effects of the cone and edge on the acoustic

Research ArticleEffects of the Cone and Edge on the Acoustic Characteristics ofa Cone Loudspeaker

Yue Hu,1,2 Xilu Zhao,2 Takao Yamaguchi,3 Manabu Sasajima,1

Tatsushi Sasanuma,1 and Akira Hara1

1Foster Electric Company, Limited, 1-1-109 Tsutsujigaoka, Akishima, Tokyo 196-8550, Japan2College of Mechanical Engineering, Saitama Institute of Technology, 1690 Fusaiji, Fukaya, Saitama 369-0293, Japan3Department of Mechanical Science and Technology, Faculty of Science and Technology, Gunma University,1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan

Correspondence should be addressed to Xilu Zhao; [email protected]

Received 8 February 2017; Accepted 15 March 2017; Published 21 May 2017

Academic Editor: Kim M. Liew

Copyright © 2017 Yue Hu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Loudspeakers are designed for reproducing the original sound field as faithfully as possible. In order to faithfully reproduce sound,it is important to understand the relationships among the physical characteristics of the loudspeaker. This paper focuses on thecone, the edge, and the behavior of air around the voice coil, which are important elements in the design of cone loudspeakers andevaluates their effects on the acoustic characteristics of the loudspeaker.

1. Introduction

People hear a variety of sounds in their everyday lives. Loud-speakers are designed for faithfully reproducing the originalsound field. Currently, the most commonly used loudspeakerdesign is the cone loudspeaker. For faithful sound repro-duction, the relationships among the physical characteristicsof the loudspeaker must be elucidated. One of the physicalcharacteristics required for an ideal loudspeaker, assuminga constant input voltage, is a flat sound pressure in thefrequency domain.

In pursuit of this goal, researchers have long studiedvibration analysis of the loudspeaker. Investigations of soundradiation by Brown [1] and Bordoni [2] treated the coneloudspeaker as a rigid body. However, the rigid body modelof the cone is valid only during the so-called pistonmotion, inwhich the vibration system components, including the cone,edge, center cap, voice coil, and spider (see Figure 1), vibratetogether as a unit in the low-frequency range. Nimura et al.[3] carried out a theoretical analysis of the vibration of thecone and used a graphicalmethod to calculate the eigenvaluesof the vibration of a conical loudspeaker cone. Kagawa [4]

calculated the eigenvalues of the membrane vibrations ofcurved cones in addition to conical loudspeaker cones andfound that curving the cone changed the eigenvalues ofthe membrane vibrations. Frankfort [5], also seeking solu-tions for the membrane vibrations of conical loudspeakercones, formulated a differential equation that accounts forbending vibration and made detailed calculations of thesound pressure frequency response, vibration patterns, anddriving-point admittance. However, Frankfort’s analysis didnot account for the effects of the edge and center cap, whichplay important roles in cone loudspeakers. Subsequently, thefinite element method (FEM) has been used for vibrationanalysis of the cone [6–11]. Kyouno et al. analyzed the soundradiation from loudspeakers, accounting for coupling of theelectrical, mechanical, and acoustic systems [12].

In these vibration analyses of cone loudspeakers, the conemaintains a piston motion in the low-frequency region. Thebiggest difficult point of these analyses is the split vibrationof the vibration plate in the middle- and high-frequencyregions. Control over the cone and the edge is very importantin the design of a loudspeaker. However, there has beeninsufficient study of this topic in research up to now.

HindawiAdvances in Acoustics and VibrationVolume 2017, Article ID 2792376, 12 pageshttps://doi.org/10.1155/2017/2792376

https://doi.org/10.1155/2017/2792376

2 Advances in Acoustics and Vibration

Frame

Plate

Magnet

Yoke

Edge

Cone

Whizzer cone

Voice coil

Spider

xy

z

Figure 1: 1/4 model of a loudspeaker.

Furthermore, these acoustic analyses, particularly in thecase of FEM, do not account for air viscosity, which mayreduce the accuracy of their estimates of sound pressurefrequency properties.

In particular, if the acoustic pathway in enclosure of smallearphone is very narrow, air viscosity exerts a significantdamping effect, reducing the estimation accuracy. Sasajimaet al. succeeded at improving the estimation accuracy in theface of this problem by accounting for air viscosity using thefinite element method [13].

Loudspeakers larger than earphones have a narrow spacebetween the voice coil and the magnetic circuit, making itvery likely that air viscosity has a significant impact and is afactor of estimation errors of the sound pressure frequency.Therefore, general acoustic analyses need to account forthe damping effect of air viscosity to accurately analyzeloudspeakers.

This paper examines cone loudspeakers, focusing inten-sively on the effects of the cone, the edge, and the narrowspace between the voice coil and the magnetic circuit, whichhave a significant impact on the vibration characteristicsand sound pressure frequency response. First, we performan experimental study of the characteristics of a cone loud-speaker to determine its vibration characteristics and soundpressure frequency response. Next, we perform a vibration-acoustic analysis of the cone loudspeaker that accounts forthe impact of air viscosity using our own independentlydeveloped acoustic analysis software. To verify the accuracyof our analysis, we compare the values from our analysis,which accounts for air viscosity, to the actual values of thevibration and sound pressure frequency properties. We alsoconsider the behavior of the air around the voice coil. Finally,we calculate the optimal Young’s modulus and density of thecone and edge and examine in detail their impact on thevibration characteristics.

2. Methods and Materials

Figure 1 illustrates the cone loudspeaker of diameter 0.16m,which is the focus of this study. Because the cone exhibits

Edge Measurement point

Voice coil

PlateYoke Pole

Cone

Whizzer cone

P

Figure 2: Important elements of the design of cone loudspeakers.

axial symmetry, we assume amodel that is 1/4 symmetricwithrespect to the 𝑥-𝑧 and 𝑦-𝑧 planes.

The defining feature of cone loudspeakers is that the coneis approximately conical in shape, as illustrated in Figure 1.The cone is supported by an edge around the outer peripheryand a spider that supports the center of the cone by holdingthe voice coil in position.The outer perimeter of the spider isanchored to the frame. The cover at the center of the cone istypically a center cap, but our model used a whizzer cone tosupport the high-frequency characteristics.

The voice coil is attached to the center of the cone,where it is supplied with the input signal current. Theseelements comprise the loudspeaker’s vibration system. Theouter periphery of the vibration system is supported by aframe.

The magnetic circuit for supplying magnetic flux to thevoice coil consists of a plate, magnet, and yoke. Magnetic fluxis supplied by themagnet through the yoke [14]. Soundwavesare generated when the driving force generated according toFleming’s left hand rule drives the cone, which is attached tothe voice coil.

A cross-sectional view of the cone loudspeaker is shownin Figure 2. The performance of a loudspeaker is largelygoverned by its cone, which radiates the sound directly.In this sense, the cone can be said to be the heart of theloudspeaker. In an idealized model, the cone is always arigid body travelling with a back-and-forth piston motion;however, in the middle- and high-frequency regions, theshape and material property values of the cone generate splitvibrations in axisymmetric and nonaxisymmetric modes,exerting a significant influence on the loudspeaker’s vibrationcharacteristics and sound pressure frequency response.

The edge, by undergoing large deformations and actingas a spring, not only facilitates movement of the cone in the𝑧-axis direction but also plays a role in reducing movementin the 𝑥-𝑦 direction along the central axis. However, theedge does not always move back and forth together with thecone. The cone and the edge can vibrate in opposite phases,affecting the vibration characteristics.

By controlling the characteristics of the cone and edge,which are important design elements of a cone loudspeaker,the vibration characteristics of the cone loudspeaker can beimproved. This study examines the vibration characteristics

Advances in Acoustics and Vibration 3

Signalsource

LaserAmplifier

Loudspeaker

FFT analyzer

PC

Figure 3: Cone vibration displacement measurement system.

of the loudspeaker by focusing attention on the materialsof the cone and edge, which can be expected to have aparticularly large impact.

In addition, the narrow space between the outer perime-ter of the yoke pole and the inner perimeter of the plate,indicated in Figure 2, increases the air viscosity, and theflow of air in this gap can also be expected to have animportant influence, which is very likely to be an error factorin the estimation of sound pressure frequency response. Thedamping effect of air viscosity, which is typically ignored inacoustic analysis, should therefore be taken into account foraccurate analysis of loudspeaker performance.

3. Experimental

The performance of a cone loudspeaker is evaluated in termsof its vibration characteristics and sound pressure frequencyproperties; therefore, in our investigation, we took actualmeasurements of these characteristics.

3.1. Measurement of Cone Vibration Displacement. The coneis the core of a loudspeaker’s vibration system, and it istherefore very important to understand the vibration charac-teristics of the cone when designing a loudspeaker’s vibrationsystem.

We measured the cone’s vibration displacement using alaser displacement meter. Measurements were taken using asweep signal of 10Hz to 5 kHz, with the measurement pointset to position P at the center of the whizzer cone, which islocated at the center of the loudspeaker cone (see Figure 2).The measurement system is shown in Figure 3.

Figure 4 shows the actual displacement characteristics atthemeasurement point. From 100Hz to 800Hz, the displace-ment can be seen to decrease nearly linearly; however, beyond800Hz, peaks occur at approximately 1 and 1.5 kHz.

3.2. Measurement of Sound Pressure Frequency Properties.We tested the sound pressure frequency of the loudspeakerby taking measurements in a reverberation room and ananechoic room, in accordance with the standards establishedin JIS Z 8732. Figure 5(a) shows a schematic of the two-roommeasurement environment, with the reverberation room onthe left and the anechoic room on the right. Figure 5(b)

10−7

10−6

10−5

10−4

Disp

lace

men

t am

plitu

de (m

)

800100 500400200 1000 2000

Frequency (Hz)

Figure 4: Comparison of the measurement result of cone vibrationdisplacement amplitude.

shows a photo of the reverberation room and anechoic roomas seen from the outside. The insides of the reverberationroom and anechoic room are shown in Figures 5(c) and 5(d),respectively.

The measurement system included an audio analyzer(Etani Electronics ASA-10 Mark II) and a microphone (Acotype 4012). The tests were performed using a sweep signalin the frequency range of 20Hz to 20 kHz. The observationpoint for measurement of the sound pressure in the anechoicroom was a position 0.8m in front of the loudspeaker,on its central axis. The measurement system is shown inFigure 6.

Themeasurement results are shown in Figure 7. An initialpeak can be seen at approximately 80Hz, and then the soundpressure frequency characteristic is generally flat from200Hzto 1 kHz. Above 1 kHz, a pattern of repeated peaks and dipscan be observed.

We were able to determine the acoustic characteristics ofthe cone loudspeaker from the results of the measurementsof the cone vibration displacement and the sound pressurefrequency properties.

However, these results by themselves do not reveal theimpact of the cone and edge on the acoustic characteristicsof the loudspeaker. This test was also inadequate in termsof the limited types of loudspeaker we were able to meas-ure.

Therefore, to elucidate the effect of the materials ofthe cone and edge on the acoustic properties of the loud-speaker, we continued our investigation as follows. First,we performed a coupled analysis of vibration and acoustics,taking air viscosity into account. Next, we compared theresults of the analysis to the experimental values to verifythe accuracy of the analysis. In lieu of an experiment,we used the acoustic analysis to reproduce the effects ofthe characteristics of the cone and edge on the acousticcharacteristics of the loudspeaker. Finally, we applied theresponse surfacemethodology to determine the optimal coneand edge materials in order to design a loudspeaker with thedesired acoustic characteristics.


Reverberation roomAnechoic room

Microphone0.8m

(a) Schematic of measurement environment

Reverberation room Anechoic room

(b) Reverberation room and anechoic room

(c) Reverberation room (d) Anechoic room

Figure 5: Measurement environment.

4. Analysis and Results

In order to account for air viscosity, which is typically ignoredin acoustic analyses of cone loudspeakers, we propose anapplication of the finite elementmethod to perform a coupledanalysis of vibration and acoustics that takes into account airviscosity.

4.1. Method of Analysis. As the acoustic element, consider athree-dimensional tetrahedral element with unknown nodaldisplacements, as illustrated in Figure 8.

Letting 𝑢𝑥, 𝑢𝑦, and 𝑢𝑧 be the displacements in the 𝑥, 𝑦,and 𝑧 directions, respectively, at any point in the element, thestrain energy �̃� can be expressed as follows:

�̃� = 12𝐸∭V𝑒

(𝜕𝑢𝑥𝜕𝑥 + 𝜕𝑢𝑦𝜕𝑦 + 𝜕𝑢𝑧𝜕𝑧 )2 𝑑𝑥 𝑑𝑦𝑑𝑧. (1)

Here, 𝐸 is the bulk modulus and V𝑒 is the volume of thetetrahedral element. Letting �̇� be the time derivative of thedisplacement, the kinetic energy �̃� is expressed as follows:

�̃� = 12∭V𝑒

𝜌 {�̇�}𝑇 {�̇�} 𝑑𝑥 𝑑𝑦 𝑑𝑧. (2)

Here, 𝜌 represents the density of the element, and 𝑇represents the transposition. In addition, the viscous energy𝐷 of the viscous compressible fluid is expressed as follows:

𝐷 =∭V𝑒

12 {𝑇𝑇} Γ𝑑𝑥 𝑑𝑦 𝑑𝑧. (3)

Here, {𝑇} is the stress vector due to viscosity, and Γ is thestrain vector.

Furthermore, using the potential energy �̃� and applyingLagrange’s equation, we obtain an equation of motion for theair elements that accounts for air viscosity:

𝑑𝑑𝑡 𝜕�̃�𝜕 {�̇�𝑒} − 𝜕�̃�𝜕 {𝑢𝑒} + 𝜕�̃�𝜕 {𝑢𝑒} − 𝜕�̃�𝜕 {𝑢𝑒} + 𝜕𝐷𝜕 {�̇�𝑒} = 0. (4)

Here, {𝑢𝑒} and {�̇�𝑒} are the nodal displacement vector andnodal velocity vector, respectively. Substituting (1)–(3) into(4), the following equation is obtained:

−𝜔2 [𝑀𝑒] {𝑢𝑒} + [𝐾𝑒] {𝑢𝑒} + [𝐶𝑒] {�̇�𝑒} = {𝑓𝑒} . (5)

Here, [𝑀𝑒], [𝐾𝑒], and [𝐶𝑒] are the element massmatrix, the element stiffness matrix, and the element damp-ing matrix, respectively. Furthermore, assuming a periodic


Reverberation room

Anechoic room

Sweep signal

Audio analyzer

0.8m

PC

Microphone

Figure 6: Sound pressure frequency measurement system.

Frequency (Hz)5000

10−2

10−1

10

102

Soun

d pr

essu

re (P

a)

50 100 500 1200 1500

Figure 7: Comparison ofmeasurement result and system character-istics of sound pressure frequency properties.

x

y

z

Node 1

Node 2

Node 3

Node 4

Figure 8: Tetrahedral element.

response with an angular frequency of 𝜔 and using {�̇�𝑒} =𝑗𝜔{𝑢𝑒}, (5) can be expressed by the following equation:

−𝜔2 [𝑀𝑒] {𝑢𝑒} + [𝐾𝑒] {𝑢𝑒} + 𝑗𝜔 [𝐶𝑒] {𝑢𝑒} = {𝑓𝑒} . (6)

Here, −𝜔2[𝑀𝑒]{𝑢𝑒} is the inertia term, [𝐾𝑒]{𝑢𝑒} is thestiffness term, 𝑗𝜔[𝐶𝑒]{𝑢𝑒} is the viscosity term, and {𝑓𝑒} is theexternal force.

For the structural elements, we use the general formula-tion:

−𝜔2 [𝑀se] {𝑢se} + [𝐾se] {𝑢se} = {𝑓se} , (7)

where {𝑢se}, {𝑓se}, [𝐾se], and [𝑀se] are the structural elements’nodal displacement vector, nodal force vector, complex stiff-ness matrix, and mass matrix, respectively. For the structuralelements, in addition to tetrahedral elements, we also usedhexahedral isoparametric elements account for incompatiblemodes.

Superimposing these equations of motion on the ele-ments in the places of interest and letting the displacementsbe the common unknown variables, the following equationof motion for the entire system, including structure andacoustics, is obtained:

−𝜔2 [𝑀] {𝑢} + [𝐾] {𝑢} + 𝑗𝜔 [𝐶] {𝑢} = {𝑓} . (8)

By solving the equation ofmotion in (8) for displacement,all nodal displacements can be determined. Furthermore, thestrain and pressure of each element can be calculated fromthe nodal displacements [13].

Therefore, we used solutions of (8) to carry out thecoupled vibration and acoustic analysis described in thispaper.

To verify the performance of the loudspeaker, we reliedprimarily on frequency response analysis. To calculate thepressure of the elements during the actual vibration andacoustic analysis, we divided up the frequency range tobe analyzed and then found the nodal displacement {𝑢}corresponding to the sine wave excitation force {𝑓} at eachfrequency. By repeating these calculations, the nodal dis-placement and element pressurewere calculated for the entirefrequency range to be analyzed, yielding frequency responsecharacteristic graphs for displacement and pressure.

4.2. Analytical Model. We created an analytical model withconditions substantially equivalent to the conditions in thereverberation room and anechoic room shown in Figure 5. Asillustrated in Figure 9(b), the analytical model is composedof three parts: the sound-absorbing elements in the outerpink areas, the air elements in the inner green areas, andthe cone loudspeaker setup in the airspace in the smallreverberation room. Figure 9(a) shows the element model ofthe loudspeaker from the middle of Figure 9(b).

Within the loudspeaker elements, there is a narrow widthof only 2 × 10−5m between the voice coil and the magneticcircuit. In order to determine the behavior of the air aroundthe voice coil in a way that accounts for the viscosity of theair, the fine air mesh shown in Figure 9(c) was created. For


(a) Loudspeaker (b) All analytical models

(c) Air between voice coil and magnetic circuit (d) No air between voice coil and magnetic circuit

Reverberation room

Loudspeaker Air Sound absorbing material

Fixed boundary condition

Air Voice coil

Plate

Magnet

Yoke

Voice coil

Yoke

Plate

Magnet

Force

Anechoic room

1.8m

0.6m

0.9m

0.3m

2.6m

Figure 9: Analytical model.

comparison, Figure 9(d) shows the state when there is no airbetween the voice coil and magnetic circuit.

As a result, the structural part of the model contained159,042 elements and 42,965 nodes, while the air part con-tained 704,942 elements and 143,585 nodes.

The sine wave excitation force F was applied to the entirecircumference of the voice coil in the direction of the centeraxis of the loudspeaker, as shown in Figure 9(d).

With respect to the boundary conditions, the threedegrees of freedom on the 𝑥-, 𝑦-, and 𝑧-axes were fixed atthe underside of the frame as shown in Figure 9(a) and atthe exterior air nodes as shown in Figure 9(b). In addition, toaccount for symmetry conditions, the displacement of nodeson the 𝑦-𝑧 and 𝑥-𝑧 planes was fixed in the 𝑥-axis and 𝑦-axisdirections, respectively.

Since the loudspeaker is made of composite materials,the material properties used in our analysis were measuredby the vibrating reed method [15] for the cone and whizzercone, by the tensile test for the edge, and by FEM staticanalysis based on load displacement measurements for thespider. Other physical property values were obtained byconsulting chronological scientific tables. Table 1 shows the

actual physical property data used in our analysis. Otherphysical property values included an effective air density of1.2 kg/m3, a bulk modulus of 1.4 × 105N/m2, a coefficientof viscosity of 1.82 × 10−5N⋅s/m2, and a speed of sound of340.0m/s.

4.3. Verification of the Accuracy of Our Analysis of ConeVibration Displacement. Figure 10 shows a comparison of thecone vibration displacement amplitudes from the measure-ment results and analysis results. The solid line representsthe measurement results, and the dashed line represents theresults calculated by considering air viscosity.

The results in Figure 10 show that the values from ouranalysis are in good agreement with the actual measuredvalues. The fact that our calculation results yield roughly thesame characteristics as the actual measured cone vibrationdisplacement values suggests that highly accurate vibrationanalysis results can be obtained accounting for air viscosity.

4.4. Verification of the Accuracy of Our Analysis of SoundPressure Frequency Properties. Figure 11 compares the results


Table 1: Material and geometric parameters.

Parts Mass density(kg/m3)

Young’s modulus(N/m2)

Poisson’sratio

Cone 3.1 × 102 1.16 × 109 0.3Edge 8.0 × 102 1.39 × 108 0.3Spider 6.60 × 102 8.90 × 107 0.3Whizzer cone 5.70 × 102 2.00 × 109 0.3Voice coil 1.87 × 103 1.29 × 1011 0.34Magnet 4.80 × 103 1.18 × 1011 0.3Plate 7.69 × 103 2.00 × 1011 0.3Yoke 7.94 × 103 2.00 × 1011 0.3Frame 9.10 × 102 1.66 × 109 0.3

Frequency (Hz)

MeasurementAnalysis

100 200 400 500 800 1000 200010−7

10−6

10−5

10−4

Disp

lace

men

t am

plitu

de (m

)

Figure 10: Comparison of themeasurement result and analysis withviscosity of cone vibration displacement amplitude.

of our calculation of the sound pressure frequency responsein the direction of the axis of the loudspeaker to the actualmeasurement results from the sweep signal excitation. Thesolid line represents themeasurement results, and the dashedline represents the results calculated by considering airviscosity.

The changes and characteristics of the sound pressurefrequency response from our analysis results are consistentwith the traditional view in the literature that more accuratecharacteristics can be obtained by analyses that account forthe viscous damping of the air [16].

The reason for this is that the narrow space between thevoice coil and the magnetic circuit is so narrow that the airviscosity in the space affects the up-and-down motion of thevoice coil. Our analysis is able to account for rotation as wellas translationalmotion of air elements along the𝑥-,𝑦-, and 𝑧-axes. It is therefore able to represent the behavior of the air inthe narrow passage, yielding calculation results that are closerto the actual measured values.

Frequency (Hz)

MeasurementAnalysis

50 100 500 1200 1500 500010−3

10−2

10−1

10

102

Soun

d pr

essu

re (P

a)

Figure 11: Comparison of measurement result and characteristicsmeasurement system pressure frequency characteristics.

(a) Vibration mode around the voice coil

(b) Vibration mode around the cone

Figure 12: Effect by vibration mode of air with viscosity.

4.5. Impact of the Narrow Space between the Voice Coil andthe Magnetic Circuit. When designing a cone loudspeaker, itis very important to consider the behavior of the air betweenthe circumference of the voice coil and the magnetic circuit.

Figure 12(a) shows the behavior of the air around the voicecoil when air viscosity is taken into account. Red indicatesareas of large displacement and blue indicates areas of small


Paper coneMetal cone

Frequency (Hz)5000100 200 400 1000 2000

10−10

10−9

10−8

10−7

10−6

10−5

10−3

10−4

Disp

lace

men

t am

plitu

de (m

)

Figure 13: Comparison vibration characteristic of metal cone andpaper cone.

displacement. When air viscosity is taken into account, thedifferences in air pressure become apparent, and, owingto the natural tendency of air to flow from areas of highpressure to areas of low pressure, the direction of the airflow from the vibration of the voice coil becomes apparent.Figure 12(b) shows the behavior of the air around the cone.The figure clearly shows the direction of the air flow aroundthe cone due to the behavior of the air around the voicecoil.

The analysis results shown in Figure 12 demonstrate thepotential for consciously designing the fine-grained charac-teristics of a loudspeaker by adjusting parameters such as theviscous resistance of the narrow acoustic pathway betweenthe voice coil and magnetic circuit.

Using this analysis method, which accounts for air vis-cosity, it is possible to visualize the movement of air andcapture the behavior of the air around the voice coil bysolving the equation of motion with the nodal displacementsas unknown quantities.

4.6. Impact of the Cone on the Acoustic Characteristics of theLoudspeaker. For purposes of comparison, we investigatedcones made of both metal and paper-based materials. Whenmetal is used, the material is highly uniform and exhibitsgreater flexural rigidity at the same thickness. On the otherhand, when paper-basedmaterials are used, the cone is easierto manufacture, but its vibration characteristics are likely tobemore complicated because its flexural rigidity is lower thanthat of metal and the random arrangement of the pulp fibersrenders the material nonuniform.

Figure 13 shows the cone displacement responses calcu-lated by this method, accounting for air viscosity. The solidand dashed lines show the displacement responses for themetal and paper cones, respectively.

The figure shows that the metal cone exhibits pistonmotion up to 5 kHz, with virtually no peaks and dips in

Soft edge materialHard edge material

Frequency (Hz)5000100 200 400 1000 2000

10−10

10−9

10−8

10−7

10−6

10−5

10−4

Disp

lace

men

t am

plitu

de (m

)

Figure 14: Comparison vibration characteristic of soft edge andhard edge material.

displacement. In the case of the paper cone, more dips andpeaks are observed, because the cone is softer andmore proneto split vibration, in which the motion of the voice coil is nottransmitted to the entire cone.

4.7. Impact of the Edge on the Acoustic Characteristics ofthe Loudspeaker. We compared the performance of an edgemade of a soft material with Young’s modulus of 6.95 ×107N/m2 and an edge made of a hard material with Young’smodulus of 2.78 × 108N/m2. Figure 14 shows the vibrationcharacteristics calculated by this method, taking into accountair viscosity. The solid and dashed lines show the displace-ment response for the soft and hard materials, respective-ly.

The results in the figure show an initial peak at 1.3 kHzin the case of the soft edge material and an initial peakat 1.7 kHz in the case of the hard edge material. In otherwords, we used our analysis method to perform a coupledanalysis of vibration and acoustics in which only the edgematerial was changed not the shape of the cone.These resultsdemonstrate the potential for consciously designing the fine-grained characteristics of a loudspeaker, such as the first-peakposition.

5. Discussion

In the previous sections of this paper, we took into accountthe viscosity of the air, verified the accuracy of the coupledanalysis of vibration and acoustics of the loudspeaker, andsaw the impact of the material of the cone and edge onthe acoustic characteristics of the loudspeaker. Extendingthis line of research, a significant remaining challenge isto design the appropriate cone and edge characteristicsto obtain the desired acoustic characteristics in the loud-speaker.


Target characteristicsOriginal speaker characteristics

Frequency (Hz)5000100 200 500 1000 2000

di

10−10

10−9

10−8

10−7

10−6

10−5

10−4

Disp

lace

men

t am

plitu

de (m

)

Figure 15: Cone vibration displacement amplitude standard analysisresult and objective function.

To address this challenge, in this section, we considerhow to optimize the cone and edge, expressing the problemmathematically as follows:

Find 𝑥 = [𝜌𝑐, 𝐸𝑐, 𝜌𝑒, 𝐸𝑒]𝑇min 𝑊 = 𝑓 (𝑥) = √ 𝑛∑

𝑖=1

𝑑𝑖2 (𝑖 = 1, 2, . . . , 𝑛)S.t. 100 [kg/m3] ≤ 𝜌𝑐 ≤ 1200 [kg/m3]

7.3 × 108 [N/m2] ≤ 𝐸𝑐 ≤ 70 × 108 [N/m2]300 [kg/m3] ≤ 𝜌𝑒 ≤ 1300 [kg/m3]3.2 × 107 [N/m2] ≤ 𝐸𝑒 ≤ 20 × 107 [N/m2] .

(9)

Here, 𝜌𝑐 is the density of the cone, 𝐸𝑐 is Young’s modulusof the cone,𝜌𝑒 is the density of the edge,𝐸𝑒 is Young’smodulusof the edge, and 𝑊 = 𝑓(𝑥) is the objective function ofoptimization. In Figure 15, the solid line represents the targetcharacteristic value, the dashed line (which was previouslyshown in Figure 10) represents the cone vibration displace-ment characteristics fromour analysis, and𝑑𝑖 (𝑖 = 1, 2, . . . , 𝑛)represents the distance from the displacement character-istic to the target characteristic value at each frequencyfrom 1 to 5 kHz. The number of sampling points was setto 𝑛 = 50.

Although paper-based materials, metal materials, andother materials have been used to construct the sound-radiating cones in loudspeakers, paper-based materials areused most often. For that reason, the constraints imposed onthe ranges of the cone’s density and Young’s modulus in (9)were set in the ranges for paper-based materials. Similarly,

Optimal solutionTarget characteristicsOriginal speaker characteristics

Frequency (Hz)5000100 200 500 1000 2000

10−10

10−9

10−8

10−7

10−6

10−5

10−4

Disp

lace

men

t am

plitu

de (m

) Figure 16: Analysis result of optimal solution.

the range of values established for the edge in (9) reflects thefact that the edge is typically constructed using cotton fabricand rubber.

We used the response surface methodology to optimizethe physical properties of the cone and edge by carryingout the following calculation steps. First, we generatedsample data in order to create the response surface forthe optimization. Next, using the sample data, we carriedout vibration analysis of the loudspeaker by changing thephysical properties of each of its parts and, based on theresults of the analysis, extracted the objective function valuesfor the optimization calculation. Finally, we established aone-to-one correspondence between sample data and thecharacteristic values, created an interpolation approximationformula by the response surface method, and then usedthe approximation formula to find the optimum solution byperforming the optimization calculation [17, 18].

Table 2 shows the sample data that was generated usingthe design variables and their upper and lower limit values,as well as the objective function value 𝑑, which is the resultobtained from the coupled analysis of vibration and acousticsin accordance with the sample data.

We used these results to create the response surface andperform the optimization, the results of which are shown inTable 3.

Figure 16 shows the results of the vibration analysis usingthe optimal solution that was obtained. The black solid linerepresents the analysis result for the optimal solution andthe red solid line represents the target characteristic values.The dashed line represents analysis results for the originalloudspeaker shown in Figure 10. The results obtained forthe optimal solution closely match the target characteristicvalues.

The optimal solution exhibits flatter characteristics com-pared to the original cone vibration displacement character-istics, which exhibited peaks at 1.5 and 3 kHz and a dip at


Table 2: Sample data and analysis results.

NumberCone

mass density(kg/m3)

ConeYoung’smodulus(108N/m2)

Edgemass density(kg/m3)

EdgeYoung’smodulus(107N/m2)

Analysis results𝑑1 100 7.3 300 3.2 0.352 100 7.3 300 12 0.333 100 7.3 300 20 0.314 650 7.3 800 3.2 0.515 650 7.3 800 12 0.606 650 7.3 800 20 0.567 1200 7.3 1300 3.2 0.678 1200 7.3 1300 12 0.619 1200 7.3 1300 20 0.6810 100 39 800 3.2 1.0611 100 39 800 12 1.0312 100 39 800 20 1.0213 650 39 1300 3.2 1.2914 650 39 1300 12 1.2515 650 39 1300 20 1.2716 1200 39 300 3.2 1.4617 1200 39 300 12 1.4318 1200 39 300 20 1.4419 100 70 1300 3.2 0.9820 100 70 1300 12 1.0121 100 70 1300 20 1.0022 650 70 300 3.2 1.5523 650 70 300 12 1.5624 650 70 300 20 1.5525 1200 70 800 3.2 1.7426 1200 70 800 12 1.7727 1200 70 800 20 1.79

Table 3: Design variables of optimization.

Initial value Optimal value𝜌𝑐 (kg/m3) 650 100𝐸𝑐 (N/m2) 39 × 108 28 × 108𝜌𝑒 (kg/m3) 800 800𝐸𝑒 (N/m2) 12 × 107 20 × 107

2.5 kHz. The characteristics also appear flatter over the entirefrequency range. The result of the objective function value𝑑 for the optimal solution was 𝑑 = 0.26, which representsan improvement of approximately 35% over the originalloudspeaker, for which the result was 𝑑 = 0.40.

In the optimal solution, the density of the cone was100 kg/m3, and Young’s modulus was 28 × 108N/m2. Theseproperties define a cone that is lighter in weight and vibratesmore easily but whose stiffness enables suppression of splitvibration in the nonaxisymmetric mode. This suggests thatcones should be constructed from light but rigid materials.

The results for the edge were a density of 800 kg/m3 andYoung’s modulus of 20 × 107N/m2. Since the edge supportsthe cone, this suggests that edges should be constructed ofmaterials with high rigidity.

In addition, the vibration mode of the cone was exam-ined.The vibrationmodes of the cones are shown in Figure 17.Red and blue indicate areas of large and small displacement,respectively. Figure 17(a) shows the vibration mode of astandard loudspeaker cone at 1.5 kHz, and Figure 17(b) showsthe vibration displacement mode of an optimal cone at1.5 kHz. In the case of the standard loudspeaker, it can beseen that the displacement is smaller at the center of the coneand greater at the periphery (the part attached to the edge),whereas the optimal cone moves with the same amount ofdisplacement.

By changing Young’s modulus and density of the coneand edge, we were able to reduce the peaks that occurdue to split vibration of the cone in the middle- and high-frequency regions above 1 kHz and achieve flatter frequencycharacteristics.


(a) Vibration mode of the standard cone (b) Vibration mode of the optimal cone

Figure 17: Analysis result of optimal solution: cone A at 1500Hz.

6. Conclusion

This study focused on the cone, the edge, and the behavior ofair around the voice coil, which are important elements in thedesign of cone loudspeakers, and evaluated their effects onthe acoustic characteristics of a loudspeaker. We performeda coupled analysis of the vibration and acoustics of a coneloudspeaker that accounted for the impact of air viscosity,which has received little attention in traditional acousticanalyses, and then sought optimal materials for the cone andedge. The following results were obtained:

(1) We took measurements of the vibration character-istics and sound pressure frequency response of acone loudspeaker in order to determine its acousticcharacteristics.

(2) We performed a coupled analysis of the vibration andacoustics of the loudspeaker that accounted for airviscosity and compared the vibration characteristicsand sound pressure frequency response to the actualmeasured values. By accounting for air viscosity, theanalysis was able to yield high accuracy.

(3) The coupled analysis of vibration and acousticsrevealed the vibration characteristics of the cone,which are influenced by the characteristics of the coneand edge. Furthermore, with respect to the behaviorof the air around the voice coil in the loudspeaker,the results of our analysis, which accounted for airviscosity, enabled us to clearly see the direction ofair movement; this is not possible with traditionalanalyses. This led to an improvement in the accuracyof the analysis.

(4) We used the response surface methodology to opti-mize the design of Young’s modulus and density ofthe cone and edge and obtained the optimumphysicalproperties of the cone and edge in order to design aloudspeaker with the desired acoustic characteristics.The optimal solution that was obtained reduced thepeaks and dips caused by split vibration of the coneand achieved flatter vibration characteristics.

With respect to future challenges, we hope to investigatethe analysis of loudspeakers of complex shapes, beyondtypical cone loudspeakers. In addition, we would like to alterthe shape of the loudspeaker components in ways that willlead to an optimum design to achieve flatter sound pressurefrequency properties.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

References

[1] W. N. Brown, “Theory of conical sound radiators,” Journal of theAcoustical Society of America, vol. 13, no. 1, pp. 20–22, 1941.

[2] P. G. Bordoni, “The conical sound source,” The Journal of theAcoustical Society of America, vol. 28, no. 2, pp. 123–126, 1945.

[3] T. Nimura, E. Matsui, K. Shibayama, and K. Kido, “Study on thecone type dynamic loudspeakers,”The Journal of the AcousticalSociety of Japan, vol. 7, no. 2, pp. 16–28, 1952.

[4] Y. Kagawa, “Natural frequency and vibration modes of curvedcone extensional vibration,”TohokuUniversityDentsuDiscourseRecord, vol. 31, no. 3, 1962.

[5] F. J. Frankfort, Vibration and sound radiation of loudspeakercones [thesis], Philips Research Reports Supplements no 2, 1975.

[6] T. Ueno, K. Takahashi, K. Ichida, and S. Ishii, “The vibrationanalysis of a cone loudspeaker by the finite element method,”The Journal of the Acoustical Society of Japan, vol. 34, no. 8, pp.470–477, 1978.

[7] K. Suzuki and I. Nomoto, “Computerized analysis and observa-tion of the vibration modes of a loudspeaker cone,” Journal ofthe Audio Engineering Society, vol. 30, no. 3, pp. 98–106, 1982.

[8] A. J. M. Kaizer, “Modeling of the nonlinear response of anelectrodynamic loudspeaker by a volterra series expansion,”AES: Journal of the Audio Engineering Society, vol. 35, no. 6, pp.421–433, 1987.

[9] A. J.M. Kaizer, “Calculation of the sound radiation of a nonrigidloudspeaker diaphragm using the finite-element method,” Jour-nal of the Audio Engineering Society, vol. 36, no. 7-8, pp. 539–551,1988.


[10] E. B. Skrodzka and A. P. Sęk, “Comparison of modal parame-ters of loudspeakers in different working conditions,” AppliedAcoustics, vol. 60, no. 3, pp. 267–277, 2000.

[11] R. M. Aarts, “Optimally sensitive and efficient compact loud-speakers,” Journal of the Acoustical Society of America, vol. 119,no. 2, pp. 890–896, 2006.

[12] N. Kyouno, T. Usagawa, T. Yamabuchi, andY. Kagawa, “Acousticresponse analysis of a cone-type loudspeaker by the finiteelement method,”The Journal of the Acoustical Society of Japan,vol. 61, no. 6, pp. 312–319, 2005.

[13] M. Sasajima, T. Yamaguchi, and A. Hara, “Acoustic analysisusing finite element method considering effects of dampingcaused by air viscosity in audio equipment,” Applied Mechanicsand Materials, vol. 36, pp. 282–286, 2010.

[14] T. Yamamoto, LoudSpeakers System, Radio Technology, Tokyo,Japan, 1977.

[15] K. Hashimoto, M. Sakane, M. Ohnami, and T. Yoshida, “Devel-opment of vibrating reed machine for measuring Young’smodulus of thin films,” Journal of the Society of MaterialsScience, Japan, vol. 44, no. 507, pp. 1456–1463, 1995.

[16] T. Saeki, LoudSpeakers & Enclosure, SeiBunDo ShinKoSha,Tokyo, Japan, 1999.

[17] X. Zhao, “Some new problems and measure with structuraloptimization software development,” in Proceedings of the Con-ference on Computational Engineering and Science (ICCS ’07),vol. 12, no. 1, pp. 129–132, Beijing, China, 2007.

[18] X. Zhao, Y.Hu, and I.Hagiwara, “Optimal design for crash char-acteristics of cylindrical thin-walled structure using origamiengineering,”The Japan Society of Mechanical Engineers, vol. 76,no. 761, pp. 10–17, 2010.

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal of

Volume 201

Submit your manuscripts athttps://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


effects of the cone and edge on the acoustic

Documents