Piezoelectric transducer with resonant modes control for parametric speaker

Transcription

1 PAPER #208 The Acoustical Society of Japan Piezoelectric transducer with resonant odes control for paraetric speaker Jun Kuroda and Yasuhiro Oikawa Departent of Interedia Art and Science, Waseda University, 3 4 Ohkubo, Shinjuku-ku, Tokyo, Japan (Received 4 October 206, Accepted for publication 7 May 207) Abstract: Paraetric speakers generate narrow directive sound field using odulated ultrasonic. Sall size paraetric speakers have a proble in sound quality. Deodulated audible sounds by paraetric speakers have poor sound pressure in low and iddle frequency bands due to the theory of the finite aplitude sound. The sound pressure of a deodulated sound onotonically increases as the frequency of the deodulated sound increases. To solve these probles, an ultrasonic transducer having a wide single peak or ore than two peaks ust be fabricated. Our previous paper proposed to use two resonant peaks which are close each other. Two close resonant peaks were obtained by two sae sized diaphrags linked each other by a specially designed rod. In this paper, a transducer was developed to better structure which has durability against strong echanical forces and two electrical inputs to control resonant odes of vibration. Durability was iproved by optiization of the junction structure. The resonant ode control was enabled by a phase difference between two electrical signals applied to piezoelectric eleents on two diaphrags. We succeeded in boosting id-range sounds, by the newly developed transducer. Keywords: Piezoelectric transducer, Ultrasonic, Paraetric speaker PACS nuber: Fx, At [doi:0.250/ast.39.]. INTRODUCTION e-ail: jun-kuroda@suou.waseda.jp Paraetric speakers generate narrow directive audible sound based on finite aplitude sonic wave theory. Paraetric speakers have been studied for various applications, for exaple, speech privacy systes, voice guidance. The governing equation of the finite aplitude sonic wave theory is presented by Westervelt (963) []. The basic idea and perforance of paraetric speakers was presented by Yoneyaa (983) [2] as the nae of audio spotlight. Although paraetric speakers have been used for various applications denoted above, they have high deand to be iproved fro point of view of the sound quality and downsizing to becoe ore popular. Sound quality is especially poor in low and iddle frequency bands. Iproving sound quality and sli sizing will proote installation of paraetric speakers into general equipents, such as personal coputers, tablets, and telephone systes. Sound signal processing, arrangeent of transducers, and nonlinear acoustical phenoena are well studied [3 9], even though there is a roo for iproveent of characteristics of piezoelectric transducers specialized to paraetric speakers. Our previous paper showed the design ethod to obtain two close resonant peaks of a piezoelectric transducer having double-linked diaphrags [0]. Design ethods are derived as two ordinary differential equations (ODEs) and an electrical-echanical equivalent circuit. These design ethod were very useful in optiizing the junction structure between two diaphrags. We fabricated piezoelectric transducers having two close resonant peaks in our previous paper. Two close resonant peaks of this designed transducer were expected to enable to boost of low and iddle range self-deodulated sounds. This paper shows the further developed design ethod to iprove the electro-acoustic efficiency and the sound pressure of the deodulated sound. Points of iproveent are, : optiization of the position, the diensions and the aterial of the junction rod, 2: addition of an electric signal input to enable control vibration odes. Two ODEs and the equivalent circuit are eployed again to design a newly developed transducer. In Sect. 2, the basic theory to design new transducer and results of nuerical calculation are denoted. In Sect. 3, the experiental results of a designed transducer are shown.

2 Fig. Scheatic of proposed ultrasonic transducer and definitions of each part. Fig. 2 Equivalent circuit of echanical coponents of transducer having double-linked diaphrags. 2. DESIGN METHODOLOGY 2.. Basic Structure of Ultrasonic Transducer Having Double-linked Diaphrags The sound pressure of a deodulated sound of a paraetric speaker onotonically increases as the frequency of the deodulated sound increases [ 3]. Therefore, it is necessary to boost the sound pressure in the vicinity of a resonant peak by iproving transducers, to obtain rich sound in id-range by paraetric speakers. More than two resonant peaks are useful in boosting the sound pressure in the vicinity of a resonant peak with aintaining the echanical quality factor (Q factor). We proposed an ultrasonic transducer which has two diaphrags linked by a resin rod. Figure shows the proposed transducer in our previous paper. In Fig., diaphrag is a uniorph-type piezoelectric transducer consisting of an adhered etal board and a piezoelectric eleent. Diaphrag 2 is a etal board. A resin rod is used to link diaphrags and 2. The resin rod ust be ade of an appropriately stiff resin to propagate the vibration to diaphrag 2. The diensions and aterials of two diaphrags are sae. Two close resonant peaks are obtained by the proposed ultrasonic transducer. By the Euler-Lagrange equation, the vibration displaceent in the thickness direction is written by two ODEs with luped constants as [0 3], þ r _ þ s þ½s ð Þ and þ r ð ÞŠ ¼ F þ r _ þ s ½s ð Þ þ r ð ÞŠ ¼ 0: ð2þ Note that and are the junction coefficients, i.e. the ratio of the su of the vibration displaceent in the junction area to that of the whole diaphrags, which depends on the junction area and the position of the rod in the diaphrags. In Eqs. () and (2),,, s, s, and r are deterined by the biharonic equation of the diaphrags. Suffixes, and ean diaphrag and 2, and rods, respectively. ðþ When the values of the junction coefficients and are approxiately equal, Eqs. () and (2) are expressed as the luped constant equivalent circuit shown in Fig. 2. Branches and 2 represent diaphrags and 2. The voltage F is the excitation force caused by the piezoelectric eleent of diaphrag. In Fig. 2, there are two resonant loops, i.e., loops and 2. Loop is the series resonant loop including diaphrags and 2. This resonance causes the first resonant peak of the current. Loop 2 coprises two parallel resonance circuits including the rod, diaphrags and 2. This resonance causes the second resonant peak of the current. In Fig. 2, the current represents the vibration velocity of the diaphrag. At the first resonant frequency (loop ), the currents in branches and 2 flow sae direction. By contrast, at the second resonant frequency (loop 2), two electrical currents flow opposite direction to each other Basic Theory of Design to Develop Transducer The transducer shown in Fig. has two points to be iproved, which are : optiization of position, diensions and aterial of junction rod, 2: addition of electric signal input to enable control vibration odes. Point is related to the durability against echanical pressures applied to the junction rod. Point 2 is related to the control of the direction of the electrical current in the echanical equivalent circuit Fig. 2. In this subsection, we denote the basic theory of these points using electroechanical equivalent circuits and analytical forulas. [Point ] The transducer shown in Fig. has one resin rod. This structure is designed to decrease the junction coefficients and, to bring two resonant frequencies near. Although this resin rod ay not endure strong echanical forces applied to rod when the high voltage is applied to the piezoelectric eleent. If soe etal rods are available, the durability against echanical forces rises. The stiffness s in the equivalent circuit of etal is uch larger than that of resin. Therefore, it is necessary to find a ethod to use etal rods with keeping a low value of the echanical reactance of the rod transfored to the side of dia- 2

3 J. KURODA and Y. OIKAWA: TRANSDUCER FOR PARAMETRIC SPEAKERS MODE CONTROL phrag, =j!s. When etal rods are used, the position of it ust not be located at the centers of the diaphrags. The junction coefficients and take axiu values when the rod is located at centers of diaphrags. On the contrary, the junction coefficients take iniu values when the rod is located around the nodes of vibration for. This is explained by equations as below. To consider the condition to suppress the junction coefficients, we abstracted the basic structure of the transducer, i.e. a bending disk and an excitation point, as show in Fig. 3. The hoogeneous and inhoogeneous equations expressing the vibration of a disk in a steady state are represented as [4 8], and r 4 g 4 0 g ¼ 0 r 4 p 4 p ¼ F Dr ðr r 0Þ; where! 0 is the wave nuber for the natural angular frequency and! is the angular frequency of the exciting force, which are defined as, ¼ h 4 D!2 and 0 ¼ h 4 D!2 0 : ð5þ The solutions of the natural ode of Eqs. (3) and (4) are written as, and Fig. 3 Scheatic of disk-shaped diaphrag and excitation force. g ¼ A 0 J 0 ð 0 rþþb 0 I 0 ð 0 rþ; p ¼ " Y ðjr r 0 jþ þ 2 # K 0ðjr r 0 jþ ds; where J 0 ð 0 rþ and Y 0 ð 0 rþ are the Bessel functions of the first and second kind (order is zero), I 0 ð 0 rþ and K 0 ð 0 rþ are the odified Bessel functions of the first and second kind (order is zero), A 0 and B 0 are the coefficients of Bessel functions, r is the absolute value of the position vector r. ð3þ ð4þ ð6þ ð7þ The vibration for of the bending disk is represented by the su of the solution Eqs. (6) and (7). The coefficients of Eq. (6) are decided by energy equations such as the Rayleigh-Ritz ethod. As shown in Fig. 3, the bending disk and the excitation area by an external force are expressed as the and, respectively. The excitation area corresponds to the junction area of the transducer. The junction coefficient is defined as, g þ s ds ¼ : ð8þ g þ s ds For siplification, we consider the case that a junction area is a point on the disk. When the disk has the circular shape, the integral of the solution g becoes zero by Eq. (A 8). Moreover, the particular solution of the inhoogeneous biharonic equation Eq. (7) is rewritten as p ¼ 8 ½Y 0ðjr r 2 0 jþ þ 2 K 0ðjr r 0 jþšds; ð9þ and p j r¼r0 ¼ 0: ð0þ Thus, Eq. (8) is rewritten as, g ds ¼ ¼ gj r¼r0 : ðþ s ds s ds The solution g of the hoogeneous equation Eq. (6) becoes zero, when the excitation area is a point which is located at nodes of the vibration displaceent. On the other hand, the solution p of the inhoogeneous equation Eq. (7) is not equal to zero. Therefore, becoes zero when the junction area is located at the node of the vibration (See Appendix). The junction coefficients of points around the nodes of vibration are close to zero by the continuity of the solutions Eqs. (6) and (9). Actual junction areas can be considered as the superposition of nubers of excitation points. Fro these consideration, it is concluded that the junction coefficient is suppressed when the rod is located at the neighborhood of the nodes of the vibration. [Point 2] Two resonant odes correspond to loops and 2 in the equivalent circuit Fig. 2, respectively. These resonant odes can be controlled by the direction of the currents _ and _ in the equivalent circuit. To control the direction of these currents, two voltage sources are needed. Thus, we consider to add a voltage source in the equivalent circuit as shown in Fig. 4. This additional voltage source eans the excitation force by the piezoelectric eleent on the 3

4 Fig. 4 Equivalent circuit of echanical coponents of transducer having double-linked diaphrags and double piezoelectric eleents. diaphrag 2. For the transducers having two piezoelectric eleents, the second ODE with luped constants is rewritten as þ r _ þ s ½s ð Þ þ r ð ÞŠ ¼ F : ð2þ The solution of Eqs. () and (2) are decided by the right hand side of these ODEs, i.e. the excitation force of the piezoelectric eleents. Equations () and (2) are rewritten for a Fourier coponent by the atrix for as, "! s þ s s þ j! r! þ r r s s þ s r r þ r!#!! 2 0 ¼ F! : ð3þ 0 F Two diaphrags of the transducer shown in Fig. have sae diensions, thus all values of echanical coponents are equal, i.e., s ¼ s ¼ s, ¼ ¼, r ¼ r ¼ r, and ¼ ¼. Equation (3) is siplified as, "! s þ s s þ j! r þ r! r s s þ s r r þ r!#! 0! 2 ¼ F! : ð4þ 0 F The ters of the left-hand side of Eq. (4) are the stiffness atrix, daping atrix and ass atrix, respectively. The natural frequencies and the natural vibration odes are calculated as the eigenvalues and eigenvectors of the stiffness atrix of Eq. (4), respectively. The resonant angular frequencies! and! 2 of the transducer are calculated by the eigenvalues of the stiffness atrix of Eq. (4) as, rffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s s þ 2s! ¼ ;! 2 ¼ : ð5þ The eigenvectors of the vibration displaceents and velocities of two diaphrags and are written as below; and _ ; _ ; ; ;! ¼ A ; ¼ j!a ; ;2 ;2 _ ;2 _ ;2! ¼ A 2 ; ð6þ ¼ j!a 2 ; ð7þ where A and A 2 are the coefficients of eigenvectors. The first and second eigenvectors correspond to the first and second eigen frequencies, respectively. The subscripts and 2 ean the first and second eigen odes, respectively. The excitation forces F and F corresponding to the natural frequencies and eigenvectors of vibration velocities are written as, F ; F ; ¼ A B ; F ;2 F ;2 ¼ A 2 B 2 ; ð8þ where coefficients B and B 2 are B ¼ s þ jr! ; B 2 ¼ s þ 2s þ jðr þ 2r Þ! : ð9þ Equations (7) and (8) ean that the first and second eigenodes of the vibration which are expressed as loops and 2, respectively, in the equivalent circuit Fig. 4. At the first natural ode, the excitation forces F ; and F ; are equal in agnitudes and phases. On the other hand, at the second natural ode, the excitation forces F ;2 and F ;2 are equal in agnitudes and opposite in phases. Note that the electrical inputs and echanical excitation forces have inphase relation, which is given by the configuration of the physical coordinates and direction in the equivalent circuit Fig. 4, and ODEs Eqs. () and (2). All solutions of Eqs. () and (2) are expressed as the superposition of the natural odes Eq. (7), which are excited by the excitation force Eq. (8) Detailed Design of Transducer We denote the detailed design of the transducer having double-linked diaphrags based on two design points above in previous subsection. Figure 7 shows the scheatic and diensions of a rod. The rod consists of a hollow bobbin-structured etal cylinder and a resin shaft. The hollow bobbin-structured cylinder and the resin shaft are ade of duraluin A207 and ABS resin, to reinforce the junction structure. Figure 8 shows the scheatics of the designed transducer. Three rods are used to connect two diaphrags. The centers of three rods are arranged on the nodes of the fundaental vibration of the diaphrags, which are located 5 fro the centers of the diaphrags. The transducer is housed in a resin box (ABS resin), which are eployed to arrange the directivity of sonic fields. 4

5 J. KURODA and Y. OIKAWA: TRANSDUCER FOR PARAMETRIC SPEAKERS MODE CONTROL Fig. 5 Plane in which the eission pattern of the sound pressure was calculated. Fig. 7 Scheatic of rod consists of etal cylinder and resin shaft. Fig. 8 Scheatic of transducer having double-linked diaphrags and double piezoelectric eleents. Fig. 6 Nuerical calculation restuls of eission pattern of sound pressure. Diaphrags and 2 have the flap vibration ode as shown in Figs. 3(a), 3(b), 3(c) and 3(d). Therefore the sound pressure of the ultrasonic radiated fro two diaphrags is dispersed to all directions. The resin box was eployed to arrange the directivity of the sound pressure. To confir the effect of the resin box, we show the calculation results of the eission patterns of the sound pressure level, which were calculated by the finite-eleentethod (FEM). Figure 5 shows the plane in which the eission patterns of the sound pressure level was calculated. Figures 6(a) and 6(b) show the nuerical calculation results of the eission patterns of the sound pressure for the first and second resonant odes. The distance between the top of the resin box and the observation point was 20 c. The solid lines and dot lines in Figs. 6(a) and 6(b) show the eission patterns when the transducer has the resin box or not, respectively. As shown in Figs. 6(a) and 6(b), the sidelobes of the sound pressure are suppressed by the resin box. It was confired that the resin box is useful in suppressing the sidelobes by this calculation results. Figure 9 shows the configuration of the electrical connection. Two etal boards of two diaphrags are electrically connected via the rods. In Fig. 9, v and v Fig. 9 Configuration of electrical connection of transducer. ean two input voltages to two piezoelectric eleents. Two diaphrags are connected to the return electrodes and the opposite sides of piezoelectric eleents are connected to two hot electrodes. Table shows the physical properties of etal cylinders, resin shafts, diaphrags and elastic supports. The elastic supports were fixed at a position of 5.0, which is the node of the fundaental natural vibration. The piezoelectric eleents were fabricated by Nihon Ceratec Co., Ltd. (Material code C). Detailed inforation, such as the elasto-piezo-dielectric (EPD) atrix, is listed in Table FEM Siulation The detailed diensions are decided by nuerical siulation by finite-eleent-ethod (FEM). FEM calculation was executed by FEMTET (Murata Software Co., Ltd.). 5

6 Table Physical properties of rod, diaphrag and elastic support. Part Material Mass density (kg/ 3 ) Young s odulus (GPa) resin rod ABS resin : etal dularluin cylinder A207 2: phosphor bronze diaphrag C520 8: elastic support polychloroprene : Table 2 Dielectric constant Physical properties of the piezoelectric board. Coupling factor Piezoelectric strain constant (0 2 /V) Elastic constant (0 2 2 /N) 33 = 0 ¼ 4;500 K r ¼ 0:6 d 3 ¼60 S ¼ 5:2 = 0 ¼ 4;700 K 3 ¼ 0:35 d 33 ¼ 280 S 33 ¼ 5:5 K 33 ¼ 0:65 d 5 ¼ 450 Fig. 0 Nuerical calculation results of frequency response of electrical adittance. Figures 0(a) and 0(b) show the frequency response of the electrical conductance and susceptance of the transducer of diaphrags and 2. The frequency response of the electrical conductance represents the su of the vibration velocity of each diaphrag. As shown in Figs. 0(a) and 0(b), both of two diaphrags have two resonant peaks. The first resonant frequency is 5.6 khz, and the second resonant frequency is 52.8 khz. Due to the location of the rods around the vibration nodes of the disk, two resonant frequencies becoe close in.2 khz. Figures (a) and (b) show the frequency response of the vibration velocity at the center of diaphrag 2. Figure (a) shows the frequency response of the vibration velocity of diaphrag 2 when the signal voltage is applied to either one of two diaphrags. Frequency range is fro 50 to 55 khz, which includes both of two resonant frequencies 5.6 khz and 52.8 khz. Figure (b) shows the frequency response of the vibration velocity when two input voltages are applied to both of two diaphrags. The Fig. Nuerical calculation results of frequency response of vibration velocity of diaphrag 2. phase difference of two electrical inputs in Fig. (b) is 0, 90 and 80 deg. As entioned in previous subsection, all vibration odes are expressed as the superposition of the eigenvectors of the vibration velocities Eq. (7), which are excited when the phase difference of two electrical inputs is 80 deg and 0 deg. In Fig., it is confired that the resonant peaks of the vibration velocity of transducer are varied, i.e. the first and second resonant peaks are enhanced when the phase difference of two electrical inputs is 80 and 0 deg. If the phase difference of two electrical inputs is gradually changed fro 0 deg to 80 deg with aintaining equal vibration agnitude, the agnitude of the first resonant peak increases and that of the second resonant peak decreases. When the phase difference of two diaphrags is not equal to either of 0 deg or 80 deg, two resonant peaks appear. For exaple, when the phase difference of two electrical inputs is 90 deg, the frequency response has two resonant peaks as shown in Fig. (b). This transition is shown as the transition fro the line of 0 deg through 90 deg to 80 deg in Fig. (b). Moreover, if the ratio of the agnitudes of two electrical inputs gradually changes fro one to zero with aintaining the phase difference as 0 deg or 80 deg, the frequency response shifts fro the single peak response to the double peaks response. This transition is shown as a shift fro the line of 0 deg or 80 deg in Fig. (b) to the line of diaphrag 2 in Fig. (a). Figures 2(a) and 2(b) show the frequency response of the sound pressure level. The distance between the top of the resin box and the observation point was 20 c. Figure 2(a) shows the frequency response of the sound pressure level when the signal voltage is applied to either one of two diaphrags. Figure 2(b) shows the frequency response of the sound pressure level when two input voltages are applied to both of two diaphrags. The phase difference of two input voltages in Fig. 2(b) is 0, 90 and 80 deg. As shown in Figs. 2(a) and 2(b), the frequency response of the sound pressure level has the sae behaviour of the vibration velocity at the center of 6

7 J. KURODA and Y. OIKAWA: TRANSDUCER FOR PARAMETRIC SPEAKERS MODE CONTROL Fig. 2 Nuerical calculation results of frequency response of sound pressure level. Fig. 4 Picture of transducer having double-linked diaphrags and double piezoelectric eleents. Fig. 5 Measured frequency response of electrical adittance. Fig. 3 Scheatic of nuerical calculation results of vibration ode. diaphrag 2. The agnitude of the second resonant peak is uch larger than that of the first peak. This difference of the agnitude of the resonant peaks is due to the vibration for of diaphrags. The vibration for of two diaphrags are different, as shown in Figs. 3(a), 3(b), 3(c) and 3(d). Thus the sound pressure levels at two resonant frequencies are different each other. Figures 3(a), 3(b), 3(c), and 3(d) show the scheatic of the calculation results of vibration for of two resonant peaks. Figures 3(a) and 3(c) show the vibration for of the first resonant ode, which is excited when the phase difference of two input voltages is 80 deg. Figures 3(b) and 3(d) show the second resonant ode, which is excited when the phases of two input voltages are equal. Two input voltages can be eployed to enforce either of two vibration for on the transducer. Figures 3(a), 3(b), 3(c) and 3(d) show the vibration odes corresponding to the first and second resonant peaks. 3. EXPERIMENT 3.. Ultrasonic Characteristics Figure 4 shows a picture of a anufactured prototype of transducer. The prototype was anufactured based on Fig. 6 Scheatic of easuring syste of LDV. Figs. 7 and 8. In Fig. 4, the edges of two diaphrags were coated by adhesives to prevent chipping of piezoelectric eleents. Figures 5(a) and 5(b) show the electrical conductance and susceptance of prototype transducer, respectively. The electrical adittance was easured the ipedance analyzer IM3570 (HIOKI E.E. CORPORATION). Input voltage was.0 V rs. Diaphrags and 2 have the first and second resonant peaks at 5.3 khz and 52.5 khz, respectively. Figure 6 shows the easuring syste of the vibration velocity of diaphrag 2. The vibration velocity at the center of diaphrag 2 was easured by Laser doppler vibroeter (LDV). Figures 7(a) and 7(b) show the experiental data of vibration velocity by LDV. Figure 7(a) shows easured vibration velocity at the center of diaphrag 2 when the input voltage is applied to 7

8 Fig. 7 Measured frequency response of vibration velocity of diaphrag 2. Fig. 9 Measured frequency response of sound pressure level of ultrasonic. Fig. 8 Scheatic of easuring syste of sound pressure level. Fig. 20 Measured frequency response of sound pressure level of self-deodulated sound. either of two diaphrags. Figure 7(b) shows the vibration velocity of diaphrag 2 when the input voltages are applied to both of two diaphrags (phase difference is set to 0, 90 and 80 deg). As shown in Fig. 7(b), the first and second resonant peaks are boosted when the phase difference of input voltages is set to 80 and 0 deg, respectively. Due to the production tolerance, the vibration velocity does not becoe the single peak response. Figure 8 shows the easuring syste of the sound pressure level of the transducer. PXI platfor (National Instruents Corporation) was eployed as the central easuring syste. The distance between the icrophone Type 4936-A-0 and the top of the box of the transducer was 20 c. Experiental conditions were; Measuring syste: Fig. 8 Roo: Anechoic chaber in Waseda University Input signal: Sweep tone ( khz, 5 V rs ), Microphone and conditioning aplifier: Type A-0 and Type 2690-A Distance between the icrophone and the top of the resin box: 20 c. Figures 9(a) and 9(b) show the experiental data of the sound pressure level of ultrasonic. As shown in Fig. 9(b), both peaks are boosted as with the vibration velocity of diaphrag Audible Sound Characteristics The frequency response of the deodulated audible sound was easured using two ultrasonic signals, i.e. a pure tone signal and a chirp signal. The audible sounds are deodulated in the air as the difference frequency sounds of the. To axiize the sound pressure level of the difference sounds, the frequency of the pure tone ust be set to one of the two resonant frequencies of the prototype transducer. We set the frequency of the pure tone to the first resonant frequency 5.5 khz. The frequency of the chirp signal was set around 52.5 kh. Thus, it is expected that the peak of the sound pressure level of the deodulated audible sound is located around 52:5 5:5 ¼ :0 [khz]. The sound pressure level of the deodulated sound was easured with changing the phase difference of input voltages applied to two diaphrags. Sound level eter NL-32 (RION Co., Ltd.) was eployed as the icrophone for audible sounds. The frequency response of the deodulated audible sound was easured as the experiental conditions as below; Measuring syste: Fig. 8 Roo: Anechoic chaber in Waseda University First signal = pure tone (frequency = 5.5 khz, phase difference = 0, 80 deg, 5 V rs ) Second signal = chirp signal (frequency = khz, phase difference = 0, 80 deg, 5 V rs ) Microphone: Sound level eter NL-32 Distance between the icrophone and the top of the resin box: 20 c. Figures 20(a) and 20(b) show the easured frequency response of the sound pressure level of the deodulated 8

9 J. KURODA and Y. OIKAWA: TRANSDUCER FOR PARAMETRIC SPEAKERS MODE CONTROL audible sound. Figure 20(a) shows the frequency response of the sound pressure level when voltage is applied to either of two diaphrags. Figure 20(b) shows the frequency response of the sound pressure level when input voltages are applied to both of two diaphrags. All of the frequency responses of the sound pressure level have the peaks around khz. In Fig. 20(b), the peaks of the sound pressure level were axiized when the phase differences of two pure tones and two chirp signals are set to 80 deg and 0 deg, respectively. These results are as expected by the ultrasonic characteristics shown in Figs. 9(a) and 9(b). 4. CONCLUSIONS In this paper, we design the ultrasonic transducer having double linked diaphrags for paraetric speakers, which enables to boost id-range audible sounds. The transducer was designed based on the basic theory, i.e. point : design to ake close the two resonant frequencies by the control of the junction coefficient between two diaphrags, point 2: design to enhance the resonant peaks by the control of the phase difference of two input signals. About point, it was derived that two resonant frequencies becoe close when junction rods are placed around the vibration nodes of the natural ode of bending disks. About point 2, it was derived that the phase difference of two input voltages is able to control vibration ode of two diaphrags. We arrived at the conclusion of theoretical consideration that the first and second resonant peaks are enhanced when the phase difference of two input voltages is 80 deg and 0 deg, respectively. Based on the theoretical consideration, prototype ultrasonic transducer was designed by these design points. It was verified by FEM and experients that the detailed design of the ultrasonic transducer fulfills the characteristics required to boost the id-range deodulated sounds. The frequency responses of the deodulated sounds were easured using two ultrasonic signals, i.e. pure tones and a chirp signals. The id-range sounds were enhanced when the phase differences of two pure tones and two chirp signals are set to 80 deg and 0 deg, respectively. REFERENCES [] P. J. Westervelt, Paraetric acoustic array, J. Acoust. Soc. A., 35, (963). [2] M. Yoneyaa and J. Fujioto, The audio spotlight: An application of nonlinear interaction of sound waves to a new type of loudspeaker design, J. Acoust. Soc. A., 73, (983). [3] T. Kaakura, M. Yoneyaa and K. Iketani, Study for practical application of paraetric speaker, J. Acoust. Soc. Jpn. (J), 4, (985) (in Japanese). [4] M. B. Moffett and R. H. Mellen, Model for paraetric acoustic sources, J. Acoust. Soc. A., 6, (976). [5] R. L. Rolleigh, Analysis of the broadband paraetric array with Gaussian priary directivity patterns, J. Acoust. Soc. A., 68, (980). [6] J. A. Gallego-Juárez and L. Gaete-Garretón, Propagation of finite-aplitude ultrasonic waves in air I. Spherically driving waves in the free field, J. Acoust. Soc. A., 73, (983). [7] P. R. Stepanishen and P. Koenigs, A tie doain forulation of the absorption liited transient paraetric array, J. Acoust. Soc. A., 82, (987). [8] P. R. Stepanishen and P. Koenigs, Matheatically trivial control of sound using a paraetric bea focusing source, J. Acoust. Soc. A., 29, (20). [9] S. Takeoka and Y. Yaasaki, Acoustic projector using directivity controllable paraetric loudspeaker array, Proc. ICA 200 (200). [0] J. Kuroda, Y. Oikawa, Y. Yaasaki, S. Sato, M. Kooda and Y. Onishi, Design of an ultrasonic piezoelectric transducer having double-linked diaphrags for paraetric speakers, Acoust. Sci. & Tech., 36, (205). [] T. Ikeda, Fundaentals of Piezoelectricity (Oxford University Press, New York, 990), 280 pp. [2] S. Takahashi and S. Hirose, Vibration-level characteristics of lead zirconate titanate ceraics, Jpn. J. Appl. Phys., 3, (992). [3] S. Sakai, Y. Shiozawa and T. Toi, Clarification of sound radiation echanis for airborne ultrasonic transducer, Acoust. Sci. & Tech., 30, (2009). [4] R. Holland, Analysis of ultiterinal piezoelectric plates, J. Acoust. Soc. A., 4, (967). [5] J. Soderkvist, An equivalent circuit description of two coupled vibrations, J. Acoust. Soc. A., 90, (99). [6] J. Saneyoshi, Y. Kikuchi and O. Nooto, Handbook of Ultrasonic Technologies (Nikkan Kogyo Shibun, Tokyo, 978), Chap. 3, pp (in Japanese). [7] Y. Urata, Analysis of the bending plate by the analytical solutions, Trans. Jpn. Soc. Mech. Eng. (Ser. C), 68, (2002) (in Japanese). [8] K. Suzuki, M. H. Isaka and K. Suzuki, Vibration characteristics of uniorph type circular piezoelectric vibrator, Dynaic and Design Conference 2007, No (2007) (in Japanese). APPENDIX: THE SUM TOTAL OF THE BENDING DISPLACEMENT OF CIRCULAR DISK It is proved that the su total of the bending displaceent of circular disk becoes zero under the plane stress condition. Integrate the both sides of Eq. (3) in the face of the circular disk, r 4 g ds 4 0 g ds ¼ 0: ða:þ When the solution g is axisyetric, Eq. (A ) is written as, g ds ¼ 2 Z R 4 rr 4 g dr; ða:2þ 0 0 where ds is the surface eleent, r is the radial coordinate, R is the radius of the circular disk. If and are both twice continuously differentiable on, Green s second identity is represented as, 9

10 ð r 2 r 2 ÞdS ða:3þ where dl is the circuference eleent, n is the noral vector of the boundary, is the boundary of the region. When is and phi is axisyetric, Eq. (A 3) is written as, Z R 2 rr 2 dr ¼ : r¼r ða:4þ Let be r 2 g, Eq. (A 4) becoes as, Z R 2 rr 4 g dr r2 g : ða:5þ 0 r¼r Substitute Eq. (A 5) into Eq. (A 2), g ds ¼ 4 r2 g : ða:6þ r¼r Here, by the boundary condition that the resultant share force is zero at the free edge, i.e., Q r2 g ¼ 0; ða:7þ r¼r where Q gr is the share force in radial direction and D is the bending stiffness. By Eqs. (A 6) and (A 7), the su total of the bending displaceent becoes g ds ¼ 0: ða:8þ 0