ABSTRACT: Statistical Energy Analysis (SEA) has some special benefits in product concept evaluation, especially

when many relatively radical ideas have to be benchmarked against e.g. a proven design. The lack of detailed geo-

metry and CAE models are typical at this early development stage of a product. SEA models are simple and quick to

build to follow even rapid generation of ideas. SEA models only need a minimum of geometry parameters and reaso-

nable assumptions of basic material parameters to provide physically reasonable, quantitative relative predictions of

the different concept ideas. The major uncertainties at this stage are often the internal damping estimates for the

different parts. Alternatively SEA models may pinpoint the need to use high enough damping for certain parts.

This paper presents some relatively simple examples from the automotive, off-shore and shipbuilding industry. The

most comprehensive example is from radical concept development for light-weight cars. The other examples illust-

rate how SEA can be used successfully for early “risk assessment” and explanation of e.g. increased sound radiation

to interior spaces even at relatively low frequencies when rib-stiffened structures are redesigned to meet e.g. lower

weight demands.

KEY WORDS: Statistical Energy Analysis; SEA; Concept Development; Product development.

1 INTRODUCTION

There are examples of engineering challenges that require

radical rethinking in product concept development. For

example, to reach expected legal limits for road vehicle CO2-

emissions around 2020-2025, weight reductions of 30-40 %

may be necessary at least for conventional combustion engine

powered cars, to reach these goals without major increase of

cost and obtain as good NVH comfort as presently for

vehicles in the premium (luxury) segment.

Road noise (structure-borne and airborne) as well as wind

noise will determine the interior sound comfort also for cars

with electric powertrains (plug-in hybrids, EVs with range

extenders, fuel-cells etc.). Road- and wind noise issues may

therefore be even more pronounced unless addressed during

the conceptual design of new low weight hybrid/electric

vehicles.

Offshore structures and ships are other examples where

weight optimisation is prioritized. Structure-borne sound

sources with high levels are often situated close to accommo-

dation or office spaces. Use of dynamic solution of detailed

FE-models is usually restricted to low frequency of economic

reasons. They are also produced late in the engineering

process when CAD geometry is mostly determined. A couple

of examples are given where SEA-modelling can illustrate

noise radiation consequences of weight optimisation of rib

stiffened plate or shell structures when only strength is

considered.

2 STATISTICAL ENERGY ANALYSIS (SEA)

The Statistical Energy Analysis (SEA) was first developed

about 50-60 years ago to deal with prediction of noise and

vibration transmission in complex structures with multi-modal

response, the fundamental theory is given in [1, 2].

The name SEA was established in the early 1960's and

means:

Statistical, since the systems being studied are members

of populations of similar design having distributions of

their dynamical parameters.

Energy is the primary variable of interest. Dynamic

variables such as displacement, pressure, etc., are derived

from the energy of vibration.

Analysis is used to say that SEA is a framework of

dynamic analysis, rather than a particular technique.

Resonance frequencies and mode shapes of higher order

modes show great sensitivity even to small variations of

geometry, construction and material properties. Modal overlap

results in high variability of frequency response functions

(FRFs) for such variations [3].

Also, FEM/BEM computer programs used for calculating

mode shapes and frequencies are rather inaccurate for higher

order modes. A statistical model of the modal parameters is

therefore natural and appropriate when the number of modes

becomes high in the frequency intervals considered.

SEA has been successfully used a long time for a number of

applications such as ships [4], buildings [5] and vehicles [6].

The SEA prediction procedure has four basic steps:

1) Modelling of the dynamic system into subsystems and

connections (junctions).

2) Determination of SEA-parameters for the model

3) Calculation of energy distribution between subsystems

4) Calculation of average response levels for subsystems

The energy storage elements are called subsystems, and

should be parts of the modelled system with similar vibratio-

nal modes. The modes are usually of the same type (flexural,

torsional, acoustical etc.) that exist in some section of the sys-

tem (an acoustic volume, a beam, a bulkhead etc.) separated

Examples of early use of Statistical Energy Analysis (SEA) for concept development

evaluation

Juha Plunt

Müller-BBM Scandinavia AB, Box 1054, SE-405 22 Gothenburg, Sweden

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014Porto, Portugal, 30 June - 2 July 2014

A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)ISSN: 2311-9020; ISBN: 978-972-752-165-4

3313

by discontinuities from the rest of the structure. Subsystems

are often reasonably easy to identify also in complex mech-

anical or acoustical systems.

These subsystems represent an entire ensemble of structures

with the same main geometry and material parameters

(volume, area, thickness, Young’s modulus, density etc.) but

no details about shape, boundary conditions etc. This can be

illustrated by Figure 1, showing an ensemble of plates that are

represented by a subsystem with exactly the same parameters!

All represented by the

same plate subsystem:

E, S and t are equal

ENSEMBLE

Figure 1. An ensemble of plates with varying shape is the

same SEA subsystem (this is the S in SEA!)

This makes SEA especially interesting for analysis of sound

and vibration properties during the development of a new

product concept, well before any detailed geometry has been

decided. This is exemplified in this paper.

The SEA parameters determined for the subsystems and

junctions by a SEA software are the following:

1) Input powers Wi to the i-th subsystem

2) Modal densities ni for the i-th subsystem

3) Internal loss factors i for the i-th subsystem.

4) Coupling loss factors ij between the i-th and the j-th

subsystems

When the SEA-parameters for all subsystems and junctions

have been determined, the set of energy balance equations

will be completely defined. This may be written as the

following matrix equation

1 1 1 12 1 1 1 1

2 21 2 2 2 2 2 2

1 2

tot N m in

tot N m in

N N N N N Ntot mN inN

n n n E W

n n n E Wf

n n n E W

(1)

or in matrix notation

m inf A E W (22a)

The matrix [A] is real, symmetric and positive definite of size

NxN where N is the number of subsystems. By appropriate

numbering of subsystems, the matrix become significantly

banded which improves the calculation speed.

.

3 LOW WEIGHT VEHICLE CONCEPT DESIGN EXAMPLE

A lightweight research project - “Collaboration as enabler for

light weight vehicles” - involving around twenty Swedish

automotive suppliers, an automotive OEM and a number of

universities developed new vehicle and system concepts that

may be able to fulfill the abovementioned targets of 30-40 %

weight reduction. The main project concepts are presented at

www.sanatt.se.

It was expected that new system and complete vehicle archi-

tectures may be necessary in order to reach 30-40% weight

reduction, while also targeting excellent performance with

respect to safety and comfort without substantially increased

cost. Optimization by change to lightweight materials within

the same conventional body and interior structure would lead

to unacceptable cost increase if both NVH and weight targets

should be reached.

The concept development process [7, 8] used in this project

was open to any new ideas while based on a number of

defined properties, their weighted importance and targeted

performance 2020-2025. Those properties were transformed

into a list of necessary functionalities, which in turn resulted

in a large matrix of “brainstormed” ideas from specialized

working groups for complete vehicle and critical systems, see

Figure 2.

Low interior noise and vibration level, which will be

discussed in this paper, is a prioritized functionality next to

low weight and passive safety. The underbody system is used

to exemplify the concept generation and analysis process used

in the project.

Fig. 3 – The idea matrix where each function for the under-

body results in a row of the matrix. The ideas are evaluated

against the property diagram and for synergies. The best

combination is used to define a possible concept design.

Combustion engines with higher energy efficiency (diesels,

direct-injection gasoline) tend to radiate more airborne sound

in the mid and high frequency range which requires improved

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3314

airborne sound insulation between the engine bay and the

interior. This may be in direct physical conflict with weight

reduction for the critical sound insulating partitions, see

Figure 3 [9]. Structure-borne transmission is generally much

more complex to analyze and design for but is not necessarily

in direct conflict with weight reduction.

Sound- and

vibration sources

Structure-

borne sound

transmission

Airborne

sound

transmission

Passenger interior

compartment

•Direct weight - NVH conflict

•Few panels contribute

•Moderate functional complexity

•No direct weight - NVH conflict

•Many panels contribute

•Considerable functional complexity

Figure 3. The relative complexity of airborne and structure-

borne sound transmission in cars is shown. Airborne sound

insulation is in a direct conflict with low weight.

Early in the concept generation process, some basic floor/

firewall ideas were compared [8] and it became obvious that a

basic, double floor/firewall concept was necessary to balance

airborne sound insulation requirements with low weight and

vehicle packaging efficiency. A summary of these SEA ana-

lysis results were presented in [8] and will not be repeated

here.

The final underbody concept is illustrated in Figure 4. It

consists of a bottom panel structure made of a stiff, light-

weight Hybrix™ [10] panel and an upper carbon-fiber reinfor-

ced composite beam and panel structure.

Figure 4. The final SåNätt underbody concept illustration

model. Upper structure of carbon-fibre reinforced composite

with different thickness for panels and beams. Lower structure

made of a stiff Hybrix™ panel and door sills of aluminium or

high strength steel.

The main concern was if the relatively stiff CF-structure

(compared to present day heavy layer mats) would result in

higher sound radiation efficiency of the firewall and foot-well

panels. The weight reduction compared to a present, conven-

tional underbody structure for a midsize premium car with all

its trim is approximately 45%.

3.1 The SEA-models

A commercially available SEA program, GSSEA-Light, was

used [11], allowing modeling with components connected in a

2D network since no 3D geometry was available anyway until

the end of the project. It has sufficient capabilities for this

type of relatively crude concept evaluation.

A SEA project defined in GSSEA-Light can also use and

compare results across multiple SEA models, making it

suitable for fast modeling and comparison of very different

concepts. The SEA model in Figure 5 is for the conventional,

present day underbody used as reference, and Figure 6 shows

the model for the final underbody concept of the SåNätt

project as shown in Figure 4.

The input data used for the baseline reference concept was

0.7 mm steel plate, 30 mm soft foam as distance holder and a

2 mm thick heavy layer mat with 2000 kg/m3 density.

The main input data for the lightweight concept were, for

the upper CFR-structure:

Carbon fiber compound AMC 8590

Front seat cross beam and tunnel 4 mm

Horizontal floor panels and firewall 2 mm

For the lower structure a Hybrix composite plate with a

bending stiffness corresponding to 1 mm steel plate but with

50% lower surface weight was used.

The initial “baseline” internal loss factors used were 5% for

the Hybrix and 2% for the CFR composite. Additional dam-

ping with loss factor of 5% for the thicker CFR composite

panels and 10% for the thinner panels were then introduced in

the SEA model to improve the structure-borne sound

transmission properties. The door sills in aluminum were

approximated by an equivalent rectangular, hollow profile.

.Figure 5. SEA-model for the baseline underbody structure.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3315

The average separation distance between the two floor/fire-

wall surfaces was set to 100 mm and airborne sound transmis-

sion was calculated for this double wall structure.

Figure 6. The SEA-model for the SåNätt lightweight

underbody concept.

The models use excitation from three power sources, one air-

borne sound source in the engine compartment, one airborne

sound source under the floor and a structure-borne point force

acting on one of the longitudinal beams, simulating input from

an engine mount or a wheel suspension bushing.

The corresponding transfer functions in 1/3rd

octaves are

calculated from these sources to the car interior compartment

and compared between both models. The target was to obtain

lower or equal transfer functions for the low-weight under-

body concept as for the conventional reference design.

3.2 The SEA-results

In summary, the comparable performance, shown in Figures 7

to 9 was obtained for airborne and structure-borne sound

transmission between the present conventional reference

design and the SåNätt underbody concept for different

amounts of damping in the CFR panels.

Figure 7 shows the estimated sound pressure level in the

interior compartment relative to conventional baseline design

for the same excitation sound pressure level in the engine

compartment. Values above zero means deterioration while

negative values means improvement of sound insulation by

the SåNätt concept. The lightweight concept is better above

approximately 250 Hz but is slightly worse for lower frequen-

cies. Airborne sound insulation is normally most important

over a few hundred Hz so this may be fully acceptable. As

expected, the additional damping of the CFR panels did not

influence the airborne sound transmission and the two curves

are perfectly overlaid.

Worse

Better

Figure 7. The predicted interior airborne sound pressure level

for the SåNätt concept (blue) relative to present baseline (red)

for airborne sound excitation in the engine compartment.

Figure 8 shows the estimated sound pressure level in the

interior compartment relative to the conventional reference for

the same sound pressure level excitation below the floor.

Worse

Better

Figure 8. The predicted interior airborne sound pressure level

for the SåNätt concept (blue and green) relative to present ba-

seline (red) for airborne sound excitation below the floor.

Figure 9 shows the estimated sound pressure level in the inte-

rior compartment relative to present, conventional “baseline”

for the same force in Z-direction on a point on the left

longitudinal beam. The case with moderately damped CFR-

panels (Baseline) seems to be much more sensitive (higher

NTF:s) in the entire frequency range than the present day

design. A “what if” case with damping added to the CFR

panels, as mentioned in Section 3.1 above, makes the concept

perform equal below 2 kHz but still worse at higher frequen-

cies. Structure borne sound contributions to interior vehicle

sound are expected to dominate at low frequencies, so this

may still be acceptable, except when higher frequency issues

like whine, clonks or rattle are of concern.

Figure 10 illustrates the response contributions to the car

interior due to radiation from the different panels for the case

with additional damping of the CFR panels. This analysis can

be used to further fine tune the underbody concept. E.g. one

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3316

can see that the dominating radiators above 1 kHz are the stif-

fer CFR cross beam surfaces.

Worse

Better

Figure 9. Predicted structure-borne NTFs for force excitation

in Z-direction on a point on the left front longitudinal beam to

the interior cavity. Two different damping inputs for the CFR

panels.

Figure 10. SEA “Transfer path analysis” into the interior com-

partment for the SåNätt concept with additional damping.

-90

-85

-80

-75

-70

-65

-60

-55

-50

Mo

bili

ty le

vel,

[dB

re.

1 m

/Ns]

Frequency , Hz

Sånätt

Baseline

Figure 11. Estimated point mobilities at the force excitation

point on the longitudinal beam for the SåNätt concept and the

baseline structure.

Figure 11 exemplifies the point mobility differences for the

longitudinal beams used in the conventional reference

baseline design and the SåNätt concept. Higher mobility

corresponds to higher power input to the structure.

The issues shown above 2 kHz for the SåNätt concept in

Figure 9 are obviously not due to higher mobility in this

frequency range for these beams. A closer comparison reveals

that the main reason is high vibration levels of the CFR cross

beam surfaces. It may also be noted that no cross-beam

structure was included in the conventional reference structure,

making the comparison above 2 kHz a bit unfair. This illustra-

tes the iterative SEA model generation process often used on

an “as needed” basis when dealing with concept comparisons.

A cross beam was introduced to the conventional concept but

results of that comparison is not shown here.

4 OFFSHORE PLATFORM EXAMPLE

A relatively large SEA model was created with the GSSEA-

Light software [11] for a topside structure of a large produc-

tion platform. The SEA model had 124 components joined

with more than 300 connectors and was excited with 4 input

power sources, representing the airborne sound and the

vertical structure-borne sound excitation of the two emer-

gency generator units respectively.

The generator sets were designed with the diesel engine and

the electric generator separately vibration isolated on a com-

mon skid. This generator unit concept may have shaft align-

ment issues, since creep and setting of the isolators may be

uneven, resulting in increased misalignment over time.

Also, since the isolators are loaded with the full torque

transferred from the engine, they cannot be made especially

soft, which limits isolation effectiveness. One may therefore

expect issues at the 1st engine order at 30 Hz due to both the

misalignment and the moderate vibration reduction by the

isolation system.

The appropriate frequency ranges for the SEA model of the

topside structure is approximately 50-60 Hz and above, see

Figure 12 that shows a sharp increase in modal density from

the 50 Hz band to the 63 Hz band. For lower frequencies,

SEA results are only indicative for a risk assessment but will

have large inherent uncertainties.

Figure 12. Number of expected vibration modes for the (sepa-

rate) ESS Generator deck as function of frequency. Note the

jump in modal density at 50-60 Hz.

Worse

Better

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3317

The SEA-model can still be used for some risk assessment

of excessive noise at frequencies as low as 30 Hz. One

particularly simple thing is to evaluate the sound radiation

efficiency of the stiffened steel deck and bulkhead structures

used. This is shown in Figure 13, which shows a dramatic

increase in the radiation efficiency at the same frequency

bands as the jump in the modal density.

The reason for this can be found by comparing the bending

wavelength of the stiffened panels with sound wavelength in

the surrounding air; one example is shown in Figure 14. We

have the expected coincidence frequency for the unstiffened

deck plate at 1600 Hz. Since the stiffened panel acts like an

orthotropic panel with smeared out additional stiffness at low

frequency, we get another coincidence frequency for the panel

at 20-25 Hz. The bending wavelength remains larger than the

wavelength in air for frequencies above this until the indivi-

dual subpanels can vibrate independently, resulting in effi-

cient sound radiation.

Figure 13. Sound radiation efficiency levels for decks radia-

ting into different interior spaces.

Sound wavelength in air

Figure 14.Bending wave length for the generator deck as

function of frequency.

Obviously, if noise levels are a concern, the vibration levels of

the deck have to be kept particularly low in this low frequency

region. The usual noise reduction methods, like vibration

insulation, floating deck covering etc. are not very effective in

this frequency range. Proper definition and fulfilment of rigo-

rous source strength requirements become especially critical

in this situation.

It is also seen from Figure 14 that the design of the bending

stiffness of the orthotropic plates may take sound radiation

properties into account to avoid increasing the bending wave-

length above the wavelength of sound. Again, simple SEA-

modeling can be used to compare suggested variants as

exemplified in the next example.

5 LIGHTWEIGHT SURFACE VESSEL EXAMPLE

A similar issue at low frequency can be illustrated by an

example of designing a lightweight steel hull for a high speed

surface vessel. The main strategy in order to reduce weight

was to use much thinner hull plating and compensate that by

using a much smaller c/c distance between the longitudinal

stringers in order to retain the static and low frequency stiff-

ness that meet strength requirements.

It was not entirely obvious how this different hull concept

would influence the transmission of structure-borne sound and

the sound radiation properties at low frequencies. For examp-

le, the radiation efficiency at low frequencies was expected to

be influenced considerably by the hull plate thickness, the

stiffener cross-sections and the c/c distances between

stiffeners of the same reasons that were discussed in the

previous section.

Figure 15 illustrates the influence on the radiation efficiency

due to different c/c distance of longitudinal stringers on a 3

mm steel hull plate with 1.2 m between web frames. Clearly

the frequency is strongly affected below which the radiation

efficiency increases rapidly.

Figure 15. Sound radiation efficiency levels for different

stiffener c/c distance. 3 mm steel plate.

Figure 16 illustrates the influence on the radiation efficiency

due to different hull plate thickness using the same 400 mm

c/c distance of the longitudinal stringers for a steel hull plate

with 1.2 m between web frames. Again the influence on low

frequency radiation efficiency is considerable.

The necessary SEA-model for these radiation efficiency

comparisons is very simple, see Figure 17. Subsystem proper-

ties such as wavelengths, mobility’s etc. are directly available

as are the connector properties such as radiation efficiencies.

It is possible using such limited SEA modeling and analysis

to use different stiffening strategies for different parts of the

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3318

ship in order to avoid low frequency structure-borne sound

issues.

Stiffer orthotropic plates may be used for hull sections

where vibration sources are acting, e.g. above the propellers

and below engine beds, to obtain lower mobility and thus

reduce vibration power input to the hull. Less stiff panels,

optimized with respect to radiation efficiency, are preferably

used for the main radiating surfaces into cabins etc. to avoid

high sound radiation at critical excitation frequencies.

Figure 16. Sound radiation efficiency levels for different

stiffener c/c distance. 3 mm steel plate.

Figure 17. Simple SEA-model to compare sound radiation

efficiency levels for different stiffener configurations and

plate thicknesses.

6 DISCUSSION AND CONCLUSIONS

Statistical Energy Analysis (SEA) is a computational method

that is especially suitable for early, quantitative sound and

vibration evaluation of crude, competing and very different

system concepts. It can of course also be used for comparative

evaluation of vastly different complete vehicle architectures

resulting in significantly larger SEA models than those shown

here.

The detailed geometry needed for global low frequency

modeling by FEM is not available at early stages when the

sound and vibration performance of radically different product

concepts is compared in order to rank the concepts or illust-

rate main parameters like surface mass, double wall thickness,

internal damping etc. needed for possible target fulfillment.

The lack of both “proper simulation=FEA” and physical

prototypes usually results in sound and vibration experts not

being involved in such early concept generation and to

missing opportunities of influencing the selected concepts in

a proper direction concerning noise and vibration.

The need to communicate target conflicts, e.g. between

weight and sound insulation in a relatively simple and

quantitative manner is imminent when the sound and vibration

expert has the opportunity to participate in early concept gene-

ration. Simplified SEA modeling will be more reliable than

qualitative, “educated” guesses or overly simplified, rough

textbook calculations since topographic complexity of the

structures can be included also in relatively simple SEA

models.

One also needs to be aware of the very limited time frame

of opportunities in most cases to influence the early, but often

decisive and major concept selection process. One has to use

the best possible analysis tools under those circumstances.

Simplified SEA models that can be rapidly built may be quite

well suited for this relatively dynamic and fast development

and selection between radically different competing ideas.

ACKNOWLEDGMENTS

The vehicle case example was developed for the project

“Collaboration as enabler for light weight vehicles” in the FFI

- Strategic Vehicle Research and Innovation program and

partly funded by VINNOVA, the Swedish Governmental

Agency for Innovation Systems.

The offshore and ship examples are extracts from different

company confidential consulting projects. These examples do

not reveal any information about the actual objects or custo-

mers; however I am still grateful for the opportunity to use the

ideas of risk assessment at such low frequencies that the SEA-

models may be considered as unreliable for prediction of

sound and vibration transmission between different parts of

the structure.

REFERENCES

[1] Lyon R H: Statistical Energy Analysis of Dynamic Systems: Theory and

Applications. MIT Press, Cambridge 1975. [2] R H Lyon, R G DeJong: Theory and Application of Statistical Energy

Analysis. 2nd Edition, Butterworth-Heinemann (1995).

[3] Plunt J: Predictability limitations of vibration transfer functions for structures with overlapping modes. Proc. Conf. Spacecraft Structures,

Materials & Mech. Testing. Noordwiik, The Netherlands, 27-29 March

1996 (ESA SP-386, June 1996). [4] Plunt J: Methods for Predicting Noise Levels in Ships, Part II. Ph.D.

Thesis, Chalmers Univ. of Technology, Gothenburg, Sweden, 1980.

[5] Craik R J M: The Prediction of Sound Transmission through Buildings Using Statistical Energy Analysis. J Sound and Vibr, 82(1982) p505-

516.

[6] Manning, J: "SEA Models To Predict Structureborne Noise In Vehicles," SAE Technical Paper 2003-01-1542, 2003.

[7] Almefelt, L: “Balancing Properties while Synthesizing a Product Con-

cept – A Method Highlighting Synergies”, Int. Conf. Engineering Design (ICED 05), Melbourne, Australia (2005).

[8] Plunt J, Easterling W: How to Handle Low Weight Vehicle Concept

Design and Conflicting NVH Targets. Proc. Inter-Noise 2012, New York City, 2012

[9] Plunt J, Wedel M: “Weight Reduction with Maintained Sound Comfort

for Automotive Structures; Possibilities and Consequences”, Proc. Inter-Noise 99, Fort Lauderdale, USA (1999).

[10] See home page of Lamera AB, www.lamera.se.

[11] GSSEA-Light, Version 1.0 from Gothenburg Sound AB, Sweden. Available at www.gothenburgsound.se.

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014

3319