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Energy Harvesting with Piezoelectric Element Using Vibroacoustic Coupling Phenomenon

Energy Harvesting with Piezoelectric Element Using Vibroacoustic Coupling Phenomenon Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 126035, 11 pages http://dx.doi.org/10.1155/2013/126035 Research Article Energy Harvesting with Piezoelectric Element Using Vibroacoustic Coupling Phenomenon 1 2 1 Hiroyuki Moriyama, Hirotarou Tsuchiya, and Yasuo Oshinoya Department of Prime Mover Engineering, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan Rolling Stock Drive Systems Department, Toshiba Transport Engineering Inc. 1 Toshiba-cho, Fuchu, Tokyo 183-8511, Japan Correspondence should be addressed to Hiroyuki Moriyama; moriyama@keyaki.cc.u-tokai.ac.jp Received 3 October 2013; Accepted 31 October 2013 Academic Editor: Abdelkrim Khelif Copyright © 2013 Hiroyuki Moriyama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper describes the vibroacoustic coupling between the structural vibrations and internal sound fields of thin structures. In this study, a cylindrical structure with thin end plates is subjected to the harmonic point force at one end plate or both end plates, and a natural frequency of the end plates is selected as the forcing frequency. The resulting vibroacoustic coupling is then analyzed theoretically and experimentally by considering the dynamic behavior of the plates and the acoustic characteristics of the internal sound field as a function of the cylinder length. The length and phase difference between the plate vibrations, which maximize the sound pressure level inside the cavity, are clarified theoretical ly. eTh theoretical results are validated experimentally through an excitation experiment using an experimental apparatus that emulates the analytical model. Moreover, the electricity generation experiment verifies that sufficient vibroacoustic coupling can be created for the adopted electricity generating system to be eeff ctive as an electric energy-harvesting device. 1. Introduction have been regarded as the representative means for harvesting acoustic energy [3]. As an example application, electricity Recently, scavenging ambient vibration energy and convert- generation using resonance phenomena in a thermoacoustic ingitintousableelectricenergyvia piezoelectricmaterials engine was investigated with aim of harvesting the work have attracted considerable attention [1]. Typical energy done in the engine. The acoustic energy spent on electricity harvesters adopt a simple cantilever congfi uration to generate generation was harvested from a resonance tube branching electric energy via piezoelectric materials, which are attached out of the engine, and the appropriate position of the reso- to or embedded in vibrational elements. High-amplitude nance tube for eecti ff vely generating electricity was described excitations reduce the fatigue life of these harvesters. u Th s, [4]. Moreover, an electricity generating device using the placing appropriate constraints on the amplitudes is one of sound of voice via piezoelectric elements was developed significant ways to improve the performance of harvesters. as a supplementary electric source for a mobile phone [5]. A cantilever beam, whose deflection was constrained by a This system is also being investigated for other applications bump stop, was modeled. The effect of electromechanical [6]. Acoustic energy is extremely small in comparison to coupling was estimated in a parametric study, where the vibration energy. However, the above-mentioned electricity placement of the bump stop and the gap between the beam is generated by sound and vibration. eTh refore, vibroacoustic and stop were chosen as parameters [2]. Acoustic energy as coupling is one way to increase acoustic energy, although it well as vibration energy to be harvested sucffi iently fills our has not yet been investigated extensively in this context. working environment. er Th moacoustic engines that exploit Vibroacoustic coupling was investigated as an architec- the inherently efficient Stirling cycle and are designed on the tural acoustic problem via a coupled panel-cavity system con- basis of a simple acoustic apparatus with no moving parts sisting of a rectangular box with slightly absorbing walls and 2 Advances in Acoustics and Vibration a simply supported panel. eTh effect of the panel characteris- T x Rigid wall tics on the decay behavior of the sound efi ld in the cavity was considered both theoretically and experimentally [7, 8]. In an F a attempt to control noise in an airplane, an analytical model a 1 F 𝜃 2 Cylindrical r 2 for investigating coupling between the sound efi ld in an 1 cavity aircraft cabin and the vibrations of the rear pressure bulkhead was proposed [9, 10]. A cylindrical structure adopted as the Circular analytical model, in which the rear pressure bulkhead at one plate 2 Circular end of the cylinder was assumed to be a circular plate, was Rigid wall plate 1 examined under various conditions. The plate was supported at its edges by springs whose stiffness could be adjusted to Figure 1: Configuration of analytical model. simulate the various support conditions. These investigations clariefi d the inu fl ence of the support conditions on the sound pressure of an internal sound efi ld coupled with the vibration terms of the vibration and acoustic characteristics. In the of the end plate. eTh authors of this study used the above- experiment, the acceleration of the plate vibrations, the phase mentioned analytical model [9, 10]ofacylindricalstructure difference between them, and the sound pressure level inside with plates at both ends to investigate the vibroacoustic the cavity are considered significant characteristics of the coupling based on the sound pressure level distribution in plate vibrations and sound eld fi . These experimental results the cavity. The acoustic characteristics under vibroacoustic demonstrate the underlying theoretical analysis based on this coupling were investigated for cases in which excitation forces model, as well as the conditions that maximize the vibration with different relative amplitudes and phases were applied to and sound pressure levels. Furthermore, the eeff ct of vibroa- both end plates [11]. In addition, the excitation frequency at coustic coupling is estimated from an electricity generation which the coupling system becomes nonperiodic owing to experiment performed with piezoelectric elements. the application of excitation forces of different frequencies to the respective end plates was investigated [12]. Finally, the eeff ctofthe excitation position with respecttothe nodal 2. Analytical Method lines on the appearance of vibration modes on the plates was investigated [13]. On the other hand, to suppress the 2.1. Equation of Motion of Plate. eTh analytical model consists above vibration and acoustic energy, which was amplified of a cavity with two circular end plates, as shown in Figure 1. by vibroacoustic coupling, an analytical model that included eTh plates are supported by translational and rotational the installation of passive devices on the vibration system springs distributed at constant intervals, and the support was proposed as the electromechanical-acoustic system. The conditions are determined by their respective spring stiffness effect was fully validated in the numerical approach owing 𝑇 ,𝑇 ,𝑅 ,and 𝑅 , where the suffixes 1 and 2 indicate plates to tuning the resonance characteristics of the shunt circuit, 1 2 1 2 1 and 2, respectively. eTh plates of radius 𝑎 and thickness in which the piezoelectric device was incorporated, to the frequency characteristics of the coupling system [14]. ℎ have a Young’s modulus 𝐸 and a Poisson’s ratio ].The In almost all such studies, vibroacoustic coupling has sound eld fi , assumed cylindrical, has the same radius as been estimated by assuming that the plate and cavity that of the plates and varying length because the resonance dimensions, as well as the phase difference between the frequency depends on the length. The boundary conditions vibroacoustics of the two plates, were xfi ed. However, the areconsidered structurally andacousticallyrigid at thelateral natural frequencies of the plate and cavity vary with the wall between the structure and sound eld fi . The coordinates dimensions of the plate and cavity, and the phase difference used are radius 𝑟 and angle 𝜃 between the planes of the plates directly aeff cts the sound eld fi when the medium is much and the cross-sectional plane of the cavity and distance 𝑧 less dense than the plate. Although the study of coupling along the cylinder axis. eTh periodic point forces 𝐹 and 𝐹 1 2 phenomena had attracted attention, an extensive parametric areapplied to plates 1and 2atdistances 𝑟 and 𝑟 and angles 1 2 study has not yet been undertaken. Hence, little is known 𝜃 and 𝜃 , respectively. eTh natural frequency of the plates is 1 2 about the influence of dimensions and phase differences on employed as theexcitationfrequency. the coupling phenomena. To formulate the plate motion, Hamilton’s principle is To develop a new electricity generation system, we adopt appliedtothe analytical model[9]: an analytical model similar to the above-mentioned cylindri- cal structure with plates at both ends, because the vibration area of the model on which piezoelectric elements can be (1) = 𝛿 ∫ (𝑇 −𝑈 −𝑈 +𝑊)𝑡𝑑 = 0, 𝑃 𝑃 𝑆 installedistwice as largeasthatincaseofasingle plate. The cylinder length is varied over a wide range, while the where 𝐻 is the Hamiltonian, 𝑇 and 𝑈 are, respectively, the harmonic point force is applied to one end plate or both 𝑃 𝑃 kinetic and potential energy of each plate, and 𝑈 is the elastic end plates, and its frequency is selected to cause the plate energy stored in the springs. 𝑊 is the work done on plates 1 to vibrate in the fundamental mode. Vibroacoustic coupling that occurs between plate vibrations and the sound el fi d in and 2 by the respective point forces and the sound pressure the cavity is investigated theoretically and experimentally in on the plates. Finally, 𝑡 and 𝑡 are two arbitrary times. 0 1 𝛿𝐻 Advances in Acoustics and Vibration 3 The flexural displacements 𝑤 and 𝑤 of plates 1 and 2, each platevibration andthe soundfield. eTh pointforce and 1 2 respectively, in terms of two sets of suitable trial functions are acoustic excitation terms are, respectively, found by substituting (3) for the plate modes into (2)below: 𝑠 𝑠 𝐹 = ∫ 𝐹 𝛿(𝑟 − 𝑟 )𝛿(𝜃 − 𝜃 )𝑋 , 1𝑛𝑚 1 1 1 𝑛𝑚 1 1 ∞ ∞ 𝑠 𝑠 𝑗(𝜔𝑡+𝜙 ) 1 (6) 𝑤 = ∑ ∑ ∑ 𝐵 𝑋 𝑒 , 1 1𝑛𝑚 𝑛𝑚 𝑠 𝑠 𝑠=0 𝑛=0 𝑚=0 𝐹 = ∫ 𝐹 𝛿(𝑟 − 𝑟 )𝛿(𝜃 − 𝜃 )𝑋 , 2 2 2 2 2𝑛𝑚 𝑛𝑚 (2) 1 ∞ ∞ 2 𝑠 𝑠 𝑗(𝜔𝑡+𝜙 ) 𝑤 = ∑ ∑ ∑ 𝐵 𝑋 𝑒 , 𝑠 𝑠 𝑠 𝑠 2 2𝑛𝑚 𝑛𝑚 𝑃 = ∫ 𝑃 𝑋 ,𝑃 = ∫ 𝑃 𝑋 . (7) 1𝑛𝑚 𝑐 𝑛𝑚 1 2𝑛𝑚 𝑐 𝑛𝑚 2 𝑠=0 𝑛=0 𝑚=0 𝐴 𝐴 1 2 𝑋 = sin (𝑛𝜃 + )( ) , (3) 𝑛𝑚 Here 𝛿 is the delta function associated with the point force 2 𝑎 on plates, 𝐴 and 𝐴 are the plate areas, and 𝑃 is the 1 2 𝑐 where 𝑛 , 𝑚 ,and 𝑠 are, respectively, the circumferential order, sound pressure at an arbitrary point on the boundary surface radial order, and symmetry index with respect to the plate of the plates. To distinguish between plates 1 and 2, the 𝑠 𝑠 vibration. 𝐵 and 𝐵 are coefficients to be determined, 𝜔 differential 𝐴𝑑(𝑟𝑑𝜃𝑑) 𝑟 is written as and in (6)and 1𝑛𝑚 2𝑛𝑚 1 2 is the angular frequency of the harmonic point force acting (7), respectively. on the plate, and 𝑡 is the elapsed time. 𝜙 and 𝜙 are the 1 2 phases of the respective plate vibrations. In this analysis, 𝜙 2.2. Coupling Equation between Plate Vibrations and Internal is set to 0 deg, and 𝜙 varies in the range of 0 deg to 180 deg. Sound Field. For simplicity, we assume that the cavity walls 𝑤 and 𝑤 are substituted for the flexural displacements in 1 2 are rigid, so that the sound eld fi in the cavity is governed by 𝑇 ,𝑈 ,𝑈 ,and 𝑊 , whose detailed expressions are obtained 𝑃 𝑃 𝑆 the wave equation consisting of the eigenfunction 𝑌 and the from [9, 10], and the variation of (1) is carried out with eigenvalue 𝑘 corresponding to a cavity mode of order 𝑁 : respect to both plates. Consequently, the extremum of the Hamiltonian yields Euler’s equations, which are the equations 2 2 ∇ 𝑌 +𝑘 𝑌 =0, (8) 𝑁 𝑁 𝑁 of motion of the respective plates. eTh motion is assumed to 𝜕𝑌 be harmonic, that is, to behave as 𝑒 ,sothat 𝑒 can be ( ) =0, (9) 𝜕 u eliminated. Consider the following: where u is the unit normal to the boundary surface 𝑆 (positive ∞ towards the outside), and the boundary condition satisfies ( 9) 𝑠 2 𝑠 [ ∑ {𝐾 (1 + 𝜂𝑗 )−𝜔 𝑀 } 󸀠 󸀠 when 𝑆 is rigid. However, if 𝑆 is not rigid but has a varying 1𝑛𝑚𝑚 𝑝 1𝑛𝑚𝑚 𝑚 =0 specific acoustic admittance, we select a Green’s function 𝐺 󸀠 to obtain a solution set for a nonuniform cavity with nonrigid 𝑚 𝑚 + ∑ {𝑇 +( )( )𝑅 }] B 󸀠 𝑒 walls for a frequency 𝜔/2𝜋 = 𝐾𝑐/2𝜋 ,where 𝐾 is an eigenvalue 𝑠𝑛 1 1 1𝑛𝑚 𝑎 𝑎 𝑚 =0 of the nonuniform cavity, and 𝑐 is the cavity speed of sound. 𝑠 𝑠 = F − P , The equation for 𝐺 is thus given by 1𝑛𝑚 1𝑛𝑚 2 2 𝑠 2 𝑠 ∇ 𝐺+𝐾 𝐺=−𝛿( p − p ) [ ∑ {𝐾 (1 + 𝜂𝑗 )−𝜔 𝑀 } 0 󸀠 󸀠 2𝑛𝑚𝑚 𝑝 2𝑛𝑚𝑚 𝑚 =0 =−𝛿(𝑟 − 𝑟 )𝛿(𝜃 − 𝜃 )𝛿(𝑧 − 𝑧 ), 0 0 0 (10) 𝑚 𝑚 𝜕𝐺 + ∑ {𝑇 +( )( )𝑅 }] B 󸀠 𝑒 ( ) =0. 𝑠𝑛 2 2 2𝑛𝑚 𝑎 𝑎 𝜕 u 󸀠 𝑆 𝑚 =0 𝑠 𝑠 =−F + P , The right-hand side is a delta function, where the measure- 2𝑛𝑚 2𝑛𝑚 ment point is p = (𝑟, 𝜃, 𝑧) if thesourcepoint is p = (4) 0 (𝑟 ,𝜃 ,𝑧 ).Expressing 𝐺 in terms of 𝑌 of (8), which satisfies 0 0 0 𝑁 𝑠 𝑠 𝑠 𝑠 where 𝐾 󸀠 , 𝐾 󸀠 and 𝑀 󸀠 , 𝑀 󸀠 are elements of thesameboundaryconditions, we nfi dthat 1𝑛𝑚𝑚 2𝑛𝑚𝑚 1𝑛𝑚𝑚 2𝑛𝑚𝑚 the symmetrical stiffness and mass matrices, respectively, 󸀠 󸀠 𝑌 (p)𝑌 (p ) because the index 𝑚 is the radial order (𝑚=𝑚 ). 𝜂 is the 𝑁 𝑁 0 𝐺( p, p )= ∑ , (11) structural dampingfactorofthe plate, and 𝐹 is a coefficient 𝑠𝑛 𝑉 𝑀 (𝑘 −𝐾 ) 𝑐 𝑁 𝑁=1 𝑁 that is determined by the indices 𝑛 and 𝑠 and is expressed as 0 at 𝑁 =𝑀̸, [9] ∫ 𝑌 (p)𝑌 (p)𝑑𝑉 =𝑉 𝑀 𝛿 ={ 𝑁 𝑀 𝑐 𝑐 𝑁 𝑉 𝑀 at 𝑁=𝑀. 𝑐 𝑐 𝑁 𝜋, at 𝑛 =0 ̸ , { (12) 𝐹 = 0, at 𝑛 = 0, 𝑠 = 0, (5) 𝑠𝑛 The dimensionless factor 𝑀 is the mean value of 𝑌 2𝜋, at 𝑛 = 0, 𝑠 = 1. 𝑁 𝑁 averaged over the cavity volume 𝑉 ,and 𝛿 is the Kronecker Both rfi st terms on the right-hand sides of ( 4)givethe delta. respective point forces, and the second terms give the acoustic Because there is no source and 𝜕𝐺/𝜕 u =0 on 𝑆 ,thespatial excitations, which also function as the coupling term between factor 𝑃 (p)of the sound pressure within and on the surface 𝑁𝑀 𝑁𝑀 𝑎𝐹 𝑗𝜙 𝑎𝐹 𝑗𝜙 𝑗𝜔𝑡 𝑗𝜔𝑡 𝑑𝐴 𝑑𝐴 𝑠𝜋 𝑑𝐴 𝑑𝐴 𝑑𝐴 𝑑𝐴 4 Advances in Acoustics and Vibration bounding the medium can be obtained from just one of the Here, substituting (2)for 𝑤 and 𝑤 and considering a modal 1 2 surface integral terms as follows: damping factor 𝜂 ,(19)can be rewrittenas 2 2 𝑠 (p ) 𝑐 0 (𝜔 +𝑗𝜂 𝜔 𝜔−𝜔 )𝑃 𝑞 𝑐 𝑞 𝑞 𝑃 (p)=− ∫ 𝐺( p, p ) , (13) 𝑐 0 0 𝜕 u ∞ ∞ 𝑆 2 (20) 𝑠 𝑠 1 2 = (− ∑ 𝐼 𝐵 𝑒 + ∑ 𝐼 𝐵 𝑒 ), 1 2 1𝑛𝑚 2𝑛𝑚 where the zero subscripts indicate differentiation and inte- 𝑐 𝑚=0 𝑚=0 gration with respect to the (𝑟 ,𝜃 ,𝑧 ) coordinates. A detailed 1 1 0 0 0 𝑠 𝑠 𝑠 𝑠 𝐼 = ∫ 𝑋 𝑌 ,𝐼 = ∫ 𝑋 𝑌 , 1 1 2 2 𝑛𝑚 𝑞 𝑛𝑚 𝑞 procedure for obtaining these equations is given in [15]. 𝑃 𝐴 𝐴 𝐴 𝐴 1 2 canalsobeexpressed as [9, 10] (21) 𝑃 𝑌 𝑁 𝑁 where 𝐴 is the total surface area of the plates, 𝐼 and 𝐼 are the 1 2 𝑃 =𝜌 𝑐 ∑ , (14) 𝑐 𝑐 spatial coupling coefficients. Moreover, substituting ( 18)for 𝑁=1 𝑃 and applying 𝐼 and 𝐼 to the integrals in (7), the acoustic 𝑐 1 2 𝑠 𝑠 where 𝜌 is thefluiddensity in thecavity, and 𝑃 is the excitation terms 𝑃 and 𝑃 can be expressed with respect 𝑐 𝑁 1𝑛𝑚 2𝑛𝑚 pressure coefficient to be determined. to an arbitrary vibration mode (𝑛, ) 𝑚 as In this investigation, the acoustic modal shape 𝑌 and ∞ ∞ 𝐼 𝑃 angular resonance frequency 𝜔 in the cavity (where the 1 𝑞 𝑠 2 𝑃 =𝜌 𝑐 𝐴 ∑ ∑ , 1𝑛𝑚 indices 𝑛 , 𝑝 ,and 𝑞 indicate the circumferential, radial and 𝑠 𝑝=1 𝑞=0 𝑞 longitudinal orders, resp.) are defined as (22) ∞ ∞ 𝐼 𝑃 2 𝑞 𝑠 2 𝑃 =𝜌 𝑐 𝐴 ∑ ∑ . 𝑞𝜋 2𝑛𝑚 𝑐 𝑠 𝑠 𝑌 = sin (𝑛𝜃 + )𝐽 (𝜆 𝑟) cos {( )𝑧}, 𝑞 𝑛 𝑝=1 𝑞=0 2 𝐿 (15) 1/2 𝑞𝜋 Finally, replacing 𝑃 in (22)withthose in (20)and 𝜔 =𝑐{𝜆 +( ) } , then inserting them in (4), we can complete the coupling equations, whose right-hand sides are where 𝐽 is the 𝑛 th-order Bessel function, and 𝜆 is the 𝑝 th solution of an eigenvalue problem for a circular sound efi ld 2 2 2 𝜌 𝑐 𝜔 𝐴 𝑠 𝑠 𝑠 𝑐 having modes (𝑛, ) 𝑝 divided by the radius. eTh boundary F − P = F + 1𝑛𝑚 1𝑛𝑚 1𝑛𝑚 conditions between the plate vibrations and sound eld fi on 𝑠 𝑠 ∞ ∞ ∞ 1 2 therespectiveplatesurfacesarefoundbyassumingcontinuity 𝐼 (𝐼 𝐵 󸀠 𝑒 −𝐼 𝐵 󸀠 𝑒 ) 1 1 2 1𝑛𝑚 2𝑛𝑚 × ∑ ∑ ∑ , of thevelocitiesonthe plates: 𝑠 2 2 󸀠 𝑀 (𝜔 +𝑗𝜂 𝜔 𝜔−𝜔 ) 𝑝=1 𝑞=0 𝑐 𝑞 𝑚 =0 𝑞 𝑞 2 2 2 2 2 𝑐 𝑐 𝜌 𝑐 𝜔 𝐴 𝑠 𝑠 𝑠 𝑐 ( ) =𝜌 𝜔 𝑤 ,( ) =−𝜌 𝜔 𝑤 , (16) 𝑐 1 𝑐 2 −F − P =−F + 2𝑛𝑚 2𝑛𝑚 2𝑛𝑚 𝜕 u 𝜕 u 𝑧=0 𝑧=𝐿 𝑠 𝑠 ∞ ∞ ∞ 1 2 𝐼 (𝐼 𝐵 𝑒 −𝐼 𝐵 𝑒 ) 󸀠 󸀠 where /𝜕 u is 0 on the lateral wall of the cylinder since the 2 1 2 𝑐 1𝑛𝑚 2𝑛𝑚 × ∑ ∑ ∑ . wall remains rigid. 𝑠 2 2 󸀠 𝑀 (𝜔 +𝑗𝜂 𝜔 𝜔−𝜔 ) 𝑝=1 𝑞=0 𝑚 =0 𝑐 𝑞 𝑞 𝑞 Because the analytical mode has two boundary surfaces, (23) we can apply (16)to(13), so that 𝑃 becomes On the right-hand sides, the second terms show the 2 2 𝑃 =− ∫ 𝜔 𝑤 + ∫ 𝜔 𝑤 . (17) 𝑐 𝑐 1 1 𝑐 2 2 acoustic excitation for plates 1 and 2, respectively. The acoustic 𝐴 𝐴 1 2 excitation terms have both 𝐼 and 𝐼 since the acoustic mode 1 2 of the sound field is coupled with the vibration modes of On the other hand, by substituting acoustic modes of three therespectiveplates. Before actual calculation, thenatural orders, 𝑛 , 𝑝 ,and 𝑞 , instead of the order 𝑁 of the cavity mode frequency of the plate must be considered in terms of the into (14), 𝑃 canalsobeexpressed as convergence of the plate vibration mode (𝑛, ) 𝑚 .Inthiscase, 𝑠 𝑠 ∞ ∞ the natural frequency is obtained as the eigenvalue of (4), 𝑃 𝑌 𝑞 𝑞 𝑃 =𝜌 𝑐 ∑ ∑ . (18) whose right-hand side is set to 0. The actual calculation is 𝑐 𝑐 𝑝=1 𝑞=0 performed by taking 15 terms for 𝑛 ,while 𝑚 issettobegreater than 13 to ensure the convergence of the natural frequency eTh equation relating ( 17)and (18) is obtained by applying andmodeshape of theplate.Using thesametruncationfor Green’s function of (11) to an arbitrary acoustic mode(𝑛,𝑝,𝑞) 𝑝 as for 𝑚 , the order accounts for acoustic modes greater as than 𝑞=15 , so that the resonance frequency containing 𝑞 canexceed theexcitationfrequency.Theplate andcavityloss 2 2 𝑠 (𝜔 −𝜔 )𝑃 𝑞 𝑞 factors are assumed to be constant: 𝜂 =𝜂 = 0.01 [9, 10]. 𝑝 𝑐 𝑠 𝑠 (19) Since 𝐵 and 𝐵 canbeobtainedsimultaneouslyfrom 𝑠 𝑠 1𝑛𝑚 2𝑛𝑚 =− (∫ 𝑌 𝑤 + ∫ 𝑌 𝑤 ). 𝑞 1 1 𝑞 2 2 (4), which have as excitation terms, the behavior of the plate 𝐴 𝐴 𝑐 1 2 𝑛𝑝 𝑛𝑝 𝑑𝐴 𝑑𝐴 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑑𝐴 𝐺𝜌 𝑑𝐴 𝐺𝜌 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝜕𝑃 𝑗𝜙 𝑗𝜙 𝜕𝑃 𝜕𝑃 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑗𝜙 𝑗𝜙 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑠𝜋 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑑𝐴 𝑑𝐴 𝑗𝜙 𝑗𝜙 𝐴𝜔 𝑑𝑆 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝜕𝑃 Advances in Acoustics and Vibration 5 8 10 5 5 5 4 4 5 153 mm 1 31 3 9 9 2 2 1: Vibration generator 6: Multifunction generator 2: Load cell 7: Power supply 3: Acceleration sensor 8: FFT analyzer 4: Condenser microphone 9: Piezoelectric element 5: Amplifier 10: Power meter (a) (b) R c (c) Figure 2: Configuration of experimental apparatus: (a) Measurement system; (b) Installation state of piezoelectric element; and (c) Electrical circuit of energy-harvesting device. 𝑉 𝑉 𝑃 1 2 V vibrations andthe soundfieldunder vibroacousticcoupling 𝐿 =10 log ,𝐿 =10 log ,𝐿 =10 log , V1 V2 𝑝 V 𝑉 𝑉 𝑃 can be determined. 0 0 0 (25) The flexural displacements 𝑤 and 𝑤 and the sound 1 2 pressure 𝑃 are, respectively, obtained from (2)and (18)by ∗ ∗ ∗ 𝑠 𝑠 where 𝑤 , 𝑤 ,and 𝑃 are the respective conjugate compo- employing 𝐵 and 𝐵 determined above. However, the 1 2 𝑐 1𝑛𝑚 2𝑛𝑚 nents. procedure to calculate 𝑃 involves substituting 𝑃 in (20) for that in (18). Solved from (24), into which 𝑤 , 𝑤 ,and 𝑃 1 2 𝑐 are inserted, over the respective entire regions, the quadric 3. Experimental Apparatus and Method velocities 𝑉 and 𝑉 and the sound pressure 𝑃 are estimated 1 2 ] Figure 2(a) shows the experimental apparatus used in this by the logarithmic values 𝐿 , 𝐿 ,and 𝐿 relative to 𝑉 = V1 V2 𝑝 V 0 −15 2 2 −10 2 4 study. The structure consists of a steel cylinder with circular 2.5 × 10 m /s and 𝑃 =4 × 10 N /m ,respectively, as aluminum end plates that are 3 mm thick. eTh cylinder has shown in (25). an inner radius of 153 mm, and this length can be varied Consider the following: from 500 to 2000 mm to emulate the analytical model. One endplate or both endplatesare subjectedtothe pointforce, 2 2 𝜔 𝜔 ∗ ∗ 𝑉 = ∫ 𝑤 𝑤 ,𝑉 = ∫ 𝑤 𝑤 , whosefrequency makesthe plateexciteinthe (0,0)mode. 1 1 1 1 2 2 2 2 2𝐴 2𝐴 𝐴 𝐴 1 1 2 2 In case of the harmonic excitation of both ends, these forces ∗ are applied to the respective plates via small vibrators, and 𝑃 = ∫ 𝑃 𝑃 𝑑𝑉 , ] 𝑐 𝑐 2𝑉 their amplitudes are controlled to be 1 N. The positions of the point forces 𝑟 and 𝑟 are normalized by radius 𝑎 and (24) 1 2 𝑑𝐴 𝑑𝐴 𝑛𝑝 6 Advances in Acoustics and Vibration are set to 𝑟 /𝑎 = 𝑟 /𝑎 = 0.4 . In the excitation experiment, 1 2 the main characteristic is the phase difference between the plate vibrations. eTh refore, acceleration sensors are installed on both plates to measure this phase difference. To estimate the internal acoustic characteristics, the sound pressure level in the cavity is measured using condenser microphones with aprobe tube.Thetipsofthe probetubes arelocated near the plates and the cylinder wall, which are the approximate locations of the maximum sound pressure level when the sound efi ld becomes resonant. To perform the electricity generation experiment, the piezoelectric element is used and is comprised of the piezo- electric part constructed of ceramics and the electrode part 0 400 800 1200 1600 2000 constructed of brass, which have the diameters of 25 and L (mm) 35 mm and the thicknesses of 0.23 and 0.30 mm, respectively. Actually, the piezoelectric elements are installed at each 𝜙= 9 0 (deg) 𝜙= 0 (deg) center of both plates, as shown in Figures 2(a) and 2(b).The 𝜙= 1 0 (deg) electric power generated by the expansion and contraction Figure 3: Variation in sound pressure level with cylinder length of the piezoelectric elements is discharged through the when phase difference changes. resistance circuit, which consists of three resistors having resistances 𝑅 , 𝑅 ,and 𝑅 ,asshown in Figure 2(c). 𝑅 and V 𝑖 𝑐 V 𝑅 are the resistances of the voltmeter and ammeter built-in 𝑖 mode. eTh excitation frequency is chosen as 𝑓 that makes the wattmeter and are 2 MΩ and 2 mΩ, respectively, while the plates vibrate in the (0, 0)mode. With respect to the plate 𝑅 is the resistance of the resistor connected outside the vibration, although the phase 𝜙 of plate 1 is xfi ed at 0 deg, the wattmeter and is 97.5 kΩ.Tograsp theeeff ct of vibroacoustic phase 𝜙 of plate 2 ranges from 0 to 180 deg, and then they are coupling on energy harvesting, the electric power is measured relatedbythe phasedieff rence 𝜙 as follows: with and without the cylinder and is estimated by the 𝜙=𝜙 −𝜙 . (26) comparison of both cases. In such a estimation, the electric 2 1 powerisnormalizedbythe vibrationpower supplied with Figure 3 shows the variations in 𝐿 with 𝐿 ,whenonly 𝑝 V the plate, which is obtained from the point force and flexural plate1isexcitedand 𝜙 is arbitrarily set to 0, 10, and 90 deg. 𝐿 𝑝 V displacement. varies only slightly over the entire range of 𝐿 when𝜙=0 deg, but varies substantially and exhibits peaks near 𝐿 = 610 , 1220, and 1830 mm when 𝜙=10 and90deg.Thevalue of 𝐿 is 𝑝 V 4. Results and Discussion lower when 𝜙=10 and 90 deg and is almost identical at all phase differences near 𝐿 = 460 ,920and1560 mm.Toestimate 4.1. Acoustic Characteristics under Vibroacoustic Coupling. In the inu fl ence of each acoustic mode on these sound efi lds, the the theoretical study, the plates are assumed to be aluminum contribution 𝐶 canbedenfi edasthe ratioofthe acoustic having a Young’s modulus 𝐸 of 71 GPa and a Poisson’s ratio ] 𝐸 energy 𝐸 stored in the specific (𝑛, ,𝑝 𝑞) mode to the total of 0.33. eTh radius 𝑎 and thickness ℎ of the plates are constant acoustic energy 𝐸 of the entire sound field: at 153 mm and 3 mm, respectively, whereas the length of the all cylindrical sound field having the same radius as that of the 𝐶 = . (27) plates varies from 100 to 2000 mm. eTh support conditions of 3 2 all the plates, which have flexural rigidity 𝐷[= 𝐸ℎ /{12(1−] )}], are expressed by the nondimensional stiffness parameters Figures 4(a) and 4(b) show 𝐶 as a function of 𝐿 for the 3 3 𝑇 (= 𝑇 𝑎 /𝐷 = 𝑇 𝑎 /𝐷) and 𝑅 (= 𝑅 𝑎/𝐷 = 𝑅 𝑎/𝐷) .These (0, 0, 𝑞) modes for values of 𝑞 between 0 and 4 for 𝜙=0 𝑛 1 2 𝑛 1 2 values are identical for both plates. If 𝑅 ranges from 0 to and 90 deg, respectively. In case of 𝜙=0 deg, 𝐶 is relatively 8 8 10 when 𝑇 is 10 , the support condition can be assumed large when 𝐿 is short. The value of 𝐶 for the (0,0,0)mode 𝑛 𝐸 from a simple support to a clamped support. eTh actual decreases as 𝐿 increases because of the increase in the value 8 1 condition adopts 𝑇 =10 and 𝑅 =10 to get closer to of 𝐶 for the (0, 0, 1)mode. eTh (0, 0, 0)mode is regarded 𝑛 𝑛 𝐸 the experimental support condition. These plates 1 and 2 are as a pumping mode that corresponds to the reciprocation subjectedtothe pointforces 𝐹 and 𝐹 ,which areset to 1N of both end plates supported by the aerostatic stiffness of 1 2 andare locatedat 𝑟 /𝑎 = 𝑟 /𝑎 = 0.4 , respectively, as well as the cavity. In the range of longer length, as the value of 𝑞 1 2 the actual excitation experiment. In particular, the analysis in increases, 𝐶 decreases relatively to 𝐶 of the (0, 0, 0)mode 𝐸 𝐸 which only one end plate is excited is carried out with taking becausethe soundfieldisaeff cted by othermodes.At 𝜙= 𝐹 as 0 N. 90 deg, 𝐶 varies dramatically when the acoustic mode that 2 𝐸 The plate and sound eld fi eigenfrequency characteristics dominates the sound eld fi changes, so that the range of 𝐿 over involved in the vibroacoustic coupling are represented by the which the influence of such a dominant mode extends can natural frequency 𝑓 corresponding to the (𝑛, ) 𝑚 mode and be clearly distinguished. eTh acoustic mode (0, 0, 𝑞) causes 𝑛𝑚 the resonance frequency 𝑓 corresponding to the (𝑛,𝑝,𝑞) 𝐿 to have peaks at 𝐿 = 610 , 1220, and 1830 mm, having 𝑞 𝑝 V L (dB) 𝑛𝑝 𝑛𝑝 𝑛𝑝 Advances in Acoustics and Vibration 7 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 400 800 1200 1600 2000 0 400 800 1200 1600 2000 L (mm) L (mm) (0, 0, 1) (0, 0, 1) (0, 0, 0) (0, 0, 0) (0, 0, 2) (0, 0, 3) (0, 0, 2) (0, 0, 3) (a) (b) Figure 4: Contribution of acoustic mode to sound field at (a) 𝜙=0 [deg] and (b) 𝜙=90 [deg]. of 𝜙 is repeated in a similar manner as 𝐿 increases to max 𝐿 = 2000 mm. Peaks in 𝐿 indicate that vibroacoustic 𝑝 V (0, 0, 1) (0, 0 ,2) (0, 0, 3) coupling between the plate vibrations and sound eld fi is promoted at approximately 90 deg. 𝑓 and 𝑓 must be 𝑛𝑚 𝑞 approximately equal for the promotion of this coupling, so that the acoustic modes involved in vibroacoustic coupling are greatly influential around the lengths at which 𝐿 peaks. 90 𝑝 V On the other hand, we confirm that the longitudinal order 𝑞 shisf ft rom1to 2and from 2to3at 𝐿 = 900 and 1560 mm where 𝐿 varies abruptly, respectively, based on the 𝑝 V distributions of the sound pressure level along the 𝑧 direction inside the cavity. As a result, the sound fields, which are classified in the ranges of 470 to 900 mm, 910 to 1560 mm, 0 400 800 1200 1600 2000 and 1570 to 2000 mm, are dominated by the (0, 0, 1) , (0, 0, 2) , L (mm) and (0, 0, 3)modes, respectively, as shown in this gur fi e where the range of the acoustic mode is classified in color. Naturally, Figure 5: Variations in phase difference with cylinder length. this classification is also thought to be due to variations in the dominant acoustic mode, whose contribution 𝐶 is maximized in Figure 4(b). modal shapes similar to that of the (0, 0)mode of the plate Figure 6 shows the sound pressure levels 𝐿 and 𝐿 , 𝑝1 𝑝2 vibrations. es Th e peaks occur at an integer 𝑞 starting from𝑞= which are measured near plates 1 and 2, respectively, as 1 with increasing 𝐿 . eTh vibration of plate 2 has a signicfi ant functions of 𝐿 . eTh theoretical level 𝐿 ,which is maximized 𝑝 V effect on the formation of the sound eld fi and vibroacoustic at each 𝐿 when the phase difference 𝜙 ranges from 0 to coupling, despite not being driven by the point force, whereas 180 deg, is also indicated to compare with the experimental results. 𝐿 and 𝐿 show peaks around 615, 1275, and the inu fl ence of 𝜙 on the acoustic characteristics has only 𝑝1 𝑝2 been described for 𝜙=0 , 10, and 90 deg. This change in 1900 mm, and these levels are almost coincident for each 𝐿 with 𝜙 indicates that the acoustic characteristics depend peak.However,theydecreaseinthe middle rangeofthose 𝑝 V strongly on the vibration of plate 2; that is, there are ranges lengths. In particular, decreases in 𝐿 are remarkable and 𝑝1 of 𝜙 that intensify or suppress coupling between the plate their differences expand around 950 and 1550 mm. 𝐿 also 𝑝 V vibrations and sound field. Here the values of 𝜙 at which 𝐿 shows peaks at 610, 1230, and 1840 mm and corresponds 𝑝 V is maximum are denoted by 𝜙 . approximately with the above lengths where 𝐿 and 𝐿 max 𝑝1 𝑝2 In Figure 5, the variations in 𝜙 with 𝐿 and the values peak. max Since the above results are derived from the investigation of 𝜙 that maximize 𝐿 are plotted by a line and circles, max 𝑝 V respectively. 𝜙 is approximately 87 deg at 𝐿 = 100 mm and based on the excitation of one end plate, furthermore, the max decreases gradually with increasing 𝐿 up to approximately excitation condition in which both plates are subjected to the same excitation force is taken to grasp the eeff ct of the 𝐿 = 460 mm, where 𝜙 suddenly increases to over 90 deg max and then decreases again with increasing 𝐿 .This behavior excitation method on vibroacoustic coupling. Figure 7 shows (deg) max 𝑛𝑝 8 Advances in Acoustics and Vibration 0 400 800 1200 1600 2000 400 800 1200 1600 2000 L (mm) L (mm) max 1 max 2 p p2 max 1 max p1 max 2 exp Figure 6: Comparison between theoretical and experimental results for sound pressure level. Figure 8: Comparison between theoretical and experimental results for phase difference. flexural displacements 𝑤 and 𝑤 is also significant in 1 2 studying the eeff ct of the plate vibrations on the sound efi ld. Here the phase differences are denoted as 𝜙 and 𝜙 max1 max2 120 when 𝑤 and 𝑤 are maximized, while they are denoted as 1 2 𝜙 and 𝜙 when 𝑤 and 𝑤 are minimized. Figure 8 min1 min2 1 2 shows 𝜙 , 𝜙 , 𝜙 ,and 𝜙 as functions of 𝐿 when max1 max2 min1 min2 only plate 1 is excited. 𝜙 is constant at 180 deg for 𝐿 max1 ranging from 100 to 390 mm and decreases abruptly up to 0 deg at 𝐿 = 400 mm. Then, remaining constant at 0 deg up to 𝐿 = 610 mm, 𝜙 increases gradually with 𝐿 and returns max1 to 180 deg at 𝐿 = 1220 mm, increasing somewhat abruptly near 𝐿 = 970 mm. Beyond 𝐿 = 1220 mm, 𝜙 is again 400 800 1200 1600 2000 max1 constant at 180 deg up to 𝐿 = 1570 mm, and this behavior L (mm) is repeated as 𝐿 increases to 𝐿 = 2000 mm. 𝜙 exhibits max2 p p2 gradual and abrupt changes similar but alternate to 𝜙 . max1 p1 For example, when 𝐿 increases, a gradual decrease occurs in 𝜙 between 𝐿 = 100 and 620 mm, and an abrupt increase max2 Figure 7: Comparison between theoretical and experimental results occurs near 𝐿 = 970 mm. Both 𝜙 and 𝜙 shift between for sound pressure level when both end plates are excited by point max1 max2 0 and 180 deg with changing 𝐿 and intersect at approximately force. 90 deg and near the length at which 𝐿 peaked in Figure 6. 𝑝 V 𝜙 and 𝜙 behave exactly alike but opposite to 𝜙 min1 min2 max1 the variations in 𝐿 corresponding to 𝜙 and the variations 𝑝 V max and 𝜙 . max2 in 𝐿 and 𝐿 that are measured in the experiment and are 𝑝1 𝑝2 In Figure 8,the theoreticalresults for 𝜙 where 𝐿 max 𝑝 V maximized when the phase difference between both point peaks and the experimental results for 𝜙 as 𝐿 and 𝐿 exp 𝑝1 𝑝2 forces ranges from 0 deg to 180 deg. Peaks in 𝐿 appear at𝐿= 𝑝 V are maximized at each 𝐿 are also plotted. 𝜙 ranges greatly exp 610, 1230, and 1840 mm. eTh se peaks are known to be caused between in-phase and out-of-phase and 𝜙 exists in the max by the (0, 0, 1) , (0,0,2),and (0, 0, 3)modes, respectively. Note process where 𝜙 changes abruptly. Then, 𝜙 lies in the exp exp that 𝐿 and 𝐿 increase greatly at 625, 1250, and 1850 mm. 𝑝1 𝑝2 light yellow areas surrounded by 𝜙 and 𝜙 in the 𝐿 max1 max2 However, 𝐿 and 𝐿 are hardly distinguished in the middle 𝑝1 𝑝2 ranges longer than the lengths when the sound pressure level rangeoflengths wherethe soundpressure levels peak,having peaks and occurs in the yellowish green areas surrounded been different in the results for the excitation of one end plate, by 𝜙 and 𝜙 on the other side. In other words, since min1 min2 as shown in Figure 6. vibroacoustic coupling is gradually weakened with increasing 𝐿 after the peaks of 𝐿 and 𝐿 , the acoustic mode involved 𝑝1 𝑝2 4.2.Plate VibrationCharacteristics underVibroacoustic Cou- in coupling shifts to that having the next order 𝑞 . pling. Since the plate vibrations influence the acoustic char- Figure 9 shows the vibration levels 𝐿 and 𝐿 of plates V1 V2 acteristics via vibroacoustic coupling, the magnitude of the 1and 2asfunctions of 𝐿 and the accelerations 𝛼 and 𝛼 of 1 2 L , L , L (dB) L , L , L (dB) p p1 p2 p p1 p2 (deg) Advances in Acoustics and Vibration 9 to that of 𝐿 and 𝐿 as showninthe coloredregions.In V1 V2 the theoretical analysis, we conrfi m that the distributions of the sound pressure level along the 𝑧 direction inside the cavity behave in a similar manner to the sound field inside a soundtubehavingsingleclosedand open ends at thelengths, around which the sound pressure level decreased in Figure 6. These distributions occur in the process of shifting acoustic modes because of changing the cylinder length and have the opposite tendency for the difference between 𝐿 and 𝐿 𝑝1 𝑝2 in thosemiddlerangesoflengths wherethe soundpressure levels peaked in Figure 6. This is derived from the difference between the tendencies of the above vibration level and 0 400 800 1200 1600 2000 acceleration with respect to both plates. If the experimental model completely emulated the theoretical model in the L (mm) flexural displacement, such a discrepancy would not take 2 1 place. In Figure 10, the experimental phase difference 𝜙 ,at exp which 𝐿 and 𝐿 are maximized, is compared with the Figure 9: Comparison between characteristics of theoretical vibra- 𝑝1 𝑝2 tion and experimental acceleration. theoretical phase differences 𝜙 , 𝜙 , 𝜙 ,and 𝜙 max1 max2 min1 min2 when both plates aresubjected to thesameexcitationforce, as shown in Figure 8.Theshisft in the (0, 0, 𝑞) modes are also represented by the changing colors. 𝜙 shifts between 0 and 180 exp 180 deg with changing 𝐿 and corresponds approximately with the 𝜙 and 𝜙 that behave uniformly in the vicinity max1 max2 of the in-phase side or the out-of-phase side. In this way, the behavior of the phase difference is very different in the excitation method. 4.3. Electricity Generation Characteristics. In this section, we consider electricity generation by the plate vibrations coupled with the sound efi ld. In this case, the electricity generation is estimated by the comparison between the electric power via the piezoelectric element and the mechanical power supplied to the plate by the vibrator that is obtained from the rela- 0 400 800 1200 1600 2000 tionship between the point force and flexural displacement at L (mm) the excitation point. Figure 11 shows the relationship between 𝜙 the electric power 𝑃 and mechanical power 𝑃 when the max 2 max 2 𝑒 𝑚 max 1 point force 𝐹 ranges from 1 to 5 N. In this case, the cylinder exp max 2 is removed; the plate vibrations do not couple with the internal sound eld fi . Although 𝑃 is considerably smaller than 𝑃 , their relationship is directly proportional. Here, the Figure 10: Comparison between theoretical and experimental relationship between 𝑃 and 𝑃 is defined as 𝑒 𝑚 results for phase difference when both end plates are excited by point force. 𝑒 𝑃 = . (28) 𝑒𝑚 Figure 12 shows variations in 𝑃 with 𝐿 ;the powers 𝑒𝑚 plates 1 and 2 are also plotted to compare with the theoretical are measured in the experimental apparatus via the cylinder plate behavior. 𝐿 is smaller than 𝐿 in the ranges of 100 to shown in Figure 2(a) when one end plate is excited by the V1 V2 610 mm, 800 to 1200 mm, and 1300 to 1820 mm, so that 𝐿 point force. Although 𝑃 of plate 1 remains almost constant V1 𝑒𝑚 and 𝐿 intersect at a number of 𝐿 and the intersections take over the entire range of 𝐿 , 𝑃 of plate 2 increases greatly at V2 𝑒𝑚 place around the lengths where 𝐿 peaks. eTh actual motion 𝐿 = 615 , 1275, and 1900 mm. It is natural that the relationship 𝑝 V of plate 1 is almost suppressed by that of the vibrator since between 𝑃 of plate 1 and 2 is derived from the behavior of 𝑒𝑚 plate1issupportedbythevibrator;hence, 𝛼 is approximately 𝛼 and 𝛼 in Figure 9. 𝑃 is considered as energy-harvesting 1 1 2 𝑒𝑚 constant over the entire 𝐿 range. However, since the motion efficiency. In an electricity generation by means of beam of plate 2 depends greatly on the behavior of the sound or plate vibrations with piezoelectric elements, coupling field, that is, the only excitation source for plate 2, 𝛼 peaks between the structural vibration and electric field, that is, at 𝐿 = 650 , 1280, and 1880 mm and is suppressed in the electromechanical coupling should be considered to relate the other ranges of 𝐿 ,suchasvariationsin 𝐿 and 𝐿 .Asa in-plane stress to the applied electric field. Considering that 𝑝1 𝑝2 result, the relative relationship of 𝛼 and 𝛼 becomes opposite the behaviors of 𝑃 for plates 1 and 2 correspond to those of 1 2 𝑒𝑚 L , L (dB) (deg) 1 2 , (m/s ) 1 2 10 Advances in Acoustics and Vibration 𝛼 and 𝛼 and the values of 𝑃 are substantially small, that is, 0.03 1 2 𝑒𝑚 the value of 𝑃 is extremely small in comparison with that of 𝑃 , we assume that the behavior of plate vibrations is hardly aeff cted by electromechanical coupling in this study. Moreover, to grasp the eeff ct of vibroacoustic coupling 0.02 on electricity generation, although 𝑃 is taken as the ratio of 𝑃 measured with and without the cylinder, 𝑃 measured 𝑒𝑚 𝑒𝑚 with the cylinder is obtained from the total value 𝑃 with respecttoplates1and2. Figure 13 shows variations in 𝑃 0.01 with 𝐿 ; the results of the excitation of both ends are also indicated to study the eeff ct of the excitation method. 𝑃 for the excitation of one end is maximized as the sound pressure level peaks due to the promotion of vibroacoustic coupling. 0 2.0 4.0 6.0 8.0 This is because the plate vibration on the nonexcitation side P (mW) contributes strongly to the electricity generation, whereas their 𝐿 ranges are limited to the narrow regions in comparison Figure 11: Relationship of electric power and mechanical power. with the sound pressure level. Since the contribution of the nonexcitation side is extremely weakened in other ranges of the above 𝐿 , the effect of the excitation side is uniformly maintained, as shown in Figure 12. On the other hand, at the 3.0 excitation of both ends, there is not such a nonexcitation side that contributes greatly to the electricity generation, so that 𝑃 never reaches values as large as those of the excitation of one end, when the sound pressure level is maximized. In 𝐿 2.0 ranges where vibroacoustic coupling is weakened, 𝑃 remains almost constant and is close to that of the excitation of one end. Under the situation of such a weakened vibroacoustic 1.0 coupling, the excitation of both ends becomes approximately twice as large as the excitation of one end in the respective total amounts of the electric power 𝑃 and mechanical power 𝑃 .Asaresult, 𝑃 shifts almost constantly with 𝐿 ,nomatter 𝑚 𝑅 0 400 800 1200 1600 2000 what theexcitationmethodis. L (mm) Hence, it follows that the excitation of one end is decidedlysuperiortothatofbothendsbyelectricity gener- Plate 2 ation efficiency, according to this estimation method. These Plate 1 results are remarkable in the viewpoint where the acoustic Figure 12: Energy-harvesting efficiency as function of cylinder energy from the sound radiation can be harvested through length. vibroacoustic coupling. However, the number or area of the piezoelectric element should be increased to improve the efficiency 𝑃 ,sothattheeeff ctofelectromechanicalcoupling 𝑒𝑚 12.0 on the plate vibration must be taken into consideration in the theoretical procedure. 10.0 8.0 5. Conclusion 6.0 To apply vibroacoustic coupling to electricity generation, as a means of harvesting energy from vibration systems, coupling 4.0 between plate vibrations and a sound eld fi was investigated theoretically and experimentally for a cylindrical structure 2.0 with thin circular end plates. The end plate was subjected to a harmonic point force. Moreover, the effect of vibroacoustic coupling on the harvest of energy was estimated from the 400 800 1200 1600 2000 electricity generating experiment. The present study focused L (mm) on promoting the vibroacoustic coupling to increase the Excitation of one end plate flexural displacements of the plates and the sound pressure Excitation of both end plates level inside the cavity. As a result of the estimation of vibroacoustic coupling Figure 13: Effect of vibroacoustic coupling on energy harvesting for from various viewpoints, the theoretical study reveals that excitations of one end plate and both end plates. −2 P P ×10 P (mW) R em e Advances in Acoustics and Vibration 11 the closeness of eigenfrequencies and the similarity of modal end plates: influence of excitation position on vibro-acoustic coupling,” Acoustical Science and Technology,vol.26, no.6,pp. shapes between the plate vibrations and sound efi ld are 477–485, 2005. indispensable for promoting coupling. The experimental [14] W. Larbi, J.-F. Deu, ¨ and R. Ohayon, “Finite element formulation results conrfi m that the theoretical estimation of increasing of smart piezoelectric composite plates coupled with acoustic the flexural displacement and sound pressure level via the uid fl ,” Composite Structures,vol.94, no.2,pp. 501–509, 2012. promotion of vibroacoustic coupling support the complicated [15] P.M.Morse andK.U.Ingard, Theoretical Acoustics ,McGraw- acoustic characteristics deduced from the theoretical results. Hill,New York,NY, USA, 1968. In particular, changes in the cylinder length shift the acoustic mode in the longitudinal order and vary periodically the phase difference between both plate vibrations. It is validated that the phase difference is greatly different in the excitation method. When vibroacoustic coupling is promoted, the electricity generation experiment verifies that the promotion of coupling causes the generation efficiency to improve in comparison with the electricity generation caused only by the plate vibration without coupling. References [1] S. R. Anton and H. A. Sodano, “A review of power harvesting using piezoelectric materials (2003–2006),” Smart Materials and Structures,vol.16, no.3,pp. 1–21,2007. [2] K.H.Mak,S.McWilliam, A. A. Popov, andC.H.J.Fox,“Per- formance of a cantilever piezoelectric energy harvester impact- ing a bump stop,” Journal of Sound and Vibration,vol.330,no. 25, pp. 6184–6202, 2011. [3] S. Backhaus and G. W. Swift, “A thermoacoustic stirling heat engine,” Nature, vol. 399, no. 6734, pp. 336–338, 1999. [4] S. Sakamoto, D. Tsukamoto, Y. Kitadani, T. Ishino, and Y. Wata- nabe, “Effect of sub-loop tube on energy conversion efficiency of loop-tube-type thermoacoustic system,” in Proceedings of the 20th International Congress on Acoustics (ICA ’10),pp. 1361– 1362, 2010 (Japanese). [5] K. Hayamizu, “Device for electric generation,” Japanese patent disclosure, 2010-200607, 2010 (Japanese). [6] K. Hayamizu, R. Ando, and Y. Takefuji, “Simultaneous provid- ing device of baseband and carrier signal using sound-gener- ated electricity,” Mobile Multimedia Communications,vol.105, no.80, pp.47–49,2005(Japanese). [7] J. Pan and D. A. Bies, “eTh eeff ct of uid-st fl ructural coupling on sound waves in an enclosure—theoretical part,” Journal of the Acoustical Society of America,vol.87, no.2,pp. 691–707, 1990. [8] J. Pan and D. A. Bies, “eTh eeff ct of uid-st fl ructural coupling on sound waves in an enclosure—experimental part,” Journal of the Acoustical Society of America,vol.87, no.2,pp. 708–717, 1990. [9] L. Cheng and J. Nicolas, “Radiation of sound into a cylindrical enclosure from a point-driven end plate with general boundary conditions,” Journalofthe Acoustical SocietyofAmerica,vol.91, no. 3, pp. 1504–1513, 1992. [10] L. Cheng, “Fluid-structural coupling of a plate-ended cylindri- cal shell: vibration and internal sound field,” Journal of Sound and Vibration,vol.174,no. 5, pp.641–654,1994. [11] H. Moriyama, “Acoustic characteristics of sound field in cylin- drical enclosure with exciting end plates,” Transactions of Japan Society of Mechanical Engineers C,vol.69, no.679,pp. 603–610, 2003 (Japanese). [12] H. Moriyama and Y. Tabei, “Acoustic characteristics inside cy- lindrical structure with end plates excited at different frequen- cies,” Journal of Visualization,vol.7,no. 1, pp.93–101, 2004. [13] H. Moriyama, Y. Tabei, and N. Masuda, “Acoustic characteris- tics of a sound field inside a cylindrical structure with excited International Journal of Rotating Machinery International Journal of Journal of The Scientific Journal of Distributed Engineering World Journal Sensors Sensor Networks Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Volume 2014 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Acoustics and Vibration Hindawi Publishing Corporation

Energy Harvesting with Piezoelectric Element Using Vibroacoustic Coupling Phenomenon

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Hindawi Publishing Corporation
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Copyright © 2013 Hiroyuki Moriyama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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10.1155/2013/126035
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 126035, 11 pages http://dx.doi.org/10.1155/2013/126035 Research Article Energy Harvesting with Piezoelectric Element Using Vibroacoustic Coupling Phenomenon 1 2 1 Hiroyuki Moriyama, Hirotarou Tsuchiya, and Yasuo Oshinoya Department of Prime Mover Engineering, Tokai University, 4-1-1 Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan Rolling Stock Drive Systems Department, Toshiba Transport Engineering Inc. 1 Toshiba-cho, Fuchu, Tokyo 183-8511, Japan Correspondence should be addressed to Hiroyuki Moriyama; moriyama@keyaki.cc.u-tokai.ac.jp Received 3 October 2013; Accepted 31 October 2013 Academic Editor: Abdelkrim Khelif Copyright © 2013 Hiroyuki Moriyama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper describes the vibroacoustic coupling between the structural vibrations and internal sound fields of thin structures. In this study, a cylindrical structure with thin end plates is subjected to the harmonic point force at one end plate or both end plates, and a natural frequency of the end plates is selected as the forcing frequency. The resulting vibroacoustic coupling is then analyzed theoretically and experimentally by considering the dynamic behavior of the plates and the acoustic characteristics of the internal sound field as a function of the cylinder length. The length and phase difference between the plate vibrations, which maximize the sound pressure level inside the cavity, are clarified theoretical ly. eTh theoretical results are validated experimentally through an excitation experiment using an experimental apparatus that emulates the analytical model. Moreover, the electricity generation experiment verifies that sufficient vibroacoustic coupling can be created for the adopted electricity generating system to be eeff ctive as an electric energy-harvesting device. 1. Introduction have been regarded as the representative means for harvesting acoustic energy [3]. As an example application, electricity Recently, scavenging ambient vibration energy and convert- generation using resonance phenomena in a thermoacoustic ingitintousableelectricenergyvia piezoelectricmaterials engine was investigated with aim of harvesting the work have attracted considerable attention [1]. Typical energy done in the engine. The acoustic energy spent on electricity harvesters adopt a simple cantilever congfi uration to generate generation was harvested from a resonance tube branching electric energy via piezoelectric materials, which are attached out of the engine, and the appropriate position of the reso- to or embedded in vibrational elements. High-amplitude nance tube for eecti ff vely generating electricity was described excitations reduce the fatigue life of these harvesters. u Th s, [4]. Moreover, an electricity generating device using the placing appropriate constraints on the amplitudes is one of sound of voice via piezoelectric elements was developed significant ways to improve the performance of harvesters. as a supplementary electric source for a mobile phone [5]. A cantilever beam, whose deflection was constrained by a This system is also being investigated for other applications bump stop, was modeled. The effect of electromechanical [6]. Acoustic energy is extremely small in comparison to coupling was estimated in a parametric study, where the vibration energy. However, the above-mentioned electricity placement of the bump stop and the gap between the beam is generated by sound and vibration. eTh refore, vibroacoustic and stop were chosen as parameters [2]. Acoustic energy as coupling is one way to increase acoustic energy, although it well as vibration energy to be harvested sucffi iently fills our has not yet been investigated extensively in this context. working environment. er Th moacoustic engines that exploit Vibroacoustic coupling was investigated as an architec- the inherently efficient Stirling cycle and are designed on the tural acoustic problem via a coupled panel-cavity system con- basis of a simple acoustic apparatus with no moving parts sisting of a rectangular box with slightly absorbing walls and 2 Advances in Acoustics and Vibration a simply supported panel. eTh effect of the panel characteris- T x Rigid wall tics on the decay behavior of the sound efi ld in the cavity was considered both theoretically and experimentally [7, 8]. In an F a attempt to control noise in an airplane, an analytical model a 1 F 𝜃 2 Cylindrical r 2 for investigating coupling between the sound efi ld in an 1 cavity aircraft cabin and the vibrations of the rear pressure bulkhead was proposed [9, 10]. A cylindrical structure adopted as the Circular analytical model, in which the rear pressure bulkhead at one plate 2 Circular end of the cylinder was assumed to be a circular plate, was Rigid wall plate 1 examined under various conditions. The plate was supported at its edges by springs whose stiffness could be adjusted to Figure 1: Configuration of analytical model. simulate the various support conditions. These investigations clariefi d the inu fl ence of the support conditions on the sound pressure of an internal sound efi ld coupled with the vibration terms of the vibration and acoustic characteristics. In the of the end plate. eTh authors of this study used the above- experiment, the acceleration of the plate vibrations, the phase mentioned analytical model [9, 10]ofacylindricalstructure difference between them, and the sound pressure level inside with plates at both ends to investigate the vibroacoustic the cavity are considered significant characteristics of the coupling based on the sound pressure level distribution in plate vibrations and sound eld fi . These experimental results the cavity. The acoustic characteristics under vibroacoustic demonstrate the underlying theoretical analysis based on this coupling were investigated for cases in which excitation forces model, as well as the conditions that maximize the vibration with different relative amplitudes and phases were applied to and sound pressure levels. Furthermore, the eeff ct of vibroa- both end plates [11]. In addition, the excitation frequency at coustic coupling is estimated from an electricity generation which the coupling system becomes nonperiodic owing to experiment performed with piezoelectric elements. the application of excitation forces of different frequencies to the respective end plates was investigated [12]. Finally, the eeff ctofthe excitation position with respecttothe nodal 2. Analytical Method lines on the appearance of vibration modes on the plates was investigated [13]. On the other hand, to suppress the 2.1. Equation of Motion of Plate. eTh analytical model consists above vibration and acoustic energy, which was amplified of a cavity with two circular end plates, as shown in Figure 1. by vibroacoustic coupling, an analytical model that included eTh plates are supported by translational and rotational the installation of passive devices on the vibration system springs distributed at constant intervals, and the support was proposed as the electromechanical-acoustic system. The conditions are determined by their respective spring stiffness effect was fully validated in the numerical approach owing 𝑇 ,𝑇 ,𝑅 ,and 𝑅 , where the suffixes 1 and 2 indicate plates to tuning the resonance characteristics of the shunt circuit, 1 2 1 2 1 and 2, respectively. eTh plates of radius 𝑎 and thickness in which the piezoelectric device was incorporated, to the frequency characteristics of the coupling system [14]. ℎ have a Young’s modulus 𝐸 and a Poisson’s ratio ].The In almost all such studies, vibroacoustic coupling has sound eld fi , assumed cylindrical, has the same radius as been estimated by assuming that the plate and cavity that of the plates and varying length because the resonance dimensions, as well as the phase difference between the frequency depends on the length. The boundary conditions vibroacoustics of the two plates, were xfi ed. However, the areconsidered structurally andacousticallyrigid at thelateral natural frequencies of the plate and cavity vary with the wall between the structure and sound eld fi . The coordinates dimensions of the plate and cavity, and the phase difference used are radius 𝑟 and angle 𝜃 between the planes of the plates directly aeff cts the sound eld fi when the medium is much and the cross-sectional plane of the cavity and distance 𝑧 less dense than the plate. Although the study of coupling along the cylinder axis. eTh periodic point forces 𝐹 and 𝐹 1 2 phenomena had attracted attention, an extensive parametric areapplied to plates 1and 2atdistances 𝑟 and 𝑟 and angles 1 2 study has not yet been undertaken. Hence, little is known 𝜃 and 𝜃 , respectively. eTh natural frequency of the plates is 1 2 about the influence of dimensions and phase differences on employed as theexcitationfrequency. the coupling phenomena. To formulate the plate motion, Hamilton’s principle is To develop a new electricity generation system, we adopt appliedtothe analytical model[9]: an analytical model similar to the above-mentioned cylindri- cal structure with plates at both ends, because the vibration area of the model on which piezoelectric elements can be (1) = 𝛿 ∫ (𝑇 −𝑈 −𝑈 +𝑊)𝑡𝑑 = 0, 𝑃 𝑃 𝑆 installedistwice as largeasthatincaseofasingle plate. The cylinder length is varied over a wide range, while the where 𝐻 is the Hamiltonian, 𝑇 and 𝑈 are, respectively, the harmonic point force is applied to one end plate or both 𝑃 𝑃 kinetic and potential energy of each plate, and 𝑈 is the elastic end plates, and its frequency is selected to cause the plate energy stored in the springs. 𝑊 is the work done on plates 1 to vibrate in the fundamental mode. Vibroacoustic coupling that occurs between plate vibrations and the sound el fi d in and 2 by the respective point forces and the sound pressure the cavity is investigated theoretically and experimentally in on the plates. Finally, 𝑡 and 𝑡 are two arbitrary times. 0 1 𝛿𝐻 Advances in Acoustics and Vibration 3 The flexural displacements 𝑤 and 𝑤 of plates 1 and 2, each platevibration andthe soundfield. eTh pointforce and 1 2 respectively, in terms of two sets of suitable trial functions are acoustic excitation terms are, respectively, found by substituting (3) for the plate modes into (2)below: 𝑠 𝑠 𝐹 = ∫ 𝐹 𝛿(𝑟 − 𝑟 )𝛿(𝜃 − 𝜃 )𝑋 , 1𝑛𝑚 1 1 1 𝑛𝑚 1 1 ∞ ∞ 𝑠 𝑠 𝑗(𝜔𝑡+𝜙 ) 1 (6) 𝑤 = ∑ ∑ ∑ 𝐵 𝑋 𝑒 , 1 1𝑛𝑚 𝑛𝑚 𝑠 𝑠 𝑠=0 𝑛=0 𝑚=0 𝐹 = ∫ 𝐹 𝛿(𝑟 − 𝑟 )𝛿(𝜃 − 𝜃 )𝑋 , 2 2 2 2 2𝑛𝑚 𝑛𝑚 (2) 1 ∞ ∞ 2 𝑠 𝑠 𝑗(𝜔𝑡+𝜙 ) 𝑤 = ∑ ∑ ∑ 𝐵 𝑋 𝑒 , 𝑠 𝑠 𝑠 𝑠 2 2𝑛𝑚 𝑛𝑚 𝑃 = ∫ 𝑃 𝑋 ,𝑃 = ∫ 𝑃 𝑋 . (7) 1𝑛𝑚 𝑐 𝑛𝑚 1 2𝑛𝑚 𝑐 𝑛𝑚 2 𝑠=0 𝑛=0 𝑚=0 𝐴 𝐴 1 2 𝑋 = sin (𝑛𝜃 + )( ) , (3) 𝑛𝑚 Here 𝛿 is the delta function associated with the point force 2 𝑎 on plates, 𝐴 and 𝐴 are the plate areas, and 𝑃 is the 1 2 𝑐 where 𝑛 , 𝑚 ,and 𝑠 are, respectively, the circumferential order, sound pressure at an arbitrary point on the boundary surface radial order, and symmetry index with respect to the plate of the plates. To distinguish between plates 1 and 2, the 𝑠 𝑠 vibration. 𝐵 and 𝐵 are coefficients to be determined, 𝜔 differential 𝐴𝑑(𝑟𝑑𝜃𝑑) 𝑟 is written as and in (6)and 1𝑛𝑚 2𝑛𝑚 1 2 is the angular frequency of the harmonic point force acting (7), respectively. on the plate, and 𝑡 is the elapsed time. 𝜙 and 𝜙 are the 1 2 phases of the respective plate vibrations. In this analysis, 𝜙 2.2. Coupling Equation between Plate Vibrations and Internal is set to 0 deg, and 𝜙 varies in the range of 0 deg to 180 deg. Sound Field. For simplicity, we assume that the cavity walls 𝑤 and 𝑤 are substituted for the flexural displacements in 1 2 are rigid, so that the sound eld fi in the cavity is governed by 𝑇 ,𝑈 ,𝑈 ,and 𝑊 , whose detailed expressions are obtained 𝑃 𝑃 𝑆 the wave equation consisting of the eigenfunction 𝑌 and the from [9, 10], and the variation of (1) is carried out with eigenvalue 𝑘 corresponding to a cavity mode of order 𝑁 : respect to both plates. Consequently, the extremum of the Hamiltonian yields Euler’s equations, which are the equations 2 2 ∇ 𝑌 +𝑘 𝑌 =0, (8) 𝑁 𝑁 𝑁 of motion of the respective plates. eTh motion is assumed to 𝜕𝑌 be harmonic, that is, to behave as 𝑒 ,sothat 𝑒 can be ( ) =0, (9) 𝜕 u eliminated. Consider the following: where u is the unit normal to the boundary surface 𝑆 (positive ∞ towards the outside), and the boundary condition satisfies ( 9) 𝑠 2 𝑠 [ ∑ {𝐾 (1 + 𝜂𝑗 )−𝜔 𝑀 } 󸀠 󸀠 when 𝑆 is rigid. However, if 𝑆 is not rigid but has a varying 1𝑛𝑚𝑚 𝑝 1𝑛𝑚𝑚 𝑚 =0 specific acoustic admittance, we select a Green’s function 𝐺 󸀠 to obtain a solution set for a nonuniform cavity with nonrigid 𝑚 𝑚 + ∑ {𝑇 +( )( )𝑅 }] B 󸀠 𝑒 walls for a frequency 𝜔/2𝜋 = 𝐾𝑐/2𝜋 ,where 𝐾 is an eigenvalue 𝑠𝑛 1 1 1𝑛𝑚 𝑎 𝑎 𝑚 =0 of the nonuniform cavity, and 𝑐 is the cavity speed of sound. 𝑠 𝑠 = F − P , The equation for 𝐺 is thus given by 1𝑛𝑚 1𝑛𝑚 2 2 𝑠 2 𝑠 ∇ 𝐺+𝐾 𝐺=−𝛿( p − p ) [ ∑ {𝐾 (1 + 𝜂𝑗 )−𝜔 𝑀 } 0 󸀠 󸀠 2𝑛𝑚𝑚 𝑝 2𝑛𝑚𝑚 𝑚 =0 =−𝛿(𝑟 − 𝑟 )𝛿(𝜃 − 𝜃 )𝛿(𝑧 − 𝑧 ), 0 0 0 (10) 𝑚 𝑚 𝜕𝐺 + ∑ {𝑇 +( )( )𝑅 }] B 󸀠 𝑒 ( ) =0. 𝑠𝑛 2 2 2𝑛𝑚 𝑎 𝑎 𝜕 u 󸀠 𝑆 𝑚 =0 𝑠 𝑠 =−F + P , The right-hand side is a delta function, where the measure- 2𝑛𝑚 2𝑛𝑚 ment point is p = (𝑟, 𝜃, 𝑧) if thesourcepoint is p = (4) 0 (𝑟 ,𝜃 ,𝑧 ).Expressing 𝐺 in terms of 𝑌 of (8), which satisfies 0 0 0 𝑁 𝑠 𝑠 𝑠 𝑠 where 𝐾 󸀠 , 𝐾 󸀠 and 𝑀 󸀠 , 𝑀 󸀠 are elements of thesameboundaryconditions, we nfi dthat 1𝑛𝑚𝑚 2𝑛𝑚𝑚 1𝑛𝑚𝑚 2𝑛𝑚𝑚 the symmetrical stiffness and mass matrices, respectively, 󸀠 󸀠 𝑌 (p)𝑌 (p ) because the index 𝑚 is the radial order (𝑚=𝑚 ). 𝜂 is the 𝑁 𝑁 0 𝐺( p, p )= ∑ , (11) structural dampingfactorofthe plate, and 𝐹 is a coefficient 𝑠𝑛 𝑉 𝑀 (𝑘 −𝐾 ) 𝑐 𝑁 𝑁=1 𝑁 that is determined by the indices 𝑛 and 𝑠 and is expressed as 0 at 𝑁 =𝑀̸, [9] ∫ 𝑌 (p)𝑌 (p)𝑑𝑉 =𝑉 𝑀 𝛿 ={ 𝑁 𝑀 𝑐 𝑐 𝑁 𝑉 𝑀 at 𝑁=𝑀. 𝑐 𝑐 𝑁 𝜋, at 𝑛 =0 ̸ , { (12) 𝐹 = 0, at 𝑛 = 0, 𝑠 = 0, (5) 𝑠𝑛 The dimensionless factor 𝑀 is the mean value of 𝑌 2𝜋, at 𝑛 = 0, 𝑠 = 1. 𝑁 𝑁 averaged over the cavity volume 𝑉 ,and 𝛿 is the Kronecker Both rfi st terms on the right-hand sides of ( 4)givethe delta. respective point forces, and the second terms give the acoustic Because there is no source and 𝜕𝐺/𝜕 u =0 on 𝑆 ,thespatial excitations, which also function as the coupling term between factor 𝑃 (p)of the sound pressure within and on the surface 𝑁𝑀 𝑁𝑀 𝑎𝐹 𝑗𝜙 𝑎𝐹 𝑗𝜙 𝑗𝜔𝑡 𝑗𝜔𝑡 𝑑𝐴 𝑑𝐴 𝑠𝜋 𝑑𝐴 𝑑𝐴 𝑑𝐴 𝑑𝐴 4 Advances in Acoustics and Vibration bounding the medium can be obtained from just one of the Here, substituting (2)for 𝑤 and 𝑤 and considering a modal 1 2 surface integral terms as follows: damping factor 𝜂 ,(19)can be rewrittenas 2 2 𝑠 (p ) 𝑐 0 (𝜔 +𝑗𝜂 𝜔 𝜔−𝜔 )𝑃 𝑞 𝑐 𝑞 𝑞 𝑃 (p)=− ∫ 𝐺( p, p ) , (13) 𝑐 0 0 𝜕 u ∞ ∞ 𝑆 2 (20) 𝑠 𝑠 1 2 = (− ∑ 𝐼 𝐵 𝑒 + ∑ 𝐼 𝐵 𝑒 ), 1 2 1𝑛𝑚 2𝑛𝑚 where the zero subscripts indicate differentiation and inte- 𝑐 𝑚=0 𝑚=0 gration with respect to the (𝑟 ,𝜃 ,𝑧 ) coordinates. A detailed 1 1 0 0 0 𝑠 𝑠 𝑠 𝑠 𝐼 = ∫ 𝑋 𝑌 ,𝐼 = ∫ 𝑋 𝑌 , 1 1 2 2 𝑛𝑚 𝑞 𝑛𝑚 𝑞 procedure for obtaining these equations is given in [15]. 𝑃 𝐴 𝐴 𝐴 𝐴 1 2 canalsobeexpressed as [9, 10] (21) 𝑃 𝑌 𝑁 𝑁 where 𝐴 is the total surface area of the plates, 𝐼 and 𝐼 are the 1 2 𝑃 =𝜌 𝑐 ∑ , (14) 𝑐 𝑐 spatial coupling coefficients. Moreover, substituting ( 18)for 𝑁=1 𝑃 and applying 𝐼 and 𝐼 to the integrals in (7), the acoustic 𝑐 1 2 𝑠 𝑠 where 𝜌 is thefluiddensity in thecavity, and 𝑃 is the excitation terms 𝑃 and 𝑃 can be expressed with respect 𝑐 𝑁 1𝑛𝑚 2𝑛𝑚 pressure coefficient to be determined. to an arbitrary vibration mode (𝑛, ) 𝑚 as In this investigation, the acoustic modal shape 𝑌 and ∞ ∞ 𝐼 𝑃 angular resonance frequency 𝜔 in the cavity (where the 1 𝑞 𝑠 2 𝑃 =𝜌 𝑐 𝐴 ∑ ∑ , 1𝑛𝑚 indices 𝑛 , 𝑝 ,and 𝑞 indicate the circumferential, radial and 𝑠 𝑝=1 𝑞=0 𝑞 longitudinal orders, resp.) are defined as (22) ∞ ∞ 𝐼 𝑃 2 𝑞 𝑠 2 𝑃 =𝜌 𝑐 𝐴 ∑ ∑ . 𝑞𝜋 2𝑛𝑚 𝑐 𝑠 𝑠 𝑌 = sin (𝑛𝜃 + )𝐽 (𝜆 𝑟) cos {( )𝑧}, 𝑞 𝑛 𝑝=1 𝑞=0 2 𝐿 (15) 1/2 𝑞𝜋 Finally, replacing 𝑃 in (22)withthose in (20)and 𝜔 =𝑐{𝜆 +( ) } , then inserting them in (4), we can complete the coupling equations, whose right-hand sides are where 𝐽 is the 𝑛 th-order Bessel function, and 𝜆 is the 𝑝 th solution of an eigenvalue problem for a circular sound efi ld 2 2 2 𝜌 𝑐 𝜔 𝐴 𝑠 𝑠 𝑠 𝑐 having modes (𝑛, ) 𝑝 divided by the radius. eTh boundary F − P = F + 1𝑛𝑚 1𝑛𝑚 1𝑛𝑚 conditions between the plate vibrations and sound eld fi on 𝑠 𝑠 ∞ ∞ ∞ 1 2 therespectiveplatesurfacesarefoundbyassumingcontinuity 𝐼 (𝐼 𝐵 󸀠 𝑒 −𝐼 𝐵 󸀠 𝑒 ) 1 1 2 1𝑛𝑚 2𝑛𝑚 × ∑ ∑ ∑ , of thevelocitiesonthe plates: 𝑠 2 2 󸀠 𝑀 (𝜔 +𝑗𝜂 𝜔 𝜔−𝜔 ) 𝑝=1 𝑞=0 𝑐 𝑞 𝑚 =0 𝑞 𝑞 2 2 2 2 2 𝑐 𝑐 𝜌 𝑐 𝜔 𝐴 𝑠 𝑠 𝑠 𝑐 ( ) =𝜌 𝜔 𝑤 ,( ) =−𝜌 𝜔 𝑤 , (16) 𝑐 1 𝑐 2 −F − P =−F + 2𝑛𝑚 2𝑛𝑚 2𝑛𝑚 𝜕 u 𝜕 u 𝑧=0 𝑧=𝐿 𝑠 𝑠 ∞ ∞ ∞ 1 2 𝐼 (𝐼 𝐵 𝑒 −𝐼 𝐵 𝑒 ) 󸀠 󸀠 where /𝜕 u is 0 on the lateral wall of the cylinder since the 2 1 2 𝑐 1𝑛𝑚 2𝑛𝑚 × ∑ ∑ ∑ . wall remains rigid. 𝑠 2 2 󸀠 𝑀 (𝜔 +𝑗𝜂 𝜔 𝜔−𝜔 ) 𝑝=1 𝑞=0 𝑚 =0 𝑐 𝑞 𝑞 𝑞 Because the analytical mode has two boundary surfaces, (23) we can apply (16)to(13), so that 𝑃 becomes On the right-hand sides, the second terms show the 2 2 𝑃 =− ∫ 𝜔 𝑤 + ∫ 𝜔 𝑤 . (17) 𝑐 𝑐 1 1 𝑐 2 2 acoustic excitation for plates 1 and 2, respectively. The acoustic 𝐴 𝐴 1 2 excitation terms have both 𝐼 and 𝐼 since the acoustic mode 1 2 of the sound field is coupled with the vibration modes of On the other hand, by substituting acoustic modes of three therespectiveplates. Before actual calculation, thenatural orders, 𝑛 , 𝑝 ,and 𝑞 , instead of the order 𝑁 of the cavity mode frequency of the plate must be considered in terms of the into (14), 𝑃 canalsobeexpressed as convergence of the plate vibration mode (𝑛, ) 𝑚 .Inthiscase, 𝑠 𝑠 ∞ ∞ the natural frequency is obtained as the eigenvalue of (4), 𝑃 𝑌 𝑞 𝑞 𝑃 =𝜌 𝑐 ∑ ∑ . (18) whose right-hand side is set to 0. The actual calculation is 𝑐 𝑐 𝑝=1 𝑞=0 performed by taking 15 terms for 𝑛 ,while 𝑚 issettobegreater than 13 to ensure the convergence of the natural frequency eTh equation relating ( 17)and (18) is obtained by applying andmodeshape of theplate.Using thesametruncationfor Green’s function of (11) to an arbitrary acoustic mode(𝑛,𝑝,𝑞) 𝑝 as for 𝑚 , the order accounts for acoustic modes greater as than 𝑞=15 , so that the resonance frequency containing 𝑞 canexceed theexcitationfrequency.Theplate andcavityloss 2 2 𝑠 (𝜔 −𝜔 )𝑃 𝑞 𝑞 factors are assumed to be constant: 𝜂 =𝜂 = 0.01 [9, 10]. 𝑝 𝑐 𝑠 𝑠 (19) Since 𝐵 and 𝐵 canbeobtainedsimultaneouslyfrom 𝑠 𝑠 1𝑛𝑚 2𝑛𝑚 =− (∫ 𝑌 𝑤 + ∫ 𝑌 𝑤 ). 𝑞 1 1 𝑞 2 2 (4), which have as excitation terms, the behavior of the plate 𝐴 𝐴 𝑐 1 2 𝑛𝑝 𝑛𝑝 𝑑𝐴 𝑑𝐴 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑑𝐴 𝐺𝜌 𝑑𝐴 𝐺𝜌 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝜕𝑃 𝑗𝜙 𝑗𝜙 𝜕𝑃 𝜕𝑃 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑗𝜙 𝑗𝜙 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑠𝜋 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝑑𝐴 𝑑𝐴 𝑗𝜙 𝑗𝜙 𝐴𝜔 𝑑𝑆 𝑛𝑝 𝑛𝑝 𝑛𝑝 𝜕𝑃 Advances in Acoustics and Vibration 5 8 10 5 5 5 4 4 5 153 mm 1 31 3 9 9 2 2 1: Vibration generator 6: Multifunction generator 2: Load cell 7: Power supply 3: Acceleration sensor 8: FFT analyzer 4: Condenser microphone 9: Piezoelectric element 5: Amplifier 10: Power meter (a) (b) R c (c) Figure 2: Configuration of experimental apparatus: (a) Measurement system; (b) Installation state of piezoelectric element; and (c) Electrical circuit of energy-harvesting device. 𝑉 𝑉 𝑃 1 2 V vibrations andthe soundfieldunder vibroacousticcoupling 𝐿 =10 log ,𝐿 =10 log ,𝐿 =10 log , V1 V2 𝑝 V 𝑉 𝑉 𝑃 can be determined. 0 0 0 (25) The flexural displacements 𝑤 and 𝑤 and the sound 1 2 pressure 𝑃 are, respectively, obtained from (2)and (18)by ∗ ∗ ∗ 𝑠 𝑠 where 𝑤 , 𝑤 ,and 𝑃 are the respective conjugate compo- employing 𝐵 and 𝐵 determined above. However, the 1 2 𝑐 1𝑛𝑚 2𝑛𝑚 nents. procedure to calculate 𝑃 involves substituting 𝑃 in (20) for that in (18). Solved from (24), into which 𝑤 , 𝑤 ,and 𝑃 1 2 𝑐 are inserted, over the respective entire regions, the quadric 3. Experimental Apparatus and Method velocities 𝑉 and 𝑉 and the sound pressure 𝑃 are estimated 1 2 ] Figure 2(a) shows the experimental apparatus used in this by the logarithmic values 𝐿 , 𝐿 ,and 𝐿 relative to 𝑉 = V1 V2 𝑝 V 0 −15 2 2 −10 2 4 study. The structure consists of a steel cylinder with circular 2.5 × 10 m /s and 𝑃 =4 × 10 N /m ,respectively, as aluminum end plates that are 3 mm thick. eTh cylinder has shown in (25). an inner radius of 153 mm, and this length can be varied Consider the following: from 500 to 2000 mm to emulate the analytical model. One endplate or both endplatesare subjectedtothe pointforce, 2 2 𝜔 𝜔 ∗ ∗ 𝑉 = ∫ 𝑤 𝑤 ,𝑉 = ∫ 𝑤 𝑤 , whosefrequency makesthe plateexciteinthe (0,0)mode. 1 1 1 1 2 2 2 2 2𝐴 2𝐴 𝐴 𝐴 1 1 2 2 In case of the harmonic excitation of both ends, these forces ∗ are applied to the respective plates via small vibrators, and 𝑃 = ∫ 𝑃 𝑃 𝑑𝑉 , ] 𝑐 𝑐 2𝑉 their amplitudes are controlled to be 1 N. The positions of the point forces 𝑟 and 𝑟 are normalized by radius 𝑎 and (24) 1 2 𝑑𝐴 𝑑𝐴 𝑛𝑝 6 Advances in Acoustics and Vibration are set to 𝑟 /𝑎 = 𝑟 /𝑎 = 0.4 . In the excitation experiment, 1 2 the main characteristic is the phase difference between the plate vibrations. eTh refore, acceleration sensors are installed on both plates to measure this phase difference. To estimate the internal acoustic characteristics, the sound pressure level in the cavity is measured using condenser microphones with aprobe tube.Thetipsofthe probetubes arelocated near the plates and the cylinder wall, which are the approximate locations of the maximum sound pressure level when the sound efi ld becomes resonant. To perform the electricity generation experiment, the piezoelectric element is used and is comprised of the piezo- electric part constructed of ceramics and the electrode part 0 400 800 1200 1600 2000 constructed of brass, which have the diameters of 25 and L (mm) 35 mm and the thicknesses of 0.23 and 0.30 mm, respectively. Actually, the piezoelectric elements are installed at each 𝜙= 9 0 (deg) 𝜙= 0 (deg) center of both plates, as shown in Figures 2(a) and 2(b).The 𝜙= 1 0 (deg) electric power generated by the expansion and contraction Figure 3: Variation in sound pressure level with cylinder length of the piezoelectric elements is discharged through the when phase difference changes. resistance circuit, which consists of three resistors having resistances 𝑅 , 𝑅 ,and 𝑅 ,asshown in Figure 2(c). 𝑅 and V 𝑖 𝑐 V 𝑅 are the resistances of the voltmeter and ammeter built-in 𝑖 mode. eTh excitation frequency is chosen as 𝑓 that makes the wattmeter and are 2 MΩ and 2 mΩ, respectively, while the plates vibrate in the (0, 0)mode. With respect to the plate 𝑅 is the resistance of the resistor connected outside the vibration, although the phase 𝜙 of plate 1 is xfi ed at 0 deg, the wattmeter and is 97.5 kΩ.Tograsp theeeff ct of vibroacoustic phase 𝜙 of plate 2 ranges from 0 to 180 deg, and then they are coupling on energy harvesting, the electric power is measured relatedbythe phasedieff rence 𝜙 as follows: with and without the cylinder and is estimated by the 𝜙=𝜙 −𝜙 . (26) comparison of both cases. In such a estimation, the electric 2 1 powerisnormalizedbythe vibrationpower supplied with Figure 3 shows the variations in 𝐿 with 𝐿 ,whenonly 𝑝 V the plate, which is obtained from the point force and flexural plate1isexcitedand 𝜙 is arbitrarily set to 0, 10, and 90 deg. 𝐿 𝑝 V displacement. varies only slightly over the entire range of 𝐿 when𝜙=0 deg, but varies substantially and exhibits peaks near 𝐿 = 610 , 1220, and 1830 mm when 𝜙=10 and90deg.Thevalue of 𝐿 is 𝑝 V 4. Results and Discussion lower when 𝜙=10 and 90 deg and is almost identical at all phase differences near 𝐿 = 460 ,920and1560 mm.Toestimate 4.1. Acoustic Characteristics under Vibroacoustic Coupling. In the inu fl ence of each acoustic mode on these sound efi lds, the the theoretical study, the plates are assumed to be aluminum contribution 𝐶 canbedenfi edasthe ratioofthe acoustic having a Young’s modulus 𝐸 of 71 GPa and a Poisson’s ratio ] 𝐸 energy 𝐸 stored in the specific (𝑛, ,𝑝 𝑞) mode to the total of 0.33. eTh radius 𝑎 and thickness ℎ of the plates are constant acoustic energy 𝐸 of the entire sound field: at 153 mm and 3 mm, respectively, whereas the length of the all cylindrical sound field having the same radius as that of the 𝐶 = . (27) plates varies from 100 to 2000 mm. eTh support conditions of 3 2 all the plates, which have flexural rigidity 𝐷[= 𝐸ℎ /{12(1−] )}], are expressed by the nondimensional stiffness parameters Figures 4(a) and 4(b) show 𝐶 as a function of 𝐿 for the 3 3 𝑇 (= 𝑇 𝑎 /𝐷 = 𝑇 𝑎 /𝐷) and 𝑅 (= 𝑅 𝑎/𝐷 = 𝑅 𝑎/𝐷) .These (0, 0, 𝑞) modes for values of 𝑞 between 0 and 4 for 𝜙=0 𝑛 1 2 𝑛 1 2 values are identical for both plates. If 𝑅 ranges from 0 to and 90 deg, respectively. In case of 𝜙=0 deg, 𝐶 is relatively 8 8 10 when 𝑇 is 10 , the support condition can be assumed large when 𝐿 is short. The value of 𝐶 for the (0,0,0)mode 𝑛 𝐸 from a simple support to a clamped support. eTh actual decreases as 𝐿 increases because of the increase in the value 8 1 condition adopts 𝑇 =10 and 𝑅 =10 to get closer to of 𝐶 for the (0, 0, 1)mode. eTh (0, 0, 0)mode is regarded 𝑛 𝑛 𝐸 the experimental support condition. These plates 1 and 2 are as a pumping mode that corresponds to the reciprocation subjectedtothe pointforces 𝐹 and 𝐹 ,which areset to 1N of both end plates supported by the aerostatic stiffness of 1 2 andare locatedat 𝑟 /𝑎 = 𝑟 /𝑎 = 0.4 , respectively, as well as the cavity. In the range of longer length, as the value of 𝑞 1 2 the actual excitation experiment. In particular, the analysis in increases, 𝐶 decreases relatively to 𝐶 of the (0, 0, 0)mode 𝐸 𝐸 which only one end plate is excited is carried out with taking becausethe soundfieldisaeff cted by othermodes.At 𝜙= 𝐹 as 0 N. 90 deg, 𝐶 varies dramatically when the acoustic mode that 2 𝐸 The plate and sound eld fi eigenfrequency characteristics dominates the sound eld fi changes, so that the range of 𝐿 over involved in the vibroacoustic coupling are represented by the which the influence of such a dominant mode extends can natural frequency 𝑓 corresponding to the (𝑛, ) 𝑚 mode and be clearly distinguished. eTh acoustic mode (0, 0, 𝑞) causes 𝑛𝑚 the resonance frequency 𝑓 corresponding to the (𝑛,𝑝,𝑞) 𝐿 to have peaks at 𝐿 = 610 , 1220, and 1830 mm, having 𝑞 𝑝 V L (dB) 𝑛𝑝 𝑛𝑝 𝑛𝑝 Advances in Acoustics and Vibration 7 1.0 1.0 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 400 800 1200 1600 2000 0 400 800 1200 1600 2000 L (mm) L (mm) (0, 0, 1) (0, 0, 1) (0, 0, 0) (0, 0, 0) (0, 0, 2) (0, 0, 3) (0, 0, 2) (0, 0, 3) (a) (b) Figure 4: Contribution of acoustic mode to sound field at (a) 𝜙=0 [deg] and (b) 𝜙=90 [deg]. of 𝜙 is repeated in a similar manner as 𝐿 increases to max 𝐿 = 2000 mm. Peaks in 𝐿 indicate that vibroacoustic 𝑝 V (0, 0, 1) (0, 0 ,2) (0, 0, 3) coupling between the plate vibrations and sound eld fi is promoted at approximately 90 deg. 𝑓 and 𝑓 must be 𝑛𝑚 𝑞 approximately equal for the promotion of this coupling, so that the acoustic modes involved in vibroacoustic coupling are greatly influential around the lengths at which 𝐿 peaks. 90 𝑝 V On the other hand, we confirm that the longitudinal order 𝑞 shisf ft rom1to 2and from 2to3at 𝐿 = 900 and 1560 mm where 𝐿 varies abruptly, respectively, based on the 𝑝 V distributions of the sound pressure level along the 𝑧 direction inside the cavity. As a result, the sound fields, which are classified in the ranges of 470 to 900 mm, 910 to 1560 mm, 0 400 800 1200 1600 2000 and 1570 to 2000 mm, are dominated by the (0, 0, 1) , (0, 0, 2) , L (mm) and (0, 0, 3)modes, respectively, as shown in this gur fi e where the range of the acoustic mode is classified in color. Naturally, Figure 5: Variations in phase difference with cylinder length. this classification is also thought to be due to variations in the dominant acoustic mode, whose contribution 𝐶 is maximized in Figure 4(b). modal shapes similar to that of the (0, 0)mode of the plate Figure 6 shows the sound pressure levels 𝐿 and 𝐿 , 𝑝1 𝑝2 vibrations. es Th e peaks occur at an integer 𝑞 starting from𝑞= which are measured near plates 1 and 2, respectively, as 1 with increasing 𝐿 . eTh vibration of plate 2 has a signicfi ant functions of 𝐿 . eTh theoretical level 𝐿 ,which is maximized 𝑝 V effect on the formation of the sound eld fi and vibroacoustic at each 𝐿 when the phase difference 𝜙 ranges from 0 to coupling, despite not being driven by the point force, whereas 180 deg, is also indicated to compare with the experimental results. 𝐿 and 𝐿 show peaks around 615, 1275, and the inu fl ence of 𝜙 on the acoustic characteristics has only 𝑝1 𝑝2 been described for 𝜙=0 , 10, and 90 deg. This change in 1900 mm, and these levels are almost coincident for each 𝐿 with 𝜙 indicates that the acoustic characteristics depend peak.However,theydecreaseinthe middle rangeofthose 𝑝 V strongly on the vibration of plate 2; that is, there are ranges lengths. In particular, decreases in 𝐿 are remarkable and 𝑝1 of 𝜙 that intensify or suppress coupling between the plate their differences expand around 950 and 1550 mm. 𝐿 also 𝑝 V vibrations and sound field. Here the values of 𝜙 at which 𝐿 shows peaks at 610, 1230, and 1840 mm and corresponds 𝑝 V is maximum are denoted by 𝜙 . approximately with the above lengths where 𝐿 and 𝐿 max 𝑝1 𝑝2 In Figure 5, the variations in 𝜙 with 𝐿 and the values peak. max Since the above results are derived from the investigation of 𝜙 that maximize 𝐿 are plotted by a line and circles, max 𝑝 V respectively. 𝜙 is approximately 87 deg at 𝐿 = 100 mm and based on the excitation of one end plate, furthermore, the max decreases gradually with increasing 𝐿 up to approximately excitation condition in which both plates are subjected to the same excitation force is taken to grasp the eeff ct of the 𝐿 = 460 mm, where 𝜙 suddenly increases to over 90 deg max and then decreases again with increasing 𝐿 .This behavior excitation method on vibroacoustic coupling. Figure 7 shows (deg) max 𝑛𝑝 8 Advances in Acoustics and Vibration 0 400 800 1200 1600 2000 400 800 1200 1600 2000 L (mm) L (mm) max 1 max 2 p p2 max 1 max p1 max 2 exp Figure 6: Comparison between theoretical and experimental results for sound pressure level. Figure 8: Comparison between theoretical and experimental results for phase difference. flexural displacements 𝑤 and 𝑤 is also significant in 1 2 studying the eeff ct of the plate vibrations on the sound efi ld. Here the phase differences are denoted as 𝜙 and 𝜙 max1 max2 120 when 𝑤 and 𝑤 are maximized, while they are denoted as 1 2 𝜙 and 𝜙 when 𝑤 and 𝑤 are minimized. Figure 8 min1 min2 1 2 shows 𝜙 , 𝜙 , 𝜙 ,and 𝜙 as functions of 𝐿 when max1 max2 min1 min2 only plate 1 is excited. 𝜙 is constant at 180 deg for 𝐿 max1 ranging from 100 to 390 mm and decreases abruptly up to 0 deg at 𝐿 = 400 mm. Then, remaining constant at 0 deg up to 𝐿 = 610 mm, 𝜙 increases gradually with 𝐿 and returns max1 to 180 deg at 𝐿 = 1220 mm, increasing somewhat abruptly near 𝐿 = 970 mm. Beyond 𝐿 = 1220 mm, 𝜙 is again 400 800 1200 1600 2000 max1 constant at 180 deg up to 𝐿 = 1570 mm, and this behavior L (mm) is repeated as 𝐿 increases to 𝐿 = 2000 mm. 𝜙 exhibits max2 p p2 gradual and abrupt changes similar but alternate to 𝜙 . max1 p1 For example, when 𝐿 increases, a gradual decrease occurs in 𝜙 between 𝐿 = 100 and 620 mm, and an abrupt increase max2 Figure 7: Comparison between theoretical and experimental results occurs near 𝐿 = 970 mm. Both 𝜙 and 𝜙 shift between for sound pressure level when both end plates are excited by point max1 max2 0 and 180 deg with changing 𝐿 and intersect at approximately force. 90 deg and near the length at which 𝐿 peaked in Figure 6. 𝑝 V 𝜙 and 𝜙 behave exactly alike but opposite to 𝜙 min1 min2 max1 the variations in 𝐿 corresponding to 𝜙 and the variations 𝑝 V max and 𝜙 . max2 in 𝐿 and 𝐿 that are measured in the experiment and are 𝑝1 𝑝2 In Figure 8,the theoreticalresults for 𝜙 where 𝐿 max 𝑝 V maximized when the phase difference between both point peaks and the experimental results for 𝜙 as 𝐿 and 𝐿 exp 𝑝1 𝑝2 forces ranges from 0 deg to 180 deg. Peaks in 𝐿 appear at𝐿= 𝑝 V are maximized at each 𝐿 are also plotted. 𝜙 ranges greatly exp 610, 1230, and 1840 mm. eTh se peaks are known to be caused between in-phase and out-of-phase and 𝜙 exists in the max by the (0, 0, 1) , (0,0,2),and (0, 0, 3)modes, respectively. Note process where 𝜙 changes abruptly. Then, 𝜙 lies in the exp exp that 𝐿 and 𝐿 increase greatly at 625, 1250, and 1850 mm. 𝑝1 𝑝2 light yellow areas surrounded by 𝜙 and 𝜙 in the 𝐿 max1 max2 However, 𝐿 and 𝐿 are hardly distinguished in the middle 𝑝1 𝑝2 ranges longer than the lengths when the sound pressure level rangeoflengths wherethe soundpressure levels peak,having peaks and occurs in the yellowish green areas surrounded been different in the results for the excitation of one end plate, by 𝜙 and 𝜙 on the other side. In other words, since min1 min2 as shown in Figure 6. vibroacoustic coupling is gradually weakened with increasing 𝐿 after the peaks of 𝐿 and 𝐿 , the acoustic mode involved 𝑝1 𝑝2 4.2.Plate VibrationCharacteristics underVibroacoustic Cou- in coupling shifts to that having the next order 𝑞 . pling. Since the plate vibrations influence the acoustic char- Figure 9 shows the vibration levels 𝐿 and 𝐿 of plates V1 V2 acteristics via vibroacoustic coupling, the magnitude of the 1and 2asfunctions of 𝐿 and the accelerations 𝛼 and 𝛼 of 1 2 L , L , L (dB) L , L , L (dB) p p1 p2 p p1 p2 (deg) Advances in Acoustics and Vibration 9 to that of 𝐿 and 𝐿 as showninthe coloredregions.In V1 V2 the theoretical analysis, we conrfi m that the distributions of the sound pressure level along the 𝑧 direction inside the cavity behave in a similar manner to the sound field inside a soundtubehavingsingleclosedand open ends at thelengths, around which the sound pressure level decreased in Figure 6. These distributions occur in the process of shifting acoustic modes because of changing the cylinder length and have the opposite tendency for the difference between 𝐿 and 𝐿 𝑝1 𝑝2 in thosemiddlerangesoflengths wherethe soundpressure levels peaked in Figure 6. This is derived from the difference between the tendencies of the above vibration level and 0 400 800 1200 1600 2000 acceleration with respect to both plates. If the experimental model completely emulated the theoretical model in the L (mm) flexural displacement, such a discrepancy would not take 2 1 place. In Figure 10, the experimental phase difference 𝜙 ,at exp which 𝐿 and 𝐿 are maximized, is compared with the Figure 9: Comparison between characteristics of theoretical vibra- 𝑝1 𝑝2 tion and experimental acceleration. theoretical phase differences 𝜙 , 𝜙 , 𝜙 ,and 𝜙 max1 max2 min1 min2 when both plates aresubjected to thesameexcitationforce, as shown in Figure 8.Theshisft in the (0, 0, 𝑞) modes are also represented by the changing colors. 𝜙 shifts between 0 and 180 exp 180 deg with changing 𝐿 and corresponds approximately with the 𝜙 and 𝜙 that behave uniformly in the vicinity max1 max2 of the in-phase side or the out-of-phase side. In this way, the behavior of the phase difference is very different in the excitation method. 4.3. Electricity Generation Characteristics. In this section, we consider electricity generation by the plate vibrations coupled with the sound efi ld. In this case, the electricity generation is estimated by the comparison between the electric power via the piezoelectric element and the mechanical power supplied to the plate by the vibrator that is obtained from the rela- 0 400 800 1200 1600 2000 tionship between the point force and flexural displacement at L (mm) the excitation point. Figure 11 shows the relationship between 𝜙 the electric power 𝑃 and mechanical power 𝑃 when the max 2 max 2 𝑒 𝑚 max 1 point force 𝐹 ranges from 1 to 5 N. In this case, the cylinder exp max 2 is removed; the plate vibrations do not couple with the internal sound eld fi . Although 𝑃 is considerably smaller than 𝑃 , their relationship is directly proportional. Here, the Figure 10: Comparison between theoretical and experimental relationship between 𝑃 and 𝑃 is defined as 𝑒 𝑚 results for phase difference when both end plates are excited by point force. 𝑒 𝑃 = . (28) 𝑒𝑚 Figure 12 shows variations in 𝑃 with 𝐿 ;the powers 𝑒𝑚 plates 1 and 2 are also plotted to compare with the theoretical are measured in the experimental apparatus via the cylinder plate behavior. 𝐿 is smaller than 𝐿 in the ranges of 100 to shown in Figure 2(a) when one end plate is excited by the V1 V2 610 mm, 800 to 1200 mm, and 1300 to 1820 mm, so that 𝐿 point force. Although 𝑃 of plate 1 remains almost constant V1 𝑒𝑚 and 𝐿 intersect at a number of 𝐿 and the intersections take over the entire range of 𝐿 , 𝑃 of plate 2 increases greatly at V2 𝑒𝑚 place around the lengths where 𝐿 peaks. eTh actual motion 𝐿 = 615 , 1275, and 1900 mm. It is natural that the relationship 𝑝 V of plate 1 is almost suppressed by that of the vibrator since between 𝑃 of plate 1 and 2 is derived from the behavior of 𝑒𝑚 plate1issupportedbythevibrator;hence, 𝛼 is approximately 𝛼 and 𝛼 in Figure 9. 𝑃 is considered as energy-harvesting 1 1 2 𝑒𝑚 constant over the entire 𝐿 range. However, since the motion efficiency. In an electricity generation by means of beam of plate 2 depends greatly on the behavior of the sound or plate vibrations with piezoelectric elements, coupling field, that is, the only excitation source for plate 2, 𝛼 peaks between the structural vibration and electric field, that is, at 𝐿 = 650 , 1280, and 1880 mm and is suppressed in the electromechanical coupling should be considered to relate the other ranges of 𝐿 ,suchasvariationsin 𝐿 and 𝐿 .Asa in-plane stress to the applied electric field. Considering that 𝑝1 𝑝2 result, the relative relationship of 𝛼 and 𝛼 becomes opposite the behaviors of 𝑃 for plates 1 and 2 correspond to those of 1 2 𝑒𝑚 L , L (dB) (deg) 1 2 , (m/s ) 1 2 10 Advances in Acoustics and Vibration 𝛼 and 𝛼 and the values of 𝑃 are substantially small, that is, 0.03 1 2 𝑒𝑚 the value of 𝑃 is extremely small in comparison with that of 𝑃 , we assume that the behavior of plate vibrations is hardly aeff cted by electromechanical coupling in this study. Moreover, to grasp the eeff ct of vibroacoustic coupling 0.02 on electricity generation, although 𝑃 is taken as the ratio of 𝑃 measured with and without the cylinder, 𝑃 measured 𝑒𝑚 𝑒𝑚 with the cylinder is obtained from the total value 𝑃 with respecttoplates1and2. Figure 13 shows variations in 𝑃 0.01 with 𝐿 ; the results of the excitation of both ends are also indicated to study the eeff ct of the excitation method. 𝑃 for the excitation of one end is maximized as the sound pressure level peaks due to the promotion of vibroacoustic coupling. 0 2.0 4.0 6.0 8.0 This is because the plate vibration on the nonexcitation side P (mW) contributes strongly to the electricity generation, whereas their 𝐿 ranges are limited to the narrow regions in comparison Figure 11: Relationship of electric power and mechanical power. with the sound pressure level. Since the contribution of the nonexcitation side is extremely weakened in other ranges of the above 𝐿 , the effect of the excitation side is uniformly maintained, as shown in Figure 12. On the other hand, at the 3.0 excitation of both ends, there is not such a nonexcitation side that contributes greatly to the electricity generation, so that 𝑃 never reaches values as large as those of the excitation of one end, when the sound pressure level is maximized. In 𝐿 2.0 ranges where vibroacoustic coupling is weakened, 𝑃 remains almost constant and is close to that of the excitation of one end. Under the situation of such a weakened vibroacoustic 1.0 coupling, the excitation of both ends becomes approximately twice as large as the excitation of one end in the respective total amounts of the electric power 𝑃 and mechanical power 𝑃 .Asaresult, 𝑃 shifts almost constantly with 𝐿 ,nomatter 𝑚 𝑅 0 400 800 1200 1600 2000 what theexcitationmethodis. L (mm) Hence, it follows that the excitation of one end is decidedlysuperiortothatofbothendsbyelectricity gener- Plate 2 ation efficiency, according to this estimation method. These Plate 1 results are remarkable in the viewpoint where the acoustic Figure 12: Energy-harvesting efficiency as function of cylinder energy from the sound radiation can be harvested through length. vibroacoustic coupling. However, the number or area of the piezoelectric element should be increased to improve the efficiency 𝑃 ,sothattheeeff ctofelectromechanicalcoupling 𝑒𝑚 12.0 on the plate vibration must be taken into consideration in the theoretical procedure. 10.0 8.0 5. Conclusion 6.0 To apply vibroacoustic coupling to electricity generation, as a means of harvesting energy from vibration systems, coupling 4.0 between plate vibrations and a sound eld fi was investigated theoretically and experimentally for a cylindrical structure 2.0 with thin circular end plates. The end plate was subjected to a harmonic point force. Moreover, the effect of vibroacoustic coupling on the harvest of energy was estimated from the 400 800 1200 1600 2000 electricity generating experiment. The present study focused L (mm) on promoting the vibroacoustic coupling to increase the Excitation of one end plate flexural displacements of the plates and the sound pressure Excitation of both end plates level inside the cavity. As a result of the estimation of vibroacoustic coupling Figure 13: Effect of vibroacoustic coupling on energy harvesting for from various viewpoints, the theoretical study reveals that excitations of one end plate and both end plates. −2 P P ×10 P (mW) R em e Advances in Acoustics and Vibration 11 the closeness of eigenfrequencies and the similarity of modal end plates: influence of excitation position on vibro-acoustic coupling,” Acoustical Science and Technology,vol.26, no.6,pp. shapes between the plate vibrations and sound efi ld are 477–485, 2005. indispensable for promoting coupling. The experimental [14] W. Larbi, J.-F. Deu, ¨ and R. Ohayon, “Finite element formulation results conrfi m that the theoretical estimation of increasing of smart piezoelectric composite plates coupled with acoustic the flexural displacement and sound pressure level via the uid fl ,” Composite Structures,vol.94, no.2,pp. 501–509, 2012. promotion of vibroacoustic coupling support the complicated [15] P.M.Morse andK.U.Ingard, Theoretical Acoustics ,McGraw- acoustic characteristics deduced from the theoretical results. Hill,New York,NY, USA, 1968. In particular, changes in the cylinder length shift the acoustic mode in the longitudinal order and vary periodically the phase difference between both plate vibrations. It is validated that the phase difference is greatly different in the excitation method. When vibroacoustic coupling is promoted, the electricity generation experiment verifies that the promotion of coupling causes the generation efficiency to improve in comparison with the electricity generation caused only by the plate vibration without coupling. References [1] S. R. Anton and H. A. Sodano, “A review of power harvesting using piezoelectric materials (2003–2006),” Smart Materials and Structures,vol.16, no.3,pp. 1–21,2007. [2] K.H.Mak,S.McWilliam, A. A. Popov, andC.H.J.Fox,“Per- formance of a cantilever piezoelectric energy harvester impact- ing a bump stop,” Journal of Sound and Vibration,vol.330,no. 25, pp. 6184–6202, 2011. [3] S. Backhaus and G. W. Swift, “A thermoacoustic stirling heat engine,” Nature, vol. 399, no. 6734, pp. 336–338, 1999. [4] S. Sakamoto, D. Tsukamoto, Y. Kitadani, T. Ishino, and Y. Wata- nabe, “Effect of sub-loop tube on energy conversion efficiency of loop-tube-type thermoacoustic system,” in Proceedings of the 20th International Congress on Acoustics (ICA ’10),pp. 1361– 1362, 2010 (Japanese). [5] K. Hayamizu, “Device for electric generation,” Japanese patent disclosure, 2010-200607, 2010 (Japanese). [6] K. Hayamizu, R. Ando, and Y. Takefuji, “Simultaneous provid- ing device of baseband and carrier signal using sound-gener- ated electricity,” Mobile Multimedia Communications,vol.105, no.80, pp.47–49,2005(Japanese). [7] J. Pan and D. A. Bies, “eTh eeff ct of uid-st fl ructural coupling on sound waves in an enclosure—theoretical part,” Journal of the Acoustical Society of America,vol.87, no.2,pp. 691–707, 1990. [8] J. Pan and D. A. Bies, “eTh eeff ct of uid-st fl ructural coupling on sound waves in an enclosure—experimental part,” Journal of the Acoustical Society of America,vol.87, no.2,pp. 708–717, 1990. [9] L. Cheng and J. Nicolas, “Radiation of sound into a cylindrical enclosure from a point-driven end plate with general boundary conditions,” Journalofthe Acoustical SocietyofAmerica,vol.91, no. 3, pp. 1504–1513, 1992. [10] L. Cheng, “Fluid-structural coupling of a plate-ended cylindri- cal shell: vibration and internal sound field,” Journal of Sound and Vibration,vol.174,no. 5, pp.641–654,1994. [11] H. Moriyama, “Acoustic characteristics of sound field in cylin- drical enclosure with exciting end plates,” Transactions of Japan Society of Mechanical Engineers C,vol.69, no.679,pp. 603–610, 2003 (Japanese). [12] H. Moriyama and Y. Tabei, “Acoustic characteristics inside cy- lindrical structure with end plates excited at different frequen- cies,” Journal of Visualization,vol.7,no. 1, pp.93–101, 2004. [13] H. Moriyama, Y. Tabei, and N. 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