spacing between the response sensors. Holographic modal

tests allow the analysis of high frequency and narrow spaced modes

(Klingele et al., 1996).

In vehicle sound and vibration analysis there is often a strong coupling

between structural and acoustic modes. Modal analysis methods have

therefore been developed that allow simultaneous measurement and analy-

sis of the acoustic and structural transfer functions (Wyckaert et al., 1996).

Modal-based damage identiﬁ cation methods are not related to sound and

vibration reﬁ nement, but are an important application of modal testing in

failure analysis and prevention (Farrar and Doebling, 1997).

In recent years experimental modal analysis has been extended to a

method called operational modal analysis (OMA). In classical experimen-

tal modal analysis the test object is built up in a laboratory and artiﬁ cially

excited. The excitation forces are therefore known and used to calculate

the required transfer functions to the driving points. For an OMA the test

object is investigated in usual operating conditions and only the responses

of the system are measured. Since in general there is no deﬁ ned information

on the location and type of the excitation, apart from the engineer’s knowl-

edge of the system, the task is to extract the structural modes from the data

without getting ‘modes’ originating from peaks of the excitations. The

advantage of this method is that the investigation can be done in the usual

environment and with excitation forces encountered when the test object

is normally operated. This method is already being used among others for

investigations of civil structures such as buildings, bridges and oil platforms,

and in the aerospace and automotive industries (Ventura and Gade, 2005).

6.6 References

Allemang R (1992), Vibrations: Analytical and Experimental Modal Analysis,

UC-SDRL-CN-20-263-662, University of Cincinnati, OH.

Avitabile P (1998–2008), Modal space – back to basics, Experimental Techniques,

Volume 22, No. 2 to Volume 32, No. 1 and ongoing, http://sem.org/TDIV-

ModalAnalysis.asp

Brown D L, Carbon G D and Ramsey K (1977), Survey of excitation techniques

applicable to the testing of automotive structures, SAE 770029.

de Siqueira L P and Nogueira F (2001), Application of modal analysis and operating

deﬂ ection shapes on the study of trucks and buses dynamic behavior, SAE

2001-01-2780.

Ewins D J (2001), Modal Testing, Theory, Practice and Application, Research

Studies Press, Letchworth, Herts, UK.

Farrar C R and Doebling S W (1997), ‘An overview of modal-based damage iden-

tiﬁ cation methods’, Proc. DAMAS Conf.

He J and Fu Z F (2001), Modal Analysis, Butterworth-Heinemann, Oxford.

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Theory and application of modal analysis in vehicle noise 141

Heylen W, Lammens S and Sas P (1997), Modal analysis theory and testing,

International Seminar on Modal Analysis, KU Leuven, Belgium.

Inman D J (2000), Engineering Vibrations, Prentice Hall, Englewood Cliffs, NJ.

Johansen S and Madsen K D (1992), Operating deﬂ ection shapes – a fast developing

technique, Proc. 10th Int. Modal Analysis Conf.

Klingele H, Steinbichler H, Freymann R, Honsberg W and Haberstok C (1996),

Holographic modal analysis for the separation of narrow-spaced eigenmodes,

Second Int. Conf. on Vibration Measurements by Laser Techniques, SPIE, Vol.

2868, pp. 412–417.

Kronast M and Hildebrandt M (2000), A high speed boom investigation of a mid-

size car using experimental vibro-acoustic modal analysis, 18th Int. Modal Analysis

Conf.

Lembregts F (1989), Parameter estimation in modal analysis, Proc. ISMA, Vol. 14,

p. 33.

Stanbridge A B and Ewins D J (1999), Modal testing using a scanning laser doppler

vibrometer, Mechanical Systems and Signal Processing, Vol. 13, pp. 255–270.

Thomson W T and Dahleh M D (1997), Theory of Vibration with Applications,

Prentice Hall, Englewood Cliffs, NJ.

Ventura C and Gade S (2005), Operational Modal Analysis course, IOMAC,

Copenhagen.

Wyckaert K, Augusztinovicz F and Sas P (1996), Vibro-acoustical modal analysis:

reciprocity, model symmetry and model validity, JASA, Vol. 100, pp.

2877–3465.

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142

Mid- and high-frequency problems in vehicle

noise and vibration reﬁ nement – statistical

energy analysis and wave approaches

X. WANG, RMIT University, Australia

Abstract: Starting from the statistical energy analysis (SEA) modal

approach, energy sharing between two oscillators and energy exchange in

a multi-degree-of-freedom system are discussed, from which the SEA

power balance equations are derived. Application procedures of the SEA

approach are presented and the main SEA assumptions are highlighted.

In order to introduce the SEA wave approach, room acoustics is reviewed;

the SEA conceptual parameters such as wave number, wave number

space, dispersion and modal density for energy storage are deﬁ ned. In

regard to energy transmission, the concepts of direct and reverberant

ﬁ elds, as well as coupling loss factors for different connections, are

illustrated; panel acoustic radiations are introduced and evaluated;

transfer matrices and insert loss of a trim lay-up are demonstrated; and

leaks and aperture effects are discussed. For energy inputs, direct ﬁ eld

impedance and input power are illustrated; distributed random loading

and turbulent boundary layer loading are deﬁ ned; and concepts of diffuse

acoustic ﬁ eld loading and propagating acoustic ﬁ eld loading are explained.

The hybrid deterministic and SEA approach is brieﬂ y introduced.

An application example of vibro-acoustic analysis on a passenger car

roof–cabin cavity system is given to validate the SEA approach.

Key words: modal oscillators, damping loss factor, phase velocity, group

velocity, wave number, dispersion property, direct ﬁ eld, reverberant

ﬁ eld, diffuse reverberant ﬁ eld, energy input, energy transmission, energy

storage, coupling loss factor, acoustic leakage, noise control treatment,

poroelastic materials, trim lay-ups, transfer matrix, insert loss, radiation

efﬁ ciency, mean energy, clamped systems, blocked subsystems, critical

wavelength, critical frequency, hybrid deterministic and SEA approach,

SEA modal approach, SEA wave approach, distributed random loading,

turbulent boundary layer loading, diffuse acoustic ﬁ eld loading,

propagating acoustic ﬁ eld loading, acoustic impedance, transmission

coefﬁ cient, mass law, non-resonant acoustic energy transmission,

resonant acoustic energy transmission, energy density, power ﬂ ow,

modal count, wavelength.

7.1 Introduction

Engineering systems with noise and vibration problems are investigated in

a model of source, path and receiver where the source can be quantiﬁ ed or

modiﬁ ed by changing its frequency content, levels, etc.; the path can be

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