Nisula et al.

Calculating sound radiation from loudspeaker enclosures using the finite element analysis

CALCULATING SOUND RADIATION FROM LOUDSPEAKER ENCLOSURES USING THE FINITE ELEMENT ANALYSISJAAKKO NISULA, JUHA HOLM, AND AKI MÄKIVIRTA
Genelec Oy, Iisalmi, Finland
jaakko.nisula@genelec.com

Walls of a loudspeaker enclosure exhibit vibrations and resonances, creating unwanted acoustic output. These vibrations are excited by the mechanical forces from the drivers and by the acoustic pressure inside the enclosure. Enclosure
vibrations cannot be satisfactorily simulated using a mechanical model that only includes the mechanical conduction ofvibration in the enclosure. Also acoustical phenomena have to be considered to get accurate results. In this paper weshow how Finite Element Analysis using a coupled model of mechanics and acoustics can predict enclosure vibrations
and their causes with good resolution.

INTRODUCTION
Ideally loudspeaker enclosure walls do not contribute tosound radiation. Enclosure wall movements can cause
audio colorations and otherwise change the performance. As materials have finite stiffness and mass,
loudspeaker enclosure walls tend to vibrate and canproduce some acoustical output. Loudspeaker manufacturers have been trying to minimize enclosure vibrations
with a number of means. [1]There are different mechanisms that generate acoustic
output from secondary sources of a loudspeaker, mainly
the enclosure walls and reflex port. The loudspeakerenclosure itself has finite mass and stiffness and therefore exhibiting resonances and deformations under load.
The reflex port introduces resonances also above thedesired Helmholtz frequency. Standing waves inside theenclosure affect the excitation of the enclosure walls
and thus also their vibrations. Internal acoustical resonances can also radiate audio through the reflex portopening. In some conditions the coincidence effect candecrease the sound attenuation through the enclosure
walls.
Calculating loudspeaker enclosure vibrations using finite element analysis is not a new idea. First references
date back to over 30 years ago [2]. However, until recently, the lack of computing power has limited the useof finite element analysis. Traditionally enclosure vibrations are studied empirically [3-5]. Numerical methods
for estimating loudspeaker enclosure vibrations havebeen studied for example by Karjalainen [6] et al. andLuan [7,8] showing the potential in solving loudspeakerenclosure vibrations numerically. When calculating
sound fields created by vibrating surfaces boundary

element method has often been used with finite element
method because of the lower computational cost of theboundary element method [9]. Even then, simplifications and idealizations of the structures and boundary
conditions have often been necessary to conduct analyses. The models presented have usually been significantly simpler than a full actual loudspeaker. Factors
limiting the accuracy of finite element simulations arereported to be the accuracy of CAD geometry and material parameters as well as the assumptions made for theanalysis. [10-12].
The aim of this paper is to study the benefits of using acoupled model over using a mechanical vibrationalmodel or acoustical modal analysis model alone. The
complexity of the modeled loudspeaker closely corresponds to that of a real loudspeaker.
1 THE ANALYSIS MODELSIn this work a case study of the resonant behavior in a
two way loudspeaker enclosure is studied with acoustical and mechanical eigenfrequency analyses and mechanical analysis with loading, as well as a coupledmodel combining both acoustical and mechanical models. The analysis models were built with cleaned-up
CAD models of the enclosure and the woofer frame.
The analyses are conducted on a model of a real prototype loudspeaker and the results are compared withacoustical measurements.
For the mechanical vibrational model the woofer magnet and the tweeter were replaced with homogenousmasses as their inner features were considered to havenegligible effect to the behavior of the whole system.

AES 51st International Conference, Helsinki, Finland, 2013 August 22–24

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Nisula et al.

Calculating sound radiation from loudspeaker enclosures using the finite element analysis

Connections between parts were considered rigid andthe enclosure was rigidly supported from the back wall.
The acoustical model consists of the air volume inside
the enclosure and a finite volume or air surrounding theenclosure.
The coupled and mechanical analysis models werehalved at the symmetry plane to reduce computationalload (Figure 1). In principle the coupled model combines mechanical and acoustical analysis models.

Figure 2. Normal velocities of eigenmodes integrated
over top, side and front surfaces. RMS scaling of eigenvectors.
2.2 Loaded mechanical analysis
The loading in the mechanical analysis is the recoil
force from the voice coil and a uniform pressure inside
the enclosure. Figure 3 shows on a log scale the normal
velocities for the same case as in Figure 2 but considering loading. The data is calculated at discrete frequencies with 50 Hz spacing. These figures indicate that theeigenmodes can have an impact on the vibration patterns of the enclosure when acoustical load is considered.

Figure 1. The analysis model for coupled analysis.
2 MECHANICAL ANALYSIS
Both eigenfrequency and frequency domain analysis
were made. The amount of radiated sound cannot bedirectly evaluated from the displacements so a roughestimation was made by integrating velocities on theenclosure external surfaces to estimate volume velocity.
2.1 Eigenfrequency analysis
The eigenfrequency analysis resulted 30 nominal frequencies between 128 and 870 Hz. A closer look reveals
that many of them appear not to be easily excited, given
the loading in the loudspeaker, and/or because they do
not seem to have the potential to generate much acoustic
output. Normal velocities integrated over top, side andfront surfaces of the enclosure at eigenfrequencies canbe seen in Figure 2. The eigenmodes are RMS scaled.
The analysis does not in any way predict how the modes
get excited in reality. This processing does demonstrate,
however, the potential of a mode to radiate sound.

Figure 3. Normal velocities with loading considered,integrated over top, side and front surfaces. 50 Hz frequency steps.
3 ACOUSTICAL ANALYSISThe acoustical model consisted of the air volume inside
the enclosure and a finite air volume surrounding theenclosure. Enclosure walls were considered rigid. An
eigenmode analysis (Table 1) found eigenfrequencies in
the inside air volume and in the reflex ports.

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Calculating sound radiation from loudspeaker enclosures using the finite element analysis

Table 1. Eigenfrequencies fe of the air inside the enclosure.
fe [Hz]possible cause

61

Helmholtz resonance

548Standing wave in the enclosure (top – bottom)

750

Standing wave in the enclosure (left – right)

762Standing wave in the enclosure (duct)

815

Open pipe resonance in reflex ports

858

Standing wave in the enclosure (diagonal)

940

Standing wave in the enclosure (diagonal)

The modes found in the analysis are typical but thebasic equations for acoustical resonances do not accurately predict the resonant frequencies with a geometry
this complex. The lowest mode frequency found is theHelmholtz resonance. The standing waves are determined by the enclosure dimensions, and in the example
case, the folded duct formed inside the enclosure structure.
Many of the eigenfrequencies found were in the region
showing problems also in the acoustical measurements.
Also this analysis fails to directly predict which modes
may cause audible problems, i.e. audible resonances, inreality.
4 COUPLED ANALYSISThe coupled model was loaded acoustically with thecone acceleration and its recoil force on the driver. The
cone was set to radiate only inside the cabinet to segregate direct sound from results.

material inside the enclosure, and this tends to emphasize acoustical resonances.

Figure 4. Simulated average sound pressure level on theexternal surface of the air domain.
5 COMPARISON TO MEASUREMENTS
An actual prototype loudspeaker matching the construction used in simulations was built and measured. The
measured cumulative spectral decay plot of the prototype loudspeaker (Figure 5) demonstrates that althoughsome problem frequencies match the simulations, amere mechanical model does not predict the real system
response. Resonances at 550 and 750 Hz match wellwith what was simulated in the coupled model.

Average sound pressure level at equal distance from theloudspeaker (Figure 4) represents the sound radiatedfrom secondary sources through enclosure walls andreflex ports.
The peak frequencies and deformations of the structure
were compared to mechanical and acoustical eigenfrequencies to resolve the contribution of acoustical andmechanical resonances on the whole.
At lower frequencies the acoustic output originates
mostly through the reflex ports. There are peaks visibleat 550, 750, 950 and 1050 Hz. In this case example,
both mechanical and acoustical resonances coincide
near 550 Hz leading to a peak in the simulated response.
The peak at 750 Hz was caused by an acoustical resonance, and radiated through the enclosure walls. Also
the reflex port pipe resonance at 800 Hz is visible, but ata lower amplitude than the peak at 750 Hz. The peaks at
950 and 1050 Hz can be linked with mechanical resonances. In this case example, there was no damping

Figure 5. Cumulative spectral decay plot of a prototypespeaker.
6 CONCLUSIONS
The results show that finite element analysis can predict
enclosure vibrations with good accuracy also in practical cases. Both mechanical and acoustical simulations ofa loudspeaker enclosure yield substantial information of
the enclosure’s resonance characteristics. Both can pinpoint some problem frequencies with good accuracy.
However, running separate mechanical and acoustical

AES 51st International Conference, Helsinki, Finland, 2013 August 22–24

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Nisula et al.

Calculating sound radiation from loudspeaker enclosures using the finite element analysis

analyses cannot predict whether it is mechanical oracoustical phenomena that dominate the sound radiatedfrom the enclosure at a given frequency.
A coupled model considers both mechanical and acoustical excitations with correct relative magnitudes, andthe sound radiation can be calculated directly from suchmodel. The coupled analysis shows good match withacoustical measurements on a real system and gives agood overall view to enclosure vibrations.
Overall, the results can be considered very informative.
It becomes clear that when looking at the total performance of a loudspeaker, the mechanical and acousticalresonances cannot be handled separately.
In our case examples, simulations indicate that there arethree main mechanisms contributing to the secondarysound radiation. Depending on the frequency, the source
mechanism is a mechanical resonance, an acousticalresonance inside the enclosure and mediated out by radiation through the walls, an open pipe resonance in thereflex port, or a combination of these.
Meaningful simulations of a loudspeaker enclosure require including all the parts that contribute significantly
to the mass and stiffness of the structure. In practice thisusually includes at least the loudspeaker drivers in addition to the enclosure. The parts in the loudspeaker construction form a system and it should be analyzed asthat. Performing separate analyses on different parts
separately makes evaluating the whole system very difficult. Our results are sufficiently accurate to enable
identification of the potential problem frequencies before building a real prototype. This can lower the number of design iterations and prototypes needed. Postprocessing the simulation data can enable understanding
the causes for problems easier than measurements of aphysical prototype. In our case example, the coupledanalysis made it possible to get a clear picture of theimpact of different design options and further improvements of the enclosure’s characteristics and therefore
the quality of acoustical response of the loudspeakersystem. In our case example, it became evident that theoverall stiffness of the enclosure dominates the performance and that there was little to be gained from small
tweaks such as, for example, optimizing the locations ofstiffening ribs.

REFERENCES[1]
J. Borwick, Loudspeaker and Headphone Handbook. Second ed. Focal Press, Oxford, UK,(1997).
[2]

S. Sakai et al. Acoustic field analysis of a loudspeaker enclosure using the finite element method. Proc. Audio Eng. Soc. 72nd Conv., Preprint#1891, (1982).

[3]

D.A. Barlow, The Sound Output of Loudspeaker
Cabinet Walls. Proc. Audio Eng. Soc. 50th convention, (1975).

[4]

W.R. Stevens, Sound Radiated from Loudspeaker Cabinets. Proc. Audio Eng. Soc. 50th convention, (1975).

[5]

S. Lipshitz, An investigation of sound radiation
by loudspeaker cabinets, Proc. Audio Eng. Soc.
90th conv., (1991).

[6]

M. Karjalainen et al., Comparison of Numerical
Simulation Models and Measured LowFrequency Behavior of Loudspeaker Enclosures.
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[7]

Yu Luan, Modeling structural acoustic propertiesof the Beolab 9 loudspeaker. Master thesis,
Technical University of Denmark, (2007).

[8]

Yu Luan, Modeling structural acoustic propertiesof loudspeaker cabinets. Ph.D. Thesis, Technical
University of Denmark, (2011).

[9]

D. J. Henwood et al., Finite element modelling ofloudspeaker diaphragms and the boundary element method for evaluating sound pressure.
Proc. Audio Eng. Soc. 82nd conv., (1987).

[10]

P. Macey, Accuracy issues in finite element simulation of loudspeakers. Proc. Audio Eng. Soc.
125th conv., (2008).

[11]

M. D. Hedges and Y. Lam, Accuracy of Fully
Coupled Loudspeaker Simulation Using COMSOL. Proc. COMSOL Conference 2009 (2009).

[12]

A. Salvatti, Virtual Acoustic Prototyping – Practical Applications for Loudspeaker Development.
Proc. Audio Eng. Soc. 129th conv., (2010).

With an efficient way to simulate the vibration and unwanted sound output from the enclosure walls, the levels of these source mechanisms can be reduced below
the threshold of audibility without making excessively
heavy-built structures. Another benefit from reliable
simulations is the decreased need for physical prototypes. This can save time and money especially whenthe enclosure is complicated to manufacture.

AES 51st International Conference, Helsinki, Finland, 2013 August 22–24

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