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Efficient Calculation of SEA Input Parameters Using a WaveBased Substructuring Technique

P. Ragnarsson1, B. Pluymers1, S. Donders2, W. Desmet1

1K.U.Leuven, Department of Mechanical Engineering,

Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

e-mail: [email protected]

2LMS International,

Interleuvenlaan 68, B-3001, Leuven, Belgium

Abstract

This paper presents a new and efficient method to calculate point mobilities from subcomponents of a full

structure. When performing dynamic analyses of subcomponents the result normally tends to strongly de-

pend on the boundary condition selected for the component. The goal is to simulate the results one would

have received if the full model would be analysed. Normally free or clamped boundary conditions are used.

In this paper waves extracted in a Wave based substructuring analysis will be used to define a more realistic

boundary condition. The results show great promise and for the three investigated cases the results are abetter match to the reference calculation than both free and clamped boundary conditions. It is especially

clear that the results are accurate for a wider frequency range. While free boundary condition can be suitable

for high frequencies and less suitable for low frequencies and clamped boundary condition can be suitable

for low frequencies but less suitable for high frequencies, the WBS boundary condition yields good results

over the entire frequency range.

1 Introduction

In modern vehicle design processes the use of Computer Aided Engineering (CAE) has increased dramat-

ically in recent years. In order to reduce time to market and to minimize the number of prototypes for avehicle manufacturer it is crucial for a design engineer to get early and accurate predictions of new designs.

Low-frequency methods such as the Finite Element Method (FEM) and the Boundary Element Method

(BEM) have, due to increasing computer speed, been able to run deterministic models to higher and higher

frequencies, while Statistical Energy Analysis (SEA) has become the standard method for both acoustic and

vibration analysis in the high-frequency range. However, there is still a frequency gap where no mature

prediction method exists today [1]. In recent years much research have been focusing on filling this gap by

extending the applicable range of the deterministic methods towards higher frequencies and by lowering the

valid frequency range for SEA.

Langley and Bremer developed a method in which deterministic and statistical techniques are combined [2].

The stiff components (long wave length) are modeled with Finite Element (FE) and the soft components

(short wave length) are modeled with SEA. In classical SEA, the input parameters are normally evaluated

using different analytical formulas e.g. for infinite plate theory. For high frequencies this theory works well,

but when trying to expand the use of SEA towards lower frequencies, other methods are needed. Lyon,

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DeJong [3] and later Manning [4] have shown that the point impedances at input and output locations can

be used to better estimate the input power and the output response. This is a well known method and for

many years measurements have been used to get a better estimation of these parameters [5]. Lately the use of

deterministic methods such as FEM to better estimate these parameters has shown promising results [6] [7].

However, to run a complete FE calculation into the mid-frequency region is highly time consuming forindustrial sized models. To really capitalize on the strength of the deterministic methods, the calculation

times must be reduced.

This paper focuses on a method to perform subcomponent modeling in order to retrieve the important SEA

parameters (input and response point mobility, coupling loss factors (CLF) etc.) without having to run the

full model. A subcomponent can be any part of the complete model of interest, in vehicle analysis this can

be e.g. a b-pillar or a section of the floor panel. The goal is to run a fast calculation of a subcomponent

and to retrieve a result that is as close as possible to the result one should expect from a calculation of the

full model. Since the subcomponent is a part of a larger system the a dominant error source will be the

boundary condition at the subcomponent interface. A classical way to treat this interface is to simplify the

boundary conditions to a free or a fixed boundary condition. This works well for high frequencies and for

low frequencies, but for the mid-frequency range neither of these boundary condition types yields accurateresults. Therefore, improved methods are needed.

Wave-Based Substructuring (WBS) [8], [9] is a powerful substructuring method which uses a set of basis

functions (waves) to describe the behavior of the interfaces. In this work it will be investigated if the WBS

formulation can be used as an efficient way to define more realistic boundary conditions. A new wave

extraction criterion is introduced to create a boundary condition which is as close as possible to the as in

assembly condition. The results will then be compared to results obtained in an analysis of a complete FE

model of the same structure.

In this paper the wave extraction is performed on the same assembly as the subcomponent originates from,

but a future step in this research project is to expand the method so an FE calculation of another similar

system can be used. In that way an already existing model (i.e. a predecessor of the new model) can be usedfor the wave extraction and the waves can then be applied to a subcomponent of the new car model already

before a full FE model of the new car has been created. This will then be a powerful method for the design

engineers to try out new subcomponents early in the design phase.

2 Theory

2.1 Classical SEA

The analysis procedure using SEA is to divide the studied structure into structural and acoustic subsystems

with a defined level of subsystem damping and coupling between the subsystems and to then solve the powerbalance equation for these subsystems for one or more applied external power input excitations. The power

balance equation for a subsystem s can be written as [3]:

Ws,in = Ws,diss +s=r

Ws,r (1)

where Ws,in is the input power, Ws,diss is the power dissipated through damping mechanisms in subsystem

s, and Ws,r is the net power transmitted from subsystem s to subsystem r , (Ws,r = Wr,s). The transmittedpower can be written as:

Ws,r = s,rEs r,sEr (2)

where s,r and r,s are the coupling loss factors between the different subsystems, is the radian frequency,

and Eis the time-averaged energy stored in the subsystem. The coupling loss factors are related to each other

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through the modal densities n(), as is shown in Equation (3). This relation is commonly known as the SEAreciprocity relation. For all cases in which the modal densities between subsystem s and subsystem r are not

the same, the coupling loss factors s,r and r,s are not reciprocal. By instead introducing the concept of a

reciprocal coupling factor r,s as the product of radial frequency, modal density, and coupling loss factor:

s,r = s,rn()s = r,sn()r = r,s (3)

the definition of modal power s as the ratio of total energy to the modal density of the subsystem can be

used. The transmitted power between subsystems can now be written as the product of the coupling factor

s,r and the difference between the modal powers of the subsystems, s and r. This means that the net

energy flow between subsystems is simply proportional to the difference in subsystem energy per modal

density in connected subsystems.

Ws,r = s,r Es

n()s

Er

n()r= s,r(s r) (4)

In a similar way the loss factor s,diss, can be introduced as a product of the radial frequency , the damping

loss factor and the modal density n():

s,diss = sn()s (5)

With this definition of loss factor, the dissipated power in a subsystem Ws,diss may also be written in terms

of the damping factor s,diss and the modal power s:

Ws,diss = sEs = sEs

n()s= s,disss (6)

As in Equations (1) - (4), the dissipated power is obtained in terms of the of modal power and a coefficient

dependent on the frequency, modal density and damping loss factor. The principal advantage of expressing

the coupling and damping in terms of reciprocal coupling factors and loss factors rather than in terms of

coupling loss factors and damping loss factors is that the linear power balance Equation (1) can now be

written in matrix form as a symmetric, positive definite matrix using the notations from Equations (1) - (6):

W1,inW2,in

...

=

1,diss +

s 1,s 1,2 1,3 . . .

1,2 2,diss +

s 2,s 2,3 . . ....

.... . .

12

...

(7)

Besides the obvious computational advantages (stability and solution speed) of solving a symmetric, positive

definite matrix, there is an additional advantage that the coupling factors and the modal powers are parameters

related to quantities that are easier to measure or calculate by FEA or other hybrid analysis methods. The

substitution of reciprocal coupling factors is also a key feature for the practical implementation of the SEA-

FEA point mobility method that is described in further detail in the following section.

2.2 SEA-FEA point mobility approach

One frequently studied parameter that is important to understand when predicting vehicle noise is the transfer

function (TF) from an excitation in one part of the vehicle to the response in another part of the vehicle. An

example of such a transfer function that can be of importance is the structural transfer function from anapplied force at an engine mount to the vibration of a floor panel or the structural-acoustic transfer function

to the sound pressure level inside the car cabin near the drivers ear. The following equations are based on

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the formulations presented by Manning in [4]. First, a dimensionless power transfer function (WTF) canbe obtained by defining a new term based on Equation (7):

WTFs,r =r

Ws,in

(8)

where WTFs,r is the power transfer function relating the modal power of the response subsystem r to the

input power of the source subsystem s. In a typical vehicle development situation the transfer function

of interest is often the transfer function between the input force and the response velocity (also called the

transfer mobility function):

TFs,r =vr

Fs(9)

This transfer mobility function TFs,r can be written as a function of the SEA power transfer function

WTFs,r by expressing the input power as a function of the drive point conductance Gs (real part of the

mobility) and the mean square of the applied force F2s . In classical SEA, the average subsystem con-ductance is used. The average is then taken over the spatial extent of the subsystem and over a band of

frequencies. However, the relation shown in Equation (10) can also be applied to single frequencies and to

individual points as long as the proper conductance is used.

Ws,in = F2s Gs (10)

The modal power r can also be defined as a function of the drive point conductance Gr and the average

mean square velocity v2r of the subsystem:

r =

2

v2r

Gr (11)

Then the square of the transfer function between an input force in subsystem s and the response in subsystem

r can be written as:

|TFs,r|2 =

vr2

Fs2=

2 r Gr

Gs

Ws,in=

2

Gs Gr WTFs,r (12)

This very useful result gives an expression for the transfer mobility function based only on the power transfer

function (SEA results) and the point conductance for both the excitation and the response point (which can

be retrieved from measurement or FEA). This result is the basis for the point mobility approach presented

in [4] and used in this paper. Calculation of the point conductance for important input and output points willbe used to enhance the results from the SEA calculations. FE calculations will also be used to enhance the

coupling factors. Manning also shows that the coupling factor for a point connection between two subsystems

can be expressed in terms of the point mobilities for these subsystems:

s,r =1

2

4 Gs Gr|Ys + Yr|2

(13)

where Y is the complex valued point mobility . In classical SEA theory the point mobility functions are

calculated from expressions for the infinite (or semi-infinite) system. The accuracy of this approach depends

on the averaging bandwidth and the modal overlap factor The modal overlap is a function of modal density

and damping and the highest value is obtained when both of these parameters are large for a certain band-

width. A wide bandwidth and/or high modal overlap result in a lower expected variance of the response from

the mean prediction, so that the accuracy of the result compared to a measurement is expected to be good.

This approach has shown to be a very accurate approximation for the high-frequency range where the modal


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overlap generally is high. This is the underlying reason that SEA has come to be accepted as the standard

tool for high-frequency CAE analysis in vehicle design engineering. For the mid- and low-frequency ranges,

where the modal overlap values tend to be smaller, the definition of the coupling factor in Equation (13) using

expressions for infinite systems may lead to large errors compared to measured response data. The reason for

this is that the smaller modal overlap is, the more the measured response (averaged over third octave bands)tends to depend on single deterministic phenomena. By instead using FEA to calculate these point mobility

functions, the transfer function accuracy can be improved for the mid- and low-frequency range. This can

of course be performed using a full FE model of the investigated system, but if the system is large e.g a car

body in white (BIW) or a full vehicle model, the computational cost will prohibit to stretch the calculations

towards higher frequencies. The point mobilities can also be calculated on only a part of the system. The

boundary conditions for the interface between the remainder and the subcomponent must then be added to

the model. Normally free or clamped boundary conditions are applied. This simplification can often lead to

results where some of the dynamic behavior of the complete system is missed. The method which will be

investigated in this paper is to apply the waves which connect the parts in Wave-Based Substructuring [8], [9]

as boundary conditions. In section 2.3 the Wave-Based Substructuring theory is explained.

2.3 Wave-Based Substructuring

Wave-Based Substructuring (WBS) is a powerful substructuring method in which the dynamic behavior of

the interfaces are described by a set of basis functions (waves). This reduces the model size compared to

conventional substructuring methods as Component Mode Synthesis (CMS) as the number of waves needed

to describe the dynamics of the interfaces is normally much smaller than the number of physical degrees

of freedom (DOF) of the interfaces [8], [9]. Instead of connecting the nodes from two neighboring subsys-

tems directly via continuity equations, WBS connects two subsystems via a set of virtual nodes (the wave

participation factors). This makes the method very suitable for subcomponent modeling.

An undamped system with no external forces can be described by the FE matrix equation:

Mx + Kx = 0 (14)

If this system is divided up into non overlapping substructures the model can be divided into internal DOFs

xi and junction DOFs xj . The equation of motion can then be divided into sub-matrices and written as:

Mii MijMji Mjj

xixj

+

Kii KijKji Kjj

xixj

=

0fj

(15)

In WBS the junction DOFs are expressed as a linear combination of a set of waves (V). These waves are

weighted with a participation factor p:

xj = V p (16)

The equation of motion, Equation (15), can then be written as:

Mii MijV

VTMji VTMjjV

xip

+

Kii KijV

VTKji VTKjjV

xip

=

0

VTfj

(17)

The waves can be seen as the modal displacement shapes defined only at the interface of the substructure

and are obtained from a modal analysis from the full FE model. Since the waves are a projection of the

orthonormal system modes on a subset of the system DOFs, orthonormality of the waves can not be guaran-

teed. Therefore some post processing of the waves must be performed aiming at orthonormalization of the

waves and at selecting the minimum number of waves needed to represent the dynamics of the interfaces.


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To avoid numerical issues in the solution of ill conditioned problems, it is therefore crucial to perform an or-

thogonalization on the interface modal displacement shapes, and to apply a selection threshold to keep only

the relevant waves. For this purpose, one can use Singular Value Decomposition (SVD) to decompose the

interface modal displacement shapes into a set of orthonormal waves and to rank these waves from the wave

with the highest singular value to the wave with the lowest singular value. The selection step then consistsof applying a threshold on the singular values, and keeping only the waves that correspond with a singular

value higher than the defined threshold. Physically, this corresponds to discarding those waves that have a

below-threshold contribution to the mathematical description of the complete vector space that describes the

interface dynamics.

In conventional WBS, the interface modal displacement shapes are obtained from a full FE analysis of the

complete system; work is ongoing to lower this requirement, for instance by obtaining the interface modal

displacement shapes only from a smaller subassembly [10].

As in all substructuring methods, continuity and equilibrium must be imposed at the boundaries. For a rigid

connection between two subsystems (a) and (b), continuity of the interface displacement and equilibrium of

the reaction forces are applied:

x(a)j = x

(b)j and f

(a)j = f

(b)j (18)

Transformed into a WBS framework the continuity conditions will be:

p(a) = p(b) and VTf(a)j = V

Tf(b)j (19)

Only rigid connections are discussed in this paper, however, the theory for elastic connections can be found

in [8] and [9].

2.4 Using WBS waves as boundary conditions for a subcomponent modeling ap-proach

Note that in conventional WBS, the orthogonalized waves are used to accurately describe the complete

interface dynamics in the assembled system. When one uses a limited number of wave degrees of freedom

instead of the (typically much larger number of) physical degrees of freedom at the interface, one obtains a

smaller description of the interface dynamics. This results in a more efficient reduction procedure of large

components with large connection interfaces and more efficient system-level calculations.

In this paper however, the aim is to develop a subcomponent modeling approach, where the WBS technique

is used to create a new type of boundary condition. This boundary condition shall describe the interface

dynamics as if the subcomponent were connected to the rest of the assembly. For this purpose, one cannotuse exactly the same selection procedure as in conventional WBS. The aim is to keep a subset of the waves,

such that in a subcomponent analysis, you describe the interface dynamics to represent an as in assembly

boundary condition. This boundary condition must be more accurate than a rigid or free-free boundary con-

ditions. If one uses all the waves (as would be done by keeping the wave selection procedure of conventional

WBS), in a component-level analysis, one has merely obtained a slightly more efficient model to predict the

free-free behavior of that component.

2.4.1 Deriving a selection criteria for the waves

As described in section 2.3, the waves connecting two components in WBS are calculated from a modal

analysis of the assembled system. This leads to a wave set with a large number of waves. Some of them

are important to the dynamic behavior of the subcomponent and some are not. The problem when using

the waves as a boundary condition without connecting the subcomponent to the remainder is that all waves


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contribute to the boundary condition with equal strength. Therefore a calculation step where the most

important waves are selected has to be performed. In this paper the selection procedure is based on the

Modal Effective Parameter (MEP). The MEP is a product of the modal displacement value at the input DOF

and the modal displacement value at the output DOF, performed for each mode [11]. This product is scaled

by the modal mass. The MEP between DOF m and DOF n for the r-th mode is defined as:

Xmn,r =m,rn,r

mr(20)

The MEP can be described as a measure of how much a specific global mode contributes to the transfer

function between two points. When all the MEPs are calculated, the modes can be sorted in a descending

order based on the MEP value. If the mode set then is truncated to contain only the modes contributing to

a certain percentage of the total cumulated MEP relating to a transfer function of interest, the global modes

most important to this transfer function are selected, see Figure 1.

Figure 1: Original mode set is rearranged so that the Mode with the highest MEP comes first.

This limited mode set is then used for the further analysis i.e for the wave extraction and wave orthonormal-

ization.

3 Numerical results

As discussed in Section 1, a fast and accurate method to perform subcomponent modeling is crucial for

using calculated point mobilities as a tool for evaluating the SEA parameters. In this paper three examples

of subcomponents are presented and the point mobility for a number of locations on each subcomponent is

calculated using different techniques. The results are then compared to a reference calculation performed on

a full BIW. The investigated sub-components are the back part of the roof, the B-pillar and a part of the spare

wheel panel, see Figure 2.

A modal analysis is then performed on the subcomponents with three different boundary conditions, namelyfree, clamped and the WBS boundary condition. Free and clamped boundary conditions are straightfor-

ward to apply, and the procedure for the calculation and application of WBS boundary condition has been

described in detail in section 2.4.1.

3.1 Calculation procedure

The model used in this paper is a Chrysler Neon BIW containing about 240 000 nodes and 230 000 elements.

From this model different parts have been cut out to simulate the different subcomponents. A modal analysis

has been performed up to 700 Hz and an FRF synthesis is performed based on the extracted modes. The FE

calculation of the modes has been performed in MSC.Nastran 2004 [12] and the model setup, preprocessing,

WBS definition, forced response analysis and postprocessing have been performed in LMS Virtual.Lab [11].

The results are then averaged both over frequency (third octave bands) and spatially (average of the responses

for all the IO points on the subcomponent, see Figure 3) .


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Figure 2: The three different subcomponents considered in the paper, extracted from the full model for a

clearer view. 1. Back of the roof, 2. B-pillar and 3. Spare wheel panel.

Figure 3: The three different subcomponents considered in the paper, with the IO points marked with yellow

dots. 1. Back of the roof (34 points), 2. B-pillar (3 points) and 3. Spare wheel panel (15 points).


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3.2 Results

First the truncation limit for the wave set has to be determined. The modes are arranged according to

descending MEP. For the back of the roof the truncation has been based on the total cumulated MEP of this

rearranged mode set. Three different values ranging from 5% to 20% have been considered and the resultingFRFs for three different WBS boundary condition cases have been compared to the reference solution. The

comparison has been performed by subtracting the reference FRF from the WBS FRF for each third octave

band from 10 Hz to 500 Hz, see Figure 4.

Figure 4: The absolute value of the difference between the reference FRF and three different realizations of

the truncation limit (5%, 10% and 20%)

In the difference plot it can be seen that the 10% truncation limit seems to give the best performance. To

further investigate this, the absolute values of the differences for all third octave bands are summed up to a

total difference. In Table 1 these summed differences confirm the conclusion from the difference plot.

Truncation limit 5% 10% 20%

Summed difference. [(m/s)/N] 0.225 0.172 0.201

Table 1: The absolute values of the differences for all third octave bands summed up to a total difference.

The 10% truncation limit is used for all three subcomponents in this study. For this application, this allows the

wave set to be reduced from about 400 waves to approximately 5-20 waves depending on the subcomponent.

Future research will focus on a more generic method to select the truncation criteria. In the next sections

the results from the numerical study are shown. The three different subcomponents are presented with thedifferent boundary conditions as well as the reference calculation. The results are presented first in a narrow

band analysis (f = 1 Hz) and then in a third octave band analysis.

3.2.1 Back of the roof

For the back of the roof 20 randomly distributed points have been used for the spatial average. The results

shown are the averaged point mobility for the back of the roof in the vertical direction (Z-direction). Global

modes corresponding to a truncation limit of 10 % of the total cumulated MEP are used to calculate the

waves.

Figures (5) and (6) show that the WBS boundary condition gives better results than the clamped and the

free boundary condition. For low frequencies (< 150 Hz), the free boundary condition greatly overestimates

the mobility and for high frequencies (> 150 Hz) the clamped boundary condition generally gives to high

mobility (can easiest be seen in the third octave plot). The WBS boundary condition however, manages to


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Figure 5: Comparison of point mobilities for different boundary conditions for the back of the roof [(m/s)/N].

The reference result is taken from a full BIW calculation.

Figure 6: Comparison of point mobilities for different boundary conditions for the back of the roof in third

octave band [(m/s)/N]. The reference result is taken from a full BIW calculation.


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capture both the low and high frequency behavior of the subcomponent. This can be seen even clearer if the

absolute value of the difference compared to the reference calculation is plotted, see Figure 7.

Figure 7: The absolute value of the difference between the reference FRF and the three different boundary

conditions, namely Free, Clamped and WBS boundary condition for the back of the roof.

3.2.2 B-pillar

For the B-pillar the results shown are the spatial average of three points on the B-pillar in the direction

perpendicular to the side of the car (Y-direction). For the WBS boundary condition the modes contributing

to 10 % of the total cumulated MEP are used to calculate the waves.

Figure 8: Comparison of point mobilities for different boundary conditions for the B-pillar [(m/s)/N]. The

reference result is taken from a full BIW calculation.

As can be seen in Figure 8 and Figure 9, the WBS boundary condition captures the real behavior of the

assembled system much better than clamped or free boundary condition. This is true for the entire frequency

range, as can bee seen in Figure 10.


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Figure 9: Comparison of point mobilities for different boundary conditions for the B-pillar in third octave

band [(m/s)/N]. The reference result is taken from a full BIW calculation.


conditions, namely Free, Clamped and WBS boundary condition for the B-pillar.


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3.2.3 Spare wheel panel

For the spare wheel panel 15 randomly distributed points have been used for the spatial average. The results

shown are averaged point mobility for the spare wheel panel in the vertical direction (Z-direction). Global

modes contributing to 10 % of the total cumulated MEP are used to calculate the waves.

Figure 11: Comparison of point mobilities for different boundary conditions for the spare wheel panel

[(m/s)/N]. The reference result is taken from a full BIW calculation.

Figure 12: Comparison of point mobilities for different boundary conditions for the spare wheel panel in

third octave band [(m/s)/N]. The reference result is taken from a full BIW calculation.

Over the full frequency range the WBS boundary condition is better or equal to both free and clamped

boundary condition, see Figure 13. However, the 10% truncation limit used for all three cases might not be

the optimal limit for this case. Future research will focus on finding a more generic method to select the

truncation limit.

4 Conclusions

In this paper, a novel approach has been presented to accurately define as in assembly boundary conditions

on individual structural subcomponents. This is important to improve the quality of a point mobility calcu-


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conditions, namely Free, Clamped and WBS boundary condition for the B-pillar.

lation on a structural component, which in turn is important for the accurate estimation of SEA parameters.

Since running industrial sized models into the mid-frequency region is often very time consuming, there is

also a need for a method were accurate results can be obtained from a smaller model. The novel approach

is based on the Wave-Based Substructuring (WBS) technique, which allows writing the interface between

substructures in terms of a limited number of basis functions (waves). In this paper, it is shown that for the

definition of accurate boundary conditions, you only need a very limited number of waves. It is demonstrated

in this paper that the most important waves can be chosen based on the Modal Effective Parameter (MEP).

The MEP is a product of the modal displacement value at the input DOF and the modal displacement value

at the output DOF, performed for each mode. This product is then scaled by the modal mass. The MEP can

be described as a measure of how much a specific global mode contributes to the transfer function between

two points.

As can be seen in the results in Section 3.2, the method to use waves from a WBS analysis as boundary

conditions shows great promise. It is clearly shown that the WBS boundary condition gives better results for

the entire investigated frequency range. While free boundary conditions can be suitable for high frequencies

and less suitable for low frequencies and clamped boundary conditions can be suitable for low frequencies

but not for high frequencies, the WBS boundary condition gives accurate results over the entire frequency

range.

There is still work to be done on finding a robust method to select the truncation limit. Another drawback

of the method as presented in this paper is that an FE calculation of the full model has to be performed in

order to extract the waves. The next step in this research project is to expand the method such that an FEcalculation of another similar system can be used. In that way an already existing model (i.e. a predecessor

of the new model) can be used for the wave extraction and the waves can then be applied to a subcomponent

of the new car model already before a full FE model of the new car has been created. This will then be a

powerful method for the design engineers to try out new subcomponents early in the design phase.

5 Acknowledgements

The European Commission is gratefully acknowledged for their support of the Marie Curie EST project

SIMVIA2 (http://www.simvia2.eu, contract nr. MEST-CT-2005-020263), from which the first author re-

ceived a Research Training grant. Furthermore, we kindly acknowledge IWT Vlaanderen for their support of

the ongoing research IWT-070337 MIDAS - next generation numerical tools for mid-frequency acoustics.


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