Energy-based vibroacoustics: SEA and beyond

Ennes SarradjGesellschaft fur Akustikforschung Dresden mbH, D-01099 Dresden, Germany, Email: ennes.sarradj@akustikforschung.de

IntroductionToday, a number of possible approaches are available forthe analysis of vibroacoustic problems. To find a re-quired result with minimum effort and cost, the choice ofthe right method is very important. Here, an overviewabout one class of approaches shall be given. Theseapproaches may be characterised by the term ”energy-based”. In contrast to the classical analysis of vibration,that is based on quantities such as force and displace-ment, these approaches are based on energy quantities(energy, energy density, power, ...). It is in principle pos-sible to achieve the same information using either one orthe other approach. But, depending on the application,energy-based methods may have some advantages. Mostnotably,

• the power - energy relation is not so sensitive to smallparameter changes,

• energy quantities can be averaged more easily,

• most often, the actual goal of a computation areenergy quantities (e.g. the sound pressure level thatis an energy quantity).

To illustrate the last two statements, consider the simpleproblem (Table 1) that may be found in similar form inmost textbooks on acoustics. Of both possible solutions,the second one is more elegant as only the informationwanted is produced.

Problem:Compute the sound pressure level (SPL) in a cavity forgiven excitation and absorption.Solution 1:• set up wave equation• apply appropriate boundary conditions• solve the equation using an analytical or a numer-

ical method• result: sound pressure distribution• average to reduce unnecessary information over-

head• result: mean SPL Lp,m in cavity

Solution 2: (energy-based)• calculate input power (if not already given)• use: Lp,m = Lw − 10 lg A

A0• note that result is an expected value

Table 1: An illustration of different approaches

Statistical energy analysisToday the statistical energy analysis (SEA) is the mostfamous energy-based method. Although similar ap-

proaches were used before in room acoustics, the actualdevelopment of SEA started in the early 1960 with theapplication to vibroacoustic problems in aerospace en-gineering. ”Statistical” means, that the variables aredrawn from statistical population and all results are ex-pected values. ”Energy” denotes that energy variablesare used and, according to Lyon[1], ”Analysis” meanshere, that SEA is more a general approach rather than aparticular technique.

The main idea in SEA is that a structure is partitionedinto coupled ”subsystems” and the stored and exchangedenergies are analysed. The original theory is based on thestudy of interaction of groups of modes. While this con-cept is clever from a mathematical point of view, it is notvery practical to use it for a quick insight. Thus, in whatfollows mathematical developments are dropped for thebenefit of those interested in an overview on SEA. Thereader interested in more in-depth treatment is referredto a recent compilation[2] as starting point.

Some basics

Consider a single subsystem – a separated part of thestructure that is to be analysed. Any excitation actingon the subsystem can be characterised by the resultingpower input Pi into the subsystem (Figure 1, left). Ifpower is injected, the subsystem stores vibrational energyWi. In practice, there will be also a power loss Pii (e.g.due to dissipation). This power loss may be related tothe stored energy by the damping loss factor ηi by:

Pii = ωηiWi. (1)

Further, if the analysis is restricted to steady state, it isclear that power input equals power loss: Pi = Pii.

Consider now a second subsystem (Figure 1, right). Ifthis subsystem were coupled to the first, the same powerbalance would hold for both subsystems i and j, respec-tively. As a result of the coupling the subsystems sharetheir vibrational energies. Power flows from subsystem ito subsystem j. From the viewpoint of subsystem i, thispower flow Pij is a power loss. A power flow exists alsoin the reverse direction (Pji) that is as a result a powergain for subsystem j. To characterise these power flows,a special quantity, the coupling loss factor ηij , is used inSEA. It is defined similarly to the damping loss factor in(1):

Pij = ωηijWi. (2)

In Figure 2, the case of four subsystems is shown as anexample. This time, there is only a power input to sub-system 1. This power input is equal to the sum power

subsystem i

Pi

Wi

Pii

Wi

Pi

Pii

Wj

Pj

Pjj

Pij

Pji

Figure 1: left: A single subsystem; right: coupled subsys-tems

W1 W2

W3

W4

P1

P11

P22

P33

P44

P12

P13

P21

P31

Figure 2: Example of four coupled subsystems

losses (due to dissipation and coupling) for that subsys-tem minus the power gains coming from the subsystems2 and 3. The power balance reads then:

P1 =P11 + P12 + P13 − P21 − P31. (3)

Similar power balances may be established for the re-maining subsystems:

0 =P22 + P21 + P23 + P24 − P12 − P32 − P42, (4)0 =P33 + P31 + P32 + P34 − P13 − P23 − P43, (5)0 =P44 + P42 + P43 − P24 − P34. (6)

As there is no power input into these subsystems, the lefthand side vanishes in the equations.

The power quantities in (3)-(6) may be substituted by thedamping and coupling loss factors and the vibrationalenergy as in (1) and (2). The resulting equations maythen be written down together as a system of equations:

ω

η11 −η21 −η31 −η41

−η12 η22 −η32 −η42

−η13 −η23 η33 −η43

−η14 −η24 −η34 η44

W1

W2

W3

W4

=

P1

000

. (7)

In the example, subsystems 1 and 4 are not coupled andconsequently the corresponding coupling loss factors are

zero. The diagonal elements of the loss factor matrix in(7) are called the total loss factors as they are the sum ofall coupling loss factors that are associated with powerlosses for the respective subsystem:

ηii = ηi +∑j,j 6=i

ηij . (8)

The loss factor matrix in (7) can be rendered symmetricby introducing the modal densities (number of modes orresonances per frequency band) of the subsystems.

What is a subsystem?

So far, no detailed definition for a subsystem was givenhere. A subsystem can be seen as a part or physical ele-ment of the structure (”the system”) that is to be anal-ysed. To be modelled as a subsystem that part or ele-ment must be capable of vibrating quite independentlyfrom other elements. The word ”quite” is emphasisedhere because as long as the element is not separated fromthe rest of the structure its vibration is not truly inde-pendent. Next, it is required for a subsystem to vibratein resonant mode. That means, if the excitation is sud-denly switched off, the vibrational energy stored in thesubsystem will decay rather then drop to zero immedi-ately (a point mass for instance is no suitable candidatefor a subsystem). Thus, a reverberant sound field existswithin the subsystem. If different wave types exist in theelement, each of the corresponding sound fields is mod-elled by one subsystem. It should be noted in passingthat the original definition for a subsystem makes themathematical treatment easier: a subsystem is a groupof ”similar” energy storage modes.

To illustrate this definition, some examples of vibro/-acoustic elements that may be treated as subsystemsshall be given here together with the necessary input datato characterise them:

• an acoustic cavity (a room): longitudinal waves –only one subsystem needed, characterised by vol-ume, fluid parameters, absorption

• a plate: bending, compression and shear waves mod-elled by three subsystems, characterised by area,thickness, material parameters, damping

• a beam: four wave types – four subsystems, char-acterised by length, shape of cross-section, materialparameters, damping

• shells, non-isotropic plates, ...

It is worth to notice that the energy stored in the sub-system is related to measurable quantities such as soundpressure level. This fact enables the practical applicationof the purely energy-based SEA equations.

The statistics in statistical energy analysis

Tough SEA is called statistical, no explicit statistics canbe seen in the above SEA equations. For the simple the-oretical approach taken here, statistical operations con-sist of a threefold average that is more implicit. First,

W1 W2

W3

P1 P2

P3

P1: W1,W2,W3

P2: W1,W2,W3

P3: W1,W2,W3

Figure 3: Procedure of experimental SEA: power is injectedto each subsystem in turn

although not mentioned yet, calculation is done alwaysfor a frequency band (octave, third-octave, ...), but onlyat the centre frequency. This is an average of frequency.Next, only one variable is used to characterise the energyin one subsystem. This corresponds to a spatial averageof the subsystem. Finally, by using very few parame-ters to characterise a subsystem there is no possibilityto restore all information from these parameters that isnecessary to describe the vibrational behaviour in detail.For instance, only area, thickness, material parametersare used as input parameters for a plate. Consequently,as long as these parameters remain the same, the shape ofthe plate does not matter in SEA. A circular, a quadraticor a trapezoidal plate map all to the same SEA model.This kind of average is called ensemble average. In modalspace, all three averages are included in the average overa group of modes. It shall be noted in passing that be-sides the estimation of mean or expected values from theaverages the estimation of variance is also possible, butis not staightforward. Generally, the variance is accept-able only at mid to high frequencies. That is why SEAis often referred to as a high-frequency method.

Procedures of experimental and predictiveSEA

In order to apply SEA, an SEA model must be prepared.The most important step in doing so is to break up thestructure, which should be analysed, into subsystems.This is often difficult to perform because the underly-ing theoretical assumptions have to be met as good aspossible and at the same time a number of practical con-siderations must be taken into account. Moreover, fora given structure there are several models possible nor-mally, so the modelling is ambiguous. After deciding fora model, the frequency range and band widths of interestmust be identified. The mentioned steps are common tothe two modes of SEA which should be detailed in thefollowing.

In experimental SEA the main idea is to determine allquantities in (7) by experiment. To this end the follow-ing procedure is implemented: Power is injected to eachsubsystem in the structure in turn. This may be donefor instance by means of a hammer, a shaker or a loud-speaker. Then, each time the energy in each subsystem

is measured (by accelerometers or microphones). As de-picted in Figure 3, for each subsystem a set of energies innow available and the following equation can be set upusing (7):P1 0 0

0 P2 00 0 P3

=

ω

η11 −η21 −η31

−η12 η22 −η32

−η13 −η23 η33

W1 W1 W1

W2 W2 W2

W3 W3 W3

. (9)

The colours here correspond to those in Figure 3. By in-verting the matrix of energies, this system may be solvedto get the coupling and damping loss factors. In practicethe matrix of energies is often bad conditioned but a num-ber of methods have been developed to cope with that.Knowing now the matrix of loss factors, main paths ofpower may be identified, the effect of modifications maybe assessed and a sensitivity analysis may be performedto find those factors that are most important for a giventransmission scenario.

The basic idea in predictive SEA is to assess the couplingloss factors theoretically. Thus, it is possible to predictthe behaviour of a structure even in an early stage ofits design when no object is available for measurements.The procedure is as follows: First, the damping loss fac-tors are estimated either from measurement, from tables,from calculations or simply from ”experience”. Then,the input power has to be determined (by experimentor calculation). Alternatively, the input power is set tounity, if only transmission loss is of interest and not ab-solute response values. The coupling loss factors maybe calculated or assessed by several different techniquesdepending on the specific case of the coupling:

• from radiation or transmission efficiencies (wave ap-proach)

• using modal approaches

• using numerical methods (e.g. Finite ElementMethod)

• coupling power proportionality ηij = ηjinj/ni

(ni, nj are the modal densities of subsystems i and j)

• ...

With all necessary input parameters available, the SEAequations may be solved and the response of the structuremay be predicted. As in experimental SEA this enablesa number of useful possibilities for analysis.

In some cases, it is useful to mix experimental and pre-dictive SEA. However, great care must be taken as dueto the ambiguous modelling this will not always lead tothe expected results.

Applications of SEA

SEA is applicable for a great number of different prob-lems. In many cases it is the only alternative left that isable to deal with high-frequency vibro-acoustic problems.

room1

room2

back1

partition

back2

floor1

floor2

left1

left2

right1

right2

ceiling1

ceiling2

10

20

30

40

50

60

63 125 250 500 1000 2000 Hz

dB

Figure 4: left:Two rooms with side walls (light concrete,15 cm), floor and ceiling (heavy concrete, 19 cm) separated bya partition wall (light concrete, 10 cm), right: SPL differencebetween the two rooms: non-resonant transmission only (red),non-resonant and resonant (green), +flanking transmission(blue)

Its use as a tool for quantifying (dominant) transmissionpaths is also very efficient. To name a few areas of ap-plication for SEA, the following should be mentioned:building acoustics (handle flanking transmission !), ve-hicle interior noise (also for railway and ships), soundpackage modelling (efficient treatment of multi-path air-borne sound transmission and absorption), (turbulenceboundary layer induced) vibration of aircraft and launchvehicles.

To further illustrate the use of SEA, some examples shallbe given here. The first example[3] is concerned withflanking transmission in building acoustics. In Figure 4,a sketch of a double room arrangement with a partitionbetween the rooms is shown. This maps to an SEA modelwith 35 subsystems, two for the two rooms and threefor the partition and for each of the walls, ceilings andfloors. Three different stages of the model can be usedfor calculation:

• two subsystems (both rooms), only non-resonanttransmission (mass law) is considered

• three subsystems (both rooms and the bendingwaves in the partition), non-resonant and resonanttransmission

• all 35 subsystems, non-resonant, resonant and flank-ing transmission

The typical dip around the critical frequency of the par-tition (400 Hz) is clearly notable in the second case, seeFigure 4. In the last case the influence of the differ-ent critical frequency of the flanking wall manifests it-self through the additional dip around that frequency(250 Hz).

Another example is the prediction of structure-bornesound transmission in an untrimmed car body shell. Thistime the model consists of 280 subsystems. In Figure5 the beam and the plate subsystems are shown. The

model is used to predict the effect that modifications ofthe body have to the transmission of sound from an exci-tation at the engine mounts into the passenger compart-ment. The modifications used are intended primarily toshow the abilities of the method and consist of adding dis-tributed mass to firewall, floor and roof in turn. Thus,it is possible to measure the change of transmission be-haviour easily. In Figure 6, both measured and predictedresults are shown. While the prediction is good for thecase of the firewall and the roof, it seems to fail for thefloor. This may be due to insufficient modelling of thespace beneath the car. Typical for SEA, it is not easy toshed light on this because there are a number of possiblereasons.

The last example is intended to demonstrate one of thereal powers of SEA: to get an estimate very quickly. Con-sider the following problem: The wall thickness of theouter housing of a pump is to be changed from 4 mmsteel to 1.5 mm steel. Assess the impact on the radiatedsound (only structure-borne excitation needs to be takeninto account). Figure 7, left shows the simple SEA modelwhich can be set up and solved within minutes. The re-sult (Figure 7), right shows that although the vibrationlevel increases, due to shift of the critical frequency theradiated sound is less in a broad frequency range.

Advanced energy based methodsThe main motivation for advanced methods is that be-sides its many advantages SEA has a number of problemsand limitations. Without going to much into detail, afew of them should be named here. First, SEA theoryrequires a number of implicit assumptions to be valid,that are often not easy to meet in practice: weak cou-pling (this refers to the problem explained in the sectionon subsystems), damping should be not too low but alsonot too high, homogeneity of the subsystems is neces-sary to render the calculation of vibrational level fromthe energy valid, sound fields have to be reverberant anddiffuse, etc. Next, as may be guessed from the car bodyexample, SEA often requires high modelling expertise.Moreover, SEA delivers no information on local distribu-

Figure 5: Graphical representation of an SEA model of acar body shell: plate (top) and beam (bottom) subsystems

Figure 6: Predicted and measured result of modifications

Excitation

Plate

Cavity

Pin

-15

-10

-5

0

5

10

100 250 630 1600 4k 8k

dB

Hz

Figure 7: SEA model for the pump housing (left), result(right): vibration level (red) and sound pressure level (green)changes

tion vibration level within the subsystems (this becomesa problem, if the variance in the spatial distribution isto high). To close this list, the modelling approach isincompatible to classical FEM / BEM which makes thepractical application of SEA in some cases too costly.

A great number of methods that circumvent some ofthese limitations and problems have been developed, butup to now only on an academic level. The following listgives an (incomplete) overview together with the mainidea behind each method:

• Wave Intensity Analysis (WIA)[4]: like SEA, butfourier decomposition of wave field intensity in asubsystem - relaxes the diffuse field assumption ofSEA

• Energy Finite Element Method (EFEM)[5]: analogywith heat conduction, see below

• (Integral) Smooth Energy Model (SEM)[6] or HighFrequency Boundary Element Method (HFBEM):boundary integral formulation using energy vari-ables, see below

• Energetic Mean Mobility Approach (EMMA)[7]:suited for treatment of heterogenous structures, es-pecially for experimental application

• Complex Envelope Distribution Analysis(CEDA)[8]: analysis using a cepstrum calcu-lated from wavenumber spectrum rather than fromfrequency spectrum (not energy-based)

• Hybrid Methods: incorporate explicit ”modal” be-haviour of components into SEA-like models, e.g.[9]

Both the energy finite element method and the high fre-quency boundary element method shall be explained herein little more detail. They both start with the same fun-damental assumption that the principle of energy conser-vation may be used to formulate an equation of energycontinuity. In contrast to SEA not a subsystem but aninfinitesimal volume is considered. The amount of vi-brational energy stored in that volume will be governedby a) power losses Pdiss (due to dissipation), b) powerinjection Pin (due to external load) and c) energy trans-port cgW through the volume boundaries (energy flowper unit area). Thus, an equation for the time-derivativeof the energy may be set up:

∂

∂tW = −∇ · (cgW ) + Pin − Pdiss, (10)

which can be seen as an energy continuity equation. Re-formulated using power density and energy density, as-sume steady state and substitute dissipated power withthe damping loss factor the equation reads:

pin = ∇ · I + ωηw. (11)

This equation may used as a basis for calculating thespatial distribution of energy density w from the knowninput power density pin. Two different approaches existto provide the necessary relation between intensity I andenergy density needed to transform (11) into a solvableequation.

The first approach assumes that the wave field consistsof superposed plane waves only:

I = −c2g

ωη∇w. (12)

If substituted into (11), the resulting equation is analogto the equation for heat conduction in a plate includingconvection:

pin = −c2g

ωη∆w + ωηw, (13)

Pin = −λ∆T + α (T − TB) . (14)

Thus, a vibration conduction coefficient c2g

ωη as well as avibration convection coefficient ωη may be defined. (13)may be solved using finite element (FE) techniques (thusthe name EFEM). In particular, using the both coeffi-cients as input parameters, existing FE codes for heatconduction may be used for the calculation. While thisis an advantage, the method possesses a serious prob-lem: because the assumption of plane does not hold forwave fields dominated by spherical waves, it is not validin these cases. However, it is not fully clear how relevantthis is for practical application.

sourcepos.(2.5,1.5)

path

start ofpath

(0,0)

(0,2)(1,1)

(2,3)

(0.3,0.5)

(4,0)

(2.8,3)

Figure 8: Large steel plate; path is used to plot the resultin Figure 9

0 1 2 3 4 5 6 710-5

10-4

10-3

10-2

10-1

f=10 kHz, =1%

f=1 kHz, =1%

f=1 kHz, =0.1%

ener

gy d

ensi

ty (

J/m

2 )

path length (m)

EFEMHF-BEMSEA

Figure 9: SEA, EFEM and HFBEM results for the plate

The second approach, that results in the HFBEM, as-sumes that the wave field may be synthesised by sphericalwaves from sources together with spherical waves withorigins at the boundary of the wave field. As a conse-quence, the energy density at some location M is givenby integration over all (primary) sources S and over allboundary sources Q:

w(M) =∫

Ω

ρ(S)G(S, M)dS

+∫

∂Ω

σ(Q)f(uMQ,−→n Q)G(Q, M)dQ. (15)

Without going into much detail, it shall be pointed outthat from this, an integral equation for the unknownsource strengths σ of the sources at the boundary maybe formulated:

σ(Q) =%γn

γ

(∫Ω

ρ(S)H(S, Q)dS

+∫

∂Ω

σ(Q′)f(uQQ′ ,−→n Q′)H(Q′, Q)dQ′)· −→n Q. (16)

This equation can be solved by a boundary element ap-proach using a collocation technique. That means, thecontinuous function σ is approximated by a stepwise con-stant function and the second integral in (16) becomes asum. Thus, the equation can be transferred into a linearsystem of equations. Once this system is solved for thesource strengths, the spatial energy density distributioncan be calculated using the discretised form of (15).

A last example[10] shall demonstrate the capabilities ofboth EFEM and HFBEM. A large steel plate (Figure 8)with a single point source (e.g. a shaker) is considered.The results from SEA, EFEM and HFBEM are shown inFigure 9 for different frequencies and damping. They areplotted along the path shown in Figure 8. While the SEAprovides no information about the spatial distribution ofthe energy density, the other methods show a somewhathigher energy level in the vicinity of the source (≈5 m).For higher damping, EFEM fails to predict the directfield correctly. The HFBEM results are less reliable forenergy equipartition in the low damping case.

SummaryIn this paper, a short overview was given on the basicideas behind methods for the treatment of vibroacousticproblems that are based on energy variables. In particu-lar, theory and application of the most popular of thesemethods, the statistical energy analysis (SEA), were ex-plained. Main advantages were shown as well as the prob-lems and limitations. As example for methods that gobeyond the limits of SEA, a brief introduction into en-ergy finite element method (EFEM) and high frequencyboundary element method (HFBEM) was given.

References[1] R. Lyon. What good is statistical energy analysis,

anyway? The Shock and Vibration Digest, 2(6),1970.

[2] F. Fahy, editor. User guide to SEA. SEANET con-sortium, 2002. http://www.seanet.be.

[3] http://www.freesea.de.

[4] R. Langley. A wave intensity technique for the anal-ysis of high frequency vibration. J. Sound Vib.,159(3):482–502, 1992.

[5] I. Moens. On the use and the validity of the energyfinite element method for high frequency vibrations.PhD thesis, K.U.Leuven, 2001.

[6] A. Le Bot. Energy transfer for high frequencies inbuilt-up structures. J. Sound Vib., 250(2):247–275,2002.

[7] G. Orefice, C. Cacciolati, and J.L. Guyader. Theenergetic mean mobility approach. In F. Fahy andW. Price, editors, IUTAM Symposium on SEA,pages 47–58, 1997.

[8] A. Carcaterra and A. Sestieri. Complex envelopedisplacment analysis: a quasi static approach to vi-brations. J. Sound Vib., 201(2):205–233, 1997.

[9] R. Langley and P. Bremner. A hybrid method forthe vibration analysis of complex structural-acousticsystems. J. Acoust. Soc. Am., 105(3):1657–1671,1999.

[10] E. Sarradj. High frequency boundary integralmethod as an alternative to sea. In Proc. X. In-ternational Congress on Sound and Vibration, 2003.