Journal of Sound and Vibration "0884# 079"3#\ 472596

PREDICTION OF STRUCTURE-BORNE SOUNDTRANSMISSION IN LARGE WELDED SHIP

STRUCTURES USING STATISTICAL ENERGYANALYSIS

P[ HYNNA\ P[ KLINGE AND J[ VUOKSINEN

Technical Research Centre of Finland\ Ship Laboratory\ P[O[ Box 003\ SF!91040 Espoo\Finland

"Received 10 February 0881\ and in _nal form 18 November 0882#

An e.cient method is presented for the prediction of structure!borne sound transmissionin large welded ship structures[ SEA "Statistical Energy Analysis# is used\ and the equationsused for the SEA parameters are also presented[ Traditionally\ the SEA method requiresa great deal of work when steel structures are modelled[ It is almost impossible to preparemodels manually for large structures such as ships[ In the method developed\ thepreprocessing programs used in the context of the _nite element method "FEM# are appliedto reduce the modelling work[ To date\ one!dimensional beam\ two!dimensional triangularand quadrilateral plate\ and three!dimensional volume elements have been implemented[The assemblage of the loss factor matrix is made in a manner analogous to the sti}nessmatrix in FEM[ In the computer implementation of the SEA program\ standard FEMtechniques are used to reduce calculation time\ including the skyline matrix technique andLDLT!matrix decomposition of the loss factor matrix[ The e}ectiveness of the presentmethod is illustrated by the computer run of the model of the passenger cruise vessel\ whichcontained over 4999 elements and 06 999 coupling branches[ It took only 06 min in the IrisIndigo R3999 workstation[ Application calculations for an echo sweeping vessel\ atimber!container carrier\ and a passenger cruise vessel are discussed\ and comparison ismade with full scale measurements[

0[ INTRODUCTION

The noise level criteria of the International Maritime Organization "IMO# 0 wereaccepted during the 0879s in many countries by local authorities\ national organizationsand shipowners[ At the same time\ requirements concerning greater ship power connectedwith reduced weight and better economy tended to increase noise levels 1[ These oftencon~icting requirements led to extensive research and development work\ producingaccumulated knowledge and possibilities for putting into practice low cost acousticallye.cient solutions 2[ Noise prediction in the ship context requires consideration of thenoise sources\ the transmission paths along the ship|s hull and the receiver spaces[Structure!borne sound transmission\ being able to carry sound to remote places\ is oftenthe main factor[ Its contribution can be predicted by using semi!empirical or analyticalmethods\ such as the wave guide method or the SEA method[

Semi!empirical methods "see\ e[g[\ references 38# used to estimate the structure!bornesound transmission in a ship require data on the most important noise sources\ includingdiesel engines\ gears\ propellers\ bow thrusters\ as well as on isolation measures\ e[g[\resilient mountings\ compliant layers and ~oating ~oors[ In these methods the estimation

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P[ HYNNA ET AL[473

of the decrease in vibration level along a ship|s hull is based on measured data[ It can beseen as a set of model rules able to transform the acoustics of previous ships into that offuture similar vessels 3[ Often\ prediction is made in octave bands with centre frequenciesfrom 20=4 Hz to 7 kHz[ For a ship which is near the statistical average used in thecalculation\ standard deviations between measured and predicted data are a maximum3=4 dB for octave bands or 2=4 dB for the A!weighted sound pressure level inaccommodation spaces 4[ The other approach is to use a statistical model of multipleregression\ in which many descriptive variables\ obtained mainly by measurementsonboard ships\ are used to predict noise spectra in the superstructure of a ship 09[ Thestandard deviation of this method is 2=93=4 dB for octave bands with centre frequenciesfrom 52 Hz to 1 kHz\ thus assuring a prediction accuracy of the order of 257 dB in the84) con_dence interval 09[ Also the optimum linear parameter estimation theory hasbeen successfully tried for predicting the air!borne noise in ship superstructures 00[ Ingeneral\ semi!empirical methods can give good results for future similar ships\ but theycan fail when predicting noise levels for new ship types because they have not the abilityto predict the e}ect of structural changes[

The waveguide model 01\ 02 can be used when estimating the structure!borne soundtransmission in the vertical section of a ship structure[ The methods 01\ 02 have beendeveloped further to include frequencies below the cut!o} frequency\ typically 099 Hz ina ship\ of the waveguide system with use of the Galerkin variational method 03[ Thus\by using these two methods\ noise prediction can be made in the octave bands with centrefrequencies from 20=4 Hz to 7 kHz[ The propagation of structure!borne sound in thelongitudinal direction is estimated in these methods by using experimentally determinedtransmission loss data "see reference 03#[ The relative importance of ~exural andlongitudinal waves in power transmission was discussed in reference 01[ The results ofthis study showed that the main power transmission was by ~exural waves in the shipstructure[ This means also that conversion to other wave types during propagation is ofless importance[ The accuracy of predicted A!weighted sound pressure levels in cabins isin most cases better than 22 dB and the standard deviation can be expected to be lessthan 0=4 dB 03[ With this method the relative importance of the various noise sourcesand structures can be evaluated[ It is also possible to determine the e}ects of structuralchanges in constructions[

SEA began to develop in 0848\ when Lyon and Smith independently made their _rstcalculations concerning power ~ow and response of linearly coupled resonators 04[ In0857 Budrin and Nikiforov proposed a method based on energy balance betweensubstructures to calculate the structure!borne sound transmission in a ship 05[ This was\according to reference 5\ the _rst attempt to apply SEA in the ship context[ Sawley wasamong the _rst to apply the SEA method to ship noise problems 06[ After 0869 a rapidperiod of development followed with new SEA applications] see\ e[g[\ references 0711[The _rst applications were to structures and scale models typical of a ship[ Thereafterlarger ship structures were analyzed and\ also\ special problems were considered in moredetail] see\ e[g[\ references 1215[ In addition\ a model of a whole ship containing 216elements was used to predict structure!borne sound transmission when predicting thesound pressure level in cabins 16[ To data\ the largest model reported has contained 0536elements and 6385 coupling branches 17[

Up to now\ e}ective methods of modelling a large steel structure for SEA applicationsor the FEM!type assemblage of the loss factor matrix and solution of the SEA equationhave not been presented[ The _rst aim of the work described in what follows here has beento reduce the amount of work required to model a large steel structure of a ship for SEAcalculations and to make possible applications for large structures comprising thousands

STRUCTURE!BORNE SOUND IN SHIPS 474

of elements[ The second aim has been to test how the SEA method works in the shipcontext when large models and even low frequencies are used[ The modelling method andthe equations of SEA parameters used are described in detail[ In addition\ applicationcalculations for a hydrographic echo sweeping vessel\ a timber!container carrier\ and apassenger cruise vessel are discussed\ and a comparison is made with measured results[ Themethods used to reduce computer time are also brie~y discussed[

1[ SOME BASIC CONCEPTS OF SEA

1[0[ GENERALIn the SEA method\ systems are considered to be divided into subsystems\ which are

linearly coupled together\ and which exchange energy via resonant vibration modes[ Inmany applications geometrical structural elements are the subsystems\ while sometimesdi}erent wave types\ e[g[\ bending or longitudinal waves\ form the subsystems[ The netamount of energy input to each subsystem\ which comes from external excitation forcesand from couplings with other subsystems\ equals the energy dissipated in this subsystem[In addition\ there are several basic assumptions 04] there is linear coupling between thesubsystems^ energy ~ows between oscillator groups via resonant vibrations in the frequencyband considered^ the oscillators are excited by broadband random excitations^ each of themodes is equally energetic within a frequency band in a subsystem^ and the ~ow of energybetween subsystems is proportional to the modal energy di}erence[

1[1[ POWER BALANCE EQUATIONSFor the two groups of oscillators or for the two!element SEA model "see Figure 0#\ where

elements are excited by statistically independent broadband sources\ the power balanceequations are 04

Pin 0 vh0E0 vh01E0 vh10E1\ Pin 1 vh1E1 vh10E1 vh01E0\ "0\ 1#where Pin i is the time!averaged input power to element i\ v is the angular frequency ofthe band of interest\ hi is the internal loss factor of element i\ hij is the coupling loss factorfrom element i to element j and Ei is the time!averaged total energy "sum of potential andkinetic energy# of element i[ In matrix form the power balance equations "0# and "1# canbe presented as

v$h0 h01h01 h10h1 h10%6E0E176Pin 0Pin 17[ "2#An important relationship in SEA is the reciprocity relationship 04\

hijni hjinj \ "3#

where hij is the coupling loss factor from element i to element j\ and ni is the modal densityof element i\ respectively for hji and nj [

In the steady state single!frequency vibration of an individual oscillator\ the input powerhas to be in balance with the power dissipated[ The time!averaged power dissipated Pd is

Figure 0[ A two!element SEA model[

P[ HYNNA ET AL[475

related to the time!averaged sum of the kinetic and the potential energy E stored in theoscillator via the damping 18\

Pd cvx 1 1zvnmx 1 1zvnEvnE:QvnEh\ "4#where cv is the viscous damping coe.cient\ x is the vibration velocity\ z is the ratio of theviscous damping coe.cient to the critical viscous damping coe.cient\ vn is the naturalangular frequency\ m is the mass of the oscillator\ E is the time!averaged stored energy\Q is the quality factor\ and h is the internal loss factor[ The concept can be extended toa collection "group# of oscillators in a frequency band 18] i[e[\

Pd vE:QvEh\ "5#where v1pf\ f is the centre frequency of the band\ and h is now the mean loss factorof the modes within the band[

In the general case of N oscillator groups\ there are N simultaneous power balanceequations[ These can be presented in matrix form\ where the loss factor matrix issymmetrical owing to the reciprocity relationship "3# as 18\ 29

K LF J0h0 sNi$ 0 h0i1n0 h01n0 h02n0 = = = h0Nn0 E0n0G GG GG GG GG GG Gh10n1 0h1 sNi$ 1 h1i1n1 h12n1 = = = h1Nn1 E1n1G Gj fG GJ Fv [[

[[[[

[[[G GG G

G GG GhN0nN = = = 0hN sNi$N hNi1nN ENnNG GG Gk lf jP0P1

gG

G

G

G

F

f

[[[hG

G

G

G

J

j

\ "6#

PN

where Ei and Pi are the time!averaged total energy and the input power of the elementnumber i\ respectively[ Noise analysis by using equation "6# requires that the angularfrequency v\ the input powers and the SEA parameters "the modal densities\ the internalloss factors and the coupling loss factors associated with oscillator groups#\ are known[These oscillator groups are subsystems\ which are suitably selected "see\ e[g[\ references04\ 18#[ In general\ weak coupling between them "hij W hi and hj # is required[ In addition\it is assumed that most of the energy exchange is by resonant modes of subsystems[ Solvingequation "6# at the angular frequency v concerned\ one obtains the time!averaged energiesE of di}erent oscillators[

1[2[ SEA PARAMETERS

1[2[0[ Modal densityThe mode count N\ which is the fundamental quantity\ is\ for a subsystem\ the numberof modes of that subsystem that resonate in the frequency band Df considered[ It maysometimes be estimated as a product of a modal density\ n" f #\ and the frequency

STRUCTURE!BORNE SOUND IN SHIPS 476

bandwidth Df[ As a derived quantity\ the modal density is de_ned as the number of modesper unit frequency 04[ Sometimes it is given per unit angular frequency as n"v#\ thesebeing related by

n" f #1pn"v#[ "7#

Theoretically derived modal densities are available in the literature "see\ e[g[\ references04\ 2921# for idealized structural elements[ Here only the analytical modal densities forthe structural elements used in the program package are given[

Modal densities of uniform beams in ~exural vibration are given by 04\ 21

n" f #L:1pcB {L:"1pf #0:1}"rS:EI#0:3\ "8#

where L is the length of the beam\ f is the frequency\ cB "1pfkbcL #0:1 is the phase velocityfor ~exural waves\ kb "I:S#0:1 is the radius of gyration\ cL is the longitudinal wave velocityfor the beam\ r is the density of the material\ S is the cross!sectional area of the beam\E is Young|s modulus of elasticity and I is the second moment of area of the beam|scross!section[

Modal densities n" f # of uniform ~at plates in ~exural vibration are given by 04

n" f #Sz01:1cLh\ "09#

where S is the surface area of the plate\ h is the thickness of the plate and cL is thefrequency!independent\ quasi!longitudinal phase speed in the plate 21\ 22

cL E:r"0 n1#0:1\ "00#

where E is Young|s modulus of elasticity\ r is the density of material and n is the Poissonratio[ For steel one obtains from equation "00# cL 4169 m s0 by using the valuesE10900 N m1\ r6=7092 kg m2 and n9=17^ for other materials see references21\ 22[ The phase speed of longitudinal waves cL is frequency independent\ whereas thephase speed of bending waves depends on frequency and is thus dispersive[ For a thin plateof thickness h the phase speed is obtained from 21\ 22 cB "1pf #0:1"B:rh#0:3 1 "0=7cLhf #0:1\where the bending sti}ness BEh2:01"0 n1#[ For example\ for a steel plate of thickness09 mm the phase speed cB 87 m s0 when f099 Hz\ and cB 209 m s0 whenf0 kHz[ Equation "09# is used also for a triangular plate element\ because modal densityis an additive property 04 and directly proportional to the surface area of the plateelement[

Modal densities n" f # of a three!dimensional acoustic volume element are given by 04

n" f # "3pf 1V:c2# "pfS:1c1# "l:7c#\ "01#

where f is the frequency\ V is the volume of the element\ c is the speed of sound in the~uid "air#\ S is the total wall area of the element and l is the total edge length of the element[The density of air and the sound velocity in air at the standard air pressure 23 at a givenCelsius air temperature and at a given relative humidity are calculated according toreferences 24\ 25[ This kind of accuracy is not of great importance\ although subroutinesneeded in other calculations were utilized also in this SEA program[

The mode count N will not deviate very much from the estimate as a product of a modaldensity n" f # and the frequency bandwidth Df\ if the modal density is great enough or thebandwidth is wide enough for the mode count estimate to be at least ten modes or so 04[If a subsystem has a small mode count\ it may still be modelled as a SEA subsystem\ eventhough the concept of modal density is inappropriate in this case 12\ p[ 087[ McCollumand Cuschieri have shown in a previous paper that SEA provides a good approximationto the mean level of the energy transmission or response even in those frequency regions

P[ HYNNA ET AL[477

where the modal density is low and the response of the structure is dominated by a fewresonances 15[ In building acoustics it has been found for plates that the con_dence limitsderived by Lyon 04 for SEA predictions considerably overestimate the actual error 26[However\ comparison with measured results gives the con_dence limits of estimates[ Inbuilding acoustics it was found that if the error of 2 dB in the prediction of soundtransmission was taken as a limit\ then the requirement would be for the mode countN n" f #Dfq 9=4 and for the modal overlap M fhn" f #q 9=3 for plates in one!thirdoctave bands\ where h is the total loss factor 26[ As to the walls and ~oors of buildingsthe modes have a large bandwidth\ which typically may be half a one!third octave bandat 099 Hz[ Thus there is a physical meaning to 9=4 modes per band\ since a mode will oftenhave a signi_cant response reaching to more than one contiguous band 26[ This alsomeans that there are interacting modes of subsystems in contiguous frequency bandsexchanging energy even when the modal density Q0[ In the ship context the globalbehaviour of a large ship hull extends to frequencies of about 0904 Hz[ Thereaftersubstructures\ such as decks and bulkheads\ are excited into resonant vibrations[ Thedrastic increase in modal density for the entire structure in a ship hull structure happensat a frequency of about 49 Hz 10\ p[ 098[

The requirement that mode count should be about ten means that su.ciently largestructural and acoustic elements should be selected and that the frequency bandwidthconsidered should be wide enough[ For example\ if it is required that the modecount within the band N n" f #Df09\ this means that the area of a plate element fromequation "09# S4z01cLh:"2Df # should be 15[4 m1 at the 49 Hz one!third octave bandand 5=7 m1 at the 52 Hz octave band for a 09 mm thick steel plate[ If in a SEA model thelength of a plate element is eight frame spacings of 9=5 m then the breadths should be 4=4 mand 0=3 m\ respectively[ These breadths are usually easily obtainable in practice in themodelling[ The bandwidth between the upper and lower limiting frequencies has thefollowing values] Df1 9=12fm for a one!third octave band and Df1 9=60fm for an octaveband _lter\ where fm is the mid!band frequency of the one!third octave or octave band[When the height between decks is 1=6 m\ then the acoustic volume element withV4=43=71=6 m2 will have n" f #1 9=04 modes:Hz\ giving N0=6 in the 49 Hzone!third octave band[ This is clearly a small number\ although acoustic volume elementsare used if the air volume is modelled or if the air!borne input power must be taken intoaccount[

When the calculation includes low frequency bands\ the mode count within a bandclearly is less than ten for many subsystems used in the modelling[ This is a relaxation ofthe Lyons| previous requirement 04[ However\ theoretically calculated estimates of modaldensity for ideal structural elements are used in the program package instead of measuredvalues[ The estimates of sound pressure or vibration velocity levels thus obtained shouldbe compared with measured values\ as has been done in building acoustics 26[ In practice\many other simpli_cations are also made during the modelling of a ship|s steel structurewith ideal triangular or quadrilateral plate and beam elements[

1[2[1[ Internal loss factorsThe loss factor is proportional to the ratio of energy dissipated per cycle to the

time!averaged energy stored 04^ see equation "4#[ Sometimes it is de_ned as the phaseangle of a complex Young|s modulus of elasticity 21\ 04[ The internal loss factor of astructural element includes several di}erent damping or energy!loss mechanisms[ Com!monly accepted forms of linear damping are structural "hysteretic or viscoelastic# dampingand acoustic radiation damping 18[ In practice\ other non!linear damping mechanismsare also present at the structural joints[ These include gas pumping\ squeeze!_lm damping

STRUCTURE!BORNE SOUND IN SHIPS 478

and frictional forces 18[ The internal loss factor of a structural element forming part ofa built!up structure is given by 18

h hs hrad hj \ "02#

where hs is the structural loss factor\ hrad is the radiation loss factor and hj is the loss factorassociated with energy dissipation at the boundaries of the structural element[ Typically\engineering structures are lightly damped and 1=4093 Q hQ 4=9091 18\ 22[ Formost structures h tends to decrease with frequency\ roughly as f0:1 22[ Theoreticalestimates of loss factors are not generally available for structural elements[ In practice\measured values are used "see\ e[g[\ references 04\ 18\ 21\ 2739#[ In this context it is veryimportant to know the measurement conditions[ This means that one must know whatcomponents are included in equation "02#[ For many engineering structures hj is zero andhrad may or may not be included[ When only hs is required\ it is measured in a vacuum[This point is not always clearly stated when measured results are presented\ although forvery thin plates the radiation loss factor is of the same order as the internal loss factor[An approximate expression for the internal loss factor of the steel plate\ where thedissipation due to sound radiation is excluded\ is given in reference 30\

h9=30f9=6\ "03#

where f is the frequency in Hz^ from equation "03# h9=905 when f099 Hz\ andh9=9921 when f0 kHz[ The radiation loss factor is given by 04

hrad r9cs:vrs \ "04#

where r9 is the ~uid density\ c is the speed of sound in the ~uid\ s is the radiation ratioof the structure\ v is the centre frequency of the band and rs is the surface mass of thestructure[

The internal loss factor for an acoustic volume element is obtained from 31

h"c:pf #ai "S:7V# ln "0 a #\ "05#

where c is the speed of sound in air\ f is the frequency\ ai is the total pure tone atmosphericabsorption coe.cient "nepers per metre#\ S is the total surface area of the volume element\V is its volume and a is the average sound absorption coe.cient of the con_ning surfacesof the element\ excluding the e}ects of sound transmission through the surfaces[ Theatmospheric absorption coe.cient "dB per metre# is calculated by using the equationspresented in reference 32[ Some representative values at a pressure of one standardatmosphere "090=214 kPa# when the air temperature is 19>C and relative humidity is 49)are ai 9=355 dB:099 m when f0 kHz\ and ai 05=0 dB:099 m when f09 kHz[Equation "05# includes the atmospheric absorption and the absorption at the con_ningsurfaces of the volume element[ The atmospheric absorption is small\ especially at lowfrequencies\ and often it is not necessary to take it into account[

1[2[2[ Coupling loss factorsThe coupling loss factor\ hij \ is related to energy ~ow from subsystem i to subsystem

j[ Theoretical expressions for couplings between di}erent structural element types areavailable] see\ e[g[\ references 04\ 20\ 21[ The coupling loss factor of a beam cantileveredto a ~at plate is given by 33

hbp "1rbcLbkbSb #1

vmb b ZpZp Zb b1

Re "Z0p #\ "06#

P[ HYNNA ET AL[489

where hbp is the coupling loss factor from the beam to the plate\ rb is the density of thebeam material\ cLb is the longitudinal wave speed in the beam\ kb "Ib :Sb #0:1 is the radiusof gyration of the beam\ Ib is the second moment of area of the beam|s cross!section\ Sbis the cross!sectional area of the beam\ v is the angular frequency\ mb is the beam mass\and Zp is the moment impedance of the plate\ given by 33

Zp 05rsk1pc1Lp:v"0 iG#\ "07#

where rs is the surface density of the plate\ kp h:z01 is the radius of gyration for theplate cross!section of thickness h\ cLp is the longitudinal wave speed in the plate\ iz0\and 34

G"3:p# ln "0=0:kpa#\ "08#

where kp is the bending wavenumber of the plate and a is the e}ective distance of the pairof point forces making up the moment on the plate[ For rectangular and circular beamcross!sections\ respectively\ ar d:2 and ac 9=48r\ where d is the side dimension of therectangular beam cross!section "in the direction of bendings# and r is the radius of thecircular beam cross!section[ The moment impedance of the beam is 33

Zb rbc1LbSbk1bkbv0"0 i#\ "19#

where rb is the density of the beam material\ Sb is the cross!section and kb is the ~exuralwavenumber of the beam[ The coupling loss factor hpb from plate to beam is calculatedby using the reciprocity relation "3#[

The coupling loss factor for a line junction\ which is often encountered in a weldedstructure such as a ship\ is obtained from 04\ 21

hij 1cBiLtij :pvSi \ "10#

where cBi is the bending wave velocity "or phase velocity# of ~exural waves in the _rstsubsystem\ L is the length of the coupling line\ tij is the power transmission e.ciencycorresponding to the wave type and the type of junction in question\ v is the centrefrequency of the band and Si is the surface area of the subsystem i[ Transmissione.ciencies tij for pure ~exural waves of direct incidence are calculated by using theequations presented in references 21\ 07[ The equations for normal incidence of ~exuralwaves to the straight!line junction presented in reference 07 for a right!angle junction\a T!junction and a right!angle cross!junction between plates are used for calculation inthe program package because of their simplicity[ Of course\ more exact equations caneasily be implemented should it be necessary to obtain better accuracy[

The coupling loss factor hij from the structural element i to the acoustic volume elementj is given by 04

hij r9cSis:vmi \ "11#

where r9 is the ~uid density\ c is the speed of sound in the ~uid\ Si is the surface area ofthe structural element i\ s is the radiation ratio of resonant vibration\ v is the centrefrequency of the band and mi is the mass of the structural element i[ In the other direction\the reciprocity relation "3# is used[ The radiation ratio is calculated in the program packagedeveloped according to Maidanik 35[

The coupling loss factor between acoustic volume elements is 04

hij cStij :3vVi \ "12#

where c is the speed of sound in the ~uid\ S is the surface area of the panel between spaces\tij is the sound intensity transmission coe.cient "sound reduction index R09 log "0:t##

STRUCTURE!BORNE SOUND IN SHIPS 480

from the source volume element i to the receiving volume element j\ v is the centrefrequency of the band and Vi is the volume of the acoustic volume element i[ Thetransmission coe.cient tij for a single simply supported panel between acoustic volumeelements is calculated by using the SEA method as presented in reference 36[ This givesbetter agreement with measured results than the mass law at frequencies near and higherthan the coincidence frequency of the panel[ Even measured results could be used\ but itwould not be e}ective from the calculation point of view[ When two volume elements areconnected by an open aperture\ then equation "12# is valid with tij 0[ Di}erent edgeconditions of panels can be taken into account by using the knowledge that\ at lowfrequencies\ the radiation resistance of a clamped panel should be twice that of a simplysupported panel\ as Maidanik has stated 35[

1[3[ ENERGY AND INPUT POWERWhen the SEA equation "6# is solved at the given angular frequency v\ one obtains the

time!averaged energies E of di}erent oscillators[ The time!averaged energy can be relatedto other quantities] e[g[\ the energy of the acoustic volume element in a di}use _eld isrelated to the mean squared sound pressure by 04

E p1V:rc1 p1rmsV:rc1\ "13#

where p1 is the mean squared pressure\ prms is the root!mean!squared "r[m[s[# pressure\ Vis the volume of the acoustic volume element\ r is the ~uid density\ c is the speed of soundand means space averaging[ In practice\ space averaging is needed because subsystemssuch as a reverberant volume are not ideal[ The time!averaged energy of a structuralelement is related to its vibrational velocity as 04

Emv1\ "14#

where v1 is the mean squared vibration velocity averaged over the element surface andm is the mass of the structural element[

The air!borne input powers of sound sources are derived by using measured valuesobtained by standardized methods based on sound pressure measurements "see\ e[g[\reference 37# or on sound intensity measurements] see\ e[g[\ reference 38[ Whenmeasured values are not available\ semi!empirical estimation formulas may be used] see\e[g[\ references 5\ 6[ The sound power level of a diesel engine can also be estimated byusing the sound pressure level it produces 49[ These air!borne input powers are used forthe acoustic volume elements where the sound sources are located[ In the ship context\these include the air!borne sound powers of the main and auxiliary engines in the machineroom and of other noise sources] for example\ ventilation fans in the ventilation machinerooms[

The structure!borne sound power is di.cult to evaluate and no standardized methodsare available 40[ However\ research workers have developed semi!empirical andtheoretical methods which may be used to obtain approximate values of structure!borneinput powers] see\ e[g[\ references 6\ 4146[ Usually the source strength is described bymeasuring the vertical vibration velocity at the machine footing or at the shell platingimmediately above the propeller[ This is insu.cient\ as the power transfer also dependson the mobility properties of the seating structure and on machine footings[ Also\ thedriving point impedance of the shell plating above the propeller a}ects the power transferfrom the propeller!induced pressure _eld into the shell plating[

In this program package the propeller input power is estimated by using semi!empiricalmethods or by using the power spectral density calculated for full scale from a model test[Measured values have better accuracy\ and hence are preferred if available[ The power

P[ HYNNA ET AL[481

spectrum density is used to estimate the velocity levels of the shell plating above thepropellers by using the theory of the response of a ship|s hull to acoustic pressure in waterinduced by the propeller 47[ The input power is then obtained approximately by usingthe driving point impedance Z for an in_nitely thin isotropic plate 21\

P 01=v =1 Re "Z#\ "15#

where v is the vibration velocity and Z\ which now is real\ is given by 21

Z7Eh2:01"0 n1#0:1"rh#0:1\ "16#

where E is Young|s modulus of elasticity\ h is the plate thickness\ n is the Poisson ratioand r is the material density[

2[ MODELLING THE STEEL STRUCTURE

In the earlier applications\ the amount of work needed to model the structure was foundto restrict the applicability of SEA for large structures such as ships[ Since in FEM verylarge steel structures are modelled by using the preprocessor programs\ an idea wasdeveloped to apply the geometrical modelling typical of FEM with SEA 48[ This _rstapproach showed that it was possible to proceed in this way by de_ning the geometry withthe aid of nodes and elements[ Later\ the method was further developed to include theindexing technique of FEM\ the skyline matrix and bandwidth optimization] that is\ theloss factor matrix was assembled as in FEM 16[ Subsequently\ beam and triangular plateelements have been added\ and intermediate nodes allowed for elements[ These intermedi!ate nodes need not be in the middle of sides of elements as in FEM[ Moreover\ calculationalgorithms were changed to allow for distorted elements[ In this way\ the modelling maybetter follow the real geometry of the steel structure[ Also bandwidth optimization basedon geometry has been added[ The following variable!number!nodes elements "see Figure1# are implemented] two! to three!noded beam element\ four! to nine!noded quadrilateralplate element\ three! to nine!noded triangular plate element and eight! to 15!nodedacoustic volume element[ At least the key nodes must be non!zero[ Intermediate nodes give

Figure 1[ Variable!number!nodes elements used in the SEA program package] "a# one!dimensional beamelement^ "b# two!dimensional quadrilateral plate element^ "c# two!dimensional triangular plate element^"d# three!dimensional acoustic volume element[ W\ Key nodes^ w\ intermediate nodes[

STRUCTURE!BORNE SOUND IN SHIPS 482

~exibility in modelling work when di}erent element types are connected in common nodes[The triangular plate element is obtained by letting one side of a plate element collapse toa point[ This is shown by giving the same node to two corners of a plate element "seeFigures 1"b# and 1"c##[

The geometry of the steel structure is de_ned with nodes and elements[ The elementboundaries are selected to join with the structural junctions[ The side length of elementsusually used is foureight frame spacings[ Only the main features of a ship are modelled\including the double bottom with longitudinal and transverse girders\ bulkheads\supporting pillars\ hull plating and decks excluding girders and small sti}eners[ Thesupporting pillars\ which are often used in passenger cruise vessels\ are modelled by usingbeam elements[ The acoustic volume elements are used in the machine room and in theother spaces where air!borne input power is brought into the structure[ Otherwise\ volumeelements are not used between decks\ their contribution to sound transmission as air!bornesound being negligible compared to the structure!borne sound transmission[ In addition\this reduces the modelling work and the size of the model[ After the geometry of the steelstructure is de_ned with the aid of nodes and elements\ the program determines thecouplings between elements[ The program also calculates the geometrical quantities andother properties of elements and couplings[

3[ ASSEMBLAGE OF LOSS FACTOR MATRIX

The symmetrical SEA equation "6# is similar to the equilibrium equation obtained inthe structural analysis when FEM is used 59\

K{U} {F}\ "17#where K is the structure sti}ness matrix corresponding to the loss factor matrix Ys \ {U}is the displacement vector corresponding to {Ei :ni}\ and {F} is the applied force vectorcorresponding to {Pi} in SEA[ In FEM programs the global sti}ness matrix is not directlyformed[ Rather\ element sti}ness matrices are formed separately and assembled into theglobal sti}ness matrix[ This is symbolically written as 59

K si

K"i#\ "18#

where K"i# is the sti}ness matrix of the ith element[ In the SEA application the assemblageof the symmetrical loss factor matrix Ys is obtained from

Ys Y"d# si

Ys "i#\ "29#

where the summation goes over all coupling branches\ which correspond to elements inFEM\

Y"d# diagonal hiini \ hii hi \ "20#

and the equation for one coupling branch is

Ys "i# $ hijnihjinj hijnihjinj % hijni$ 00 00%[ "21#4[ PRACTICAL IMPLEMENTATION

In the current implementation of the SEA program\ as many subroutines of a FEMprogram 50 as possible\ 49 altogether\ have been used[ The SEA program part itself now

P[ HYNNA ET AL[483

contains 48 subroutines and about 8399 lines of FORTRAN 66 code "without commentlines#\ and about 03 999 lines in all[ At the moment the FEM program package containsabout 024 999 lines of code including comments and the SEA part[

The FEM!type approach requires a list of coupling branch elements[ The assemblageof local loss factor matrices into the global loss factor matrix is made with the aid of thislist[ Before the list is made\ bandwidth optimization based on geometry is applied toarrange the elements into suitable order[ This usually results in so dense a loss factor matrixthat bandwidth optimization is not necessary[ Geometrical optimization also reduces thetime required to determine the couplings between elements\ because elements can beconnected to each other within certain distances in space[ Here bandwidth optimizationbased on geometry is made in the longitudinal direction of the ship[

Since the loss factor matrix is symmetric\ only the upper half of the matrix needs to bestored[ For e}ective storage and fast computation it is useful to store only the minimumnumber of zero matrix elements[ One way of storing this kind of matrix is the skylinetechnique\ which is commonly used in FEM programs 59[ The loss factor matrix isbanded especially if large structures are modelled[ When using the skyline technique\the solution of equation "6# requires about 01nm

1K operations\ where n is the order of the

matrix and mK is the average half!bandwidth "see reference 59#[ The loss factor matrixis not really inverted\ only its LDLT!factorization is determined by using Gausselimination[

The overall solution proceeds as follows[ After some initial checks are made\ e[g[\ fortriangular and quadrilateral plate elements\ a check is made to ensure that all nodes arein the same plane within a given tolerance[ Also the geometrical dimensions of elementsare checked to be within tolerances[ After checks\ the average global co!ordinates ofelements are calculated and elements are sorted into suitable order\ that is geometricaloptimization is made[ Then coupling branches are determined[ Subsequently\ thefrequency!independent properties of the elements and coupling branches are calculated[Then the frequency!dependent properties are calculated in one!third octave bands withcentre frequencies from 49 Hz to 09 kHz[ Thereafter the global loss factor matrix isassembled and the energies per modal density of elements are solved[ The program packagecontains an error handler that is used to check that the calculated values are within theallowed range during the computer run[ If a discrepancy is observed\ this gives an errormessage with location information[

5[ SOUND PRESSURE LEVELS IN ACCOMMODATION SPACES

Both the air!borne and structureborne sound transmission\ as well as local soundsources\ are included in the prediction of sound pressure levels[ The air!borne sound isincluded as sound power levels both for space elements and for cabins[ For structure!bornesound\ the sound radiated into the space is determined by the velocity levels\ radiationratios and dimensions of the structures facing the cabin[ The calculation follows the schemepresented in references 3\ 03[ The velocity levels of a ship|s steel structure are determinedafter solving the SEA equation "6#[ Thereafter measured or semi!empirically estimatedtransmission loss spectra are used to obtain the velocity levels of the cabin surfaces[ Byusing measured or estimated sound radiation ratio spectra for cabin surfaces\ thecontribution of structure!borne sound radiated into a cabin is calculated[ The soundabsorption is estimated by using measured or estimated sound absorption spectra for everysurface instead of using the average 9=4 s reverberation time typical of cabins[ A _xedreverberation time does not allow alternative design calculations for di}erent space typesincluding hard walled spaces[ The sound pressure level in a space is calculated by using

STRUCTURE!BORNE SOUND IN SHIPS 484

di}use _eld approximation in octave bands with centre frequencies from 52 Hz to 7 kHz\while in addition the A!weighted sound pressure level is determined[

6[ APPLICATION EXAMPLES

6[0[ HYDROGRAPHIC ECHO SWEEPING VESSEL

The SEA program was applied for the _rst time to a hydrographic echo sweeping vesselbuilt by Rauma!Repola "now part of Finnyards Ltd# in Finland "see Figure 2#[ The maindimensions of the ship are length overall 21=8 m\ width 09=3 m and draught 1=7 m[ Thepropulsion is produced by a pair of main engines\ each generating 119 kW when therotational frequency is 0499 min0\ coupled through reduction gears to two controllablepitch propellers rotating at 269 min0[ Accommodation is for 08 persons in 01 cabins[ Atthe shipyard the input data for the SEA calculation was prepared manually\ de_ning thesteel structure with the aid of nodes and elements[ The whole steel structure was modelledwith 130 plate elements and 75 space elements 48\ supplement[ The coupling branches\in all 0037\ were calculated by the SEA program[ In the design stage the shipyardindependently calculated the sound pressure levels by using empirical methods andmeasured these during a sea trial[ The practical implementation of the program versionwas discussed and the SEA estimates were compared with measured and empiricallycalculated values 48\ supplement[ The mean absolute di}erence between SEA estimatesand measured A!weighted sound pressure levels LpA was "3=02 0=0# dB "x 2 s:zn# wheneight spaces of di}erent type were evaluated[ In the octave bands with centre frequenciesfrom 52 Hz to 7 kHz the mean absolute di}erence varied between 2=2 and 7=4 dB[ Theindividual error curves and the arithmetic mean error are presented in Figure 3[ The meanarithmetic error for LpA was 1=0 dB and the standard deviation 4=9 dB[

Figure 2[ The hydrographic echo sweeping vessel[

P[ HYNNA ET AL[485

Figure 3[ SEA estimate compared with measured octave band sound pressure level Lp on board thehydrographic echo sweeping vessel[ **\ The eight individual curves^ * *\ arithmetic mean value[ Themeasurements were made by the shipyard[

6[1[ TIMBER!CONTAINER CARRIERThe second version of this SEA program package was developed to include the skyline

technique and bandwidth optimization which are widely used in similar problems in FEM[The improved SEA program was applied to analyze a timber!container carrier "see Figure4# with the main dimensions length overall 020 m\ width 19 m and draught 5=7 m 16[ Theaccommodation was located in the afterbody of the ship[ The model of the steel structurewas prepared with the aid of the preprocessing program[ The model of the whole steelstructure of this ship comprised 1334 elements[ The SEA program determined 6829coupling branches between these elements[ The ship was built in Spain at the shipyardAstilleros Reunidos Del Nervion\ S[A[ The shipyard independently measured the soundpressure levels on board the ship during a sea trial[ During the trial the ship was in theballast condition and the main propulsion machinery was running at the maximumcontinuous rating "74)#[ The meteorological conditions were good] sea state and windforce varied from 0 to 1 on the Beaufort scale "17 March 0889#[ The estimated soundpressure levels were in reasonable agreement with measured results[ The mean absolutedi}erence x between SEA estimates and measured A!weighted sound pressure levels LpAwas 4=42 9=6 dB "x 2 s:zn# when 14 spaces of di}erent type were evaluated[ Theindividual error curves in the octave bands with centre frequencies from 52 Hz to 7 kHzand the arithmetic mean error are presented in Figure 5[ For these 14 spaces the arithmeticmean error for LpA was 0=8 dB and the standard deviation 5=3 dB[

6[2[ PASSENGER CRUISE VESSELThe latest version of the program package has been applied to the passenger cruise vessel

Crown Jewel with the main dimensions length overall 053 m\ draught 4=3 m\ and mouldedbreadth 11=4 m "see Figure 6#[ The number of passenger decks is eight and the number ofcabins 309[ The ship was built in Spain at the Shipyard Union Naval de Levante\ S[A[ Thewhole steel structure of this ship was modelled for vibration analysis by using FEM[ Thissame model\ with some alterations\ was used for this SEA calculation[ The model wasgenerated by using the preprocessor[ Many simplifying assumptions were made to reducethe number of elements and modelling work[ The SEA model consisted of 4032 elements[These included 2096 plate elements\ 0246 triangular plate elements\ 500 beam elements and57 acoustic volume elements[ Not all of the beam elements were necessary for the SEA

STRUCTURE!BORNE SOUND IN SHIPS 486

Figure 4"a#[ The timber!container carrier Igor Ilinskiy^ "b# the SEA model of the timber!container carrier[

Figure 5[ SEA estimates compared with measured octave band sound pressure levels Lp on board thetimber!container carrier Igor Ilinskiy[ **\ The 14 individual curves^ * *\ arithmetic mean value[ Themeasurements were made by the shipyard[

P[ HYNNA ET AL[487

Figure 6"a#[ The passenger cruise vessel Crown Jewel^ "b# the SEA model of the passenger cruise vessel*theviewing angle is selected so that the decks are distinguishable[

calculation[ In the node mesh used to de_ne the geometry of the steel structure with theseelements were 6390 nodes[ The couplings between elements\ 06 338 altogether\ weredetermined by the program[ In the model real material thicknesses were used in accordancewith the steel structure[

The propulsion machinery consisted of four main engines\ four auxiliary engines andtwo highly skewed propellers[ For the main engines and the auxiliary engines the measuredvibration velocity level under resilient mountings was available[ These _gures weretransformed to power levels with the aid of the point mobility "see section 1[3#[ Theair!borne input powers were obtained by using sound pressure levels measured at adistance of 0 m[ The propeller input power was obtained by using the power spectraldensity calculated for full scale from a model test by using equations "15# and "16#[ Theair!borne noise of ventilation fans in the ventilation rooms on di}erent decks was includedin the calculation model[ The structure!borne noise of resiliently mounted fans wasestimated to be negligible compared to the noise of other sources[ The air!borne soundpower into the engine casing was taken into account and estimated with the estimatedsound pressure levels[

The acoustical data on cabins and other accommodation areas was obtained partly frommeasured data and partly from semi!empirical data[ The local air!borne sound power ofair ventilation units in cabins and other areas was included as measured values[

6[2[0[ Estimated sound pressure levelsThe sound pressure levels on board the passenger cruise vessel Crown Jewel were

estimated at 85 places on di}erent decks where it was assumed to be necessary to checkthe conformity with the requirements agreed[ Clearly\ the in~uence of structure!bornesound transmission was seen in the calculated estimates\ both in the horizontal and in thevertical directions[ In the cabins facing the engine casing and the ventilation rooms highernoise levels\ as expected\ were estimated owing to the in~uence of more structure!borne

STRUCTURE!BORNE SOUND IN SHIPS 488

sound[ The spaces located near propellers or above the machine rooms showed highervalues\ as expected[ Also\ the relative estimated di}erences between the con_ning placeswere in agreement with experience[ When making repeated computer runs\ the e}ectivenessof some noise abatement measures was estimated[ Whatever the absolute accuracy of theestimates\ the relative di}erences provided good guidance during design[ The shipyardindependently measured the sound pressure levels on board the ship during sea trials inJune and July 0881[ During the trials the main propulsion machinery was running at themaximum continuous rating "74)#[ The meteorological conditions were good duringtrials[ In June the wind force was 3 and sea state 2 "moderate sea# on the Beafort scale[In July the wind force and sea state were both 1[ The ship was not fully completed duringtrials[ In some cases the measurements were not made in the places for which the soundpressure levels were estimated[ Hence only 11 measurement points could be evaluated[The mean absolute di}erence between SEA estimates and measured A!weighted soundpressure levels LpA was 2=92 9=4 dB "x 2 s:zn# when 11 spaces of a di}erent typewere evaluated[ The individual error curves in the octave bands with centre frequenciesfrom 52 Hz to 7 kHz and the arithmetic mean error are presented in Figure 7[ For these11 spaces the arithmetic mean error for LpA was 0=5 dB and the standard deviation was2=3 dB[

6[2[1[ Calculated vibration velocity level di}erence between decksThe performance of this SEA method in predicting the vibration velocity level di}erence

between decks in the vertical direction was tested numerically by exciting only one elementin the double bottom under one of the main engines "see Figure 8#[ The shape of the inputpower spectrum was similar to that of the main engine spectrum\ but its level was increasedto obtain reasonable response[ The selection of the place of the input element alsosimulates the situation faced in practice when one applies a semi!empirical method[ Thecalculated vibration level di}erences DL between decks vary considerably "see Figure 09#[Vibration is more easily transmitted from the double bottom up to the _rst deck viaprimarily vertical bulkheads surrounding the machine room and shell platings\ rather thanin the horizontal direction along the complicated double!bottom structure\ where thetransmission loss per frame is higher "see Figure 00"b##[ The transmission path was veri_edby comparing the calculated vibration velocity levels of the elements concerned[ Thissame explanation is also valid concerning the di}erence between the double bottom and

Figure 7[ SEA estimates compared with measured octave band sound pressure levels Lp on board the passengercruise vessel Crown Jewel[ **\ The 11 individual curves^ * *\ arithmetic mean value[ The measurements weremade by the shipyard[

P[ HYNNA ET AL[599

Figure 8[ A cross!section of the SEA model of the passenger cruise vessel showing the element "darkened#used for the power input when the horizontal and vertical vibration transmission properties were numericallytested] viewed from behind double bottom\ engine casing\ location of main engines\ and only decks 0\ 1\ and2^ elements of the engine foundation are removed for clarity[

deck 1[ The vibration energy is more evenly distributed over longer distances\ becausestructure!borne sound travels around the discontinuities along many di}erent paths in thestructure[ This is clearly seen when one looks at the averaged vibration level di}erencesper deck between the double bottom and deck 1 or 6 "see Figure 09#[

The decrease in vibration level TL in dB along a ship|s hull according to thesemi!empirical method of Janssen and Buiten is given by 3

TLm00=99=46 for machineryfor propeller 1 n0 4\101:n\ if nQ 3if ne 31\ "22#where m is the number of transverse frames between the source and the receiver\ and nis the number of the deck on which the receiver is located[ If the source is on deck number0 or 1\ instead of the tanktop\ the corresponding 4 or 09 dB must be subtracted[ Equation"22#\ based on empirical data\ gives a di}erence of 4 dB per deck when nE 3\ 3=3 dB whenn4\ and thereafter a little less[ The accuracy decreases rapidly when m exceeds 14 3[The same type of result is obtained with this model in the vertical direction[ The largevibration level di}erences\ up to 04=6 dB per deck\ are explained by the fact that the pillarssupporting decks in large public areas and in some accommodation areas transmit verylittle ~exural vibration[ It seems that the calculation method is able to predict reasonablythe di}erences caused by structural details in vibration transmission[ The results have not

Figure 09[ Calculated vibration level di}erences DL "dB# at octave bands between the decks of the passengercruise vessel Crown Jewel\ when noise was input into only one element in the double bottom at frame no[ 55[Average DL between two decks at frame numbers 09 "w#\ 49 "e#\ 029 "t#\ and 069 "T#^ average DL betweendouble bottom and deck 0 at frame number 029 "Q#^ average DL between double bottom and deck 1 at framenumbers 49 "r# and 029 "R#\ average DL between double bottom and deck 6 at frame numbers 49 "E#\ 89 "q#and 029 "W#[ "Frame spacing9=54 m[#

STRUCTURE!BORNE SOUND IN SHIPS 590

Figure 00[ The calculated horizontal transmission loss TLH of structure!borne sound at octave bands"dB:frame# along the decks of the passenger cruise vessel Crown Jewel\ when noise was input into one elementin the double bottom at the frame number 55[ Reference elements were selected from equivalent positions justright of the centreline on the starboard side "frame spacing9=54 m#[ "a# Values shown are TLH across a totalof 7 frames "q#\ 21 frames "r#\ 61 frames "w# and 019 frames "e#^ also shown are values calculated fromequations "23a\ b# "equivalent black symbols#[ "b# Average TLH along double bottom "Q#\ deck 0 "W#\ deck 1"R#\ deck 2 "E#\ deck 3 "q#\ deck 4 "t#\ deck 5 "r#\ deck 6 "e# and deck 7 "w#[

been veri_ed by vibration measurements\ as the ship was manufactured abroad and thelaboratory could not a}ord these measurements[

6[2[2[ Calculated decrease in vibration velocity level along decksThe same test run as in section 6[2[1 was used to obtain the vibration velocity level

decrease horizontally along the double bottom and decks "see Figure 00#[ The averagedecrease in vibration velocity level in the double bottom was 0=6 dB per frame at the 52 Hzoctave band\ decreasing thereafter to the value of 0=2 dB per frame at the 7 kHz octaveband[ The mean frame spacing in this ship is 9=54 m[ Near the source\ the decrease in thedouble bottom is almost twice the decrease on the decks[ The structure of the doublebottom is very complicated and there are many small sized structural elements formingmore junctions\ as compared to those for the decks[ The bulkheads bounding the machineroom\ the engine casing\ ventilation rooms\ stairs and lift wells from discontinuities in thevibration transmission path[ The values on decks 02 are fairly near the empirical valueof 0=9 dB per frame given by equation "22# for machinery[ On deck 2 between frames 47and 89 there is the engine casing and ventilation room^ otherwise\ the deck is supportedby pillars[ This means that in the structure near the centreline there are only straight!through connections and connections with pillars[ This is seen as a lower decrease value[Thereafter the vibration energy is distributed again more evenly[ The behaviour is similaron decks 37[ These numerically obtained values can be compared with those used in the

P[ HYNNA ET AL[591

waveguide method\ where the transmission loss TLH in dB in the horizontal directionacross a total of m frames at frequency f based on full!scale measurements is given by03\ 51

TLH 8 mb0\19b0 "m19#b1\19b0 29b1 "m49#b2\ 19EmE 19mE 49me 499 "23a#where

b0 9=6\ b1 9=099=050 log f\ b2 9=9219=941 log f[ "23b#

The horizontal transmission loss TLH calculated from equations "23a# and "23b# gives9=6 dB per frame for the _rst 19 frames[ The values in Figure 00 are in close agreementwith the empirical data of equations "23a# and "23b#[ One considerable di}erence is thatwhen using this SEA method\ the values are\ in contrast\ lower when the frequency isincreasing[ This may be due to excessively low damping at higher frequencies because noacoustic volume elements were used\ excluding the machine room\ engine casing andventilation rooms[ The other reason is that the contributions of the in!plane compressionaland shear mode types were ignored[ They have an e}ect when structure!borne soundpropagates over large distances 52[ However\ this is not of great practical importance\because just the low frequency sound is usually the main factor when estimating soundpressure levels in the ship context[ In addition\ in the places far away from noise sources\the local noise sources dominate] e[g[\ air distribution devices in cabins[ One should alsobear in mind that the test ship here was a very special passenger cruise ship\ whencomparing the estimated values with those obtained empirically or from semi!impiricalformulas[ In the waveguide method\ the transmission loss of structure!borne sound isdetermined analytically in a vertical ship cross!section between the same two frames[ Thehorizontal transmission loss is determined by using experimentally determined values[ Thisis because most noise problems are encountered in spaces just above the machine roomsor the propeller\ and it is of less importance to predict accurately a noise level in a placefar away horizontally if it is known to be below any noise requirements 51[

6[3[ GENERAL REMARKS ON SOUND PRESSURE LEVEL DIFFERENCESIn the SEA method the following quantities may be erroneous] input power estimates

of structure!borne or air!borne sound of sound sources\ the SEA parameters\ the simpli_edmodel of a steel structure\ and the simpli_ed sound wave _eld model\ including here onlythe ~exural waves[ The conventional acoustic part used to estimate the sound pressurelevels in the accommodation includes erroneous estimates] e[g[\ sound transmission lossbetween steel structure and cabin surfaces\ sound radiation ratios\ sound absorption andlocal air!borne input powers[ Errors due to these erroneous input data may partly canceleach other out[ Di}erences in the _nal construction from the drawings according to whichthe calculations were made may have some e}ect[ The e}ects of workmanship are alsore~ected in the measured sound pressure levels[

One di.culty is to obtain high quality input data\ which is essential for the accuracyof the estimates[ This is especially true for structure!borne input powers 40 or di}erentkinds of machinery and propellers[ The acoustical properties of any noise abatementmeasures should be known[ These include\ e[g[\ vibration!damping properties of~oating!~oor constructions\ sound radiation ratios of panels\ sound!absorption properties\and insertion loss of resilient mountings of wall and ceiling panels used[ Measured valuesare preferred\ if available\ rather than those estimated by using semi!empirical or empiricalmethods[

STRUCTURE!BORNE SOUND IN SHIPS 592

During sea trials the ship is usually not yet _nished\ and hence not all of the pointsincluded in the estimates are measured or included[ Thus the measured di}erences betweenthe SEA estimates and measured sound pressure levels re~ect the in~uence of all possibleerrors in the whole estimation scheme[ The mean absolute di}erences of this SEA methodare of the same order at octave bands from 014 Hz up to 7 kHz\ while at the lowest bandof 52 Hz the mean absolute di}erence is about 1 dB larger[

6[4[ COMPUTER TIMEThe SEA model of the passenger cruise vessel which contains 4032 elements and 06 338

coupling branches has been calculated using three computers[ The charged CPU "centralprocessor unit# time\ including the reading of input data and all the subsequentcalculations\ was about 7 h 34 min\ while the elapsed real time was about 02 h on a MicroVaxII[ Subsequently\ the program package was installed on a UNIX!based workstation\Iris Indigo[ The equivalent CPU time was now about 27 min and the elapsed real time wasabout 32 min[ With the workstation Iris Indigo R3999 the equivalent CPU time was05=2 min and the real time 05=3 min[ The given times depend on the device con_gurationand on the computer load during the run time and\ as such\ give only an indication ofrequired computer resources and capabilities[ These _gures can be compared with the CPUtime of 2 h 7 min in a Micro VaxII which elapsed during the calculation with the SEAmodel of the timber!container carrier containing 1334 elements and 6829 couplingbranches[ Additional information on the computational e}ectiveness of the skylinetechnique with bandwidth optimization is presented\ e[g[\ in references 16\ 59[

7[ DISCUSSION

With the applications it was shown that the principles of modelling the geometry of alarge welded structure such as those in FEM are also applicable when the SEA approachis used[ It was also shown that the methodology typical of FEM for the assemblage andsolving of the SEA equation is very e}ective[ The geometrical optimization for reducingthe bandwidth of the loss factor matrix also proved to be e}ective in the SEA programpackage developed for ship applications[ It also reduces the computation time when thecouplings between elements are determined[

In the special applications considered\ the SEA method proved to be applicable to theanalysis of the structure!borne sound transmission in large ship structures\ even whenmany approximations were made during modelling of the structure[ Comparison withsemi!empirical estimation methods showed that the SEA method was able to predictreasonably the structure!borne sound "vibration# transmission loss when only thesubsystem of ~exural sound waves was included[ This supports the _ndings of Nilsson|sstudy 01] i[e[\ that the main power ~ow is by ~exural waves in a ship structure[ In theSEA method the prediction is made both in the horizontal and in the vertical directionof propagation\ whereas in the waveguide method the sound transmission in the horizontaldirection is estimated semi!empirically with data based on measured results[ It is obviousthat the SEA method is able to predict the relative in~uence of structural changes anddetails on structure!borne sound transmission[

The fact that modal densities n" f # are low is often seen as a problem[ This may be partlycircumvented by using su.ciently large elements\ as must be done for the economicalmodelling of large structures[ If a subsystem has a small mode count\ it may still bemodelled as a SEA subsystem\ even though the concept of modal density is inappropriatein this case 12[ SEA provides a good approximation to the mean level of the energytransmission or response even in those frequency regions where the modal density is low

P[ HYNNA ET AL[593

and the response of the structure is dominated by a few resonances 15[ In buildingacoustics it was found that if the error of 2 dB in the prediction of sound transmissionwas taken as a limit\ then the requirement would be for the mode count N n" f #Dfq 9=4and for the modal overlap M fhn" f #q 9=3 for plates in one!third octave bands\ whereh is the total loss factor 26[ The results of this paper also support the fact that inengineering calculations the SEA calculation can be extended to lower frequencies despitelow modal densities[ However\ thorough research is required to establish the uncertaintyin the applications of the SEA method at low frequencies\ where there are only a fewinteracting modes per frequency band[

The importance of in!plane vibration transmission has been emphasized by manyauthors "see\ e[g[\ references 12\ 52\ 14#\ although so far\ in many applications of the SEAmethod\ only relatively small models have been used[ In the applications considered in thispaper\ the SEA method worked reasonably even for the large model of the passenger cruiseship without the inclusion of the in!plane vibration transmission[ For engineering purposesit seems feasible to develop further this program package\ including more developed theoryand new element types for new applications[ Implementing acoustic volume elementswhere the ~uid is a liquid would allow the inclusion of the e}ects of liquid storage tanksand the water surrounding the ship|s hull in the model 13[ However\ it is worth notingthat the modelling accuracy must be adequate for describing the phenomena considered[As such\ the SEA program package developed can very well be used in the estimation ofsound pressure levels in the ship context for octave!band frequencies from 52 Hz to 7 kHz[The mean absolute di}erence between estimated and measured LpA was in the range of2=94=4 dB "44 measurement points# when three di}erent types of ship were analyzed[General limits of accuracy cannot be given based on these limited statistics[

The SEA approach is especially applicable to new ship types when the structure!bornesound transmission and the e}ects of structural details are to be estimated[ This methodestimates only the transfer function from the point of excitation to the receiving point[However\ problems are encountered when the structure!borne input powers aredetermined by using the usually available information on measured vibration velocities atthe machine foundations or at the footings measured either at the workshop or in situ onboard the ship[ The accuracy of input power estimates is of vital importance for thecalculated SEA estimates of vibration or sound pressure levels[ After the vibration levelsof the steel structure are obtained by using the SEA method\ the sound pressure levels inaccommodation spaces are estimated by using the same methods as in the wave guidemethod or in semi!empirical methods[ Semi!empirical methods are cheap to apply and aresuitable if scant data is available\ as in the early design stage[ Hence the choice betweenmethods depends on the design stage as well as on the data available[

ACKNOWLEDGMENTS

The authors are grateful to Mr M[ Vahteri who worked at Rauma!Repola|s RaumaShipyard and helped in the preparation of input data of the hydrographic echo sweepingvessel[ They also thank Mr P[ Olkinuora of Finnyards Ltd for his consent to publish dataand for the photograph of this ship[

The authors would like to express their gratitude to Mr G[ Asuar and to the personnelof the shipyard Astilleros Reunidos del Nervion\ S[A[ in Spain for their permission topublish the results of the SEA calculations and sound pressure level measurements of thetimber!container carrier[ The authors acknowledge the assistance provided by Mr P[ Arkkefrom Deltamarin\ Ltd "part of Elomatic Ltd# in Finland in the preparation of data forthe SEA calculations of this ship[

STRUCTURE!BORNE SOUND IN SHIPS 594

The authors would like to express their gratitude to Mr J[ Poblet and to the personnelof the shipyard Union Naval de Levante\ S[A[ in Spain for their patient help during thepreparation of data for the SEA calculations of the passenger cruise vessel[ For themeasured results they also deserve considerable thanks[ The authors owe thanks to MrH[ Bache r and the personnel of E}John International in Finland for valuable help duringthe SEA calculations and for their permission to publish the data[

The authors are indebted to Mr Jyrki Kullaa Lic[ Tech[\ Mr Sauli LiukkonenM[Sc["Tech[#\ Mr Jukka Airaksinen M[Sc["Tech[#\ Mr Kai Katajama ki M[Sc["Tech[# andMr Jari Kivela from VTT for preparing the model of the passenger cruise vessel with thepreprocessor and for help during the programming and computer runs[ The authors Klingeand Vuoksinen specialize in the _nite element method and have written the computer codeof the FEM package[ Mr Hynna is responsible for the acoustic part as well as for the SEAprogram code[ The _nancial support provided by the Division for ManufacturingTechnology of VTT is gratefully acknowledged[

REFERENCES

0[ INTERNATIONAL MARITIME ORGANIZATION 0871 Noise Levels on Board Ships] Code on NoiseLevels on Board Ships and Recommendation on Methods of Measuring Noise Levels at ListeningPosts[

1[ D[ MAJUMDER and H[ GOETHE 0872 Schi} + Hafen 24\ 2326[ Statistical investigations into noisesituation in deep!sea shipping "in German#[

2[ H[ F[ STEENHOEK 0877 ICMES |76\ Proceedings of the Fourth International Symposium on MarineEngineering Systems {{Application of Technological Advances||[ Stockholm\ Sweden] Inter!national Cooperation on Marine Engineering Systems "ICMES#[ Marine engineering and noisecontrol[

3[ J[ H[ JANSSEN and J[ BUITEN 0862 Proceedings of INTER!NOISE 62[ On acoustical designingin naval architecture[

4[ J[ BUITEN 0866 Proceedings of the International Symposium on Shipboard Acoustics 0865[Amsterdam] Elsevier[ Experiences with structure!borne sound transmission in sea!goingships[

5[ J[ PLUNT 0879 Doctoral Thesis\ Chalmers University of Technology\ Gothenburg\ Sweden[Methods for predicting noise levels in ships[ Experiences from empirical and SEA calculationmethods\ part I] noise level prediction methods for ships\ based on empirical data[

6[ R[ W[ FISHER\ C[ B[ BURROUGHS and D[ L[ NELSON 0872 The Society of Naval Architects andMarine Engineers\ Technical and Research Bulletin 226[ Design guide for shipboard airbornenoise control[

7[ Handbook for Shipboard Airborne Noise Control 0863 U[S[ Coast Guard Technical Publication962!9099[

8[ Guide to Ship Noise Control 0875[ Tokyo] Nippon Kaiji Kyokai[09[ E[ SZCZERBICKI and A[ SZUWARZYNSKI 0872 Archives of Acoustics 2\ 008029[ Prediction of

octave noise spectra in accommodations in the superstructure of a ship[00[ P[ CALCAGNO\ R[ MALTESE and F[ PINAZZI 0875 Shipboard Acoustics\ Proceedings of the 1nd

International Symposium on Shipboard Acoustics\ ISSA |75[ Dordrecht] Martinus Nijho}[Applications of two mathematical approaches to predict airborne noise levels is shipsuperstructures[

01[ A[ C[ NILSSON 0866 Journal of Sound and Vibration 44\ 6080[ Attenuation of structure!bornesound in superstructures on ships[

02[ A[ C[ NILSSON 0867 Journal of Sound and Vibration 50\ 3459[ Reduction of structure!bornesound in simple ship structures] results of model tests[

03[ A[ C[ NILSSON 0873 Journal of Sound and Vibration 83\ 300318[ A method for the predictionof noise and velocity levels in ship constructions[

04[ R[ H[ LYON 0864 Statistical Energy Analysis of Dynamical Systems] Theory and applications[Cambridge\ Massachusetts] MIT Press[

05[ S[ V[ BUDRIN and A[ S[ NIKIFOROV 0858 Joint Published Research Service Translation\ JPRS36[792[ Propagation and damping of sound vibrations on ships[

P[ HYNNA ET AL[595

06[ R[ J[ SAWLEY 0858 Proceedings of the Symposium on Stochastic Processes in Dynamical Problems[New York] American Society of Mechanical Engineers[ The evaluation of a shipboard noise andvibration problem using statistical energy analysis[

07[ J[ ODEGAARD JENSSEN 0866 Proceedings of the International Symposium on Shipboard Acoustics0865[ Amsterdam] Elsevier[ Calculation of structureborne noise transmission in ships using thestatistical energy analysis approach[

08[ Y[ IRIE and S[ TAKAGI 0867 Proceedings of INTER!NOISE 67[ Structure!borne noisetransmission in steel structure like a ship[

19[ K[ FUKUZAWA and C[ YASUDA Proceedings of INTER!NOISE 68[ Studies on structure!bornesound in ship[

10[ J[ PLUNT 0879 Doctoral Thesis\ Chalmers University of Technology\ Gothenburg\ Sweden[Methods for predicting noise levels in ships[ Experiences from empirical and SEA calculationmethods\ part II] prediction of structure!borne sound transmission in complex structures withthe SEA method[

11[ T[ YOSHIKAI\ K[ HATTORI and T[ SATO 0870 Proceedings of INTER!NOISE 70[ Prediction ofnoise level on board ship using statistical energy analysis[

12[ J[ TRATCH JR[ 0874 M[Sc[ "Eng[# Thesis\ Massachusetts Institute of Technology[ Vibrationtransmission through machinery foundation and ship bottom structure[

13[ Y[ SHIMOMURA 0874 Ocean Engineering Thesis\ Massachusetts Institute of Technology[ The e}ectof a liquid storage tank on sound transmission through ship structures[

14[ R[ H[ LYON 0875 Noise Control Engineering Journal 15\ 1116[ In!plane contribution tostructural noise transmission[

15[ M[ D[ MCCOLLUM and J[ M[ CUSCHIERI 0889 Journal of the Acoustical Society of America 77\03790374[ Bending and in!plane wave transmission in thick connected plates using statisticalenergy analysis[

16[ P[ HYNNA\ P[ KLINGE and M[ NIEMINEN 0877 ICMES |76\ Proceedings of the Fourth InternationalSymposium on Marine Engineering Systems {{Application of Technological Advances||[Stockholm\ Sweden] International Cooperation on Marine Engineering Systems "ICMES#[Statistical energy analysis with _nite element model for noise prediction in ships[

17[ K[ IINO and I[ HONDA 0881 Proceedings of the 4th International Symposium on Practical Designof Ships and Mobile Units\ PRADS |81\ vol[ 0\ 05370550[ "J[ B[ Caldwell and G[ Ward\ editors#[London] Elsevier Applied Science[ Total noise prediction system for a passenger cruise ship[

18[ M[ P[ NORTON 0878 Fundamentals of Noise and Vibration Analysis for Engineers[ Cambridge]Cambridge University Press[

29[ F[ J[ FAHY 0871 in Noise and Vibration "R[ G[ White and J[ G[ Walker\ editors#\ 054075[Chichester] Ellis Horwood[ Statistical Energy Analysis[

20[ F[ D[ HART and K[ C[ SHAH 0860 National Aeronautics and Space Administration Report NASACR!0662[ Compendium of modal densities for structures[

21[ L[ CREMER\ M[ HECKL and E[ E[ UNGAR 0862 Structure!borne Sound] Structural Vibrations andSound Radiation at Audio Frequencies[ Berlin] Springer!Verlag[

22[ F[ FAHY 0874 Sound and Structural Vibration] Radiation\ Transmission and Response[ London]Academic Press[

23[ ISO 1422[ Standard atmosphere[24[ G[ S[ K[ WONG and T[ F[ W[ EMBLETON 0874 Journal of the Acoustical Society of America 66\

06090601[ Variation of the speed of sound in air with humidity and temperature[25[ G[ S[ K[ WONG 0875 Journal of the Acoustical Society of America 79\ 01920193[ Characteristic

impedance of humid air[26[ R[ J[ M[ CRAIK\ J[ A[ STEEL and D[ I[ EVANS 0880 Journal of Sound and Vibration 033\ 84096[

Statistical energy analysis of structure!borne sound transmission at low frequencies[27[ J[ C[ SUN and E[ J[ RICHARDS 0874 Journal of Sound and Vibration 092\ 098006[ Prediction of

total loss factors of structures\ I] theory and experiments[28[ J[ C[ SUN\ H[ B[ SUN\ L[ C[ CHOW and E[ J[ RICHARDS 0875 Journal of Sound and Vibration 093\

132146[ Prediction of total loss factors of structures\ part II] loss factors of sand!_lled structure[39[ H[ B[ SUN\ J[ C[ SUN and E[ J[ RICHARDS Journal of Sound and Vibration 095\ 354368[ Prediction

of total loss factors of structures\ part III] e}ective loss factors in quasi!transient conditions[30[ Y[ IRIE and T[ NAKAMURA 0874 Bulletin of the Marine Engineering Society in Japan 02\ 5961[

Prediction of structure borne sound transmission using statistical energy analysis[31[ S[ UOSUKAINEN and K[ PESONEN 0872 Kari Pesonen Consulting Engineers Ltd\ Structure!borne

sound research\ Research Report 0[ The calculation of structure!borne and airborne sound ofindustrial buildings using the statistical energy analysis "in Finnish#[

STRUCTURE!BORNE SOUND IN SHIPS 596

32[ ISO:DP 8502:0 0875 Acoustics*Prediction of environmental noise*Part 0] calculation of theabsorption of sound by the atmosphere[

33[ R[ H[ LYON and E[ EICHLER 0853 Journal of the Acoustical Society of America 25\ 02330243[Random vibration of connected structures[

34[ I[ L[ VER and C[ I[ HOLMER 0877 in Noise and Vibration Control "L[ L[ Beranek\ editor#\ 169250[Washington\ D[C[] Institute of Noise Control Engineering^ revised edition[ Interaction of soundwaves with solid structures[

35[ G[ MAIDANIK 0851 Journal of the Acoustical Society of America 23\ 798715[ Response of ribbedpanels to reverberant acoustic _elds[ Erratum] 0864 Journal of the Acoustical Society of America46\ 0441[

36[ M[ J[ CROCKER and A[ J[ PRICE 0858 Journal of Sound and Vibration 8\ 358375[ Soundtransmission using statistical energy analysis[

37[ ISO 26392636[ Acoustics*Determination of sound power levels of noise sources[38[ ISO 8503!0[ Acoustics*Determination of sound power levels of noise sources using sound

intensity*Measurement at discrete points[49[ W[ HEMPEL and T[ SEIDL 0869 Motortechnische Zeitschrift "MTZ# 20\ 042045[ Statistic

investigations into diesel engine noises "in German#[40[ T[ TEN WOLDE and G[ R[ GADEFELT 0876 Noise Control Engineering Journal 17\ 403[

Development of standard measurement methods for structureborne sound emission[41[ A[ C[ NILSSON and N[ P[ TYVAND "editors# 0870 NORDFORSK\ Miljo va rdsserien Publication

0870] 1[ Noise sources in ships\ I] propellers[42[ J[ PLUNT and J[ ODEGAARD "editors# 0872 NORDFORSK\ Miljo va rdsserien Publication 0872] 1[

Noise sources in ships\ II] diesel engines[43[ B[ PETERSSON and J[ PLUNT 0879 Chalmers University of Technology\ Gothenburg\ Sweden\

Department of Building Acoustics Report 79!08[ Structure!borne sound transmission frommachinery to foundations[

44[ J[ W[ VERHEIJ 0871 Doctoral Thesis\ Delft University of Technology\ Delft\ The Netherlands[Multi!path sound transfer from resiliently mounted shipboard machinery[

45[ J[ BUITEN "editor# 0875 Shipboard Acoustics\ Proceeding of the 1nd International Symposium onShipboard Acoustics\ ISSA |75 "and Supplement#[ Dordrecht] Martinus Nijho}[

46[ R[ G[ WHITE 0871 in Noise and Vibration "R[ G[ White and J[ G[ Walker\ editors#\ 574601[Chichester] Ellis Horwood\ Vibration control "II#[

47[ J[ PLUNT and D[ MCQUEEN 0879 Acustica 36\ 0510[ A theory of the response of a ship|s hullto propeller!generated acoustic pressure[

48[ P[ HYNNA 0875 in Shipboard Acoustics\ Proceeding of the 1nd International Symposium onShipboard Acoustics\ ISSA |75\ 122132 "and Supplement\ 424439# "J[ Buiten\ editor#[Dordrecht] Martinus Nijho}[ A literature survey concerning propeller as a noise source andprediction methods of structure!borne noise in ships[

59[ K[!J[ BATHE 0871 Finite Element Procedures in Engineering Analysis[ Englewood Cli}s\ NewJersey] Prentice!Hall[

50[ M[ K[ HAKALA 0875 VTT Symposium 57\ The 3th Marine Technology Symposium\ ShipVibration\ Noise + Hydrodynamics\ 0203 January\ Espoo\ Technical Research Centre of Finland"VTT#[ Solution of ship vibration problems*current situation and future prospects[

51[ A[ C[ NILSSON 0872 Danish Acoustical Institute\ The Danish Academy of Technical SciencesReport no[ 092[ A method for the prediction of noise and velocity levels in ship constructions[

52[ R[ E[ POWELL and J[ E[ MANNING 0877 Proceedings of the 48th Shock and Vibration Symposium\SANDIA REPORT SAND77!1362C[ Albuquerque\ New Mexico] Sandia National Labora!tories[ The importance of non!resonant and in!plane vibration transmission in statistical energyanalysis[