# Energy Harvesting with Piezoelectric Element Using Vibroacoustic Coupling Phenomenon

Authors: Hiroyuki Moriyama; Hirotarou Tsuchiya; Yasuo Oshinoya
Journal: Advances in Acoustics and Vibration
 (2013)

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## Abstract

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 theoretically. The 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 effective as an electric energy-harvesting device.


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## Body

## 1. Introduction

Recently, scavenging ambient vibration energy and converting it into usable electric energy via piezoelectric materials have attracted considerable attention [1]. Typical energy harvesters adopt a simple cantilever configuration to generate electric energy via piezoelectric materials, which are attached to or embedded in vibrational elements. High-amplitude excitations reduce the fatigue life of these harvesters. Thus, placing appropriate constraints on the amplitudes is one of significant ways to improve the performance of harvesters. A cantilever beam, whose deflection was constrained by a bump stop, was modeled. The effect of electromechanical coupling was estimated in a parametric study, where the placement of the bump stop and the gap between the beam and stop were chosen as parameters [2]. Acoustic energy as well as vibration energy to be harvested sufficiently fills our working environment. Thermoacoustic engines that exploit the inherently efficient Stirling cycle and are designed on the basis of a simple acoustic apparatus with no moving parts have been regarded as the representative means for harvesting acoustic energy [3]. As an example application, electricity generation using resonance phenomena in a thermoacoustic engine was investigated with aim of harvesting the work done in the engine. The acoustic energy spent on electricity generation was harvested from a resonance tube branching out of the engine, and the appropriate position of the resonance tube for effectively generating electricity was described [4]. Moreover, an electricity generating device using the sound of voice via piezoelectric elements was developed as a supplementary electric source for a mobile phone [5]. This system is also being investigated for other applications [6]. Acoustic energy is extremely small in comparison to vibration energy. However, the above-mentioned electricity is generated by sound and vibration. Therefore, vibroacoustic coupling is one way to increase acoustic energy, although it has not yet been investigated extensively in this context.Vibroacoustic coupling was investigated as an architectural acoustic problem via a coupled panel-cavity system consisting of a rectangular box with slightly absorbing walls and a simply supported panel. The effect of the panel characteristics on the decay behavior of the sound field in the cavity was considered both theoretically and experimentally [7, 8]. In an attempt to control noise in an airplane, an analytical model for investigating coupling between the sound field in an aircraft cabin and the vibrations of the rear pressure bulkhead was proposed [9, 10]. A cylindrical structure adopted as the analytical model, in which the rear pressure bulkhead at one end of the cylinder was assumed to be a circular plate, was examined under various conditions. The plate was supported at its edges by springs whose stiffness could be adjusted to simulate the various support conditions. These investigations clarified the influence of the support conditions on the sound pressure of an internal sound field coupled with the vibration of the end plate. The authors of this study used the above-mentioned analytical model [9, 10] of a cylindrical structure with plates at both ends to investigate the vibroacoustic coupling based on the sound pressure level distribution in the cavity. The acoustic characteristics under vibroacoustic coupling were investigated for cases in which excitation forces with different relative amplitudes and phases were applied to both end plates [11]. In addition, the excitation frequency at which the coupling system becomes nonperiodic owing to the application of excitation forces of different frequencies to the respective end plates was investigated [12]. Finally, the effect of the excitation position with respect to the nodal lines on the appearance of vibration modes on the plates was investigated [13]. On the other hand, to suppress the above vibration and acoustic energy, which was amplified by vibroacoustic coupling, an analytical model that included the installation of passive devices on the vibration system was proposed as the electromechanical-acoustic system. The effect was fully validated in the numerical approach owing to tuning the resonance characteristics of the shunt circuit, in which the piezoelectric device was incorporated, to the frequency characteristics of the coupling system [14].In almost all such studies, vibroacoustic coupling has been estimated by assuming that the plate and cavity dimensions, as well as the phase difference between the vibroacoustics of the two plates, were fixed. However, the natural frequencies of the plate and cavity vary with the dimensions of the plate and cavity, and the phase difference directly affects the sound field when the medium is much less dense than the plate. Although the study of coupling phenomena had attracted attention, an extensive parametric study has not yet been undertaken. Hence, little is known about the influence of dimensions and phase differences on the coupling phenomena.To develop a new electricity generation system, we adopt an analytical model similar to the above-mentioned cylindrical structure with plates at both ends, because the vibration area of the model on which piezoelectric elements can be installed is twice as large as that in case of a single plate. The cylinder length is varied over a wide range, while the harmonic point force is applied to one end plate or both end plates, and its frequency is selected to cause the plate to vibrate in the fundamental mode. Vibroacoustic coupling that occurs between plate vibrations and the sound field in the cavity is investigated theoretically and experimentally in terms of the vibration and acoustic characteristics. In the experiment, the acceleration of the plate vibrations, the phase difference between them, and the sound pressure level inside the cavity are considered significant characteristics of the plate vibrations and sound field. These experimental results demonstrate the underlying theoretical analysis based on this model, as well as the conditions that maximize the vibration and sound pressure levels. Furthermore, the effect of vibroacoustic coupling is estimated from an electricity generation experiment performed with piezoelectric elements.


## 2. Analytical Method

### 2.1. Equation of Motion of Plate

The analytical model consists of a cavity with two circular end plates, as shown in Figure1. The plates are supported by translational and rotational springs distributed at constant intervals, and the support conditions are determined by their respective spring stiffness T1,T2,R1, and R2, where the suffixes 1 and 2 indicate plates 1 and 2, respectively. The plates of radius a and thickness h have a Young’s modulus E and a Poisson’s ratio ν. The sound field, assumed cylindrical, has the same radius as that of the plates and varying length because the resonance frequency depends on the length. The boundary conditions are considered structurally and acoustically rigid at the lateral wall between the structure and sound field. The coordinates used are radius r and angle θ between the planes of the plates and the cross-sectional plane of the cavity and distance z along the cylinder axis. The periodic point forces F1 and F2 are applied to plates 1 and 2 at distances r1 and r2 and angles θ1 and θ2, respectively. The natural frequency of the plates is employed as the excitation frequency.Figure 1
Configuration of analytical model.To formulate the plate motion, Hamilton’s principle is applied to the analytical model [9]:
(1)δH=δ∫t0t1(TP-UP-US+W)dt=0,
where H is the Hamiltonian, TP and UP are, respectively, the kinetic and potential energy of each plate, and US is the elastic energy stored in the springs. W is the work done on plates 1 and 2 by the respective point forces and the sound pressure on the plates. Finally, t0 and t1 are two arbitrary times.The flexural displacementsw1 and w2 of plates 1 and 2, respectively, in terms of two sets of suitable trial functions are found by substituting (3) for the plate modes into (2) below:
(2)w1=∑s=01∑n=0∞∑m=0∞B1nmsXnmsej(ωt+ϕ1),w2=∑s=01∑n=0∞∑m=0∞B2nmsXnmsej(ωt+ϕ2),(3)Xnms=sin(nθ+sπ2)(ra)m,
where n, m, and s are, respectively, the circumferential order, radial order, and symmetry index with respect to the plate vibration. B1nms and B2nms are coefficients to be determined, ω is the angular frequency of the harmonic point force acting on the plate, and t is the elapsed time. ϕ1 and ϕ2 are the phases of the respective plate vibrations. In this analysis, ϕ1 is set to 0 deg, and ϕ2 varies in the range of 0 deg to 180 deg. w1 and w2 are substituted for the flexural displacements in TP,UP,US, and W, whose detailed expressions are obtained from [9, 10], and the variation of (1) is carried out with respect to both plates. Consequently, the extremum of the Hamiltonian yields Euler’s equations, which are the equations of motion of the respective plates. The motion is assumed to be harmonic, that is, to behave as ejωt, so that ejωt can be eliminated.Consider the following:(4)[∑m′=0∞{K1nmm′s(1+jηp)-ω2M1nmm′s}.+∑m′=0∞aFsn{T1+(ma)(m′a)R1}]B1nm′sejϕ1=F1nms-P1nms,[∑m′=0∞{K2nmm′s(1+jηp)-ω2M2nmm′s}.+∑m′=0∞aFsn{T2+(ma)(m′a)R2}]B2nm′sejϕ2=-F2nms+P2nms,
where K1nmm′s, K2nmm′s and M1nmm′s, M2nmm′s are elements of the symmetrical stiffness and mass matrices, respectively, because the index m′ is the radial order (m=m′). ηp is the structural damping factor of the plate, and Fsn is a coefficient that is determined by the indices n and s and is expressed as [9]
(5)Fsn={π,atn≠0,0,atn=0,s=0,2π,atn=0,s=1.Both first terms on the right-hand sides of (4) give the respective point forces, and the second terms give the acoustic excitations, which also function as the coupling term between each plate vibration and the sound field. The point force and acoustic excitation terms are, respectively,
(6)F1nms=∫A1F1δ(r-r1)δ(θ-θ1)XnmsdA1,F2nms=∫A2F2δ(r-r2)δ(θ-θ2)XnmsdA2,(7)P1nms=∫A1PcXnmsdA1,P2nms=∫A2PcXnmsdA2.
Here δ is the delta function associated with the point force on plates, A1 and A2 are the plate areas, and Pc is the sound pressure at an arbitrary point on the boundary surface of the plates. To distinguish between plates 1 and 2, the differential dA(rdθdr) is written as dA1 and dA2 in (6) and (7), respectively.


### 2.2. Coupling Equation between Plate Vibrations and Internal Sound Field

For simplicity, we assume that the cavity walls are rigid, so that the sound field in the cavity is governed by the wave equation consisting of the eigenfunctionYN and the eigenvalue kN corresponding to a cavity mode of order N:
(8)∇2YN+kN2YN=0,(9)(∂YN∂u)S=0,
where u is the unit normal to the boundary surface S (positive towards the outside), and the boundary condition satisfies (9) when S is rigid. However, if S is not rigid but has a varying specific acoustic admittance, we select a Green’s function G to obtain a solution set for a nonuniform cavity with nonrigid walls for a frequency ω/2π=Kc/2π, where K is an eigenvalue of the nonuniform cavity, and c is the cavity speed of sound. The equation for G is thus given by
(10)∇2G+K2G=-δ(p-p0)=-δ(r-r0)δ(θ-θ0)δ(z-z0),(∂G∂u)S=0.
The right-hand side is a delta function, where the measurement point is p=(r,θ,z) if the source point is p0=(r0,θ0,z0). Expressing G in terms of YN of (8), which satisfies the same boundary conditions, we find that
(11)G(p,p0)=∑N=1∞YN(p)YN(p0)VcMN(kN2-K2),(12)∫VcYN(p)YM(p)dVc=VcMNδNM={0atN≠M,VcMNatN=M.
The dimensionless factor MN is the mean value of YN2 averaged over the cavity volume Vc, and δNM is the Kronecker delta.Because there is no source and∂G/∂u=0 on S, the spatial factor Pc(p) of the sound pressure within and on the surface bounding the medium can be obtained from just one of the surface integral terms as follows:
(13)Pc(p)=-∫SG(p,p0)∂Pc(p0)∂u0dS0,
where the zero subscripts indicate differentiation and integration with respect to the (r0,θ0,z0) coordinates. A detailed procedure for obtaining these equations is given in [15]. Pc can also be expressed as [9, 10]
(14)Pc=ρcc2∑N=1∞PNYNMN,
where ρc is the fluid density in the cavity, and PN is the pressure coefficient to be determined.In this investigation, the acoustic modal shapeYnpqs and angular resonance frequency ωnpq in the cavity (where the indices n, p, and q indicate the circumferential, radial and longitudinal orders, resp.) are defined as
(15)Ynpqs=sin(nθ+sπ2)Jn(λnpr)cos{(qπL)z},ωnpq=c{λnp2+(qπL)2}1/2,
where Jn is the nth-order Bessel function, and λnp is the pth solution of an eigenvalue problem for a circular sound field having modes (n,p) divided by the radius. The boundary conditions between the plate vibrations and sound field on the respective plate surfaces are found by assuming continuity of the velocities on the plates:
(16)(∂Pc∂u)z=0=ρcω2w1,(∂Pc∂u)z=L=-ρcω2w2,
where ∂Pc/∂u is 0 on the lateral wall of the cylinder since the wall remains rigid.Because the analytical mode has two boundary surfaces, we can apply (16) to (13), so that Pc becomes
(17)Pc=-∫A1Gρcω2w1dA1+∫A2Gρcω2w2dA2.
On the other hand, by substituting acoustic modes of three orders, n, p, and q, instead of the order N of the cavity mode into (14), Pc can also be expressed as
(18)Pc=ρcc2∑p=1∞∑q=0∞PnpqsYnpqsMnpqs.The equation relating (17) and (18) is obtained by applying Green’s function of (11) to an arbitrary acoustic mode (n,p,q) as
(19)(ωnpq2-ω2)Pnpqs=-ω2Vc(∫A1Ynpqsw1dA1+∫A2Ynpqsw2dA2).
Here, substituting (2) for w1 and w2 and considering a modal damping factor ηc, (19) can be rewritten as
(20)(ωnpq2+jηcωnpqω-ω2)Pnpqs=Aω2Vc(-∑m=0∞I1B1nmsejϕ1+∑m=0∞I2B2nmsejϕ2),(21)I1=1A∫A1XnmsYnpqsdA1,I2=1A∫A2XnmsYnpqsdA2,
where A is the total surface area of the plates, I1 and I2 are the spatial coupling coefficients. Moreover, substituting (18) for Pc and applying I1 and I2 to the integrals in (7), the acoustic excitation terms P1nms and P2nms can be expressed with respect to an arbitrary vibration mode (n,m) as
(22)P1nms=ρcc2A∑p=1∞∑q=0∞I1PnpqsMnpqs,P2nms=ρcc2A∑p=1∞∑q=0∞I2PnpqsMnpqs.Finally, replacingPnpqs in (22) with those in (20) and then inserting them in (4), we can complete the coupling equations, whose right-hand sides are
(23)F1nms-P1nms=F1nms+ρcc2ω2A2Vc×∑m′=0∞∑p=1∞∑q=0∞I1(I1B1nm′sejϕ1-I2B2nm′sejϕ2)Mnpqs(ωnpq2+jηcωnpqω-ω2),-F2nms-P2nms=-F2nms+ρcc2ω2A2Vc×∑m′=0∞∑p=1∞∑q=0∞I2(I1B1nm′sejϕ1-I2B2nm′sejϕ2)Mnpqs(ωnpq2+jηcωnpqω-ω2).On the right-hand sides, the second terms show the acoustic excitation for plates 1 and 2, respectively. The acoustic excitation terms have bothI1 and I2 since the acoustic mode of the sound field is coupled with the vibration modes of the respective plates. Before actual calculation, the natural frequency of the plate must be considered in terms of the convergence of the plate vibration mode (n,m). In this case, the natural frequency is obtained as the eigenvalue of (4), whose right-hand side is set to 0. The actual calculation is performed by taking 15 terms for n, while m is set to be greater than 13 to ensure the convergence of the natural frequency and mode shape of the plate. Using the same truncation for p as for m, the order accounts for acoustic modes greater than q=15, so that the resonance frequency containing q can exceed the excitation frequency. The plate and cavity loss factors are assumed to be constant: ηp=ηc=0.01 [9, 10]. Since B1nms and B2nms can be obtained simultaneously from (4), which have as excitation terms, the behavior of the plate vibrations and the sound field under vibroacoustic coupling can be determined.The flexural displacementsw1 and w2 and the sound pressure Pc are, respectively, obtained from (2) and (18) by employing B1nms and B2nms determined above. However, the procedure to calculate Pc involves substituting Pnpqs in (20) for that in (18). Solved from (24), into which w1, w2, and Pc are inserted, over the respective entire regions, the quadric velocities V1 and V2 and the sound pressure Pν are estimated by the logarithmic values Lv1, Lv2, and Lpv relative to V0=2.5×10-15 m2/s2 and P0=4×10-10 N2/m4, respectively, as shown in (25).Consider the following:(24)V1=ω22A1∫A1w1w1*dA1,V2=ω22A2∫A2w2w2*dA2,Pν=12Vc∫VcPcPc*dVc,(25)Lv1=10logV1V0,Lv2=10logV2V0,Lpv=10logPvP0,
where w1*, w2*, and Pc* are the respective conjugate components.


## 2.1. Equation of Motion of Plate

The analytical model consists of a cavity with two circular end plates, as shown in Figure1. The plates are supported by translational and rotational springs distributed at constant intervals, and the support conditions are determined by their respective spring stiffness T1,T2,R1, and R2, where the suffixes 1 and 2 indicate plates 1 and 2, respectively. The plates of radius a and thickness h have a Young’s modulus E and a Poisson’s ratio ν. The sound field, assumed cylindrical, has the same radius as that of the plates and varying length because the resonance frequency depends on the length. The boundary conditions are considered structurally and acoustically rigid at the lateral wall between the structure and sound field. The coordinates used are radius r and angle θ between the planes of the plates and the cross-sectional plane of the cavity and distance z along the cylinder axis. The periodic point forces F1 and F2 are applied to plates 1 and 2 at distances r1 and r2 and angles θ1 and θ2, respectively. The natural frequency of the plates is employed as the excitation frequency.Figure 1
Configuration of analytical model.To formulate the plate motion, Hamilton’s principle is applied to the analytical model [9]:
(1)δH=δ∫t0t1(TP-UP-US+W)dt=0,
where H is the Hamiltonian, TP and UP are, respectively, the kinetic and potential energy of each plate, and US is the elastic energy stored in the springs. W is the work done on plates 1 and 2 by the respective point forces and the sound pressure on the plates. Finally, t0 and t1 are two arbitrary times.The flexural displacementsw1 and w2 of plates 1 and 2, respectively, in terms of two sets of suitable trial functions are found by substituting (3) for the plate modes into (2) below:
(2)w1=∑s=01∑n=0∞∑m=0∞B1nmsXnmsej(ωt+ϕ1),w2=∑s=01∑n=0∞∑m=0∞B2nmsXnmsej(ωt+ϕ2),(3)Xnms=sin(nθ+sπ2)(ra)m,
where n, m, and s are, respectively, the circumferential order, radial order, and symmetry index with respect to the plate vibration. B1nms and B2nms are coefficients to be determined, ω is the angular frequency of the harmonic point force acting on the plate, and t is the elapsed time. ϕ1 and ϕ2 are the phases of the respective plate vibrations. In this analysis, ϕ1 is set to 0 deg, and ϕ2 varies in the range of 0 deg to 180 deg. w1 and w2 are substituted for the flexural displacements in TP,UP,US, and W, whose detailed expressions are obtained from [9, 10], and the variation of (1) is carried out with respect to both plates. Consequently, the extremum of the Hamiltonian yields Euler’s equations, which are the equations of motion of the respective plates. The motion is assumed to be harmonic, that is, to behave as ejωt, so that ejωt can be eliminated.Consider the following:(4)[∑m′=0∞{K1nmm′s(1+jηp)-ω2M1nmm′s}.+∑m′=0∞aFsn{T1+(ma)(m′a)R1}]B1nm′sejϕ1=F1nms-P1nms,[∑m′=0∞{K2nmm′s(1+jηp)-ω2M2nmm′s}.+∑m′=0∞aFsn{T2+(ma)(m′a)R2}]B2nm′sejϕ2=-F2nms+P2nms,
where K1nmm′s, K2nmm′s and M1nmm′s, M2nmm′s are elements of the symmetrical stiffness and mass matrices, respectively, because the index m′ is the radial order (m=m′). ηp is the structural damping factor of the plate, and Fsn is a coefficient that is determined by the indices n and s and is expressed as [9]
(5)Fsn={π,atn≠0,0,atn=0,s=0,2π,atn=0,s=1.Both first terms on the right-hand sides of (4) give the respective point forces, and the second terms give the acoustic excitations, which also function as the coupling term between each plate vibration and the sound field. The point force and acoustic excitation terms are, respectively,
(6)F1nms=∫A1F1δ(r-r1)δ(θ-θ1)XnmsdA1,F2nms=∫A2F2δ(r-r2)δ(θ-θ2)XnmsdA2,(7)P1nms=∫A1PcXnmsdA1,P2nms=∫A2PcXnmsdA2.
Here δ is the delta function associated with the point force on plates, A1 and A2 are the plate areas, and Pc is the sound pressure at an arbitrary point on the boundary surface of the plates. To distinguish between plates 1 and 2, the differential dA(rdθdr) is written as dA1 and dA2 in (6) and (7), respectively.


## 2.2. Coupling Equation between Plate Vibrations and Internal Sound Field

For simplicity, we assume that the cavity walls are rigid, so that the sound field in the cavity is governed by the wave equation consisting of the eigenfunctionYN and the eigenvalue kN corresponding to a cavity mode of order N:
(8)∇2YN+kN2YN=0,(9)(∂YN∂u)S=0,
where u is the unit normal to the boundary surface S (positive towards the outside), and the boundary condition satisfies (9) when S is rigid. However, if S is not rigid but has a varying specific acoustic admittance, we select a Green’s function G to obtain a solution set for a nonuniform cavity with nonrigid walls for a frequency ω/2π=Kc/2π, where K is an eigenvalue of the nonuniform cavity, and c is the cavity speed of sound. The equation for G is thus given by
(10)∇2G+K2G=-δ(p-p0)=-δ(r-r0)δ(θ-θ0)δ(z-z0),(∂G∂u)S=0.
The right-hand side is a delta function, where the measurement point is p=(r,θ,z) if the source point is p0=(r0,θ0,z0). Expressing G in terms of YN of (8), which satisfies the same boundary conditions, we find that
(11)G(p,p0)=∑N=1∞YN(p)YN(p0)VcMN(kN2-K2),(12)∫VcYN(p)YM(p)dVc=VcMNδNM={0atN≠M,VcMNatN=M.
The dimensionless factor MN is the mean value of YN2 averaged over the cavity volume Vc, and δNM is the Kronecker delta.Because there is no source and∂G/∂u=0 on S, the spatial factor Pc(p) of the sound pressure within and on the surface bounding the medium can be obtained from just one of the surface integral terms as follows:
(13)Pc(p)=-∫SG(p,p0)∂Pc(p0)∂u0dS0,
where the zero subscripts indicate differentiation and integration with respect to the (r0,θ0,z0) coordinates. A detailed procedure for obtaining these equations is given in [15]. Pc can also be expressed as [9, 10]
(14)Pc=ρcc2∑N=1∞PNYNMN,
where ρc is the fluid density in the cavity, and PN is the pressure coefficient to be determined.In this investigation, the acoustic modal shapeYnpqs and angular resonance frequency ωnpq in the cavity (where the indices n, p, and q indicate the circumferential, radial and longitudinal orders, resp.) are defined as
(15)Ynpqs=sin(nθ+sπ2)Jn(λnpr)cos{(qπL)z},ωnpq=c{λnp2+(qπL)2}1/2,
where Jn is the nth-order Bessel function, and λnp is the pth solution of an eigenvalue problem for a circular sound field having modes (n,p) divided by the radius. The boundary conditions between the plate vibrations and sound field on the respective plate surfaces are found by assuming continuity of the velocities on the plates:
(16)(∂Pc∂u)z=0=ρcω2w1,(∂Pc∂u)z=L=-ρcω2w2,
where ∂Pc/∂u is 0 on the lateral wall of the cylinder since the wall remains rigid.Because the analytical mode has two boundary surfaces, we can apply (16) to (13), so that Pc becomes
(17)Pc=-∫A1Gρcω2w1dA1+∫A2Gρcω2w2dA2.
On the other hand, by substituting acoustic modes of three orders, n, p, and q, instead of the order N of the cavity mode into (14), Pc can also be expressed as
(18)Pc=ρcc2∑p=1∞∑q=0∞PnpqsYnpqsMnpqs.The equation relating (17) and (18) is obtained by applying Green’s function of (11) to an arbitrary acoustic mode (n,p,q) as
(19)(ωnpq2-ω2)Pnpqs=-ω2Vc(∫A1Ynpqsw1dA1+∫A2Ynpqsw2dA2).
Here, substituting (2) for w1 and w2 and considering a modal damping factor ηc, (19) can be rewritten as
(20)(ωnpq2+jηcωnpqω-ω2)Pnpqs=Aω2Vc(-∑m=0∞I1B1nmsejϕ1+∑m=0∞I2B2nmsejϕ2),(21)I1=1A∫A1XnmsYnpqsdA1,I2=1A∫A2XnmsYnpqsdA2,
where A is the total surface area of the plates, I1 and I2 are the spatial coupling coefficients. Moreover, substituting (18) for Pc and applying I1 and I2 to the integrals in (7), the acoustic excitation terms P1nms and P2nms can be expressed with respect to an arbitrary vibration mode (n,m) as
(22)P1nms=ρcc2A∑p=1∞∑q=0∞I1PnpqsMnpqs,P2nms=ρcc2A∑p=1∞∑q=0∞I2PnpqsMnpqs.Finally, replacingPnpqs in (22) with those in (20) and then inserting them in (4), we can complete the coupling equations, whose right-hand sides are
(23)F1nms-P1nms=F1nms+ρcc2ω2A2Vc×∑m′=0∞∑p=1∞∑q=0∞I1(I1B1nm′sejϕ1-I2B2nm′sejϕ2)Mnpqs(ωnpq2+jηcωnpqω-ω2),-F2nms-P2nms=-F2nms+ρcc2ω2A2Vc×∑m′=0∞∑p=1∞∑q=0∞I2(I1B1nm′sejϕ1-I2B2nm′sejϕ2)Mnpqs(ωnpq2+jηcωnpqω-ω2).On the right-hand sides, the second terms show the acoustic excitation for plates 1 and 2, respectively. The acoustic excitation terms have bothI1 and I2 since the acoustic mode of the sound field is coupled with the vibration modes of the respective plates. Before actual calculation, the natural frequency of the plate must be considered in terms of the convergence of the plate vibration mode (n,m). In this case, the natural frequency is obtained as the eigenvalue of (4), whose right-hand side is set to 0. The actual calculation is performed by taking 15 terms for n, while m is set to be greater than 13 to ensure the convergence of the natural frequency and mode shape of the plate. Using the same truncation for p as for m, the order accounts for acoustic modes greater than q=15, so that the resonance frequency containing q can exceed the excitation frequency. The plate and cavity loss factors are assumed to be constant: ηp=ηc=0.01 [9, 10]. Since B1nms and B2nms can be obtained simultaneously from (4), which have as excitation terms, the behavior of the plate vibrations and the sound field under vibroacoustic coupling can be determined.The flexural displacementsw1 and w2 and the sound pressure Pc are, respectively, obtained from (2) and (18) by employing B1nms and B2nms determined above. However, the procedure to calculate Pc involves substituting Pnpqs in (20) for that in (18). Solved from (24), into which w1, w2, and Pc are inserted, over the respective entire regions, the quadric velocities V1 and V2 and the sound pressure Pν are estimated by the logarithmic values Lv1, Lv2, and Lpv relative to V0=2.5×10-15 m2/s2 and P0=4×10-10 N2/m4, respectively, as shown in (25).Consider the following:(24)V1=ω22A1∫A1w1w1*dA1,V2=ω22A2∫A2w2w2*dA2,Pν=12Vc∫VcPcPc*dVc,(25)Lv1=10logV1V0,Lv2=10logV2V0,Lpv=10logPvP0,
where w1*, w2*, and Pc* are the respective conjugate components.


## 3. Experimental Apparatus and Method

Figure2(a) shows the experimental apparatus used in this study. The structure consists of a steel cylinder with circular aluminum end plates that are 3 mm thick. The cylinder has an inner radius of 153 mm, and this length can be varied from 500 to 2000 mm to emulate the analytical model. One end plate or both end plates are subjected to the point force, whose frequency makes the plate excite in the (0,0) mode. In case of the harmonic excitation of both ends, these forces are applied to the respective plates via small vibrators, and their amplitudes are controlled to be 1 N. The positions of the point forces r1 and r2 are normalized by radius a and are set to r1/a=r2/a=0.4. In the excitation experiment, the main characteristic is the phase difference between the plate vibrations. Therefore, 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 a probe tube. The tips of the probe tubes are located near the plates and the cylinder wall, which are the approximate locations of the maximum sound pressure level when the sound field becomes resonant.Configuration of experimental apparatus: (a) Measurement system; (b) Installation state of piezoelectric element; and (c) Electrical circuit of energy-harvesting device.
(a)
(b)
(c)To perform the electricity generation experiment, the piezoelectric element is used and is comprised of the piezoelectric part constructed of ceramics and the electrode part constructed of brass, which have the diameters of 25 and 35 mm and the thicknesses of 0.23 and 0.30 mm, respectively. Actually, the piezoelectric elements are installed at each center of both plates, as shown in Figures2(a) and 2(b). The electric power generated by the expansion and contraction of the piezoelectric elements is discharged through the resistance circuit, which consists of three resistors having resistances Rv, Ri, and Rc, as shown in Figure 2(c). Rv and Ri are the resistances of the voltmeter and ammeter built-in the wattmeter and are 2 MΩ and 2 mΩ, respectively, while Rc is the resistance of the resistor connected outside the wattmeter and is 97.5 kΩ. To grasp the effect of vibroacoustic coupling on energy harvesting, the electric power is measured with and without the cylinder and is estimated by the comparison of both cases. In such a estimation, the electric power is normalized by the vibration power supplied with the plate, which is obtained from the point force and flexural displacement.


## 4. Results and Discussion

### 4.1. Acoustic Characteristics under Vibroacoustic Coupling

In the theoretical study, the plates are assumed to be aluminum having a Young’s modulusE of 71 GPa and a Poisson’s ratio ν of 0.33. The radius a and thickness h of the plates are constant at 153 mm and 3 mm, respectively, whereas the length of the cylindrical sound field having the same radius as that of the plates varies from 100 to 2000 mm. The support conditions of the plates, which have flexural rigidity D[=Eh3/{12(1-ν2)}], are expressed by the nondimensional stiffness parameters Tn(=T1a3/D=T2a3/D) and Rn(=R1a/D=R2a/D). These values are identical for both plates. If Rn ranges from 0 to 108 when Tn is 108, the support condition can be assumed from a simple support to a clamped support. The actual condition adopts Tn=108 and Rn=101 to get closer to the experimental support condition. These plates 1 and 2 are subjected to the point forces F1 and F2, which are set to 1 N and are located at r1/a=r2/a=0.4, respectively, as well as the actual excitation experiment. In particular, the analysis in which only one end plate is excited is carried out with taking F2 as 0 N.The plate and sound field eigenfrequency characteristics involved in the vibroacoustic coupling are represented by the natural frequencyfnm corresponding to the (n,m) mode and the resonance frequency fnpq corresponding to the (n,p,q) mode. The excitation frequency is chosen as f00 that makes the plates vibrate in the (0,0) mode. With respect to the plate vibration, although the phase ϕ1 of plate 1 is fixed at 0 deg, the phase ϕ2 of plate 2 ranges from 0 to 180 deg, and then they are related by the phase difference ϕ as follows:
(26)ϕ=ϕ2-ϕ1.Figure3 shows the variations in Lpv with L, when only plate 1 is excited and ϕ is arbitrarily set to 0, 10, and 90 deg. Lpv varies only slightly over the entire range of L when ϕ=0 deg, but varies substantially and exhibits peaks near L=610, 1220, and 1830 mm when ϕ=10 and 90 deg. The value of Lpv is lower when ϕ=10 and 90 deg and is almost identical at all phase differences near L=460, 920 and 1560 mm. To estimate the influence of each acoustic mode on these sound fields, the contribution CE can be defined as the ratio of the acoustic energy Enpq stored in the specific (n,p,q) mode to the total acoustic energy Eall of the entire sound field:
(27)CE=EnpqEall.Figure 3
Variation in sound pressure level with cylinder length when phase difference changes.Figures4(a) and 4(b) show CE as a function of L for the (0,0,q) modes for values of q between 0 and 4 for ϕ=0 and 90 deg, respectively. In case of ϕ=0 deg, CE is relatively large when L is short. The value of CE for the (0,0,0) mode decreases as L increases because of the increase in the value of CE for the (0,0,1) mode. The (0,0,0) mode is regarded as a pumping mode that corresponds to the reciprocation of both end plates supported by the aerostatic stiffness of the cavity. In the range of longer length, as the value of q increases, CE decreases relatively to CE of the (0,0,0) mode because the sound field is affected by other modes. At ϕ=90 deg, CE varies dramatically when the acoustic mode that dominates the sound field changes, so that the range of L over which the influence of such a dominant mode extends can be clearly distinguished. The acoustic mode (0,0,q) causes Lpv to have peaks at L=610, 1220, and 1830 mm, having modal shapes similar to that of the (0,0) mode of the plate vibrations. These peaks occur at an integer q starting from q=1 with increasing L. The vibration of plate 2 has a significant effect on the formation of the sound field and vibroacoustic coupling, despite not being driven by the point force, whereas the influence of ϕ on the acoustic characteristics has only been described for ϕ=0, 10, and 90 deg. This change in Lpv with ϕ indicates that the acoustic characteristics depend strongly on the vibration of plate 2; that is, there are ranges of ϕ that intensify or suppress coupling between the plate vibrations and sound field. Here the values of ϕ at which Lpv is maximum are denoted by ϕmax.Contribution of acoustic mode to sound field at (a)ϕ=0 [deg] and (b) ϕ=90 [deg].
(a)
(b)In Figure5, the variations in ϕmax with L and the values of ϕmax that maximize Lpv are plotted by a line and circles, respectively. ϕmax is approximately 87 deg at L=100 mm and decreases gradually with increasing L up to approximately L=460 mm, where ϕmax suddenly increases to over 90 deg and then decreases again with increasing L. This behavior of ϕmax is repeated in a similar manner as L increases to L=2000 mm. Peaks in Lpv indicate that vibroacoustic coupling between the plate vibrations and sound field is promoted at approximately 90 deg. fnm and fnpq must be 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 Lpv peaks. On the other hand, we confirm that the longitudinal order q shifts from 1 to 2 and from 2 to 3 at L=900 and 1560 mm where Lpv varies abruptly, respectively, based on the 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, and 1570 to 2000 mm, are dominated by the (0,0,1), (0,0,2), and (0,0,3) modes, respectively, as shown in this figure where the range of the acoustic mode is classified in color. Naturally, this classification is also thought to be due to variations in the dominant acoustic mode, whose contribution CE is maximized in Figure 4(b).Figure 5
Variations in phase difference with cylinder length.Figure6 shows the sound pressure levels Lp1 and Lp2, which are measured near plates 1 and 2, respectively, as functions of L. The theoretical level Lpv, which is maximized at each L when the phase difference ϕ ranges from 0 to 180 deg, is also indicated to compare with the experimental results. Lp1 and Lp2 show peaks around 615, 1275, and 1900 mm, and these levels are almost coincident for each peak. However, they decrease in the middle range of those lengths. In particular, decreases in Lp1 are remarkable and their differences expand around 950 and 1550 mm. Lpv also shows peaks at 610, 1230, and 1840 mm and corresponds approximately with the above lengths where Lp1 and Lp2 peak.Figure 6
Comparison between theoretical and experimental results for sound pressure level.Since the above results are derived from the investigation based on the excitation of one end plate, furthermore, the excitation condition in which both plates are subjected to the same excitation force is taken to grasp the effect of the excitation method on vibroacoustic coupling. Figure7 shows the variations in Lpv corresponding to ϕmax and the variations in Lp1 and Lp2 that are measured in the experiment and are maximized when the phase difference between both point forces ranges from 0 deg to 180 deg. Peaks in Lpv appear at L=610, 1230, and 1840 mm. These peaks are known to be caused by the (0,0,1), (0,0,2), and (0,0,3) modes, respectively. Note that Lp1 and Lp2 increase greatly at 625, 1250, and 1850 mm. However, Lp1 and Lp2 are hardly distinguished in the middle range of lengths where the sound pressure levels peak, having been different in the results for the excitation of one end plate, as shown in Figure 6.Figure 7
Comparison between theoretical and experimental results for sound pressure level when both end plates are excited by point force.


### 4.2. Plate Vibration Characteristics under Vibroacoustic Coupling

Since the plate vibrations influence the acoustic characteristics via vibroacoustic coupling, the magnitude of the flexural displacementsw1 and w2 is also significant in studying the effect of the plate vibrations on the sound field. Here the phase differences are denoted as ϕmax1 and ϕmax2 when w1 and w2 are maximized, while they are denoted as ϕmin1 and ϕmin2 when w1 and w2 are minimized. Figure 8 shows ϕmax1, ϕmax2, ϕmin1, and ϕmin2 as functions of L when only plate 1 is excited. ϕmax1 is constant at 180 deg for L 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, ϕmax1 increases gradually with L and returns to 180 deg at L=1220 mm, increasing somewhat abruptly near L=970 mm. Beyond L=1220 mm, ϕmax1 is again constant at 180 deg up to L=1570 mm, and this behavior is repeated as L increases to L=2000 mm. ϕmax2 exhibits gradual and abrupt changes similar but alternate to ϕmax1. For example, when L increases, a gradual decrease occurs in ϕmax2 between L=100 and 620 mm, and an abrupt increase occurs near L=970 mm. Both ϕmax1 and ϕmax2 shift between 0 and 180 deg with changing L and intersect at approximately 90 deg and near the length at which Lpv peaked in Figure 6. ϕmin1 and ϕmin2 behave exactly alike but opposite to ϕmax1 and ϕmax2.Figure 8
Comparison between theoretical and experimental results for phase difference.In Figure8, the theoretical results for ϕmax where Lpv peaks and the experimental results for ϕexp as Lp1 and Lp2 are maximized at each L are also plotted. ϕexp ranges greatly between in-phase and out-of-phase and ϕmax exists in the process where ϕexp changes abruptly. Then, ϕexp lies in the light yellow areas surrounded by ϕmax1 and ϕmax2 in the L ranges longer than the lengths when the sound pressure level peaks and occurs in the yellowish green areas surrounded by ϕmin1 and ϕmin2 on the other side. In other words, since vibroacoustic coupling is gradually weakened with increasing L after the peaks of Lp1 and Lp2, the acoustic mode involved in coupling shifts to that having the next order q.Figure9 shows the vibration levels Lv1 and Lv2 of plates 1 and 2 as functions of L and the accelerations α1 and α2 of plates 1 and 2 are also plotted to compare with the theoretical plate behavior. Lv1 is smaller than Lv2 in the ranges of 100 to 610 mm, 800 to 1200 mm, and 1300 to 1820 mm, so that Lv1 and Lv2 intersect at a number of L and the intersections take place around the lengths where Lpv peaks. The actual motion of plate 1 is almost suppressed by that of the vibrator since plate 1 is supported by the vibrator; hence, α1 is approximately constant over the entire L range. However, since the motion of plate 2 depends greatly on the behavior of the sound field, that is, the only excitation source for plate 2, α2 peaks at L=650, 1280, and 1880 mm and is suppressed in the other ranges of L, such as variations in Lp1 and Lp2. As a result, the relative relationship of α1 and α2 becomes opposite to that of Lv1 and Lv2 as shown in the colored regions. In the theoretical analysis, we confirm 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 sound tube having single closed and open ends at the lengths, 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 Lp1 and Lp2 in those middle ranges of lengths where the sound pressure levels peaked in Figure 6. This is derived from the difference between the tendencies of the above vibration level and acceleration with respect to both plates. If the experimental model completely emulated the theoretical model in the flexural displacement, such a discrepancy would not take place.Figure 9
Comparison between characteristics of theoretical vibration and experimental acceleration.In Figure10, the experimental phase difference ϕexp, at which Lp1 and Lp2 are maximized, is compared with the theoretical phase differences ϕmax1, ϕmax2, ϕmin1, and ϕmin2 when both plates are subjected to the same excitation force, as shown in Figure 8. The shifts in the (0,0,q) modes are also represented by the changing colors. ϕexp shifts between 0 and 180 deg with changing L and corresponds approximately with the ϕmax1 and ϕmax2 that behave uniformly in the vicinity 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.Figure 10
Comparison between theoretical and experimental results for phase difference when both end plates are excited by point force.


### 4.3. Electricity Generation Characteristics

In this section, we consider electricity generation by the plate vibrations coupled with the sound field. 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 relationship between the point force and flexural displacement at the excitation point. Figure11 shows the relationship between the electric power Pe and mechanical power Pm when the point force F1 ranges from 1 to 5 N. In this case, the cylinder is removed; the plate vibrations do not couple with the internal sound field. Although Pe is considerably smaller than Pm, their relationship is directly proportional. Here, the relationship between Pe and Pm is defined as
(28)Pem=PePm.Figure 11
Relationship of electric power and mechanical power.Figure12 shows variations in Pem with L; the powers are measured in the experimental apparatus via the cylinder shown in Figure 2(a) when one end plate is excited by the point force. Although Pem of plate 1 remains almost constant over the entire range of L, Pem of plate 2 increases greatly at L=615, 1275, and 1900 mm. It is natural that the relationship between Pem of plate 1 and 2 is derived from the behavior of α1 and α2 in Figure 9. Pem is considered as energy-harvesting efficiency. In an electricity generation by means of beam or plate vibrations with piezoelectric elements, coupling between the structural vibration and electric field, that is, electromechanical coupling should be considered to relate the in-plane stress to the applied electric field. Considering that the behaviors of Pem for plates 1 and 2 correspond to those of α1 and α2 and the values of Pem are substantially small, that is, the value of Pe is extremely small in comparison with that of Pm, we assume that the behavior of plate vibrations is hardly affected by electromechanical coupling in this study.Figure 12
Energy-harvesting efficiency as function of cylinder length.Moreover, to grasp the effect of vibroacoustic coupling on electricity generation, althoughPR is taken as the ratio of Pem measured with and without the cylinder, Pem measured with the cylinder is obtained from the total value Pe with respect to plates 1 and 2. Figure 13 shows variations in PR with L; the results of the excitation of both ends are also indicated to study the effect of the excitation method. PR for the excitation of one end is maximized as the sound pressure level peaks due to the promotion of vibroacoustic coupling. This is because the plate vibration on the nonexcitation side contributes strongly to the electricity generation, whereas their L ranges are limited to the narrow regions in comparison 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 excitation of both ends, there is not such a nonexcitation side that contributes greatly to the electricity generation, so that PR never reaches values as large as those of the excitation of one end, when the sound pressure level is maximized. In L ranges where vibroacoustic coupling is weakened, PR remains almost constant and is close to that of the excitation of one end. Under the situation of such a weakened vibroacoustic 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 Pe and mechanical power Pm. As a result, PR shifts almost constantly with L, no matter what the excitation method is.Figure 13
Effect of vibroacoustic coupling on energy harvesting for excitations of one end plate and both end plates.Hence, it follows that the excitation of one end is decidedly superior to that of both ends by electricity generation efficiency, according to this estimation method. These results are remarkable in the viewpoint where the acoustic energy from the sound radiation can be harvested through vibroacoustic coupling. However, the number or area of the piezoelectric element should be increased to improve the efficiencyPem, so that the effect of electromechanical coupling on the plate vibration must be taken into consideration in the theoretical procedure.


## 4.1. Acoustic Characteristics under Vibroacoustic Coupling

In the theoretical study, the plates are assumed to be aluminum having a Young’s modulusE of 71 GPa and a Poisson’s ratio ν of 0.33. The radius a and thickness h of the plates are constant at 153 mm and 3 mm, respectively, whereas the length of the cylindrical sound field having the same radius as that of the plates varies from 100 to 2000 mm. The support conditions of the plates, which have flexural rigidity D[=Eh3/{12(1-ν2)}], are expressed by the nondimensional stiffness parameters Tn(=T1a3/D=T2a3/D) and Rn(=R1a/D=R2a/D). These values are identical for both plates. If Rn ranges from 0 to 108 when Tn is 108, the support condition can be assumed from a simple support to a clamped support. The actual condition adopts Tn=108 and Rn=101 to get closer to the experimental support condition. These plates 1 and 2 are subjected to the point forces F1 and F2, which are set to 1 N and are located at r1/a=r2/a=0.4, respectively, as well as the actual excitation experiment. In particular, the analysis in which only one end plate is excited is carried out with taking F2 as 0 N.The plate and sound field eigenfrequency characteristics involved in the vibroacoustic coupling are represented by the natural frequencyfnm corresponding to the (n,m) mode and the resonance frequency fnpq corresponding to the (n,p,q) mode. The excitation frequency is chosen as f00 that makes the plates vibrate in the (0,0) mode. With respect to the plate vibration, although the phase ϕ1 of plate 1 is fixed at 0 deg, the phase ϕ2 of plate 2 ranges from 0 to 180 deg, and then they are related by the phase difference ϕ as follows:
(26)ϕ=ϕ2-ϕ1.Figure3 shows the variations in Lpv with L, when only plate 1 is excited and ϕ is arbitrarily set to 0, 10, and 90 deg. Lpv varies only slightly over the entire range of L when ϕ=0 deg, but varies substantially and exhibits peaks near L=610, 1220, and 1830 mm when ϕ=10 and 90 deg. The value of Lpv is lower when ϕ=10 and 90 deg and is almost identical at all phase differences near L=460, 920 and 1560 mm. To estimate the influence of each acoustic mode on these sound fields, the contribution CE can be defined as the ratio of the acoustic energy Enpq stored in the specific (n,p,q) mode to the total acoustic energy Eall of the entire sound field:
(27)CE=EnpqEall.Figure 3
Variation in sound pressure level with cylinder length when phase difference changes.Figures4(a) and 4(b) show CE as a function of L for the (0,0,q) modes for values of q between 0 and 4 for ϕ=0 and 90 deg, respectively. In case of ϕ=0 deg, CE is relatively large when L is short. The value of CE for the (0,0,0) mode decreases as L increases because of the increase in the value of CE for the (0,0,1) mode. The (0,0,0) mode is regarded as a pumping mode that corresponds to the reciprocation of both end plates supported by the aerostatic stiffness of the cavity. In the range of longer length, as the value of q increases, CE decreases relatively to CE of the (0,0,0) mode because the sound field is affected by other modes. At ϕ=90 deg, CE varies dramatically when the acoustic mode that dominates the sound field changes, so that the range of L over which the influence of such a dominant mode extends can be clearly distinguished. The acoustic mode (0,0,q) causes Lpv to have peaks at L=610, 1220, and 1830 mm, having modal shapes similar to that of the (0,0) mode of the plate vibrations. These peaks occur at an integer q starting from q=1 with increasing L. The vibration of plate 2 has a significant effect on the formation of the sound field and vibroacoustic coupling, despite not being driven by the point force, whereas the influence of ϕ on the acoustic characteristics has only been described for ϕ=0, 10, and 90 deg. This change in Lpv with ϕ indicates that the acoustic characteristics depend strongly on the vibration of plate 2; that is, there are ranges of ϕ that intensify or suppress coupling between the plate vibrations and sound field. Here the values of ϕ at which Lpv is maximum are denoted by ϕmax.Contribution of acoustic mode to sound field at (a)ϕ=0 [deg] and (b) ϕ=90 [deg].
(a)
(b)In Figure5, the variations in ϕmax with L and the values of ϕmax that maximize Lpv are plotted by a line and circles, respectively. ϕmax is approximately 87 deg at L=100 mm and decreases gradually with increasing L up to approximately L=460 mm, where ϕmax suddenly increases to over 90 deg and then decreases again with increasing L. This behavior of ϕmax is repeated in a similar manner as L increases to L=2000 mm. Peaks in Lpv indicate that vibroacoustic coupling between the plate vibrations and sound field is promoted at approximately 90 deg. fnm and fnpq must be 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 Lpv peaks. On the other hand, we confirm that the longitudinal order q shifts from 1 to 2 and from 2 to 3 at L=900 and 1560 mm where Lpv varies abruptly, respectively, based on the 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, and 1570 to 2000 mm, are dominated by the (0,0,1), (0,0,2), and (0,0,3) modes, respectively, as shown in this figure where the range of the acoustic mode is classified in color. Naturally, this classification is also thought to be due to variations in the dominant acoustic mode, whose contribution CE is maximized in Figure 4(b).Figure 5
Variations in phase difference with cylinder length.Figure6 shows the sound pressure levels Lp1 and Lp2, which are measured near plates 1 and 2, respectively, as functions of L. The theoretical level Lpv, which is maximized at each L when the phase difference ϕ ranges from 0 to 180 deg, is also indicated to compare with the experimental results. Lp1 and Lp2 show peaks around 615, 1275, and 1900 mm, and these levels are almost coincident for each peak. However, they decrease in the middle range of those lengths. In particular, decreases in Lp1 are remarkable and their differences expand around 950 and 1550 mm. Lpv also shows peaks at 610, 1230, and 1840 mm and corresponds approximately with the above lengths where Lp1 and Lp2 peak.Figure 6
Comparison between theoretical and experimental results for sound pressure level.Since the above results are derived from the investigation based on the excitation of one end plate, furthermore, the excitation condition in which both plates are subjected to the same excitation force is taken to grasp the effect of the excitation method on vibroacoustic coupling. Figure7 shows the variations in Lpv corresponding to ϕmax and the variations in Lp1 and Lp2 that are measured in the experiment and are maximized when the phase difference between both point forces ranges from 0 deg to 180 deg. Peaks in Lpv appear at L=610, 1230, and 1840 mm. These peaks are known to be caused by the (0,0,1), (0,0,2), and (0,0,3) modes, respectively. Note that Lp1 and Lp2 increase greatly at 625, 1250, and 1850 mm. However, Lp1 and Lp2 are hardly distinguished in the middle range of lengths where the sound pressure levels peak, having been different in the results for the excitation of one end plate, as shown in Figure 6.Figure 7
Comparison between theoretical and experimental results for sound pressure level when both end plates are excited by point force.


## 4.2. Plate Vibration Characteristics under Vibroacoustic Coupling

Since the plate vibrations influence the acoustic characteristics via vibroacoustic coupling, the magnitude of the flexural displacementsw1 and w2 is also significant in studying the effect of the plate vibrations on the sound field. Here the phase differences are denoted as ϕmax1 and ϕmax2 when w1 and w2 are maximized, while they are denoted as ϕmin1 and ϕmin2 when w1 and w2 are minimized. Figure 8 shows ϕmax1, ϕmax2, ϕmin1, and ϕmin2 as functions of L when only plate 1 is excited. ϕmax1 is constant at 180 deg for L 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, ϕmax1 increases gradually with L and returns to 180 deg at L=1220 mm, increasing somewhat abruptly near L=970 mm. Beyond L=1220 mm, ϕmax1 is again constant at 180 deg up to L=1570 mm, and this behavior is repeated as L increases to L=2000 mm. ϕmax2 exhibits gradual and abrupt changes similar but alternate to ϕmax1. For example, when L increases, a gradual decrease occurs in ϕmax2 between L=100 and 620 mm, and an abrupt increase occurs near L=970 mm. Both ϕmax1 and ϕmax2 shift between 0 and 180 deg with changing L and intersect at approximately 90 deg and near the length at which Lpv peaked in Figure 6. ϕmin1 and ϕmin2 behave exactly alike but opposite to ϕmax1 and ϕmax2.Figure 8
Comparison between theoretical and experimental results for phase difference.In Figure8, the theoretical results for ϕmax where Lpv peaks and the experimental results for ϕexp as Lp1 and Lp2 are maximized at each L are also plotted. ϕexp ranges greatly between in-phase and out-of-phase and ϕmax exists in the process where ϕexp changes abruptly. Then, ϕexp lies in the light yellow areas surrounded by ϕmax1 and ϕmax2 in the L ranges longer than the lengths when the sound pressure level peaks and occurs in the yellowish green areas surrounded by ϕmin1 and ϕmin2 on the other side. In other words, since vibroacoustic coupling is gradually weakened with increasing L after the peaks of Lp1 and Lp2, the acoustic mode involved in coupling shifts to that having the next order q.Figure9 shows the vibration levels Lv1 and Lv2 of plates 1 and 2 as functions of L and the accelerations α1 and α2 of plates 1 and 2 are also plotted to compare with the theoretical plate behavior. Lv1 is smaller than Lv2 in the ranges of 100 to 610 mm, 800 to 1200 mm, and 1300 to 1820 mm, so that Lv1 and Lv2 intersect at a number of L and the intersections take place around the lengths where Lpv peaks. The actual motion of plate 1 is almost suppressed by that of the vibrator since plate 1 is supported by the vibrator; hence, α1 is approximately constant over the entire L range. However, since the motion of plate 2 depends greatly on the behavior of the sound field, that is, the only excitation source for plate 2, α2 peaks at L=650, 1280, and 1880 mm and is suppressed in the other ranges of L, such as variations in Lp1 and Lp2. As a result, the relative relationship of α1 and α2 becomes opposite to that of Lv1 and Lv2 as shown in the colored regions. In the theoretical analysis, we confirm 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 sound tube having single closed and open ends at the lengths, 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 Lp1 and Lp2 in those middle ranges of lengths where the sound pressure levels peaked in Figure 6. This is derived from the difference between the tendencies of the above vibration level and acceleration with respect to both plates. If the experimental model completely emulated the theoretical model in the flexural displacement, such a discrepancy would not take place.Figure 9
Comparison between characteristics of theoretical vibration and experimental acceleration.In Figure10, the experimental phase difference ϕexp, at which Lp1 and Lp2 are maximized, is compared with the theoretical phase differences ϕmax1, ϕmax2, ϕmin1, and ϕmin2 when both plates are subjected to the same excitation force, as shown in Figure 8. The shifts in the (0,0,q) modes are also represented by the changing colors. ϕexp shifts between 0 and 180 deg with changing L and corresponds approximately with the ϕmax1 and ϕmax2 that behave uniformly in the vicinity 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.Figure 10
Comparison between theoretical and experimental results for phase difference when both end plates are excited by point force.


## 4.3. Electricity Generation Characteristics

In this section, we consider electricity generation by the plate vibrations coupled with the sound field. 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 relationship between the point force and flexural displacement at the excitation point. Figure11 shows the relationship between the electric power Pe and mechanical power Pm when the point force F1 ranges from 1 to 5 N. In this case, the cylinder is removed; the plate vibrations do not couple with the internal sound field. Although Pe is considerably smaller than Pm, their relationship is directly proportional. Here, the relationship between Pe and Pm is defined as
(28)Pem=PePm.Figure 11
Relationship of electric power and mechanical power.Figure12 shows variations in Pem with L; the powers are measured in the experimental apparatus via the cylinder shown in Figure 2(a) when one end plate is excited by the point force. Although Pem of plate 1 remains almost constant over the entire range of L, Pem of plate 2 increases greatly at L=615, 1275, and 1900 mm. It is natural that the relationship between Pem of plate 1 and 2 is derived from the behavior of α1 and α2 in Figure 9. Pem is considered as energy-harvesting efficiency. In an electricity generation by means of beam or plate vibrations with piezoelectric elements, coupling between the structural vibration and electric field, that is, electromechanical coupling should be considered to relate the in-plane stress to the applied electric field. Considering that the behaviors of Pem for plates 1 and 2 correspond to those of α1 and α2 and the values of Pem are substantially small, that is, the value of Pe is extremely small in comparison with that of Pm, we assume that the behavior of plate vibrations is hardly affected by electromechanical coupling in this study.Figure 12
Energy-harvesting efficiency as function of cylinder length.Moreover, to grasp the effect of vibroacoustic coupling on electricity generation, althoughPR is taken as the ratio of Pem measured with and without the cylinder, Pem measured with the cylinder is obtained from the total value Pe with respect to plates 1 and 2. Figure 13 shows variations in PR with L; the results of the excitation of both ends are also indicated to study the effect of the excitation method. PR for the excitation of one end is maximized as the sound pressure level peaks due to the promotion of vibroacoustic coupling. This is because the plate vibration on the nonexcitation side contributes strongly to the electricity generation, whereas their L ranges are limited to the narrow regions in comparison 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 excitation of both ends, there is not such a nonexcitation side that contributes greatly to the electricity generation, so that PR never reaches values as large as those of the excitation of one end, when the sound pressure level is maximized. In L ranges where vibroacoustic coupling is weakened, PR remains almost constant and is close to that of the excitation of one end. Under the situation of such a weakened vibroacoustic 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 Pe and mechanical power Pm. As a result, PR shifts almost constantly with L, no matter what the excitation method is.Figure 13
Effect of vibroacoustic coupling on energy harvesting for excitations of one end plate and both end plates.Hence, it follows that the excitation of one end is decidedly superior to that of both ends by electricity generation efficiency, according to this estimation method. These results are remarkable in the viewpoint where the acoustic energy from the sound radiation can be harvested through vibroacoustic coupling. However, the number or area of the piezoelectric element should be increased to improve the efficiencyPem, so that the effect of electromechanical coupling on the plate vibration must be taken into consideration in the theoretical procedure.


## 5. Conclusion

To apply vibroacoustic coupling to electricity generation, as a means of harvesting energy from vibration systems, coupling between plate vibrations and a sound field was investigated theoretically and experimentally for a cylindrical structure 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 electricity generating experiment. The present study focused on promoting the vibroacoustic coupling to increase the flexural displacements of the plates and the sound pressure level inside the cavity.As a result of the estimation of vibroacoustic coupling from various viewpoints, the theoretical study reveals that the closeness of eigenfrequencies and the similarity of modal shapes between the plate vibrations and sound field are indispensable for promoting coupling. The experimental results confirm that the theoretical estimation of increasing the flexural displacement and sound pressure level via the promotion of vibroacoustic coupling support the complicated acoustic characteristics deduced from the theoretical results. 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.



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*Source: 126035-2013-12-23.xml*