Determination of optimal exciter deployment for panel speakers using the genetic algorithm

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SOUND AND VIBRATION

www.elsevier.com/locate/jsvi Journal of Sound and Vibration 269 (2004) 727–743

Determination of optimal exciter deployment for panel

speakers using the genetic algorithm

Mingsian R. Bai*, Bowen Liu

Department of Mechanical Engineering, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsin-Chu 300, Taiwan, ROC

Received 29 August 2002; accepted 23 January 2003

Abstract

A large multi-exciter panel speaker has been constructed in this work. In order to achieve the best design, an optimization procedure using the genetic algorithm (GA) has been developed. A total numerical model was first established for simulation, where the electrical system, the mechanical system, and the acoustical coupling in the panel speaker are accounted for within a coupled framework. Performance indices including the frequency response, the sound power, and the directional response are calculated. The simulation model also serves as the basis for the optimal design that aims at achieving omni-directional responses at high efficiencies. A GA-based optimization scheme was exploited to search for the positions of exciters and the delays of input signals which render the optimal performance. The optimal design was verified by experimental investigations. The results indicate that the optimal configuration indeed produced better performance in termsof efficiency and omni-directionality than the non-optimal one.

1. Introduction

Panel speakers have attracted some research interest in recent years. The basic structure of a panel speaker generally consists of a panel and one or several inertia exciters (Fig. 1). The advantages of panel loudspeakers compared with conventional loudspeakers are compactness, omni-directionality, linear on-axis, attenuation, insensitivity to room conditions, bi-polar radiation, good linearity, and so forth [1]. Potential applications of panel speaker encompass multimedia, high-ﬁdelity audio systems, public addressing systems, projection screens, LCD monitor speakers, and so forth.

*Corresponding author. Fax: +886-3-5720634. E-mail address:[email protected] (M.R. Bai).

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Despite all the merits claimed by the supporters of panel speakers, there is still one un-resolved problem that needs to be addressed before we find ubiquitous use of the newly advent device. Although panel loudspeakers have less beaming problem in high frequency, they generally suffer from another problem of efficiency due to hydrodynamic short circuit below coincidence frequency [2]. The physical constraint pertaining to the panel speaker hinders itself from being an ideally omni-directional and full-range device. Thismotivatesthe development of a systematic yet practical optimization scheme in this paper that seeks to best trade-off the omni-directionality and the efficiency by choosing appropriate exciter positions and electronic compensation.

A numerical model was ﬁrst developed as the basis of the optimization procedure. In contrast to the simpliﬁed approach used in Ref. [2] which neglected the effect of acoustic coupling, this paper treats the electrical, mechanical, and acoustical systems as a coupled system. Various approaches dealing with sound–structure interactions can be found in literature [3–5]. In this work, impedance matricesof the exciters, the panel, and the medium are combined into a total impedance matrix for the coupled analysis. To simplify the calculation, the assumed-modes method is used in generating a structure model of the panel. Electro-mechanical analogy is also utilized to model the exciters. A discrete version of the Rayleigh’s integral is used to account for the effect of acoustic loading. The frequency response, the sound power and the directional response of the panel speaker can be calculated using this numerical model.

On the basis of the numerical model, an optimization procedure was then developed to reach the best compromise between the omni-directionality and efficiency. Instead of the ‘‘golden aspect ratio’’ used in the conventional isotropic panel speaker design, the optimization pro-cedure was exploited to find the best positions to deploy exciters and the best electronic compensation (pure delays in our work) to the input signal, such that the beaming problem at high frequenciesisalleviated, with maximal output of acoustic power. In thispaper, the genetic algorithm (GA), an optimization technique based on the law of the evolution of species by natural selection, is adopted in the optimization procedure [6]. The thusobtained optimal design was verified experimentally in this paper. The results show that the optimal design indeed produced better performance in terms of the efficiency and the omni-directionality than the non-optimal design. The results will be discussed and summarized in the conclusion. Panel Suspension Coil Magnet Exciter

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2. Modelling of panel speakers

In this section, details of the dynamic model of the panel–exciter system are given, with the acoustic coupling taken into account. The impedance approach is presented for the analysis of the coupled system.

2.1. Coupled system analysis of a multiply excited panel speaker

Consider a ﬂuid-loaded thin panel depicted inFig. 2. The panel isdivided into N elementswith the same area. Assume that the panel is subjected to external concentrated forces due to exciters. Let f ¼ f 1 f2 ? f_NT and v ¼ u ₁ u₂ ? u_NT be the force vector and the velocity vector associated with the center of each element on the panel surface. By certain discretization scheme, there exists the following relation between f and v [3–5]:

Zmv ¼ f Zav; ð1Þ

where Zm isthe mechanical impedance matrix of the panel and Za isthe radiation impedance

matrix. Note that all forces are expressed in the concentrated from. Hence,

ðZ_mþZaÞv ¼ f: ð2Þ

Next, electro-mechanical analogy is employed for modelling the exciters. The exciters are assumed to be ﬂoating and the magnets of the exciters serve as a proof masses to produce inertia force. The exciter can be modelled by the equivalent circuit (mobility analogy) in Fig. 3(a). In thisﬁgure, Zc¼ Rcþ joLEisthe electrical impedance of the voice coil. Bl isthe motor constant of the voice

coil. Cs and Rs are the compliance and the damping, respectively, between the magnet and the

panel. Mm is the mass of the magnet assembly. Mc isthe massof the voice coil. In Fig. 3(b),

the equivalent circuit isreﬂected to the mechanical side, where fb isthe equivalent blocked

force reﬂected to the mechanical side, Zx isthe equivalent impedance of the exciter, and ZLoad is

the loading experienced by the exciter, including the mechanical loading and the acoustic loading. The relation between fb and f is

f ¼ fb Zxu: ð3Þ

Panel

f2

fN

f1

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With some algebraic manipulations, the impedance Zx can be expressed as

Zx ¼ joMcþ joMm

ðjoÞ

2_R

sCsLEþ ðjoÞðRcRsCsþ B2l2Csþ LEÞ þ Rc

ðjoÞ3MmCsLEþ ðjoÞ2ðMmCsRcþ CsLERsÞ þ ðjoÞðB2l2Csþ RcRsCsþ LEÞ þ Rc

: ð4Þ The blocking force fb can also be related to the input voltage source Eg as

fb¼

ðjoÞEgBlMmCsZx

ðjoÞ2MmMcCsZcþ ðjoÞðB2l2þ RsZcÞðMmþ McÞCsþ ðMmþ McÞZc

; ð5Þ

where Zc ¼ Rcþ joLE; asdeﬁned previously, and Zx isgiven in Eq. (4). Note that the above

expression has a blocking zero at DC, which means that the inertia shaker is not able to supply a force to the panel at DC. Thisequation isonly valid for one exciter. If we consider N exciters mounted on the panel, Eq. (3) should be modiﬁed into a matrix form

f ¼ fb Zxv; ð6Þ where fb¼ f b1 fb2 ? f_bNT; Zx ¼ Zx1 0 ? 0 0 Zx2 & ^ ^ _{& &} ₀ 0 ? 0 Z_xN 2 6 6 6 4 3 7 7 7 5: ð7Þ (a) (b) V i Zc Cs 1/Rs 1 : Bl 1/ZLoad Mm Mc Eg u f fb ZLoad Zx u’ f ’

Fig. 3. Electro-mechanical analogy of a panel speaker: (a) equivalent circuit (mobility analogy) and (b) equivalent circuit reﬂected to mechanical side (impedance analogy). The symbols f and u in the ﬁguresdenote, respectively, the force and the velocity.

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Substituting Eq. (6) into Eq. (2) gives

ðZ_mþZaÞv ¼ fb Zxv: ð8Þ

Let Z ¼ ðZ_mþZaþZxÞ be the total impedance matrix. We can ﬁnally arrive at the succinct relation

Zv ¼ fb ð9Þ

from which one may derive the surface velocity of the panel from the known input voltage. 2.2. Mechanical impedance matrix of the panel (Zm)

Without ﬂuid loading, the relation between the concentrated force vector f and the velocity vector v can be written as

f ¼ Zmv: ð10Þ

In this paper, the assumed-modes method is employed to evaluate the mechanical impedance matrix Zm [7]. Consider a rectangular plate of the dimension Lx Ly: Using the assumed-modes

method, we express the displacement of the plate as [7]

wðx; y; tÞ ¼X

c i¼1

f_iðx; yÞqiðtÞ; ð11Þ

wherec isthe number of modes, f_iðx; yÞ isthe ith admissible function of the panel, and qiðtÞ isthe

generalized co-ordinate. The admissible functions can be found by analytical methods or numerical methodssuch asthe ﬁnite-element method [8].

The strain energy of the plate is U ¼D 2 Z Ly 0 Z Lx 0 ½w2 xxðx; y; tÞ þ w2yyðx; y; tÞ þ 2nwxxðx; y; tÞwyyðx; y; tÞ þ 2ð1 nÞw2 xyðx; y; tÞ dx dy; ð12Þ where D ¼ Eh 3 12ð1 n2_Þ ð13Þ

is the bending stiffness of the plate. E; n; and h are Young’s modulus, the Poisson ratio, and the thickness of the panel. The subscripts of w indicate differentiation of w with respect to that subscript. The kinetic energy is given by

T ¼1 2 Z Ly 0 Z Lx 0 mw2_tðx; y; tÞ dx dy; ð14Þ where m is the surface mass density. The virtual work done by the exciting force f ðx; y; tÞ is

dW ¼ Z Ly 0 Z Lx 0 f ðx; y; tÞdwðx; y; tÞ dx dy: ð15Þ

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Using the assumed-modes method, we can rewrite Eqs. (12), (14) and (15) as U ¼1 2 Xc i¼1 Xc j¼1 kijqiðtÞqjðtÞ; ð16Þ

where kij is the modal mass,

kij ¼ D

Z Ly 0

Z Lx 0

½f_i;xxðx; yÞf_j;xxðx; yÞ þ f_i;yyðx; yÞf_j;yyðx; yÞ

þ 2nf_i;xxðx; yÞf_j;yyðx; yÞ þ 2ð1 nÞf_i;xyðx; yÞf_j;xyðx; yÞ dx dy; ð17Þ

T ¼1 2 Xc i¼1 Xc j¼1 mij’qiðtÞ’qjðtÞ; ð18Þ

where mij is the modal stiffness,

mij ¼ m

Z Ly 0

Z Lx 0

f_iðx; yÞf_jðx; yÞ dx dy;

dV ¼X c i¼1 fidqiðtÞ; ð19Þ where fi¼ Z Ly 0 Z Lx 0 f ðx; y; tÞf_iðx; yÞ dx dy: Deﬁne the Lagrangian L ¼ T U : The Lagrange’sequation reads[9]

@ @t @L @’qi @L @qi ¼ fi; i ¼ 1; y;c: ð20Þ

Substituting Eqs. (16), (18) and (19) into Eq. (20) leads to the following matrix differential equation:

*

M.q þ *Kq ¼ f; ð21Þ

where *M isthe modal massmatrix and *K is the modal stiffness matrix. From Eq. (21), we can identify the modal mechanical impedance matrix of panel

*Zm¼

*K o2_M*

jo : ð22Þ

On the other hand, Eq. (11) can be expressed in matrix notations

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where u ¼ f₁ðx1; y1Þ f2ðx1; y1Þ ? f_cðx₁; y₁Þ f₁ðx2; y2Þ f2ðx2; y2Þ ? f_cðx₂; y₂Þ ^ ^ _& ^ f₁ðxc; ycÞ f2ðxc; ycÞ ? fcðxc; ycÞ 2 6 6 6 4 3 7 7 7 5 ð24Þ

isthe modal matrix which isfor convenience made to be square (number of elements=number of modes) and normalized into an orthogonal matrix, i.e., uT_{u ¼ uu}T_{¼ I: Therefore,}

the mechanical impedance in the physical space and that in the modal space can be related by

Zm¼ u *ZmuT: ð25Þ

2.3. Radiation impedance matrix ðZaÞ

Let fabe equivalent concentrated forces due to acoustic pressure, acting on each element. There exists the following relation:

fa¼ Zav: ð26Þ

Many methodsare available for calculating Za. A simple technique described in Ref.[10]isused in our work to obtain the matrix Za:

Za¼ Sr0c 1 ejk ffiffiffiffiffiffi S=p p _jkS 2p ejkr12 r12 ? jkS 2p ejkr1N r1N jkS 2p ejkr21 r21 1 ejk ffiffiffiffiffiffi S=p p ? ^ ^ ^ ? ^ jkS 2p ejkrN1 rN1 ? ? 1 ejk ffiffiffiffiffiffi S=p p 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 : ð27Þ

S isthe area of each element, rmnisthe distance from the center of the element n to the point m

(m; n ¼ 1; y; N).

2.4. Evaluation of the sound pressure and the sound power

The farﬁeld pressure can be calculated using the propagation matrix E:

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where p is the farﬁeld pressure vector and v isthe surface velocity vector of the panel [10]. For bafﬂed radiators, E ¼ jr0ckS 2p ejkr11 r11 ejkr12 r12 ? e jkr1N r1N ejkr21 r21 ejkr22 r22 ? e jkr2N r2N ^ ^ ? ^ ejkrM1 rM1 ejkrM2 rM2 ? e jkrMN rMN 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 ; ð29Þ

where rmn isthe distance from the center of the element n to the ﬁeld point m (m; n ¼ 1; y; M).

The radiated sound power can be calculated as

W ¼ vHRv; ð30Þ

where R ¼ Re Zf ag=2; and the radiation resistance matrix R isa positive deﬁnite matrix [10]:

R ¼o 2_rS2 4pc 1 sinðkr12Þ kr12 ? sinðkr1N Þ kr1N sinðkr21Þ kr21 1 ? ^ ^ ^ _& ^ sinðkrN1Þ krN1 ? ? 1 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 : ð31Þ

2.5. Numerical simulation of the system response

In this section, simulations are conducted to verify the forgoing model of the panel speaker. Assume that the panel is of the dimension 0:27 m 0:27 m: The core material of panel isPU foam. The parameters of the exciter and the panel measured from a real 25 mm voice coil are listed in

Table 1. Although in principle admissible functions of any kind can be used in the assumed-modes method, the eigenfunctions of the simply supported plate are adopted in this paper because they are practical for efﬁcient computation involved in the ensuing optimization. For a simply supported plate of dimension Lx Ly; material constants D and m; the resonance

frequenciesare[11] omn¼ ffiffiffiffi D m s ðmp=LxÞ2þ ðnp=LyÞ2 ; m; n ¼ 1; 2; 3; y; ð32Þ where m and n are integers. The eigenfunctions of the panel are

f_mnðx; yÞ ¼ ffiffiffiffiffiffiffiffiffiffiffi2 LxLy

p sinðmpx=LxÞsinðnpy=LyÞ; m; n ¼ 1; 2; 3; y: ð33Þ

The panel is divided into 121 elements, as shown inFig. 4. Assume that two exciters are mounted on the 58th and the 64th elementsinFig. 4. The voice-coil resistance is 4 O and the input voltage

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is2 Vr:m:s:; which amounts to 1 W input power. The sound pressure level at a distance of 1 m

on the central axisfrom the panel, calculated using the aforementioned numerical model, isplotted against frequency in Fig. 5. The approach neglecting the acoustic loading is also calculated for comparison. The effect due to acoustic loading is evidenced from the results: the peaks are decreased (damping) and the resonance frequencies are lowered (mass loading) when the acoustic coupling is incorporated into the model. For the same panel speaker, the sound power and the directional response are also calculated and shown in Figs. 6 and 7. These are not optimized results, but are only meant to show the capability of the developed numerical model. 11 22 33 44 55 66 77 88 99 110 121 10 21 32 43 54 65 76 87 98 109 120 9 20 31 42 53 64 75 86 97 108 119 8 19 30 41 52 63 74 85 96 107 118 7 18 29 40 51 62 73 84 95 106 117 6 17 28 39 50 61 72 83 94 105 116 5 16 27 38 49 60 71 82 93 104 115 4 15 26 37 48 59 70 81 92 103 114 3 14 25 36 47 58 69 80 91 102 113 2 13 24 35 46 57 68 79 90 101 112 1 12 23 34 45 56 67 78 89 100 111 0.27 m 0.27m

Fig. 4. Mesh structure of the panel discretized into 121 elements. Table 1

Parametersof the panel and the exciter Parameters Panel Young’smodulusE ¼ 2:28 109_N=m2 Bending stiffness D ¼ 26:625 N m Area density m ¼ 0:741 kg=m2 Dimension=0:27 m 0:27 m 0:005 m Poisson ratio n ¼ 0:33

Exciter Impedance of voice coil Zc¼ 4 þ jo32 106O Motor constant Bl ¼ 2:35 Wb/m

Compliance of coil suspension Cs¼ 297 106m=N Damping of panel suspension Rs¼ 0:257 N s=m Mass of magnet Mm¼ 37 103kg

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1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 40 50 60 70 80 90 100 110 120 Frequency (Hz)

Sound Power Spectral Density (dB re 1e-12 Watt)

Without acoustic loading With acoustic loading

Fig. 6. Simulation of the sound power spectral density of the panel speaker. The results pertaining to the condition with acoustic loading (solid line) and without acoustic loading (dot line) are compared.

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 40 50 60 70 80 90 100 110 120 Frequency (Hz)

Sound Pressure Level (dB re 2e-5Pa)

Without acoustic loading With acoustic loading

Fig. 5. Simulation of the sound pressure level of the panel speaker. The results pertaining to the condition with acoustic loading (solid line) and without acoustic loading (dotted line) are compared.

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3. Optimization using GA

In this section, a systematic procedure using GA intended for optimal design of the panel speakers is presented. The goal is to maximize omni-directionality and the efficiency by adjusting the positions to mount exciters and electronic delay to each exciter. GA is a search technique based on the evolution theory. A typical GA procedure consists of a string representation (genes) of the nodes in the search space, a fitness function to evaluate, three genetic operators for generating new search nodes, and a stochastic assignment to control the genetic operators. GA is particularly effective in non-convex optimization owing to itsmultiple-starting-pointsnature. In this paper, we wish to find a design with omni-directional responses, applicable in a wide frequency range. Thus, the center frequencies (from 31.5 Hz to 16 kHz) of octave band filters[12]

are chosen for the calculation in optimization. The ﬂow chart of the GA procedure in Fig. 8

consists of the following steps:

(1) Initialization: An initial population of search nodes is randomly generated. In this paper, the design variables are the positions of exciters and the delay of input signal to each exciter. From

Fig. 4, there are 121 possible positions of exciters. We also restrict the sample delay of the input signal to be 0pNp90: The sampling rate is assumed as 25 kHz. Assume that three exciters are

50 ₃₀ 210 60 240 90 270 120 300 150 330 180 0 500 Hz 50 ₃₀ 210 60 240 90 270 120 300 150 330 180 0 1k H z 100 dB 100 dB 50 ₃₀ 210 60 240 90 270 120 300 150 330 180 0 2k Hz 50 ₃₀ 210 60 240 90 270 120 300 150 330 180 0 8k Hz 100 dB 100 dB deg deg deg deg

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used in the simulation, among which two of them have the same delay and are located at symmetrical positions. The positions and the delays are then encoded into binary strings called chromosomes, asshown inFig. 9(a). The population includes100 genesand the iteration number of GA isset to be 200.

START

K<Iteration

END

Initialization & encoding

Reproduction Fitness function Crossover Mutation Decoding K=0 K++ No Yes

Fig. 8. Flowchart of the GA-based optimization procedure.

(a) (b) 10110010….. P2: 10011110….. P1: 10111110….. P2: 10010010….. P1: 11110110…..* * 10111110…..* *

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(2) Fitness function: Deﬁne the spatial ﬂatness function G ¼ 1 K XK j¼1 XL i¼1

giðojÞ2 LgmeanðojÞ2

; ð34Þ

where K isthe number of frequencies, L isthe number of the ﬁeld pointsalong a semi-circle, giðojÞ

is the sound pressure of the point i along the semi-circle at the frequency oj; and gmeanðojÞ isthe

mean of the sound pressures at the frequency oj: A small value of spatial ﬂatness function

corresponds to good omni-directionality. Next, deﬁne the efﬁciency function Y ¼ 1

K XK

j¼1

W ðojÞ; ð35Þ

where W ðojÞ isthe sound power at the frequency oj: A large value of efﬁciency function ismost

desirable, though this generally contradicts the requirement of the omni-directionality. Now, combining the spatial flatness and the efficiency yields the fitness function for the GA procedure:

Q ¼ Y =G: ð36Þ

(3) Reproduction: Based on the ﬁtness values of the strings in the population, a new pool of population of strings is produced for the subsequent genetic operations. Strings with larger ﬁtness valuesare more likely to reproduce.

(4) Crossover: The crossover operator allows the reproduction of new strings through combination of parts of strings. A simple crossover operation is done by swapping parts of a pair

0 20 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Iteration number Fitness Value

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of strings to form a new pair of strings. Pairs of strings are randomly selected for mating, and the splice point of a string where the swapping takes place is also randomly selected (Fig. 9(a)).

(5) Mutation: Mutation is the sporadic alteration of chromosomes. Mutation is performed by inverting a bit in the binary code (Fig. 9(b)). The position at which the bit is inverted is randomly selected with a small probability.

(6) Repeat Steps(2)–(5) until a convergence limit or a pre-speciﬁed number of iterationshas been reached.

Fig. 10 shows the learning curve of the GA procedure applied to our problem. The ﬁtness function starts to converge after approximately 100 iterations. As the genes are decoded into physical variables, the optimal positions are found to locate at the 63rd, 47th and 69th elements with sample delays N ¼ 0; 3 and 3, respectively.

4. Experimental investigations

To verify the proposed GA-based optimization technique, experimental investigations are carried out in the laboratory. To minimize the effect of room response, the experiment data are measured in an anechoic room. Fig. 11 shows the experimental arrangement. Recall that, in simulation, the panel is assumed to be simply supported, which is difﬁcult to realize in practice. Instead, an adhesive tape is used in the experiment to ﬁx the boundary of the panel.

In the experiment, ISO 3745 was employed for measuring the sound power in the anechoic room [13]. Directional response of the panel speaker was measured by using an automated turntable depicted inFig. 12. A stepping motor controlled by a PC rotates the turntable on which the panel speaker was mounted. The motor rotates from 0 _{to 180} _{with 1} _{increments. The}

measuring microphone is positioned at a distance of 2 m away from the turntable. The voltage input to each exciter is2 Vr:m:s:: A digital signal processor (DSP, TMS320C31) was utilized to

Baffle Tape

1m

Microphone

Panel

Exciters

Fig. 11. Experimental arrangement of the panel speaker. An adhesive tape is used to suspend the panel within a rigid bafﬂe.

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produce the electronic delay (N ¼ 3), as required in the optimal design, of the input signal to the exciters. The optimal positions of the 63rd, 47th and 69th elements found in the GA procedure were selected in the experiment to mount the exciters. For comparison, a configuration where the exciters are mounted on an arbitrarily chosen ‘‘non-optimal’’ positions, the 61th, 38th and 82nd elements, without delay was also tested.Fig. 13compares the directional response of the optimal design to that of the non-optimal design. From the results, it is observed that the optimal design produced an improved omni-directionality ascompared to the non-optimal design. In particular, thisisreflected in the flatnessof the beam pattern in 500 and 1 kHz. In 2 and 4 kHz, the optimal design appears to generate a pattern with an angle wider than the non-optimal design. Table 2

compares the sound power measured by ISO3745 between the optimal design and the non-optimal design. The non-optimal design was found to radiate unanimously higher acoustic output than the non-optimal design at all frequencies. Not only this case but also several other choices of exciter setting, as not presented here, produced inferior performance than the optimal conﬁguration. Therefore, it isconcluded that the GA-based optimization procedure indeed has produced a design of panel speaker with improved performance in terms of omni-directionality and efﬁciency.

DAS Microphone

PC Motor control line

2m Baffle

Panel speaker

Motor

Fig. 12. Experimental arrangement for measuring the directional response of the panel speaker. A turntable is rotated by a stepping motor. A data acquisition system is used to measure the signal from the microphone. The overall signal processing activity is monitored by a personal computer.

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5. Conclusions

The principal outcome of this work can be summarized in two aspects. First, a fully coupled model of the panel speaker has been established for simulation. Second, a GA-based procedure has been developed for obtaining the optimal design. The present simulation model takes into account the acoustic loading on the light panel structures. The impedance matrices of the exciter, the panel and the medium are combined into one matrix in the formulation. The assumed-modes

50 100 30 210 60 240 90 270 120 300 150 330 180 0 50 100 30 210 60 240 90 270 120 300 150 330 180 0 75 150 30 210 60 240 90 270 120 300 150 330 180 0 75 150 30 210 60 240 90 270 120 300 150 330 180 0 Non-optimal Optimal 500 Hz 1k Hz 2k Hz 4k Hz deg deg deg deg dB dB dB dB

Fig. 13. Comparison of directional responses between the optimal design (dashed line) and a non-optimal design (solid line) of the panel speaker. The directional responses are measured at the frequencies 500 Hz, 1, 2 and 4 kHz.

Table 2

Comparison of the experimental results of the sound power between the optimal design and the non-optimal design (dB re. 1 1012_W)

500 Hz 1 kHz 2 kHz 4 kHz

Optimal 92.5 94.7 95.3 91.6

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method was used in the model, which provides an efficient means for response computation. On the basis of the simulation model, a search scheme was exploited to optimize, by using the GA, the efficiency and omni-directionality of panel speaker. The GA procedure produces the optimal positions to mount exciters and the electronic delay to the input signals. The thus obtained optimal configuration of the panel speaker has been verified by experiments. The experimental result indicates that the performance was enhanced by means of the optimization design approach.

As a limitation of the present work, eigenfunctions of simply supported plates are used in the assumed-modes expansion. It is usually difﬁcult to derive a dynamic model for more general problems by using assumed-modes method which is only practical for simple boundary conditions. A ﬁnite-element-based model that is able to handle complex boundary conditions is currently being developed to improve the design optimization.

Acknowledgements

Thanks are due to the illuminating discussions with NXT, New Transducers Ltd., UK. The work wassupported by the National Science Council (NSC) in Taiwan, under the Project Number NSC 89-2212-E009-057.

References

[1] H. Azima, NXT up against wall, Audio Magazine (1998) 34–41.

[2] M.R. Bai, T. Huang, Development of panel loudspeaker system: design, Evaluation and enhancement, Journal of the Acoustical Society of America 109 (2001) 2751–2761.

[3] M.C. Junger, D. Feit, Sound, Structure, and Their Interaction, The MIT Press, Cambridge, MA, 1986. [4] F. Fahy, Sound and Structural Vibration, Academic Press, London, 1985.

[5] L. Cremer, M. Heckl, Structure-born Sound, Springer, Berlin, 1988.

[6] R. Caponetto, L. Fortuna, G. Muscato, G. Nunnari, Search of optimal realization matrix for ﬁlter implementation by using a genetic algorithm, IEEE Transactions on Systems, Man and Cybernetics 20 (2) (1993) 471–474. [7] B.H. Tongue, Principles of Vibration, Oxford University Press, Oxford, UK, 1996.

[8] Y. Kagawa, T. Tsuchiya, N. Wakatsuki, Equivalent circuit representation of a vibrating structure with piezoelectric transducers and the stability consideration in the active damping control, Int. J. of Smart Material and Structures10 (2001) 389–394.

[9] F.S. Tse, I.E. Morse, R.T. Hinkle, Mechanical Vibrations Theory and Applications, Allyn & Bacon, Newton, MA, 1978.

[10] A.P. Berkhoff, Sensor Scheme Design for Active Structural Acoustic Control, Journal of thw Acoustical Society of America 108 (2000) 1037–1045.

[11] A.W. Leissa, Vibration of Plates, Acoustical Society of America, Woodbury, NY, 1993. [12] A.D. Pierce, Acoustics, McGraw-Hill, New York, 1981.