Vibro-acoustic analysis procedures for the evaluationof the sound insulation characteristicsof agricultural machinery cabins

W. Desmet, P. SasDepartment of Mechanical Engineering, division PMA, K.U.Leuven, Belgiume-mail: wim.desmet@mech.kuleuven.ac.be

AbstractOver the last few years, customer demands regarding acoustic performance, along with the tightening oflegal regulations on noise emission levels and human exposure to noise, have made the noise and vibrationproperties into important design criteria for agricultural machinery cabins. In this framework, bothexperimental analysis procedures for prototype testing as well as reliable numerical prediction tools for earlydesign assessment are compulsory for an efficient optimisation of the cabin noise and vibration comfort.This paper describes an experimental procedure for the in-situ assessment of the air-borne sound insulationcharacteristics of a cabin using a two-microphone sound intensity probe. In addition, several numericalapproaches, which are based on the finite element and boundary element method, are discussed in terms oftheir practical use for air-borne sound insulation predictions. To illustrate the efficiency and reliability of thevarious vibro-acoustic analysis procedures, all experimental and numerical procedures have been appliedand evaluated for the case of a cabin scale model.

1. Introduction

The high noise and vibration levels, to whichdrivers of agricultural machinery are often exposedfor long periods of time, have a significant part inthe driver’s fatigue and may lead to substantialhearing impairment and health problems. For thesereasons, the noise and vibration comfort has becomean important criterion in the design of the driver’scabin and a determining factor in the acceptanceand sales potential of agricultural machinery.

Therefore, it is essential for an optimal cabindesign to have time and cost effective analysis toolsfor the assessment of the noise and vibrationcharacteristics of various design alternatives at boththe early design stages and the prototype testingphase.

Two types of dynamic cabin excitations mainlycause the interior noise in a driver’s cabin. Air-borne excitation consists of sound that impinges onthe exterior of the cabin and introduces cabinvibrations, which transmit sound to the interior.Typical air-borne noise sources in agriculturalmachinery such as harvesters are the engine, theheader, the threshing unit and auxiliary equipmentlike pumps and compressors.

Structure-borne excitation consists of dynamicforces, which are directly transmitted to the cabinthrough the cabin suspension. These transmittedforces introduce cabin vibrations, which in turngenerate interior noise.

This paper outlines both an experimental analysistool as well as some numerical modelling proce-dures, which can be used for the evaluation of theair-borne sound insulation characteristics of a cabin.

The tool for the in-situ measurement of the air-borne sound insulation is based on sound powermeasurements using a two-microphone soundintensity probe. It adopts a reciprocal measurementscheme in that the exterior sound power, radiated bythe cabin due to a loudspeaker excitation in thecabin interior, is measured instead of the interiorcabin sound field due to external air-borne soundexcitation.

In addition to the experimental tool for physicalprototype testing, some numerical counterparts forvirtual prototype testing are addressed. Variousnumerical modelling procedures, which are basedon the finite element and boundary element method,are discussed in terms of their accuracy andassociated computational loads for calculating theair-borne sound insulation properties of a cabin.

In order to illustrate the efficiency and reliability ofthe various vibro-acoustic analysis procedures, allexperimental and numerical procedures have beenapplied and evaluated for the case of a cabin scalemodel.

2. Cabin scale model

The various sound insulation analysis tools, whichare discussed throughout the paper, have beenevaluated for a simplified scale model of an agri-cultural machinery cabin. Compared to a realdriver’s cabin, the geometry of the scale model hasbeen kept simple in order to minimise the computa-tional efforts, involved with the numerical analysistools. This simplification doesn’t compromise thegenerality of the conclusions, drawn from the scalemodel evaluation, since geometrical complexity isnot a limiting factor for the applicability of thefinite element and boundary element method. Thescale model has been designed such that, in asimilar way as a real-life driver’s cabin, it exhibitsalready a fairly high structural modal density andstructural damping in the low-frequency audiorange.

The cabin scale model is a box-typeconstruction, which consists of 5 flat, trapezoidalshaped panels, as shown in figure 1.

Figure 1: cabin scale model

All panels have different thicknesses and are madeof different types of plexiglass material to simulatethe distinct structural parts of a cabin (roof, floor,doors, windscreen, …), which have differentdynamic characteristics. The material andgeometrical properties of the scale model panels aresummarised in table 1. The panels are gluedtogether along their common ribs. Note that none of

the panel connections is right angled. The scalemodel is placed in small grooves in a rubber mat,which is placed on the rigid floor of a semi-anechoic test room. The remaining slits between thepanels and the grooves are filled with silicone toacoustically seal the interior cavity.

panel E

(N/m2) density (kg/m3)

thickness(m)

area (m2)

A 2.3 109 1138 3.0 10-3 0.48 B 4.6 109 1208 2.3 10-3 0.68 C 1.8 109 1217 6.0 10-3 0.62 D 1.8 109 1217 6.0 10-3 0.62

top 2.9 109 1309 2.3 10-3 0.64

Table 1: properties of the scale model panels

3. Experimental analysis tool

3.1 Insertion loss measurement

A conventional way to experimentally identify theair-borne sound insulation properties of a driver’scabin is to measure the sound transmission loss ofthe various cabin components in a standardtransmission suite test [1]. In such a test, thecomponent under investigation is placed in anaperture between two reverberation rooms, i.e. thesource room and the receiving room. A diffusesound field is generated in the source room and theincident sound power on the component is deducedfrom the spatially averaged mean square soundpressure in the source room. In a similar way, thetransmitted sound power is measured in thereceiving room. The transmission loss of thecomponent is then defined as the ratio between theincident and the transmitted sound power. In thisway, the quantification of the sound insulationproperties of the various cabin componentsinvolves a large series of time consuming measure-ments in a special-purpose laboratory environment.

An alternative way to quantify the air-borne soundinsulation properties of a driver’s cabin is themeasurement of the cabin insertion loss, which canbe performed in-situ and doesn’t require thedisassembly of the cabin.

The insertion loss (IL) of a cabin is defined asthe logarithmic ratio between the sound powerWsource, radiated by an acoustic source in a certainenvironment (see fig. 2(a)), and the sound powerWcabin, radiated in that environment by the

considered cabin when the same acoustic source isplaced in the cabin interior (see fig. 2(b)),

=

cabin

source

W

Wlog.10IL (1)

Figure 2: insertion loss measurement layout

The total sound power, radiated by an acousticsource, can be derived from the sound intensityvector field. The instantaneous intensity at a certainposition r in a sound field, at a certain moment t intime and in a certain direction d is defined as

)t,(v).t,(p)t,(I dd rrr = (2)

where p is the acoustic pressure and vd is the fluidparticle velocity in direction d. This intensitydenotes the instantaneous net sound power thatflows per unit area through an infinitesimal surface,which is located at position r and has a normalvector, oriented in direction d.

For practical purposes, it is common use todefine the time-averaged intensity

∫∞→

=T

0d

Td dt).t,(v).t,(p

T

1lim)(I rrr (3)

The total time-averaged sound power W, radiated byan acoustic source, is then defined as

∫=S

n )(dS).(IW rr (4)

where S is a closed surface, that encloses theacoustic source and that comprises no other acousticsources (or sinks). In denotes the time-averagedintensity in the direction, normal to the surface S.

Note that the sound power W is independent of thegeometrical shape of the surface S, provided that nodamping is present in the fluid.

It follows from eq. (4) that the radiated sound powercan be obtained from sound intensity measurements.The most commonly used sound intensity measure-ment procedure utilises a pressure gradient soundintensity probe [2-5]. The probe consists of twoclosely spaced pressure microphones, as shown infigure 3.

According to Euler’s relation, the fluid particlevelocity can be expressed as

∫∞− ∂

∂−=t

d d.d

),(p1)t,(v ττ

ρr

r (5)

in which ρ represents the ambient fluid density. Thepressure gradient at the mid position r=½(r1+r2)between both microphones is approximatelymeasured as

r

)t,(p)t,(p

d

)t,(p 2211

∆rrr −≈

∂∂

(6)

where ∆r=||r1-r2|| is the (small) distance betweenboth microphones. The pressure at the mid positionis approximated as

2

)t,(p)t,(p)t,(p 2211 rr

r+≈ (7)

Figure 3: sound intensity measurement layout

Substitution of eqs. (5), (6) and (7) into eq. (3) leadsto the approximation of the time-averaged intensityin the direction of the line joining the twomicrophones

∫ ∫

−

+

≈

∞−∞→

T

0

t

11

222211

T

d

dtd),(p

),(p

r..2

)t,(p)t,(p

T

1lim

)(I

ττ

τ∆ρ r

rrr

r

(8)

∆r

p1 p2

d

rr

r

12

Wsource Wcabin

(a) (b)

In most practical sound intensity probe implemen-tations, the sound intensity is not directlydetermined from its time domain formulation in eq.(8), but from its equivalent formulation in thefrequency domain, in which the spectrum of theintensity is defined as

{ }r..

),,(GIm),(I

21ppd

21

∆ωρω

ωrr

r ≈ (9)

where Im{Gp1p2} is the imaginary part of the crossspectrum between the two microphone signals,which can be readily obtained from the two pressuremeasurements using a dual channel FFT analyser.

The microphone distance ∆r is a determiningfactor in the sound intensity measurement accuracy.In order to minimise the approximation errorsinvolved in eqs. (6) and (7), the microphones shouldbe as close as possible. Selecting a too smallseparation distance results, however, in a very weakdifference between both microphone signals, so thatthe influence of measurement noise can becomedominant and can unacceptably influence themeasurement accuracy. For these reasons, a fairlylarge microphone distance - typically a fewcentimetres – is used to measure sound fields, ofwhich the energy is mainly distributed in the low-frequency range, where wavelengths are large,whereas a small distance – typically a fewmillimetres – is used for high-frequency measure-ments. For a detailed discussion of the measurementerrors involved with a two-microphone soundintensity probe, the reader is referred to [2-5].

The surface integral in eq. (4), which determines theradiated sound power of an acoustic source, can beapproximated either by measuring the intensity atsome discrete points [6] or by scanning [7]. In thelatter scanning procedure, some hypothetical non-overlapping measurement planes are defined aroundthe source, such that these planes form together aclosed surface, which entirely encloses the radiatingsound source. For each of the measurement planes,a space-averaged normal intensity spectrum niI ismeasured by moving the intensity probe continuous-ly over the measurement plane during a certainamount of time in such a way that the axis of theprobe is always perpendicular to the measurementplane. The frequency spectrum of the total radiatedsound power is then obtained as the sum of theaveraged intensity spectra, multiplied with thesurface area Si of the associated measurementplanes,

∑=

≈N

1iini S).(I)(W ωω (10)

where N is the number of measurement planes.By applying this procedure for the measurement

of the frequency spectra of the aforementionedsound powers Wsource and Wcabin, the insertion loss ofa cabin can be determined, according to eq. (1).

3.2 Insertion loss of the scale model

The procedure for insertion loss measurements asoutlined above has been applied for the cabin scalemodel, described in section 2. Five rectangularmeasurement planes are defined. These scanningplanes are located at a distance of 150 mm from thetop plate or one of the four sidewalls of the scalemodel. The five planes, together with a part of therigid floor of the semi-anechoic test room, for whichthe normal velocity is zero and hence doesn’trequire an intensity scan, form a closed surface, thatcompletely comprises the cabin scale model.

A loudspeaker is placed inside the scale model(see figure 1) to provide a dynamic excitation with abroadband spectrum. The measurement of theradiated sound power of the loudspeaker with andwithout the cabin scale model results in theinsertion loss spectrum, shown in figure 4.

Figure 4: insertion loss of the cabin scale model

A positive insertion loss indicates that some noisereduction is obtained by surrounding the acousticsource with the considered cabin, whereas anegative insertion loss indicates a noise ampli-fication by the cabin. Note also that the insertionloss is a good parameter for quantifying thereciprocal sound insulation properties, in that the

frequency (Hz)

inse

rtio

n lo

ss (

dB)

0

30

20

10

0

-10

-20

-30500400300200100

frequency components of an external air-bornecabin excitation, which have positive insertion lossvalues, are reduced by the cabin, while thefrequency components with a negative insertion lossappear at amplified levels in the spectrum of thecabin interior noise.

It follows from figure 4 that the overall air-bornesound insulation quality of the cabin scale model isfairly poor in the low-frequency range and that thesource excitation components below 100 Hz andaround 245 Hz, 330 Hz and 455 Hz are evenamplified by the scale model.

4. Numerical prediction tools

4.1 Introduction

A major imperative for reducing the design time andcost is to apply predictive engineering methods,which enable the evaluation of various design alter-natives already at the early design stages. For acous-tic performance evaluation, predictive methods mustencounter the following problem formulation.

Small-amplitude sound fields in a homogeneousfluid are governed by the acoustic wave equation

2

2

22

t

)t,(p

c

1)t,(p

∂∂=∇ r

r (11)

where ∇ 2 is the Laplace operator and where c is thespeed of sound in the fluid.

For a time-harmonic sound field with angularfrequency ω, the steady-state acoustic pressure canbe expressed in a complex exponential notationp(r,t)=Re{p(r,ω).ejωt}, of which the complex ampli-tude p(r,ω) is governed by the Helmholtz equation

0),(p.k),(p 22 =+∇ ωω rr (12)

where k=ω/c is the acoustic wavenumber.It follows from eq. (5) that the complex exponentialnotation of the steady-state fluid particle velocity indirection d is vd(r,t)=Re{vd(r,ω).ejωt}, with

d

),(pj),(vd ∂

∂= ωρω

ω rr (13)

Based on eq. (3), the corresponding time-averagedintensity can be expressed as

{ }),(v).,(pRe2

1),(I *

dd ωωω rrr = (14)

where * denotes the complex conjugate.

The main numerical prediction tools for solvinglow-frequency (vibro-)acoustic problems are basedon the finite element (FE) and the boundary element(BE) method.

The finite element method is a commonly usedmethod for predicting the steady-state pressure fieldin a closed cavity volume [8-10]. To uniquelydefine a pressure field, governed by the Helmholtzequation (12), a boundary condition must bespecified at each location on the closed boundarysurface of the cavity volume. Most often theboundary conditions impose either a prescribednormal fluid velocity,

),(vn

),(pjn ωω

ρωr

r =∂

∂ (15)

or a prescribed normal impedance relation betweenthe pressure and the normal fluid velocity to modelthe damping effect of a locally reacting absorbentlining on (a portion of) the boundary surface,

n

),(p),(Zj),(p n

∂∂= ω

ρωωω rr

r (16)

In the FE method, the cavity volume is discretizedinto small elements and nodes are defined at somediscrete locations in each element. Within eachelement, the pressure field is approximated in termsof simple prescribed shape functions Na, which arelocally defined within an element,

[ ] { }aaaa p.N)(p).(N),(p =≈ ∑ ωω rr (17)

The element shape functions are defined such thatthe vector of unknown shape function contributions{pa} consists of the pressure approximations at thenodal locations.

By transforming the Helmholtz equation (12)into a weighted residual or a variational formulationand taking into account the boundary conditions(15) and (16), a matrix equation in the unknownnodal pressures is obtained,

[ ] [ ] [ ]( ){ } { }aaa2

aa Fp.MCjK =−+ ωω (18)

The acoustic stiffness and mass matrices [Ka] and[Ma] are sparsely populated, symmetric matriceswith real, frequency-independent coefficients. Thesparsely populated, symmetric damping matrix [Ca]has usually complex, frequency-dependent coeffi-cients, since this matrix results from the impedance

boundary condition (16), in which the normalimpedance nZ is usually complex and frequency-dependent. The excitation vector {Fa} results fromthe normal velocity boundary condition (15) andfrom possible external acoustic excitation sourcesinside the cavity volume.

The boundary element method is a commonlyused alternative method for solving acousticproblems, especially for predicting the steady-statepressure field in unbounded fluid domains. Themethod is based on a boundary integral formulation,which relates the steady-state pressure at anyposition in the fluid domain to the distributions ofsome acoustic variables on the boundary surface ofthe considered problem. In this way, the BE methodfollows a two-step procedure. In the first step, thedistributions of the boundary surface variables aredetermined. In the second step, the steady-statepressure and associated fluid velocity and intensityvectors at any position in the (unbounded) fluiddomain can be obtained from the boundary integralformulation, using the boundary surface resultsfrom the first step.

The indirect BE method is commonly used forpredicting the steady-state pressure field, radiatedfrom a thin-walled (open or closed) vibratingsurface. The indirect boundary integral formulation[11] relates the pressure at any position r in the(unbounded) acoustic fluid that surrounds the thinboundary surface S to the distributions of a singlelayer potential σ(ra,ω) and a double layer potentialµ(ra,ω) on that boundary surface,

∫

−∂

∂=

Sa

aa

aa )(dS

),,(G).,(n

),,(G.),(

),(p rrrr

rrr

rωωσ

ωωµω (19)

The single layer potential σ is the difference innormal pressure gradient between both sides of thethin boundary surface and the double layer potentialµ is the pressure difference between both sides,

n

),(p

n

),(p),( aa

a ∂∂−

∂∂=

−+ ωωωσ rrr (20)

),(p),(p),( aaa ωωωµ −+ −= rrr (21)

The positions +ar and −

ar indicate the boundarysurface positions at the positive and negative side ofthe normal direction n. The function G(r,ra,ω) in(19) is the Green’s kernel function,

a

jk

a 4

e),,(G

a

rrrr

rr

−=

−−

πω (22)

which represents the free-field steady-state pressureat position r due to a time-harmonic point sourceexcitation at position ra with angular frequency ω.

In a similar way as is done in the FE method, theindirect BE method is based on the discretization ofthe boundary surface into small elements and thesingle and double layer potentials are approximatedin terms of locally defined element shape functions.When the boundary conditions consist of prescribednormal velocity (eq. (15)) and normal impedanceconditions (eq. (16)), the single layer potential ateach boundary surface position is either zero or atleast related to the double layer potential throughthe normal impedance relation. In this case, thedouble layer potential is the only independent boun-dary surface variable, which is approximated as

[ ]{ }aaaa .N)().(N),( µωµωµ µµ =≈ ∑ rr (23)

Transforming the indirect boundary integralformulation (19), together with the boundary condi-tions (15) and (16), into a variational formulationand imposing the stationarity condition on theassociated energy functional yield a matrix equationin the unknown nodal double layer potentials [12],

[ ] { } { }µµω F.)(H a = (24)

The excitation vector {Fµ} results from the normalvelocity boundary condition (15) and from possibleexternal acoustic excitation sources in the fluiddomain. The normal impedance boundary condition(16) appears in the model matrix [H(ω)]. Thismodel matrix is symmetric, but, in contrast with anFE model, it is a fully populated matrix withcomplex, frequency-dependent coefficients, whichresult from more complicated numerical integra-tions than it is the case for FE model coefficients.Moreover, for exterior radiation problems with aclosed boundary surface, the boundary integralformulation suffers from non-uniqueness of thesolution at some particular (irregular) frequencies,which requires special numerical treatment tocircumvent this problem [13].

Compared with an FE model, the size of a BEmodel is significantly smaller, since only theproblem boundary surface must be discretized intoelements. However, the smaller model size usually

doesn’t result in smaller computational efforts. Asmall, but fully populated and frequency-dependentBE model must be constructed and solved at eachfrequency of interest. The construction of an FEmodel, however, is computationally efficient, sinceit usually involves only an assembly of frequency-independent submatrices. Moreover, efficientmatrix solvers are available for large, but sparselypopulated, symmetric FE models.

As a result, the FE method is usually the mostcomputationally efficient method for solvingacoustic problems, defined in a bounded fluiddomain, while the strength of the BE methodbecomes apparent for solving radiation problems inunbounded fluid domains.

In the FE and the BE method, the dynamicvariables within each element are expressed interms of simple (polynomial) shape functions. Inorder to represent the spatial variation of theacoustic response accurately within each element, asubstantial amount of elements is required. In thisrespect, a rule of thumb states that at least 10(linear) elements per wavelength are needed. Thismay result in large models, whose size increases forincreasing frequency, since the acoustic wavelengthλ=2π/k=2πc/ω decreases with frequency. Since thesubsequent computational efforts increase also, theuse of FE and BE methods is practically restrictedto the low-frequency range.

4.2 Numerical modelling approaches

For the numerical prediction of the low-frequencyinsertion loss of a driver’s cabin, as defined in eq.(1), three different modelling approaches, which arebased on the FE and the BE method, can befollowed.

4.2.1 fully coupledTo determine the insertion loss, the sound power,radiated by the cabin due to an acoustic excitationin the cabin interior (see fig 2(b)) must becalculated. In principle, the mutual fluid-structurecoupling interaction between the cabin structuralvibrations and the acoustic pressures in both thecabin interior fluid and the exterior fluid must betaken into account. For this purpose, a structural FEmodel for the cabin vibrations can be coupled withan acoustic indirect BE model of type (24) for theinterior and exterior pressure fields,

=

−+

µµρω

ωωω

F

Fw.)(H

L

LMCjKs

a

s

2Tc

cs2

ss (25)

[Ks], [Cs] and [Ms] are the structural stiffness,damping and mass matrices, {Fs} is the structuralexcitation vector and {ws} is the vector of unknownnodal displacement degrees-of-freedom of the cabinstructure [14]. The coupling matrix [Lc] results fromthe pressure loading effect of the interior andexterior sound fields on the cabin structural displa-cements, while the transpose of the coupling matrixarises in the acoustic part of the coupled model (25)to ensure that the normal fluid displacements alongthe fluid-cabin interface equal the normal structuraldisplacements [15].

The coupled FE/indirect BE model (25) can bedirectly solved at each frequency of interest for theunknown nodal structural displacements andacoustic double layer potentials. The latterpotentials allow then the calculation of the pressureat any field point in the fluid domains using theindirect boundary integral formulation (19). Accor-ding to eq. (4), the total sound power, radiated bythe cabin, can then be obtained by defining a closedmesh of field points that completely encloses thecabin, and by evaluating the normal intensities onthis mesh surface, according to eq. (14).

Especially for low-frequency predictions, thesize of the coupled model (25) can be reducedwithout substantial loss of accuracy by projectingthe structural displacements onto a modal base ofin-vacuo normal modes,

{ } [ ] { }sss .w φΦ= (26)

The columns of matrix [Φs] are normalised modesand vector {φs} comprises their contribution factorsto the structural displacement. The resultingcoupled model becomes then

=

−+

µ

Φµφ

ρωωΦ

Φωω

F

F.)(H

L

LIC~

jK~

sT

a

s

2Tc

cT

s2

ss (27)

[ ]sK~

is a diagonal matrix, containing the squared

natural frequencies 2sω of the considered modes.

The modal damping matrix [ ]sC~

is usuallyconstructed as a diagonal matrix with coefficientseither of type ss2 ωξ , in which sξ represents amodal viscous damping coefficient, or of type

ωωη /2ss , in which sη represents a modal hystere-

tic damping coefficient.

Since a structural and an acoustic problem must besolved simultaneously, the size of a coupled FE/BEmodel (25), even in its semi-modal form (27),becomes very large. For this reason and due to thefrequency dependence of the acoustic part of themodel, the computational efforts, involved withsolving a fully coupled model for a real-life driver’scabin in a wide frequency range, become prohibi-tively large, even with the nowadays availablepowerful computer resources.

4.2.2 semi-coupledAlthough the cabin structural vibrations mutuallyinteract with both the interior and the exteriorpressure fields, the strength of the fluid-structurecoupling interaction between the cabin vibrationsand the interior pressure is significantly larger,since the cabin interior is a closed fluid cavity witha relatively small volume. Therefore, a two-stepsemi-coupled modelling approach can be followed,in which the mutual coupling interaction betweenthe cabin vibrations and the exterior pressure fieldis neglected. This approach is computationally moreefficient than the fully coupled approach, howeverat the expense of some accuracy loss.

In the first step of the semi-coupled approach,the coupled dynamic response of the cabin structureand the interior fluid due to an acoustic sourceexcitation in the cabin interior is calculated, whilethe effect of the exterior pressure field isdisregarded. For this purpose, a structural FE modelfor the cabin vibrations can be coupled with anacoustic FE model of type (18) for the cabin interiorpressure field [9],

[ ]

=

a

s

a

s

F

F

p

w.A (28)

with

[ ]

−+−+=

a2

aaTc

2cs

2ss

MCjKK

KMCjKA

ωωρωωω

The coupling matrix [Kc] results from the pressureloading effect of the interior sound field on thecabin structural displacements, while the couplingmatrix ρω2[Kc]

T arises in the acoustic part of thecoupled model (28) to ensure that the normal fluiddisplacements along the fluid-cabin interface equalthe normal structural displacements.

In a similar way as described in the previoussubsection, applying a modal expansion can reducethe size of the coupled model (28). The mostappropriate mode selection for a modal expansion is

the use of the modes of the coupled system.However, since the coupled FE/FE model (28) is nolonger symmetric, the calculation of coupled modesrequires a non-symmetric eigenvalue calculation,which is very time consuming and which makes it apractically impossible procedure for many real-lifevibro-acoustic problems. A possible alternative is toapply a component mode synthesis technique. Inthis technique, the structural displacements areprojected onto a base of uncoupled structuralmodes, i.e. modes of the structure without fluidpressure loading (see eq. (26)). The fluid pressuresare projected onto a base of uncoupled acousticmodes Φa, i.e. acoustic modes with rigid boundaryconditions along the fluid-structure interface,

{ } [ ] { }aaa .p φΦ= (29)

The resulting coupled model becomes then

[ ]

=

aTa

sTs

a

s

F

F.A

~

ΦΦ

φφ

(30)

with

[ ]

−+−+=

a2

aasTc

Ta

2ac

Tss

2ss

IC~

jK~

K

KIC~

jK~

A~

ωωΦΦρωΦΦωω

The advantage of using uncoupled modes is thatthese modes result from symmetric and computa-tionally efficient eigenvalue problems. It should benoted, however, that uncoupled acoustic modeshave a zero displacement component, normal to thefluid-structure coupling interface, which impliesthat a large number of high-order uncoupledacoustic modes is required to accurately representthe normal displacement continuity along the fluid-structure interface.

In the second step of the semi-coupled approach,the radiated sound power is calculated using a BEmodel of type (24). The mesh of the BE modelconsists of the exterior envelope of the structural FEmesh. The normal structural velocities on thisenvelope are extracted from the results, obtainedfrom the first step. These structural velocities,which serve as normal velocity boundary conditionsfor the exterior radiation problem, are then used toconstruct the excitation vector {Fµ} of the BEmodel. Finally, the resulting double layer potentialson the BE mesh are used to calculate the normalintensities on an enclosing field point mesh and theassociated radiated sound power, as described insection 4.2.1.

4.2.3 uncoupledBy neglecting the mutual coupling interaction of thecabin structural vibrations with both the interior andthe exterior pressure fields, a three-step uncoupledmodelling approach is obtained, which is thecomputationally most efficient, but least accurateapproach.

In the first step, an uncoupled acoustic FE modelof type (18) is used to calculate the cabin interiorpressure field due to an acoustic source excitation.For this pressure calculation, the cabin structure iseither assumed to be rigid or its flexibility can betaken into account in an approximate way bymodelling the cabin structure as a normal impe-dance boundary condition for the interior pressurefield (see eq. (16)). The normal impedance valuescan be obtained from a separate structural FEcalculation, in which a uniform pressure loading isapplied to a structural FE mesh of the cabin and theresulting normal structural velocities are evaluated.

In the second step, the cabin structural displace-ments are calculated from an uncoupled FE model,in which the pressures at the fluid-cabin interface,obtained in the first step, are used as external forceexcitations.

The third step is completely similar to thesecond step of the semi-coupled approach, in thatthe radiated sound power is obtained from anacoustic BE model, in which the cabin structuralvibrations from the second step are used as normalvelocity inputs.

4.3 Cabin scale model validations

4.3.1 preliminary considerationsAs described in section 2, the cabin scale model isplaced on a rubber mat, which rests on the rigidfloor of a semi-anechoic test room. Although therubber mat has a finite normal impedance, it isassumed in the numerical modelling that the rubbermat is rigid, having an infinitely large normalimpedance, which is a fair assumption at lowfrequencies.

For the numerical calculation of the low-frequency insertion loss of the cabin scale model, anacoustic point source excitation is adopted, whichradiates into the acoustic half-space above the rigidfloor of the semi-anechoic test room (see fig. 5(a)).The radiated sound power Wsource can be determinedin an analytical way. The pressure at position r inthe half-space due to a time-harmonic point sourceexcitation at position ra with angular frequency ωcan be expressed as p(r,ra,ω)=A.GH(r,ra,ω), where A

is the source strength and where GH is [16]

2

jkR

1

jkR

aH R

e

R

e),,(G

21 −−+=ωrr (31)

with a1 -R rr= the distance between the point

source position ra and position r and with

a2 -R rr’= the distance between the source

position and position r’, which is the image positionof r with respect to the rigid floor (see fig. 5(a)).

Figure 5: modelling layout for cabin scale model

By defining a hemisphere around the source, asshown in fig. 5(a), and by using eqs. (13) and (14)to derive the normal intensities on the hemispherefrom the analytical formulation of the half-spacepressure field, the total radiated sound power Wsource

is obtained, according to eq. (4).In view of determining the insertion loss of the

cabin scale model (see eq. (1)), the previouslydescribed modelling approaches are used tocalculate the sound power Wcabin, radiated by thecabin into the acoustic half-space due to an acousticpoint source excitation inside the cabin (see fig.5(b)).

4.3.2 validation of the various approaches

numerical modelsFor the calculation of the structural vibrations, thefive panels of the cabin scale model have beendiscretized into four-noded third-order plateelements, yielding a structural FE mesh with a totalof 22991 unconstrained degrees-of-freedom. Thewavelength of bending waves in flat panels equals

422

s

24

b)1(3

Et4

νωρπλ

−= (32)

with t the panel thickness, E the modulus of elasti-city, ρs the plate density and ν the Poisson’s ratio.

Wsource Wcabin

(a) (b)

hemisphere

rigid floor

PR

R

1

2

P’

Q

Based on the panel properties in table 1 and a ruleof thumb, that states that at least 5 third-order plateelements are needed per wavelength, the structuralFE mesh, shown in figure 6, may provide accuratedynamic displacement predictions up to 350 Hz.

In all validation calculations, a modal projectionof type (26) is used for the structural displacements.The modal base contains 331 normal modes, whichconstitute all the in-vacuo structural modes of thescale model with natural frequencies ωs less than400 Hz. To include structural damping effects in thestructural FE model, a modal hysteretic dampingcoefficient ηs of 0.03 is defined for each mode.

The acoustic part of the fully coupled model (27) isbased on a BE mesh of four-noded linear fluidelements. Since low-frequency acoustic wave-lengths are much larger than structural bendingwavelengths, acoustic meshes could be coarser thanstructural meshes. However, it is advisable forcoupled vibro-acoustic predictions to use anacoustic mesh with a similar mesh density as thecorresponding structural FE mesh in order to allowan accurate modelling of the continuity of thenormal fluid and structural displacements along thefluid-structure coupling interface. Therefore, thenodes of the acoustic BE mesh coincide with thenodes of the FE mesh of the scale model structure,yielding a total of 3923 acoustic degrees-of-freedom. Note that the rigid floor of the semi-anechoic testroom is not explicitly included in theBE mesh. Instead of function G, defined in eq. (22),the function GH, defined in eq. (31), is used asGreen’s kernel function in the indirect boundaryintegral formulation (19). In that way, the effect ofthe rigid floor is implicitly taken into account in theBE model and the surface integral in (19) must onlybe evaluated along the surface of the scale modelstructure.

In the semi-coupled and uncoupled modellingapproaches, the calculation of the pressure field inthe scale model interior cavity is based on anacoustic FE mesh of eight-noded linear volumeelements. Again, a similar mesh density is adoptedas in the structural FE mesh, yielding a volumemesh with a total of 24389 nodal pressure degrees-of-freedom. In the calculations, the nodal pressuresare projected onto a modal base of type (29). Thebase contains 40 modes, which constitute alluncoupled cavity modes with natural frequencies ωa

less than 780 Hz. Note that this frequency span ofthe acoustic modal base is larger than the 400 Hzfrequency span of the structural modal base. As

mentioned before, the higher-order uncoupledacoustic modes are taken into account to accuratelyrepresent the normal displacement continuity alongthe interface between the scale model structure andthe interior cavity.

In the first step of the uncoupled approach, theinterior pressure field is calculated with theaforementioned acoustic FE model. To approximatethe effect of the dynamic flexibility of the scalemodel, normal impedance boundary conditions oftype (16) have been used in the FE model. Asoutlined in section 4.2.3, the frequency- and space-dependent values of the normal impedances resultfrom the aforementioned structural FE model, inwhich the normal structural velocities are calculatedfor a uniform pressure loading on the scale model.

For the calculation of the radiated sound power,the normal intensities are evaluated on a hemi-spherical mesh of 1241 field points around the scalemodel, as shown in figure 6.

In all acoustic models, the air is modelled as afluid with speed of sound c=340 m/s and fluiddensity ρ=1.2 kg/m3.

Figure 6: meshes for cabin scale model predictions

prediction accuracyThe three numerical modelling approaches havebeen used to evaluate the air-borne insertion loss ofthe cabin scale model in the frequency range from60 Hz to 500 Hz with a resolution of 10 Hz.

The uncoupled structural results are obtainedwith the MSC/NASTRAN v70.5 software, while theacoustic and coupled vibro-acoustic results areobtained with the LMS/SYSNOISE Rev. 5.4software. All calculations have been performed onan HP-C3000 Unix-workstation (400 MHz singleprocessor, 1 GB memory, SPECint95=31.8,SPECfp95=52.4).

Figure 7: cabin scale model insertion loss(dashed: measured, solid : predicted (upper: fully

coupled, middle: semi-coupled, lower: uncoupled))

Figure 7 compares the measured insertion loss ofthe cabin scale model with the numerical results,obtained with the three modelling approaches. Thisfigure illustrates that the fully coupled modellingapproach provides accurate prediction results in thefrequency range below 350 Hz. As could beexpected, the accuracy drops above this frequency,

since the frequency span of the structural modalbase is limited to 400 Hz and since the structural FEmesh becomes too coarse for higher frequencies.

It can be noticed that the numerically identifiedfrequencies of poor insulation (e.g. around 245 Hzand 330 Hz) are slightly smaller than the measuredfrequencies. This may be due to small errors on theair density and speed of sound and due to smallgeometrical errors. The latter is caused by the factthat the boundary surface of the cavity volume inthe numerical models coincides with the middlesurfaces of the panels, whereas the actual cavitydimensions are slightly smaller due to the finitethickness of the panels.

Compared with the fully coupled approach, theaccuracy of the semi-coupled approach is slightlysmaller, but still reasonable in the frequency rangebelow 350 Hz, whereas the accuracy of theuncoupled approach is fairly poor.

computational efficiencyAs discussed in section 4.1, BE models must beconstructed and solved at each frequency of interest,while the construction of FE models mainly invol-ves computations, which must only be performedonce. Table 2 lists the loads of the computations,which must be repeated for each frequency in thescale model calculations and in the evaluation of theradiated sound power on the hemispherical fieldpoint mesh.

approach fully semi uncoupledstep 1 634 21 0.68step 2 303 0.54step 3 303Wcabin 74 74 74

Table 2: computational loads(in seconds CPU time per frequency)

In addition to these variable loads, the calculation ofthe structural and acoustic modal bases required,respectively, 270 and 104 CPU seconds. The semi-coupled approach required an additional fixed loadof 1082 CPU seconds, which is mainly needed forconstructing the coupling matrix [Kc] in eq. (30).

It can be concluded from figure 7 and table 2 thatthe semi-coupled approach provides the bestbalance between prediction accuracy and computa-tional effort.

Besides the computational efforts, the humanefforts, involved with generating the elementmeshes, should not be disregarded in this

frequency (Hz)

inse

rtio

n lo

ss (

dB)

0

30

20

10

0

-10

-20

-30500400300200100

measuredsemi-coupled

frequency (Hz)

inse

rtio

n lo

ss (

dB)

0

30

20

10

0

-10

-20

-30500400300200100

measureduncoupled

frequency (Hz)

inse

rtio

n lo

ss (

dB)

0

30

20

10

0

-10

-20

-30500400300200100

measuredfully coupled

framework. The fully coupled approach requiresonly the generation of a boundary surface mesh,which may serve both as a structural FE and anacoustic BE mesh, while the semi-coupled and theuncoupled approaches require also the generation ofa cavity volume mesh.

5. Conclusions

This paper describes how a two-microphone soundintensity probe can be used for the in-situ measure-ment of the air-borne insertion loss of driver’scabins (of agricultural machinery). In addition, threenumerical modelling approaches, which are basedon the finite element and boundary element method,are discussed in terms of their practical use for low-frequency air-borne insertion loss predictions.

It is concluded from validations on a cabin scalemodel that the semi-coupled approach provides agood balance between prediction accuracy andcomputational load. The first step in this approachconsists of calculating the coupled vibro-acousticresponse of the cabin structure and the interior fluidusing a coupled structural FE/acoustic FE model. Inthe second step, the structural vibrations on theexterior surface of the cabin structure, obtained inthe first step, are used as velocity inputs for anacoustic BE model to calculate the radiated pressurefield in the fluid domain, exterior to the cabin.

Acknowledgements

This research was partly funded by the BelgianMinistry of Small Entreprises, Traders andAgriculture (DG 6).

W. Desmet is a Postdoctoral Fellow of the Fundfor Scientific Research – Flanders (Belgium)(F.W.O.).

References

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