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Comparison of Numerical Simulation Models and Measured Low-Frequency Behavior of a Loudspeaker 4722 (P3-3)
MattiKarjalainen VeijoIkonenHelsinki University of Technology Espoo Finland Tampere University of Technology Tampere Finland
AnttiJdrvinen PanuMaijalaHelsinki University of Technology Espoo Finland Helsinki University of Technology Espoo Finland
LauriSavioja AnttiSuutalaHelsinki University of Technology Espoo Finland Tampere University of Technology Tampere Finland
JuhaBackman SeppoPohjolainenNokia Mobile Phones Salo Finland Tampere University of Technology Tampere Finland
Presented at AUDIOthe 104th Convention1998 May 16-19Amsterdam
®
Thispreprinthas been reproducedfrom the author'sadvancemanuscript,withoutediting,correctionsor considerationbythe ReviewBoard. TheAES takesno responsibilityfor thecontents.
Additionalpreprintsmay be obtainedby sendingrequestandremittanceto the AudioEngineeringSociety,60 East42nd St.,New York,New York 10165-2520, USA.
Al/rights reserved.Reproductionof thispreprint, orany portionthereof, is not permittedwithoutdirectpermissionfrom theJournalof the Audio EngineeringSociety.
AN AUDIO ENGINEERING SOCIETY PREPRINT
Comparison of Numerical Simulation Modelsand Measured Low-Frequency Behavior
of a Loudspeaker
Matti Karjalainen 1, Veijo Ikonen 2, Antti J_irvinen l,
Panu Maijala l, Lauri Savioja l, Antti Suutala 2,Juha Backman 1,3, and Seppo Pohjolainen 2
1Helsinki University of TechnologyEspoo, Finland
2 Tampere University of TechnologyTampere, Finland
3Nokia Mobile Phones, $alo, Finland
matt i. karj alainen_hut, f i, i77309_cc, tut. f i, antt i. j arvinen_hut, fi
panu.maij ala_hut, fi, lauri, savioj a_hut, fi, s 129948_alpha. cc. tut. f i
j uha. backman_nmp, nokia, eom, seppo, pohj olainen©cc, tut. f i
http://acoustics, hut .fi/
ABSTRACT
The vibroacoustic behavior below lkHz of a prototype closed-box loud-
speaker has been studied in detail by comparing measurements and elementmodel simulations. Sound fields were measured using a microphone array
of 90 electret capsules and vibrations using a laser vibrometer and accel-erometers. Simulations have been carried out using analytical, finite and
boundary element, and finite difference methods. The enclosure conditions
were varied from fixed wall case buried in sand to free-standing empty box
and to free-standing damped box. Two positions of the driver in the front
plate were examined (only the 'rim' position is documented here). Theapplicability of each modeling technique is discussed 1.
1Special thanks for support are due to'.Kaarina Melkas, Nokia Research Center, Tampere, FinlandJorma Salmi, Gradient Oy, J_rvenpg_&, FinlandAki M&kivirta and Ari Varla, Genelec Oy, Iisalmi, FinlandJukka Linjama, VTT Manufacturing Technology, Espoo, FinlandTechnology Development Centre Finland (TEKES)
1
INTRODUCTION
The design of a loudspeaker has traditionally been an iterative process
based on approximate rules, experience from prior designs, and finally trial
and error by constructing and modifying prototypes. Computer-basedmethods have helped in exploring basic features of driver to enclosure
matching and crossover network design. However, not much have been
published on the use of more advanced computer-based methods and tools
in loudspeaker design.
The detailed behavior of a loudspeaker consisting of an enclosure and
driver(s) is very complex and escapes analytical mathematical solutions.Approximate (semianalytic) approaches may turn out to be useful, how-ever, especially within a limited frequency range and when the geometry
of the system is simple enough, e.g., a shoebox design. At low to mid fre-
quencies, lumped element models may be used both for the electroacousticand vibroacoustic subsystems. In more complex cases the system should
be considered as a complex electro-vibro-acoustic system that is not easily
partitioned to any simple submodels.
The progress in computer-based numeric simulation of complex spatially
distributed systems using various element methods has raised the question
of how useful they might be for practical loudspeaker design [1]. These
modeling techniques include the finite element method (FEM), the boundary
element method (BEM), and the finite difference time domain (FDTD)method. In advanced forms they can be used for simulating any linear
and time-invariant (and with limitations nonlinear) vibroacoustic systemsat low frequencies. Low frequencies means here that the element size in the
model mesh, and thus the number of spatially discrete elements, limits the
highest useful frequency of the simulation. Loudspeaker design at low to
mid frequencies is in principle a good application for such methods.Several commercial or experimental tools are available for element-based
vibroacoustic modeling and simulation, such as SYSNOISE [2], I-DEAS
Vibroacoustics [3] and Comet/Acoustics [4]. Other FEM/BEM tools that
are not specifically tuned to acoustic problems are ABAQUS [5], ANSYS
[6], and MSC/NASTRAN [7].The problem of using the FEM/BEM programs, at least from the point
of view of loudspeaker design, is that they are expensive, need powerfulcomputers to work fast, the construction of the model is tedious, and the
availability of material data (acoustic and mechanical parameters) is poor.
Experimental programs, e.g., from academic institutions, often lack docu-mentation and continuing support. Thus, such simulation and design tools
are not widely used in loudspeaker design, and information on their useful-
ness as well as comparison of their properties is practically non-existing.
As the progress in this field is fast, it is important to be prepared toutilize such tools whenever they turn out to be productive. Potentially,
computer-based design tools promise to make the product developmenttime faster whenever the designer can start from an approximate model and
rapidly go through variations and experimentations using software model-
ing up to a prototype which, when actually built, works closely enough as
expected. Even more ambitiously, the computer may automatically run
through some optimization steps to search for the best match to givenspecifications and targe t criteria.
In this study our interest was focused to the applicability of the element-based simulation and modeling tools to basic loudspeaker design. We se-
lected a case that is simple enough, yet practical and realistic. Thus we
specified and constructed a closed box enclosure with a single driver ele-ment so that it was simple to vary some interesting parameters such as the
driver position, the stiffness of the enclosure walls, and the damping mater-ial inside the box. Next the actual vibroacoustic behavior of our case was
measured extensively with such parametric variations. A microphone array
with 90 electret capsules was constructed to measure the sound field insideand outside the loudspeaker box as impulse responses of electric excitationof the driver. A laser vibrometer and accelerometers were used to obtain
vibration data of the walls and the cone of the driver so that an almost
complete picture of the behavior below about 1 kHz was captured in thisdata. The acoustical and mechanical parameters of the materials (MDF for
construction and fiber wool for damping) were also measured.
The next step of the study was to model the loudspeaker using variousFEM, BEM, and FDTD software tools. The computational models werebuilt first and the measured or estimated material parameters were given tothe models. The simulation results of the models are shown in this article
and compared with the measured behavior. Finally, the models were handtuned to match better to the measured data. This yields new material
parameters that work better than the measured ones in modeling similarcases, but may not be generalizable to very different cases. The results
of simulations and measurements are also compared to simpler analytic or
semianalytic models of the same loudspeaker.
After presenting the results we will discuss the usefulness of the models
and how they could be improved. Directions of further studies are shownas well.
I CASE STUDY: A CLOSED-BOX LOUDSPEAKER
To study a problem of reasonably low complexity, yet interesting from a
practical point of view, we designed and constructed a closed-box proto-
type loudspeaker of medium size (600 x 400 x 250 mm 3) and easy enoughto modify and measure. The structure of the enclosure is shown in the
drawings of Fig. 1. The front panel (facing up in the figure) is removable
and is built in two variations, one with a driver element symmetrically in
the middle and another one with an asymmetric speaker positioning (rim
case), as shown in the top drawing of Fig. 1. The loudspeaker element wasa 6.5 inch driver of type SEAS P17 REX.
The enclosure was made of 20 mm MDF, all panels being rigidly coupledat their edges. In our simulations and measurements the behavior of the
box was studied both as buried in sand and as freestanding to allow wallsto vibrate. In both cases it was studied as an empty box and with damping
material (Partek mineral wool) on the back wall inside the box (Fig. 1).
1.1 Conceptual Model of the Loudspeaker
h'om a vibroacoustic point of view the loudspeaker works as follows. Thedriver element converts its electrical excitation into the movement of the
diaphragm. This has coupling to the air outside and inside the cabinet,transmitting a wave to both parts of the system. Another vibroacoustic
coupling of interest is from the interior sound field to the walls of the
enclosure, making them to vibrate and, due to this vibration, to radiateexternal sound field in addition to the radiation of the driver diaphragm.
The driver element has also a direct mechanical coupling to the front panel
and through it indirectly to all other panels of the enclosure. If the wallswere rigid, only the driver diaphragm vibration would be of interest. In
practice, however, the vibration of enclosure walls is not negligible andshould be included in detailed simulation of the system. Furthermore, the
acoustic loading of cabinet interior on the driver diaphragm movement hasan effect on its radiation to the external field. All these effects influence
the magnitude and phase response of the loudspeaker and its directivity
4
pattern. We have assumed, however, that the transmission of transversal
waves through walls that is important in sound insulation, is not prominent
in loudspeaker enclosures.
2 VIBROACOUSTIC MEASUREMENT SYSTEM
In order to be able to evaluate how realistic the numeric simulation results
of the loudspeaker case are, we decided to construct a system for extensivevibroacoustic measurements. It consists of an array of miniature micro-
phones to collect acoustic responses and a combination of a laser vibrometer
and vibration sensors (accelerometers) to probe the mechanical vibrationsof the loudspeaker system. A computer system was programmed to col-lect the sound field and mechanical vibration data in the form of impulse
responses of the loudspeaker element excitation.
2.1 Microphone Array
A microphone array of 90 small electret capsules was constructed so thatit fits to the interior of the loudspeaker cabinet, see Figs. 2 and 3. A fi'ame/
of metal tube was used to support row and column wires, spaced by 40
mm x 40 mm, as shown in Fig. 2. At each wire crossing an electret mi-
crophone (Hosiden 2823) and a cascaded diode was attached as depicted in
Fig. 3. Digitally controlled analog multiplexers were used to select one ofthe column wires and one of the row wires at a time. Only a single electret
capsule, activated by current through the load resistor (R), is functional at
a time to capture the sound pressure field and to transduce it to the micro-phone preamplifier. Thus the multiplexed microphone array can be used
to measure acoustic responses in the spatially distributed mesh positions,
point by point, both inside and outside the box.The lower cutoff frequency of the microphone/amplifier combination was
30 Hz and the response was found flat within 1 dB in the most interesting
measurement range 100 Hz - 2 kHz of our study so that only the slightly
varying gains of individual capsules needed compensation.
2.2 Vibration Measurements
For vibration measurements a laser vibrometer (Polytec OFV3001) andacceleration probes were used. A mesh of 40 mm was measured, point by
point, to obtain the vibration responses of all walls. A number of points inthe cone of the driver were also registered.
Point mobility measurements were made by applying impact testing to
the walls of the enclosure as well as isolated pieces of MDF plates cor-
responding to the walls of the enclosure. Following equipment were used:impact hammer with Brfiel &: Kjeer 8200 force transducer, B&:K 4393 ac-
celerometer with two B&K 2635 charge amplifiers. HP 3565 S analyzer and
STAR-software were used for modal damping determination.
2.3 Data Acquisition System and Analysis Tools
The measurement system was based on the QuickSig signal processing en-
vironment [8], developed in the Laboratory of Acoustics and Audio SignalProcessing, Helsinki University of Technology. Impulse response measure-
ments were carried out using random phase fiat spectrum (RPFS) excitationsignal of typically 8192 samples at a sampling rate of 22050 Hz, averaged
typically over 10 repetitions. This is in practice equivalent to the more
commonly used MLS (maximum length sequence) measurements. The fre-
quency range of interest, from the viewpoint of element-based modelingin this study, is only up to 1-2 kHz. The signal-to-noise ratio of acousticmeasurements was in all conditions better than 40 dB so that its effect to,
e.g., magnitude responses is negligible.
The same data acquisition system was also used in vibration measure-
ments (except in impact testing). Further signal analysis of acoustic andvibration data was carried out in MATLAB.
3 MEASURED LOUDSPEAKER BEHAVIOR
Typical acoustic responses, as measured inside the enclosure, are shown
in Fig. 4. Subplots (a) and (b) show the impulse response from driverterminals to sound pressure in one mesh point, r14c5 (row 14/column 5),
fol' a sand-supported, (a) undamped vs (b) mineral wool damped, enclosure.
Subplot (c) shows the magnitude responses for the sand-supported and free-standing cases without interior damping, and subplot (d) the corresponding
magnitude responses for the case with 10 cm wool at the back panel.The impulse response in Fig. 4a illustrates a long ringing of interior
resonances in the undamped case. The corresponding ringing is radically
shorter in a damped case, see Fig. 4b. The same information is presented in
6
the frequency domain in subplots 4c and 4d. The former one shows the res-
onances and antiresonances in the undamped enclosure as measured in the
sand supported and free-standing case, respectively. The mode frequenciesexhibit strong resonances. The difference between these two curves is sur-
prisingly small. Only minor extra effects are introduced in the free-standing
case, such as seen around 200 Hz. The same is true also for the damped
case of Fig. 4d except that resonances and antiresonances are effectivelysmoothed out.
The vibration of an enclosure wall is characterized in Fig. 5 based on
three different measurements. Figure 5a illustrates the accelerance (accel-eration/force) of an isolated wall panel near corner, as measured by impact
testing. This information can be used to estimate the parameters of MDF
for vibroacoustic modeling. Figure 5b shows the corresponding behavior
when the side wall (600 mm x 400 mm) is excited at point 260 mm fromfront panel and 297 mm from top plate. This can be compared further withthe velocity of the same point as response to an electric excitation of the
driver element (Fig. 5c). Only the lowest mode (200 Hz) has prominenteffect to external sound field radiation.
4 ANALYTICAL AND SEMIANALYTICAL MOD-ELING
An accurate analytical solution of coupled vibroacoustic equations for a
loudspeaker is out of question. Yet it is possible to try a simplified and ap-proximate solution, especially at relatively low frequencies. In this section
we will try this approach since the loudspeaker in our study has a relativelyregular shape.
4.1 Analytical Modeling Techniques
The first approximation of a closed loudspeaker enclosure is obtained when
the walls of the enclosure are considered to be rigid. The element is modeled
as a simple piston with given velocity. The volume inside the enclosure
is denoted by fL Its boundary, the walls of the enclosure, are denoted by
c912-- 0_iUc0f_2, where 0f22 is the surface of the piston and cO_1 refers to theother wall surfaces. The spatial variable is denoted by x and the frequency
f is given as angular frequency w -- 2_r/. The pressure field p(x,w) insidethe enclosure as a function of frequency is given by the solution of the
7
Hehnholtz equation [14, Chapter 6]
V2p + k2p = 0 (1)
with boundary conditions
- 0, x & cqf_lOn --
___ (2)On -- -ipowVn, x E Of_2
Here k = _- is a wave number, c = 343 m/s is the speed of sound in air andc
n is the normal of the boundary pointing away from the fluid· In boundary
conditions _ is the directional derivative of pressure in the direction of the
normal n. Ill the boundary condition for the piston area, i is the imaginary
unit, po = 1.21 kg/m 3 is the density of air in equilibrium state and Vn(X,w)is the velocity of the piston in the direction of its surface's normal n. In
this case the velocity vn is considered to be constant in 0fl2. This boundaryvalue problem can be solved using Green's function Gw, which is the solution
of the equation [14]
v_a_+ k_a_= _(x- x0) (3)with homogenous boundary conditions
OG_On =0' xeOm (4)
Here 5(x - xo) is the Dirac delta function and point x0 is considered as a
source point. Using the eigenfunctions _N and eigenvalues kN of the Helm-
holtz equation (1) with boundary condition _ = 0, the Green's functioncall be expressed as a series [14, Chapter 9.4]
a_(_,xo)= _ _N(_)_(x0)
For a rectangular enclosure with dimensions l_ × ly x l, the eigenfunctionsare of tlle form [14]
_n_._n.(x)=cos(k_x)cos(kyy)cos(kzz) (6)
where k_ = '*-/_-,ky = _-_-_and k_ = _f. The coefficients %, ny and n, are· ? y · . z
non-negative integers, creating triplets of numbers, that are used to index
the eigenfunctions and corresponding eigenvalues
k,2 n_n. (nJ_ 2 (ny_r_ 2 (nz_r_ 2=\t_/ +kty] +\_; (7)
The solution of the boundary value problem given by Eq. 1 and Eq. 2 can
be obtained by using the integral equation [14]
=f /( ,xo)a (x,xo)dxo+
pl( ,x0j ]ds0 (8)Since Eq. 1 is homogenous, the boundary condition for the Green's func-
tion (4) is homogenous and using boundary condition for pressure (2), thisequation simplifies to the form
p(w, x) =/orb -iwpoGw(x, xo)vn(w, xo)dso (9)
In order to obtain numerical results from these equations, the Green's func-
tion was approximated by using only a finite number of terms in the sum-mation. The form used for the Green's function was
G (x,xo)= E 2 _ 2
Since only a finite number of terms was used in the summation, the in-
tegration in Eq. 9 could be taken inside the summations. The coordinatesystem was chosen so that the enclosure was in the positive octant of the
coordinate space and that the corner of the enclosure was in origin. The
element, or the piston, was placed on the wall that lies on xy-plane. Thedimensions of the studied enclosure were as show in Fig. 1. The circu-
lar piston element's centre was at point x_ = 125 mm, Ye = 125 mm and
z = 0 (the 'rim' position). The radius of the element was r = 75 mm. Theactual numerical computations were made using MATLAB. The integra-
tions over the elements, considered as a flat piston surface, were computed
numerically using MATLABs 'quad8' function.
The transfer function between the piston velocity vn(w) and pressureat some measurement point xm can be computed by dividing computed
pressure p(w, Xm) by velocity vn(w). In the case studied here, the frequencydependent part of the velocity can be taken out of the integral in the Eq.9, so the transfer function can be obtained from this equation by setting
vn(w) = 1. The chosen measurement point was 120 mm from the back
plate and r12c2 in the microphone mesh. The frequency response of the
computed transfer function is represented in Fig. 6 in comparison with
the corresponding measured response. The overall fit is good except the
damping of resonances since in the analytical model the walls were assumed
totally rigid. Other minor deviations could be reduced by adjustmentsof enclosure interior measures, except at higher frequencies above 700 Hz
where for example the piston assumption of driver diaphragm movement
is not valid anymore. (The deviation at very low frequencies is due to the
high-pass characteristics of the measurement microphone.)
4.2 Weakly Coupled Heuristics
A full description of the coupled vibroacoustic problem of an loudspeaker
enclosure using a Green's function expansion for both the acoustical wave
and the mechanical bending waves is numerically very inefficient due to the
large number of terms needed. However, some heuristic deductions enableus to achieve a very efficient approximation for the effect of coupling of the
lowest modes, which typically are also the only ones which are of importance
when studying the sound radiation by the loudspeaker enclosure [9, 10].Here we take as our aim to develop a first-order approximation of the
vibration, which implies the following assumptions about the mechanisms:
· The starting points for the calculations are the acoustical modes in a
rigid enclosure with locally reacting walls and mechanical modes of anenclosure in a vacuum.
· The vibrational properties of the driver (diaphragm mass, suspensioncompliance, losses, etc.) can be taken into account as a variation of
the impedance of the surface, as included in the boundary conditionsin the discussion above.
· The driving mechanism for the acoustical modes is a volume velocitysource corresponding to the driver and for the mechanical modes a
point force corresponding to the recoil of the driver.
· The coupling between acoustical and mechanical vibrations is assumed
to be weak, so only first-order coupling (acoustical --->mechanical,mechanical -+ acoustical) is taken into account. The mechanical vi-
brations of adjacent panels must be assumed to be strongly coupled.
· The field outside the enclosure can be ignored for the following reas-
ons: outside sound pressure on the surfaces is at least one order of
10
magnitude smaller than the pressure inside at the eigenfrequencies,and there is less frequency and place dependence in the field.
The bending wave equation applicable to the enclosure walls, assumedto consist of thin plates, is of the form
O2u EK 2 c94u
69t2 = P OX4 (11)
where E is the modulus of elasticity and K 2 a geometry-dependent coeffi-cient.
In a finite rectangular plate the boundary condition which is of interest
when analysing the acoustically excited vibrations is the simply supportededge which can rotate, but cannot have transverse displacement; the other
possible boundary condition, clamped edge, where the edge cannot haveeither rotation or transverse displacement has a significantly higher mech-
anical impedance, so neither the mechanical force nor the sound pressure
excite these modes as efficiently.
The simply supported boundary condition, which yields the lowest res-
onance frequency, corresponds to a situation where adjacent enclosure sur-faces move to opposite directions, thus enabling the rotation of the edge,
and there is no torque on the edge. A clamped boundary condition requires
that there is something to provide the torque needed to prevent the edge
from rotating, which is possible only if the adjacent surfaces moves in the
same direction (Fig. 7).The modes corresponding to the clamped boundary conditions also have
significantly higher frequencies than the simply supported modes, the lowest
modal frequency of completely clamped plate being about twice that of asimply supported plate [11]. The situation where clamped modes were
the most appropriate description would be the one where the enclosurevibration is driven by a homogeneous pressure field or by a purely one-
dimensional standing wave, but this situation arises only at low frequencieswhere there are no clamped eigenfrequencies.
In the following discussion we assume the enclosure to be rectangular.
For simply supported edges the modal shapes are given by equation
= A sin (m + X)_x sin (_ + 1)_y (12)L_ Ly
where z is the displacement and Lx and Ly are plate dimensions and m and
11
n non-negative integers. The eigenfrequencies are defined by equation
0.453 cLh[\ L_ / +, Ly } L_Ly
where h is the thickness of plate and cz is the speed of longitudinal wavein the plate given by equation
i E (la)CL= p(1--Y2)
where l_ is the Poisson coefficient. Thus, even if the modal shapes of thesimply supported plate are similar to those of the acoustical modes of a
rectangular cavity, the relationships between modal frequencies are funda-
mentally different, and the modal density at low frequencies (i.e., in the firstoctaves above the lowest mode) is very low as compared to the acousticalmode density.
The assumptions made of the modal shape have implications on thecoupling between various modes. If losses are small, thc vibration can be
efficiently transmitted from one enclosure surface to another at the edge
only if modal indices corresponding to the wave component along the edges
are equal. Similar orthogonality is valid also for the coupling of mechan-ical modes corresponding to the simply supported edges and the acoustical
modes. Bending modes corresponding to other edge boundary conditions
need also hyperbolic functions for their description, so their inner productwith the acoustical modes is non-zero also for unequal modal indices, so all
the modes have some, although small, coupling. However, as stated earlier,
the practical significance of these modes is small.The plane wave decomposition for the sound field inside the enclosure
enables the enclosure surface vibrations to be described as a superposition
of vibrations excited by plane waves. To achieve this we must determine the
impedance of each mode and the force distribution caused by the incident
wave. The vibration velocity can be then formally written as their quotient
[12]:
m Zm
The modal impedances can be determined by writing the standing wave
as a superposition of travelling waves in opposite directions, and using thisdecomposition to describe the modal impedance as a sum of travelling-wave
impedances. The exiting force is given by the sound pressure.
12
The importance of the orthogonality of the coupling is that instead ofusing the full Green's function expansion for both the acoustical and mech-
anical modes, it is necessary to only determine the modal frequencies of
the bending wave modes with the modal indices corresponding to the mostsignificant acoustical modes.
Another interesting problem, discussed here only briefly, is the effect
of the coupling to the modal frequencies. An analysis of losslessly coupled
simple vibrating systems [13] indicates that coupling increases the frequencydifference between the two resonators. Similar results hold also for systems
consisting of coupled acoustical and mechanical standing waves. The coup-
ling strength, which determines the amount of change in the resonancefrequencies, can be determined from the bandwidths and initial frequencydifferences of the resonances.
5 VIBROACOUSTIC MODELING TECHNIQUES
Numerical modeling is, in principle, a solution to any problem that can
be formulated precisely enough. There are, however, limitations of compu-tational resources such as finite memory, accuracy, and computation time
that restrict the applicability of element-based numerical methods. Al-
though they are becoming more and more relaxed with rapid development
of computer hardware, yet they will remain one of the limiting factors.Another and in practice a very important restriction is the accuracy of
available material (and structure) parameters. Acoustic properties of ab-sorbent materials are seldom known precisely, and even less information is
available about the dynamic parameters of enclosure construction materi-
als. These materials are not very homogeneous so that the variation range
of parameters should be known, not only the values from a single samplemeasurement.
In this chapter we will present the basic principles of techniques forelement-based modeling that have been applied in our study. These meth-
ods include the finite element method (FEM), the boundary element method
(BEM), and the finite difference time domain method (FDTD), especiallyits waveguide mesh formulation.
13
5.1 Finite Element Method in Acoustics
The finite element method (FEM) is a popular method for solving partial
differential equations (PDEs). A PDE is transformed into an integral equa-tion, the solntion domain f is discretized with a mesh and the solution is
approximated at the nodes of the mesh by means of element functions.
5.1.1 Derivation of FEM
In this chapter a finite element method for solving the following PDE ispresented. The internal acoustic field of a loudspeaker box is modeled by
using inhomogeneous Helmholtz equation (16). The MDF walls of the boxare acoustically very hard and they have been modeled using an impedance
boundary condition (17). Because the simulation is carried out at low
frequencies, the loudspeaker element can be modeled as a simple piston by
means of a velocity boundary condition (18).
V2p + k2p = O, x _ f (16)Bp -piw
On - Z(w) p' x _ Of! (17)BpOn -- piwv,, x E 092 (18)
Here p is fluid density, Z(w) acoustical impedance and v, normal velocky.
From now on, boundary conditions (17) and (18) are combined as
Bp -piwcon-- Z 'p- pi_vv' p e Of (19)
It should be noted that in this notation both Z and v are functions of place
and frequency, Z = Z(x,_v) and v = v,(x,_v). Instead of looking for anexact solution of this equation and the associated boundary conditions an
approximate solution is searched for. First the so called weak formulation
of Eq. 16 is derived [15]: both sides of the equation are multiplied with an
arbitrary test function w _ V, where V is a suitable function space, in fact
H _(fi) [17], and then the equation is integrated over _:
+ = 0 (20)After Green's theorem [17] is applied
14
The boundary condition can now be taken into account, which leads to anintegral equation
/pice )fa(-Vp. Vw+k2pw)d_-]oa_-_-p+icov wdF=0 (22)
The original PDE can now be replaced by its weak formulation. An N-
dimensional subspace of V, VN, is chosen and the weak formulation of PDE
is projected into this subspace. This means that f_ is divided into finite
elements and the geometry of the domain is described with the vertices of
the elements. Then a basis function _biof subspace VN is chosen for each
node xi, i = 1,..., N, such that _i(xi) = 1, ¢i(xj) = 0, j _ i. The solutionfor Eq. 22 is approximated as
N
p _ Z pj(ce)_j(x) (23)j=l
This is the Galerkin method. Because Eq. 22 is valid for all the test
functions v, it is also valid for basis functions qbj. Substituting the trial Eq.23 and basis functions into Eq. 22, a system of linear equations to solve
the unknowns pi, (j = 1,..., N) is obtained:
N
Z [fo(-V(pj,j)vi+k2(pj j) ,)dig]j=l
- \zpj_j + =j=l
and after simplification
j=l_ [f_(V_j. _i--k2(/_j_i)pjdig "]- fo_l Piceq_JqJiPjdPz ]
= fOa2 icevq_i dr (25)
5.1.2 Matrix representation of FEM
Let us define the following acoustic mass, damping and stiffness matrices
M = (M/j), C = (C/j), K = (Kij), source vector F = (fi) and pressurevector P = (Pi) respectively as:
= fa V4i. V¢j digKij
15
Cij = _al P2_i_j dF (26)
fi = faa2 v_i dF
Eq. 25 can now be expressed in a matrix form
NP + iwCP - co2MP = ipwF (27)
This is a system of linear equations which can be solved for pi, i = 1,..., N
on all frequencies of interest using standard linear algebra.
5.2 Boundary Element Method
The boundary element method (BEM) is another approach to solving PDEs.The PDE is transformed into an integral equation which consists of bound-
ary integrals only. As a consequence, the three-dimensional acoustical prob-
lem is reduced to a two-dimensional one. When the problem is discretized,a system of linear equations is obtained. In addition to boundary nodes,
BEM can be used to calculate the solution for Eq. 16 in an arbitrary pointof region a.
5.2.1 Derivation of BEM for Acoustic Problem
In this chapter a direct boundary element method, or collocation method,
for solving the Hehnholtz equation (16-18) is presented [18], [19].
Assuming that the equation has a solution in a, the following integralequation is valid for all functions p* regular enough:
fn(V2p+ k2p)p*da = 0 (28)
Applying Green's theorem leads to
f (v"p + k2p)p*da
Opp,_-/ok2p ,da+ c r-fow w*dx (29)
Applying Green's theorem once more results in
_o..O*
_ rjaVp. Vp* da = - j_oap_ n dP +/apV2p ' da (30)
16
Based on Eq. 29 and Eq. 30 the basic integral equation of BEM can bewritten as:
_,(POP*_nnOnnpop ,\)fa (V2p* + k2p*)pd_ = _a dF (31)
Next p* is chosen to be the Green's function of the differential operatorV 2 + k2. This means that p* is the solution of the PDE
V2p * + k2p* = -507 ) (32)
in an infinite domain, 5 is the Dirac delta function. Because of the proper-ties of the delta function, Eq. 31 can be used to calculate a value for p, at
any point r/E _h
_,(POP*_nn_nnOpp*hp(v)= - / (33)In the theory of BEM the Green's function is often called the fundamentalsolution. For three-dimensional Helmholtz operator the fundamental solu-
tion associated with point r/is
e-lkll_-xll
p;(x) - 4_11v _ xl[ (34)
It can be assumed that p is almost constant in a small neighbourhood of the
boundary point _, Uc(_), and the left side of Eq. 31 can be approximated:
ffi(Vp*( )+ k2p*(_) )p(_) df_ _ C(_) (35)
where
¢(_) = lim_0Ju_/"(e)(V2p* (_) + k2p*(_))p(_) d_ (36)
If the boundary is smooth enough, it can be proved that C(_) = -½p(_).At points _ of boundary I' = 09 can Eq. 31 be formulated as:
Op*l_p(_) + _P_n dI'= _ _-_PnP*dI' (37)
This equation consists only of values of p and its normal derivative atthe boundary I'. This equation is solved in the same way as in FEM:
the boundary is approximated with surface elements and the solution issearched at nodes. Let 0_2 be divided into N disjoint parts:
N
=Zr, (38)i=1
17
For simplicity, the following notation is used:
ap,Q, Op* (39)At every boundary point xi Eq. 37 is
12p(xi) + _ /r pQ* dF = _ fr. Qp* dF (40)j=l J j=l J
Here p* is the fundamental solution associated with node xi and Q* =op.[ respectively. It depends on the selected boundary elements how theOn
integrals over boundary parts Fi are computed. If constant elements areused, Eq. 40 becomes simpler:
N N
l_p(xi) + Zp(xJ)/r Q;dF= Z Q(x,)/r p;dF (41)j=l J j=l -/
Finally, the boundary condition (19) is substituted into Eq. 41:
N N (_pi w ),l_p(xi)+ _ p(xj)fr Q? dF = _ --_-p(xj)- piwv frjpdr (42)j=l J j=l
and the system of linear equations is obtained (i = 1,..., N):
_p(xi) N ( piw ,\ Ar+ Y_.p(xJ) fr Q_ +-_-pij dC= _-piWV/r P_ dP (43)j=l J j=l J
5.2.2 Matrix representation of BEM
With the notations (i,j = 1,..., N)
piw ,\Hij -- fr i (Q* +-_-pij dP, i _ j
Hii fPi (Q* piw ,\ 1 (44)= + -Fp,) ar +N
fi = E-vk p'dFj=l at j
Pi = p(xi)
Eq. 41 can be written in matrix form
HP = piwF (45)
Using the fundamental solution Pl and the values of P Eq. 16 can be solvedat every inner point r/6 _:
N 1 v_Ofp:dP]f'r ]P(tl) _-' - r [p(xJ) /r Q: + Piw (_p(xj) + (46)j=l
18
5.3 Coupled FEM/BEM
When the coupling between the structure, i.e., the loudspeaker box andthe acoustic field is taken into consideration, the situation becomes more
complex. FEM can be used to simulate the vibration of walls under acous-
tical excitation and this structm'al model can be coupled with acoustical
FEM/BEM model. In practice the coupling is described with a couplingmatrix T. Here a structural FEM model has been coupled with an acous-tical FEM model
/fsU - w2MsU = -TP (47)KP + iwCP - w2MP = piwF + pw2TTU
Ks and /VI_are the structural stiffness and mass matrices and U is the
structural displacement vector. When the structural FEM-model is coupled
with an acoustical BEM model, the following system of matrix equationsis obtained:
KsU - w2MsU --- -TP (48)HP piwF + pw2TrU
5.4 Finite difference schemes
Finite difference time domain (FDTD) methods are found a possible solu-
tion for acoustic problems such as room acoustics simulation [20, 21]. Here
we study its applicability to loudspeaker modeling.The main principle in the finite difference methods is that derivatives
are replaced by corresponding differences [22]. There are various techniquesavailable but for the wave equation it is suitable to use the so called center-
scheme, such that
dp(t) _ p(t + At) - p(t - At) (49)dt - 2At
For this purpose the wave equation is presented in the time domain:
a 2P (50)c2V2P=After applying the previous discretization technique (shown in Eq. 49)
twice both for space and time, in a one-dimensional case Eq. 50 results
into the following form:
p(x + Ax, t) - 2p¢, t) + p(x - Ax, t) cZ = (51)Ax 2
p(x, t + At) -- 2p¢, t) + pC, t - at)At 2
19
where the sound pressure p is a function of both time and place. This
scheme call easily be expanded also to higher dimensions. Spatial di-mensions mw by separated and discretized individually. Thus in a three-
dimensional case, to the left-hand side of Eq. 51 similar terms are addedconcerning spatial differences Ay and Az.
The difference scheme in Eq. 51 is explicit. In practice it means that
the sound pressure values for the next time step t + At can be calculatedpurely from the data of time t and earlier.
As the finite difference schemes are often calculated in the time domain
the results can be visualized easily and the propagation of wavefronts in
the space under study are clearly seen. Another advantage of time domain
calculation of impulse responses is the ability to use the results directly for
auralization purposes, i.e., the simulation results can be easily listened to.
There are also drawbacks in the finite difference schemes. Traditionallythe space discretization has been done such that resulting elements are cube
shaped in a rectangular mesh. That causes both dispersion and magnitude
error at higher frequencies. Due to that limitation the valid frequency range
of the FDTD method is somewhat lower than in the corresponding FEM.In practice for an FDTD grid at least 10 nodes per wavelength are needed.
5.4.1 Waveguide Mesh Method
The waveguide mesh method is an FDTD scheme. Its background is in
digital signal processing. The method was first developed for physical mod--eling of musical instruments [23]. The method is computationally efficien_
and with one-dimensional systems, such as flutes or strings, even real-time
applications are easily possible [24, 25].
A waveguide mesh is a regular army of discrete space digital 1-D wave-guides arranged along each perpendicular dimension, interconnected at
their crossings as illustrated in Fig. 8 which represents a two-dimensional
waveguide mesh. Two conditions must be satisfied at a lossless junction
connecting 2N lines of equal impedance [26]:
1. the stun of inputs equals the sum of outputs, (flows add to zero),/=2N /=2N
E = E p;- (52)i=1 i=1
2. the signals in each crossing waveguide are equal at the junction, (con-tinuity of impedance).
Pi = Pi, Vi, j (50)
20
where p/+ represents the incoming signal in the digital waveguide i and pi isthe outgoing signal in the same waveguide. The actual value of a waveguide
is the sum of its input and output.
Pi = P+ + Pi (54)
Since that value is the same in all waveguides connected to the node this
value is also the value of the node p. The digital waveguide between two
nodes implements a unit delay, such that what goes out from a waveguide
gets in to its opposite end at the next time step.
P_ ('_) = P_,opposin_(n - 1) (55)
Based on these conditions a difference equation can be derived for the nodes
of an N-dimensional rectangular mesh:
1 2N
pk( )= - l) - - 2) (56)where p represents the sound pressure at a junction at time step n, k is the
position of the junction to be calculated and I represents all the neighbors
of k. As one can see in this formulation the incoming and outgoing signals
(p+,p-) have been eliminated and only the actual value p of a node isneeded.
This waveguide mesh equation is equivalent to a difference equationderived from the wave equation by discretizing time and space as shown in
Eq. 51. The discretization is done such that At = 1 time step, Ax = Ay =
Az = 1 grid unit and the wave propagation speed:
1 Ax
c- at (57)The real update frequency of a three-dimensional mesh is:
CrealV_
A- dx (5s)where c_.,at represents the speed of sound in the medium and dx is the
actual unit distance Ax between two nodes. That same fi'equency is also
the sampling frequency of the resulting impulse response.Boundary conditions are presented as relative impedances to the air
such that impedance 1 represents the impedance of air corresponding to an
opening. Another choice for setting the boundary conditions is by using
21
digital filters as presented in [27]. In Fig. 8 the boundaries are filters
having a transfer function H(z). When compared to FEM models this isan advantage especially in more complex cases where non-linear or time-variant boundaries are needed.
A more detailed study on deriving the mesh equations and boundary
conditions is presented in [28].In the waveguide mesh method the error caused by cube-shaped ele-
ments (as described in the previous section) can be reduced, e.g., by using
tetrahedral elements [29] or some interpolation technique [30]. In this studywe have used cube-shaped elements.
6 COMPARISON OF MEASURED AND SIMULAT-
ED BEHAVIOR
In this section we show the results of simulating the behavior of the closedbox loudspeaker using element-based numerical modeling tools.
6.1 FEM and BEM simulations
The internal acoustic field of the loudspeaker box has been simulated byusing both finite and boundary element methods. All calculations have
been carried out using vibroacoustic software SYSNOISE, rev. 5.3, and a
300 MHz DEC Alpha workstation. FEM models have 702 nodes and BEMmodels have 394 nodes. Both models use the same mesh but for BEM
all internal nodes have been removed. It is often required that an elementmesh should have at least six nodes per wavelength. In this case the models
used are valid up to 1100 Hz.
Both FEM and BEM simulations of an empty enclosure, Figs. 9 and10, show accurate results in comparison with measured behavior of the
loudspeaker, except at frequencies above 600-700 Hz. Magnitude responses
in a point 120 mm from back plane and mesh position r12c2 (see Fig. 2)are shown.
Although it is possible to use FEM modeling of absorbent materials in
SYSNOISE, it did not yield satisfactory results and instead the damping
material has been modeled using impedance boundary condition (Eq. 17).
It was necessary to use frequency-dependent impedances, which were tunedby hand. Figure 11 shows that the simulation is fairly successful with the
5 cm wool case, but the stronger damping effect of the 10 cm wool (not
22
shown here) might need another approach. In addition to the backplane ofthe cabin, the absorbing boundary condition was used on the side walls to
the height of the mineral wool.
The coupling effect between structure and fluid was not completelymodeled. The vibration of a side plate in an undamped enclosure due to
interior sound field was simulated using SYSNOISE. Three kinds of plate
support principles were tried: clamped, simply supported, and free edges.
None of these yielded results accurate enough. In section 4.2 simple support
was suggested but our simulations matched best to measurements (see Fig.
12) when the free-edge assumption was applied, which is not easily mo-tivated. The damping of structural vibrations was also found inadequate.
Thus the boundary conditions and the material parameters for the struc-tural FEM model as well as mechanical coupling from the driver element
need further attention. However, it seems possible to use coupled element
modeling techniques with these simulations.
6.2 Waveguide mesh simulations
The simulations made in this study used a three-dimensional waveguide
mesh covering the interior space of the loudspeaker. Currently this method
is capable of simulating only uncoupled systems, and thus only the insidesound field of the loudspeaker was simulated.
The loudspeaker was modeled as a rectangular cabinet and the element
was a cylinder acting like a piston sound source. The simulations weremade with a 10 mm spatial discretization resulting in ca. 65.000 meshnodes. The simulations were carried out on an SGI Octane workstation.
In Fig. 13 an example of visualized time-domain simulation is shown.
The figure presents a two-dimensional slice inside the enclosure 230 mmfrom the back plate. The excitation has been a Gaussian pulse and the
driver element was located at the rim position. In the figure the primary
wavefront is approaching the bottom of the cabin and behind that the firstreflections from the side walls can be seen. Using the FDTD method it
is easy to visualize the temporal evolution of the sound field inside and
outside the loudspeaker cabinet.
Figures 14 and 15 show the results of waveguide mesh simulations of the
loudspeaker interior in the same point that was used in section 6.1 for FEMand BEM. Boundary conditions were varied so that in Fig. 14 the empty
loudspeaker was given relative impedance of the walls set to frequency-
23
independent value of 100. The results match the measured magnituderesponse fairly well up to about 550 Hz.
When absorption material was added the relative impedance of the back-plane of the enclosure was changed to be close to 1 depending on the thick-
ness of the absorbing material. At the same time also the location of the
backplane ;vas changed such that it was at the height of the surface of the
mineral wool. Figure 15 shows the simulation result when 50 mm thickmineral wool was at the back wall. The response curve shows a useful
match with measured response up to about 350 Hz.
7 DISCUSSION AND CONCLUSIONS
The aim of this study has been to apply element-based vibroacoustic sim-
ulations to the modeling of a closed-box loudspeaker in order to test theapplicability of these methods in loudspeaker design. Simulation results
are validated by comparing them with measured behavior of a real speaker.
Good agreement of modeled and measured as well as semianalytic results
in simple configurations, such as in Figs. 6, 9, and 10, confirms generalvalidity of the approach and the measurements.
First the measurement system, especially the electret microphone array,
was found very important for obtaining extensive and reliable data. Thisdata is stored on a CD-ROM for further experiments and analysis work.
The first observation is that semianalytic and simplified models may
yield surprisingly accurate simulation results if the enclosure is simple
enough, as shown in Fig. 6 and ref. [10]. The second finding was that allelmnent-based methods yielded accurate enough internal sound field simu-
lations at low frequencies (below 500-600 Hz) for an undamped enclosure.The anomalies at higher frequencies may be due to inaccuracies of model
parameters, the non-piston behavior of the driver element, the relatively
small size of FEM/BEM-meshes (5 cm discretization), and the fact thatthe magnet of the driver was not included in the models.
FEM and BEM based models outperformed the accuracy of the wave-
guide mesh (difference method) in these simulations. In principle the wave-guide mesh should be as accurate up to frequencies of about 5 kHz. One
problem with this method was the regular mesh with 1 cm discretization
whereby points of computation (including the point of observation) werediscretized to the nearest available point in the mesh.
The sinmlation of a more realistic case, that of a damped enclosure
24
(Figs. 11 and 15) were not as accurate. We did not succeed to use SYS-NOISE in a straightforward way of modeling the coupling of the air space
and the damping wool. The wool was given as an equivalent impedancecondition and its placement only on one wall required some hand4uning
of the model parameters to obtain a fairly good match to measured sound
field. The FEM/BEM method yielded better results than the waveguidemesh method since it had a frequency-dependent complex impedance of
the damping material, while the waveguide mesh technique used a simple
resistive impedance.The vibration behavior of the enclosure plates due to driver excitation
was simulated by the FEM/BEM method in SYSNOISE. Some of the sim-
ulations were encouraging although the results are not accurate enough
and the model only partially included the vibroacoustic couplings within
the system. This problem is important since the wall vibrations at lowfrequencies are of interest to the overall radiation of the loudspeaker. Es-
pecially the lowest modes should be simulated accurately enough.The internal sound field is of less interest in a closed box than in a
vented box structure since it can radiate only through walls or due to driver
cone loading. The simulation of the latter effect requires that the acoustic
impedance matching of the driver and the enclosure is known.
Finally, only the external sound field is of interest to the listener. It canbe solved with the element methods if the driver cone behavior as well as
wall vibrations are modeled accurately enough. In this study we did not try
this since the piston model of the driver diaphragm is not accurate enoughand the wall vibrations are not known precisely enough.
The computational efficiency of the simulations was fairly good: a typ-
ical run-time was about 15 minutes for the uncoupled FEM/BEM problems
and a few minutes for waveguide methods. Although even faster simula-tion would be wished for fast experimentation, such a speed is certainlytolerable.
As a conclusion, the simulations with element-based models in our study
have shown that the vibroacoustic behavior of a relatively simple loud-speaker can be simulated precisely and efficiently enough for the sound
fields. Yet some crucial behavior, especially the mechanical vibration of
walls and tile detailed behavior of the driver cone are not detailed enoughto be useful. Our next step is to develop these crucial parts further, to ex-
tend the modeling to vented box loudspeakers, and to compute the external
sound field based on this knowledge.
25
REFERENCES
[1] Bank G., and W,'ight J., "Loudspeaker Enclosures" in Loudspeaker and
Headphone Handbook, (ed. J. Borwick), Focal Press, Oxford, 1997.
[2] SYSNOISE home page: http://www, lmsintl, com.
[3] I-DEAS (SDRC) home page: http://www, sdrc. corn
[4] Comet/Acoustics home page: http://www, autoa, corn
[5] ABAQUS home page: http://www.hks, corn
[6] ANSYS home page: http://www, ansys, corn
[7] MSC/NASTRAN home page: http ://www.raacsch. corn
[8] Karjalainen M., "DSP Software Integration by Object-Oriented Pro-
gramming, A Case Study of QuickSig," IEEE ASSP Magazine, April1990.
[9] Backman J., "Effect of Panel Damping on Loudspeaker Enclosure Vi-bration,'' AES 101st Convention, Los Angeles, Nov. 8 - 11, 1996, pre-
print n:o 4395 (C-5).
[10] Backman J., "Computing the Mechanical and Acoustical Resonancesin a Loudspeaker Enclosure," AES 102nd Convention, Munich, March
22 - 25, 1997, preprint n:o 4471 (J6).
[11] Blevins R. D., Formulas for Natural Frequency and Mode Shape, VanNostrand Reinhold Company, New York, 1979, pp. 258 & 261.
[12] Fahy F., Sound and Structural Vibration, Academic Press, London1985, pp. 221 - 226.
[13] Richardson E. G., Technical Aspects of Sound, vol. I, Elsevier Publish-
ing Company, Amsterdam, 1953, pp. 3 - 4.
[14] Morse P. M., and Ingard K. U., Theoretical acoustics, Princeton Uni-versity Press, USA, 1965.
[15] Zienkiewicz O. C., and Taylor R_ L., The Finite Element Method Vol.i: Basic formulation and linear problems, 4. ed. McGraw-Hill, London,1989.
26
[16] Zienkiewicz O. C., and Taylor R. L., The Finite Element Method Vol. 2:Solid and fluid mechanics, dynamics and non-linearity, 4. ed. McGraw-Hill, London, 1991.
[17] Schatz A. H., Thom_e V., and Wendland W. L., Mathematical Theory
of Finite and Boundary Element Methods, (DMV Seminar; Bd. 15),Birkh/iuser, Basel, 1990.
[18] Brebbia C. A., Zelles J. C. F., and Wrobel L. C., Boundary ElementTechniques: Theory and Applications in Engineering, Springer Verlag,Berlin, 1984.
[19] Brebbia C. A., and Ciskowsld, R. D. (ed.) Boundary Element Methods
in Acoustics, Computational Mechanics Publications, Southampton,1991.
[20] Botteldooren D., "Finite-difference time-domain simulation of low-frequency room acoustic problems," The Journal of the Acoustical
Society of America, vol. 98, no. 6, pp. 3302-3308, 1995.
[21] Savioja L., Backman J., J/irvinen A., and Takala T., "Waveguide meshmethod for low-frequency simulation of room acoustics," in Proc. 15th
Int. Congr. Acoust., Trondheim, Norway, June 1995, vol. 2, pp. 637-640.
[22] Strikwerda J., Finite Difference Schemes and Partial Differential
Equations, Wadsworth&Brooks, Pacific Grove, CA, 1989.
[23] Smith J. O., "Physical modeling using digital waveguides," ComputerMusic J., vol. 16, no. 4, pp. 74-87, 1992 Winter.
[24] Viilim/iki V., Huopaniemi J., Karjalainen M., and Janosy Z., "Physicalmodeling of plucked string instruments with application to real-time
sound synthesis," Journal of the Audio Engineering Society, vol. 44,no. 5, pp. 331-353, 1996 May.
[25] Jaffe D., and Smith J. O., "Extensions of the Karplus-Strong plucked
string algorithm," Computer Music J., vol. 7, no. 2, pp. 56-69, 1983
Summer, Reprinted in C. Roads (ed.), The Music Machine (MIT Press,Cambridge, MA, USA, 1989), pp. 481-49.
27
[26] Van Duyne S., and Smith J. O., "Physical modeling with the 2-d
digital waveguide mesh," in Proc. 1993 Int. Computer Music Conf.,Tokyo, Japan, Sept. 1993, pp. 40-47.
[27] Huopaniemi J., Savioja L., and Karjalainen M., "Modeling of reflec-
tions and air absorption in acoustical spaces -- a digital filter designapproach," in Proceedings of IEEE 1997 Workshop on Applications of
Signal Processing to Audio and Acoustics_ Mohonk, New Paltz_ NewYork, Oct. 19-22 1997.
[28] Savioja L., Karjalainen M., and Takala T., "DSP formulation of a
finite difference method for room acoustics simulation," in Proc. 1996
IEEE Nordic Signal Processing Symp., Espoo, Finland, Sept. 1996, pp.455-458.
[29] Van Duyne S., and Smith J. O., "The tetrahedral digital waveguide
mesh," in Proc. 1995 IEEE Workshop on Applications of Signal Pro-
cessing to Audio and Acoustics, New Paltz, NY, USA, Oct. 1995.
[30] Savioja L., and V_limSki V., "Improved discrete-time modeling of
multi-dimensional wave propagation using the interpolated digitalwaveguide mesh," in Proceedings of the International Conference on
Acoustics, Speech and Signal Processing, Mfinchen, Germany, April19-24 1997, vol. 1, pp. 459-462.
28
The element is centralized on the front plate.
In the case of rim speaker, the dimensions
are according to next picture. [ _ 120,/ 1_4t _.,]
The material of the plates [ _ --'is 20mm MDF.
I \_ { \
x speaker
5/ _ 110
speaker (rim case)
The inner dimensions of the cabinet are 600mm, 400mm and 250 mm.
Figure 1: Mounting of the loudspeaker (asymmetric position) and the closed
box loudspeaker with microphone array and damping wool inside.
29
Microphone grid:
· o -_ _ _ _,_ distance between .o -_ 1J -" _ :2microphonesis 40mm r0c0
a · · _, · a 7
'4
r14c5 _metal frame single microphone
Figure 2: The structure and dimensions of the electret microphone array.
Electret microphone capsulesr ....... 2 + Bias
_- _ _ _}- _ _-;°_'_ Preamplifier
....: o "_ o-: \ o
__ _ Digitallycontrolledanalog switches
Figure 3: Principle of the electret microphone array, the multiplexer, and
the amplifier circuit.
30
0.05 a) undamped 0.05 b) damped,kl,
0 _ ......................... 0 _'
-0,05 -0.05
0 0.1 0,2 0,3 sec 0 0.1 0.2 0.3 sec
dB I I _l I I I I I I
/_\ A A A C) undamped
20 __a nd_k_ /__/_'__ /__
-20
I I I I
100 200 300 400 500 600 700 800 900 f/Hz
dB i i J i i i i i i
sand d)damped
20-_ e_
0 fr
-20
I I I I I I I I I
100 200 300 400 500 600 700 800 900 f/Hz
Figure 4: Examples of responses inside the enclosure: (a) and (b) impulse
responses from the electric excitation of the driver element to sound pres-
sure in microphone mesh point r14c5 (see Fig. 2), 12 cm from back plane
for (a) empty enclosure in sand and (b) damped enclosure with 10 cm wool,
in sand; (c) magnitude response in the same position for undamped enclos-
ure in sand and free standing; (d) magnitude response in the same position
for damped enclosure in sand and free standing.
31
0 200 400 600 800 1000 1200
I I I I
........................ z .........................
-20 _ i L
0 200 400 600 800 1000 1200
40 , I
-200 200 400 600 800 1000 1200
Figure 5: Results of vibration measurements; top: accelerance (accelera-
tion/force) of an isolated side plate with impact test at a corner of the plate;
middle: accelerance of the side plate of the freely supported undamped en-
closure at point 260 mm from front panel and 297 mm from top plate
(driver side); bottom: side plate veloc!ty at the same point, as a responseto electrical excitation of the driver element.
32
' ' I' ' I '1 I ' , I ,l
! ' ! ' i
II I II I ii ii I i!
,I i!
II
ii
0 1O0 200 300 400 500 600 700 800 900 Hz
Figure 6: Analytic modeling: magnitude response (diaphragm velocity tosound pressure) in point 120 mm from back plate and r12c2 in the micro-phone mesh in sand-supported undampedenclosure as measured (solid line)and computed from analytic modeling with rigid walls (dash-dot line).
Figure 7: Lowest vibrational mode corresponding to a clamped (left) anda simply supported (right) boundary condition.
33
H(z_ ! _.r'--"_ _ ,
t t t
q
Figure 8: A two-dimensional waveguide mesh, which consists of one-
dimensional digital waveguides interconnected at their crossings. At bound-
aries there are filters H(z) implementing the reflection characteristics ofeach surface.
34
dB
o 'A .... .......... ,,............. :..:,..̂........,o 'I..._..I,...[A.._........._,.̂....._..,.l...r_/.......
........, _!1.:_ .,/il _1:i..i,,.\-20
'1" "!! l.-30 ............ -- ....
' : i (!_{'' .!! r-40.............. i........... i........ :............. !...... : .......
._o.......................i.......i.....................'ii'_/': ,'6(_(30 200' 360 4()0 500 600 760 800 9()0 HZ
Figure 9: FEM modeling: magnitude response (diaphragm velocity tosound pressure) in point 120 mm from back plate and r12c2 in the micro-phone mesh as measured in sand-supported undampedenclosure as com-puted from FEM modeling (solid line) and measured (dash-dot line) re-sponse.
dB' _! _ , !
! '_: /_ - !i)_'. i' i : :o ;...._:... _ . . i......... ; ............... ! ....... i.......
.' , _ :: ,-" ,_ - -,I-.; ....... : ,I........ : ......
: . ', :r i I'\'
-20 . _.. . ..,. .......... .i ..
-_o :: !\ ! !-40 : : ......i i j....
.-_o......;............................. i...... i ...I.i ................
_B ' ' ' ' '-6 0 260 3()0 460 500 600 700 800 900 Hz
Figure 10: BEM modeling: magnitude response (diaphragm velocity tosound pressure) in point 120 mm from back plate and r12c2 in the micro-phone mesh as measured in sand-supported undamped enclosure as com-puted from BEM modeling (solid line) and measured (dash-dot line) re-sponse.
35
dB _ _ _ _ _ _
-10 ......... / '...... '.il ..... '......... : ....... i ............... .........: / !i _ ! : i !
.......... / ' ..... i' ...... i........ :........ i........ i .......i ' ii i ! ! i
-20 ,. '.'_'"¢' .... i_ '"!....... i........ ' ........ '........i i" i i i i
..... ! ........ ". ' )......... i........... i........... :.........! ! _.' ' i /.^
-30..... : ..... :....._, : : : . ........ :._.. ..:
/
: ......... : ..... /::: ......... :........-40........................... :...
....... _..... i ....... _...... :..... i ........ i ........ i ...... i ........
i i i i i i i1O0 2[)0 300 400 500 600 700 800 900 Hz
Figure 11: FEM modeling: magnitude response (diaphragm velocity to
sound pressure) in point 120 mm from back plate and r12c2 in the micro-phone mesh as measured in damped enclosure with 50 mm mineral wool as
computed fi'om FEM modeling (solid line) and measured (dash-dot line)response.
_' :. .....':"' i ....i'i ....... . ...................
.... i/ :'i:" ...."'1 I' ' : I :i ' '
-30Ii.- ........... :'_" I
_:'_!....t.!?,'./! _ !¢I
'501 250 3()0 400 5()0 6()0 7()0 8()0 950 HZ
Figure 12: Vibration modeling: magnitude response, diaphragm velocityto side wall velocity, at point 260 mm from front plate and 297 mm fi'om
top plate (driver side) in undamped enclosure: computed (solid line) and
measured (dash-dot line) response.
36
.'!"....... . ....
. '..,
,'.· . .: ·... : .. ,..........; .,': '"....:. .....
"i,,' i : ..........
. .,..' ...,.".." "' ' '..', '.'.'.·.' .''......
;.. ".,....../i·i .' ......._: i :!.... ?:
· ' · '" "'" '·""''".i
! ? _............i i: ..... ......... /_o
1 ' ._' !.'....... ..:'. ." .: .........
oq...i' i.",'_?_ ..' ." .........I ..":...... ,.';;,!_f_,_ :.......... ..' /- 0.2
-l_J,' ..... :,, :7_?_:_!4 ,..' :,'.... ,.,
.'...... ;::': '::....... ,. .." ._'/0.3
.? '..:?'.....,.., ..'" '.
-0,1 ' " '.:"'. ,..../n,5
0,2
0.3
Figure 13: Finite difference time domain simulation of a wavefront insidethe enclosure. A two-dimensional slice at the height of 230 mm from back
plate is visualized. The primary wavefront is approaching the bottom ofthe enclosure and behind that the first reflections from the sidewalls can
be seen.
37
dE ..... ! !
.,o......_....l..X,........f..::_.._:,..:...,...........,:_............l_,..v,.,..,.[.....̂...' / i¥ ' ! \,_l,,: :,' : I/ _: ':I II
._o...._. .-.x:,..........i..._._:::_...,..>,.._^......:,7.......i_...I.,\.
-4o .......:.........i W!..:.........:......!...........i.(...,'--.........:..._.......
-so.........i........;.........:......................................v..:......................: i .
'6hi0''_0 i i i200 300 400 5()0 600 700 O00 O00 HZ
Figure 14: Waveguide mesh modeling: magnitude response (diaphragmvelocity to sound pressure) in sand-supported undamped enclosure in point
120 mm from back plate and r12c2 in the microphone mesh as computed
from modeling (solid line) and measured (dash-dot line) response.
dB !
0 ......... ; ................ - ...............................................
; /% : ; /'1
-lo ..... : / ........ / ....... !....... _..........;..........;.........>.': /! i i !
-2o..... :...... !",' '.'"::.......... i .........::........ ::....i ,: :_'!_:__<",_- _'_
_30
:1 /
-40 ..... !..... !.. ;......................;....................i...........
'5_ O0 i i i200 300 400 500 600 700 800 900 Hz
Figure 15: Waveguide mesh modeling: magnitude response (diaphragm
velocity to sound pressure) in point 120 mm from back plate and r12c2in the microphone mesh in damped enclosure with 50 mm mineral wool as
computed from modeling (solid line) and measured (dash-dot line) response.
38