Energy Harvesting with Piezoelectric Element Using Vibroacoustic Coupling Phenomenon

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.

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 Figure 1. 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]: 𝛿𝐻=𝛿^𝑡1_𝑡0𝑇_𝑃−𝑈_𝑃−𝑈_𝑆+𝑊𝑑𝑡=0, (1) where 𝐻 is the Hamiltonian, 𝑇_𝑃 and 𝑈_𝑃 are, respectively, the kinetic and potential energy of each plate, and 𝑈_𝑆 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 displacements 𝑤₁ 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: 𝑤₁=^1∞_𝑠=0^∞_𝑛=0𝑚=0𝐵^𝑠_1𝑛𝑚𝑋^𝑠_𝑛𝑚𝑒^{𝑗(𝜔𝑡+𝜙1)},𝑤₂=^1∞_𝑠=0^∞_𝑛=0𝑚=0𝐵^𝑠_2𝑛𝑚𝑋^𝑠_𝑛𝑚𝑒^{𝑗(𝜔𝑡+𝜙2)},𝑋^𝑠_𝑛𝑚=sin𝑛𝜃+𝑠𝜋2𝑟𝑎^𝑚, (2)(3) where 𝑛, 𝑚, and 𝑠 are, respectively, the circumferential order, radial order, and symmetry index with respect to the plate vibration. 𝐵^𝑠_1𝑛𝑚 and 𝐵^𝑠_2𝑛𝑚 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 𝑇_𝑃,𝑈_𝑃,𝑈_𝑆, 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 𝑒^𝑗𝜔𝑡, so that 𝑒^𝑗𝜔𝑡 can be eliminated.

Consider the following: ^∞_𝑚′=0𝐾^𝑠_{1𝑛𝑚𝑚′}1+𝑗𝜂_𝑝−𝜔²𝑀^𝑠_{1𝑛𝑚𝑚′}+^∞_𝑚′=0𝑎𝐹_𝑠𝑛𝑇₁+𝑚𝑎𝑚^′𝑎𝑅₁𝐁^𝑠_1𝑛𝑚′𝑒^𝑗𝜙1=𝐅^𝑠_1𝑛𝑚−𝐏^𝑠_1𝑛𝑚,_∞𝑚′=0𝐾^𝑠_{2𝑛𝑚𝑚′}1+𝑗𝜂_𝑝−𝜔²𝑀^𝑠_{2𝑛𝑚𝑚′}+^∞_𝑚′=0𝑎𝐹_𝑠𝑛𝑇₂+𝑚𝑎𝑚^′𝑎𝑅₂𝐁^𝑠_2𝑛𝑚′𝑒^𝑗𝜙2=−𝐅^𝑠_2𝑛𝑚+𝐏^𝑠_2𝑛𝑚, (4) where 𝐾^𝑠_{1𝑛𝑚𝑚′}, 𝐾^𝑠_{2𝑛𝑚𝑚′} and 𝑀^𝑠_{1𝑛𝑚𝑚′}, 𝑀^𝑠_{2𝑛𝑚𝑚′} are elements of the symmetrical stiffness and mass matrices, respectively, because the index 𝑚^′ is the radial order (𝑚=𝑚^′). 𝜂_𝑝 is the structural damping factor of the plate, and 𝐹_𝑠𝑛 is a coefficient that is determined by the indices 𝑛 and 𝑠 and is expressed as [9] 𝐹_𝑠𝑛=𝜋,at𝑛≠0,0,at𝑛=0,𝑠=0,2𝜋,at𝑛=0,𝑠=1. (5)

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, 𝐹^𝑠_1𝑛𝑚=_𝐴1𝐹₁𝛿𝑟−𝑟₁𝛿𝜃−𝜃₁𝑋^𝑠_𝑛𝑚𝑑𝐴₁,𝐹^𝑠_2𝑛𝑚=_𝐴2𝐹₂𝛿𝑟−𝑟₂𝛿𝜃−𝜃₂𝑋^𝑠_𝑛𝑚𝑑𝐴₂,𝑃^𝑠_1𝑛𝑚=_𝐴1𝑃_𝑐𝑋^𝑠_𝑛𝑚𝑑𝐴₁,𝑃^𝑠_2𝑛𝑚=_𝐴2𝑃_𝑐𝑋^𝑠_𝑛𝑚𝑑𝐴₂. (6)(7) Here 𝛿 is the delta function associated with the point force on plates, 𝐴₁ and 𝐴₂ are the plate areas, and 𝑃_𝑐 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 eigenfunction 𝑌_𝑁 and the eigenvalue 𝑘_𝑁 corresponding to a cavity mode of order 𝑁: ∇²𝑌_𝑁+𝑘²_𝑁𝑌_𝑁=0,𝜕𝑌_𝑁𝜕𝐮_𝑆=0, (8)(9) 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 ∇²𝐺+𝐾²𝐺=−𝛿𝐩−𝐩₀=−𝛿𝑟−𝑟₀𝛿𝜃−𝜃₀𝛿𝑧−𝑧₀,𝜕𝐺𝜕𝐮_𝑆=0. (10) 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 𝑌_𝑁 of (8), which satisfies the same boundary conditions, we find that 𝐺𝐩,𝐩₀=^∞_𝑁=1𝑌_𝑁(𝐩)𝑌_𝑁𝐩₀𝑉_𝑐𝑀_𝑁𝑘²_𝑁−𝐾²_𝑉𝑐,𝑌_𝑁(𝐩)𝑌_𝑀(𝐩)𝑑𝑉_𝑐=𝑉_𝑐𝑀_𝑁𝛿_𝑁𝑀=𝑉0at𝑁≠𝑀,_𝑐𝑀_𝑁at𝑁=𝑀. (11)(12) The dimensionless factor 𝑀_𝑁 is the mean value of 𝑌²_𝑁 averaged over the cavity volume 𝑉_𝑐, and 𝛿_𝑁𝑀 is the Kronecker delta.

Because there is no source and 𝜕𝐺/𝜕𝐮=0 on 𝑆, the spatial factor 𝑃_𝑐(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: 𝑃_𝑐(𝐩)=−_𝑆𝐺𝐩,𝐩₀𝜕𝑃_𝑐𝐩₀𝜕𝐮₀𝑑𝑆₀, (13) 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]. 𝑃_𝑐 can also be expressed as [9, 10] 𝑃_𝑐=𝜌_𝑐𝑐^2∞_𝑁=1𝑃_𝑁𝑌_𝑁𝑀_𝑁, (14) where 𝜌_𝑐 is the fluid density in the cavity, and 𝑃_𝑁 is the pressure coefficient to be determined.

In this investigation, the acoustic modal shape 𝑌^𝑠_𝑛𝑝𝑞 and angular resonance frequency 𝜔_𝑛𝑝𝑞 in the cavity (where the indices 𝑛, 𝑝, and 𝑞 indicate the circumferential, radial and longitudinal orders, resp.) are defined as 𝑌^𝑠_𝑛𝑝𝑞=sin𝑛𝜃+𝑠𝜋2𝐽_𝑛𝜆_𝑛𝑝𝑟cos𝑞𝜋𝐿𝑧,𝜔_𝑛𝑝𝑞=𝑐𝜆²_𝑛𝑝+𝑞𝜋𝐿^21/2, (15) where 𝐽_𝑛 is the 𝑛th-order Bessel function, and 𝜆_𝑛𝑝 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: 𝜕𝑃_𝑐𝜕𝐮_𝑧=0=𝜌_𝑐𝜔²𝑤₁,𝜕𝑃_𝑐𝜕𝐮_𝑧=𝐿=−𝜌_𝑐𝜔²𝑤₂, (16) where 𝜕𝑃_𝑐/𝜕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 𝑃_𝑐 becomes 𝑃_𝑐=−_𝐴1𝐺𝜌_𝑐𝜔²𝑤₁𝑑𝐴₁+_𝐴2𝐺𝜌_𝑐𝜔²𝑤₂𝑑𝐴₂. (17) 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), 𝑃_𝑐 can also be expressed as 𝑃_𝑐=𝜌_𝑐𝑐^2∞∞_𝑝=1𝑞=0𝑃^𝑠_𝑛𝑝𝑞𝑌^𝑠_𝑛𝑝𝑞𝑀^𝑠_𝑛𝑝𝑞. (18)

The equation relating (17) and (18) is obtained by applying Green’s function of (11) to an arbitrary acoustic mode (𝑛,𝑝,𝑞) as 𝜔²_𝑛𝑝𝑞−𝜔²𝑃^𝑠_𝑛𝑝𝑞𝜔=−²𝑉_𝑐𝐴1𝑌^𝑠_𝑛𝑝𝑞𝑤₁𝑑𝐴₁+_𝐴2𝑌^𝑠_𝑛𝑝𝑞𝑤₂𝑑𝐴₂. (19) Here, substituting (2) for 𝑤₁ and 𝑤₂ and considering a modal damping factor 𝜂_𝑐, (19) can be rewritten as 𝜔²_𝑛𝑝𝑞+𝑗𝜂_𝑐𝜔_𝑛𝑝𝑞𝜔−𝜔²𝑃^𝑠_𝑛𝑝𝑞=𝐴𝜔²𝑉_𝑐−^∞_𝑚=0𝐼₁𝐵^𝑠_1𝑛𝑚𝑒^𝑗𝜙1+^∞_𝑚=0𝐼₂𝐵^𝑠_2𝑛𝑚𝑒^𝑗𝜙2,𝐼₁=1𝐴_𝐴1𝑋^𝑠_𝑛𝑚𝑌^𝑠_𝑛𝑝𝑞𝑑𝐴₁,𝐼₂=1𝐴_𝐴2𝑋^𝑠_𝑛𝑚𝑌^𝑠_𝑛𝑝𝑞𝑑𝐴₂, (20)(21) where 𝐴 is the total surface area of the plates, 𝐼₁ and 𝐼₂ are the spatial coupling coefficients. Moreover, substituting (18) for 𝑃_𝑐 and applying 𝐼₁ and 𝐼₂ to the integrals in (7), the acoustic excitation terms 𝑃^𝑠_1𝑛𝑚 and 𝑃^𝑠_2𝑛𝑚 can be expressed with respect to an arbitrary vibration mode (𝑛,𝑚) as 𝑃^𝑠_1𝑛𝑚=𝜌_𝑐𝑐²𝐴^∞∞_𝑝=1𝑞=0𝐼₁𝑃^𝑠_𝑛𝑝𝑞𝑀^𝑠_𝑛𝑝𝑞,𝑃^𝑠_2𝑛𝑚=𝜌_𝑐𝑐²𝐴^∞∞_𝑝=1𝑞=0𝐼₂𝑃^𝑠_𝑛𝑝𝑞𝑀^𝑠_𝑛𝑝𝑞. (22)

Finally, replacing 𝑃^𝑠_𝑛𝑝𝑞 in (22) with those in (20) and then inserting them in (4), we can complete the coupling equations, whose right-hand sides are 𝐅^𝑠_1𝑛𝑚−𝐏^𝑠_1𝑛𝑚=𝐅^𝑠_1𝑛𝑚+𝜌_𝑐𝑐²𝜔²𝐴²𝑉_𝑐×^∞_𝑚′^∞₌₀^∞_𝑝=1𝑞=0𝐼₁𝐼₁𝐵^𝑠_1𝑛𝑚′𝑒^𝑗𝜙1−𝐼₂𝐵^𝑠_2𝑛𝑚′𝑒^𝑗𝜙2𝑀^𝑠_𝑛𝑝𝑞𝜔²_𝑛𝑝𝑞+𝑗𝜂_𝑐𝜔_𝑛𝑝𝑞𝜔−𝜔²,−𝐅^𝑠_2𝑛𝑚−𝐏^𝑠_2𝑛𝑚=−𝐅^𝑠_2𝑛𝑚+𝜌_𝑐𝑐²𝜔²𝐴²𝑉_𝑐×^∞_𝑚′^∞₌₀^∞_𝑝=1𝑞=0𝐼₂𝐼₁𝐵^𝑠_1𝑛𝑚′𝑒^𝑗𝜙1−𝐼₂𝐵^𝑠_2𝑛𝑚′𝑒^𝑗𝜙2𝑀^𝑠_𝑛𝑝𝑞𝜔²_𝑛𝑝𝑞+𝑗𝜂_𝑐𝜔_𝑛𝑝𝑞𝜔−𝜔². (23)

On the right-hand sides, the second terms show the acoustic excitation for plates 1 and 2, respectively. The acoustic excitation terms have both 𝐼₁ 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: 𝜂_𝑝=𝜂_𝑐=0.01 [9, 10]. Since 𝐵^𝑠_1𝑛𝑚 and 𝐵^𝑠_2𝑛𝑚 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 displacements 𝑤₁ and 𝑤₂ and the sound pressure 𝑃_𝑐 are, respectively, obtained from (2) and (18) by employing 𝐵^𝑠_1𝑛𝑚 and 𝐵^𝑠_2𝑛𝑚 determined above. However, the procedure to calculate 𝑃_𝑐 involves substituting 𝑃^𝑠_𝑛𝑝𝑞 in (20) for that in (18). Solved from (24), into which 𝑤₁, 𝑤₂, and 𝑃_𝑐 are inserted, over the respective entire regions, the quadric velocities 𝑉₁ and 𝑉₂ and the sound pressure 𝑃_𝜈 are estimated by the logarithmic values 𝐿_𝑣1, 𝐿_𝑣2, and 𝐿_𝑝𝑣 relative to 𝑉₀=2.5×10⁻¹⁵ m²/s² and 𝑃₀=4×10⁻¹⁰ N²/m⁴, respectively, as shown in (25).

Consider the following: 𝑉₁=𝜔²2𝐴_1𝐴1𝑤₁𝑤^∗₁𝑑𝐴₁,𝑉₂=𝜔²2𝐴_2𝐴2𝑤₂𝑤^∗₂𝑑𝐴₂,𝑃_𝜈=12𝑉_𝑐𝑉𝑐𝑃_𝑐𝑃^∗_𝑐𝑑𝑉_𝑐,𝐿_𝑣1𝑉=10log¹𝑉₀,𝐿_𝑣2𝑉=10log²𝑉₀,𝐿_𝑝𝑣𝑃=10log^𝑣𝑃₀, (24)(25) where 𝑤^∗₁, 𝑤^∗₂, and 𝑃^∗_𝑐 are the respective conjugate components.

3. Experimental Apparatus and Method

Figure 2(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.

(a)

(b)

(c)

Figure 2 

Configuration of experimental apparatus: (a) Measurement system; (b) Installation state of piezoelectric element; and (c) Electrical circuit of energy-harvesting device.

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 Figures 2(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 𝑅_𝑣, 𝑅_𝑖, and 𝑅_𝑐, as shown in Figure 2(c). 𝑅_𝑣 and 𝑅_𝑖 are the resistances of the voltmeter and ammeter built-in the wattmeter and are 2 MΩ and 2 mΩ, respectively, while 𝑅_𝑐 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 modulus 𝐸 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 𝑇_𝑛(=𝑇₁𝑎³/𝐷=𝑇₂𝑎³/𝐷) and 𝑅_𝑛(=𝑅₁𝑎/𝐷=𝑅₂𝑎/𝐷). These values are identical for both plates. If 𝑅_𝑛 ranges from 0 to 10⁸ when 𝑇_𝑛 is 10⁸, the support condition can be assumed from a simple support to a clamped support. The actual condition adopts 𝑇_𝑛=10⁸ and 𝑅_𝑛=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 frequency 𝑓_𝑛𝑚 corresponding to the (𝑛,𝑚) mode and the resonance frequency 𝑓_𝑛𝑝𝑞 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: 𝜙=𝜙₂−𝜙₁. (26)

Figure 3 shows the variations in 𝐿_𝑝𝑣 with 𝐿, when only plate 1 is excited and 𝜙 is arbitrarily set to 0, 10, and 90 deg. 𝐿_𝑝𝑣 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 𝐿_𝑝𝑣 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 𝐶_𝐸 can be defined as the ratio of the acoustic energy 𝐸_𝑛𝑝𝑞 stored in the specific (𝑛,𝑝,𝑞) mode to the total acoustic energy 𝐸_all of the entire sound field: 𝐶_𝐸=𝐸_𝑛𝑝𝑞𝐸_all. (27)

Figure 3 

Variation in sound pressure level with cylinder length when phase difference changes.

Figures 4(a) and 4(b) show 𝐶_𝐸 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, 𝐶_𝐸 is relatively large when 𝐿 is short. The value of 𝐶_𝐸 for the (0,0,0) mode decreases as 𝐿 increases because of the increase in the value of 𝐶_𝐸 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, 𝐶_𝐸 decreases relatively to 𝐶_𝐸 of the (0,0,0) mode because the sound field is affected by other modes. At 𝜙=90 deg, 𝐶_𝐸 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 𝐿_𝑝𝑣 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 𝐿_𝑝𝑣 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 𝐿_𝑝𝑣 is maximum are denoted by 𝜙_max.

(a)

(b)

Figure 4 

Contribution of acoustic mode to sound field at (a) 𝜙=0 [deg] and (b) 𝜙=90 [deg].

In Figure 5, the variations in 𝜙_max with 𝐿 and the values of 𝜙_max that maximize 𝐿_𝑝𝑣 are plotted by a line and circles, respectively. 𝜙_max is approximately 87 deg at 𝐿=100 mm and decreases gradually with increasing 𝐿 up to approximately 𝐿=460 mm, where 𝜙_max suddenly increases to over 90 deg and then decreases again with increasing 𝐿. This behavior of 𝜙_max is repeated in a similar manner as 𝐿 increases to 𝐿=2000 mm. Peaks in 𝐿_𝑝𝑣 indicate that vibroacoustic coupling between the plate vibrations and sound field is promoted at approximately 90 deg. 𝑓_𝑛𝑚 and 𝑓_𝑛𝑝𝑞 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 𝐿_𝑝𝑣 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 𝐿_𝑝𝑣 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 𝐶_𝐸 is maximized in Figure 4(b).

Figure 5 

Variations in phase difference with cylinder length.

Figure 6 shows the sound pressure levels 𝐿_𝑝1 and 𝐿_𝑝2, which are measured near plates 1 and 2, respectively, as functions of 𝐿. The theoretical level 𝐿_𝑝𝑣, 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. 𝐿_𝑝1 and 𝐿_𝑝2 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 𝐿_𝑝1 are remarkable and their differences expand around 950 and 1550 mm. 𝐿_𝑝𝑣 also shows peaks at 610, 1230, and 1840 mm and corresponds approximately with the above lengths where 𝐿_𝑝1 and 𝐿_𝑝2 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. Figure 7 shows the variations in 𝐿_𝑝𝑣 corresponding to 𝜙_max and the variations in 𝐿_𝑝1 and 𝐿_𝑝2 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 𝐿_𝑝𝑣 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 𝐿_𝑝1 and 𝐿_𝑝2 increase greatly at 625, 1250, and 1850 mm. However, 𝐿_𝑝1 and 𝐿_𝑝2 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 displacements 𝑤₁ and 𝑤₂ is also significant in studying the effect of the plate vibrations on the sound field. Here the phase differences are denoted as 𝜙_max1 and 𝜙_max2 when 𝑤₁ and 𝑤₂ are maximized, while they are denoted as 𝜙_min1 and 𝜙_min2 when 𝑤₁ and 𝑤₂ are minimized. Figure 8 shows 𝜙_max1, 𝜙_max2, 𝜙_min1, and 𝜙_min2 as functions of 𝐿 when only plate 1 is excited. 𝜙_max1 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, 𝜙_max1 increases gradually with 𝐿 and returns to 180 deg at 𝐿=1220 mm, increasing somewhat abruptly near 𝐿=970 mm. Beyond 𝐿=1220 mm, 𝜙_max1 is again constant at 180 deg up to 𝐿=1570 mm, and this behavior is repeated as 𝐿 increases to 𝐿=2000 mm. 𝜙_max2 exhibits gradual and abrupt changes similar but alternate to 𝜙_max1. For example, when 𝐿 increases, a gradual decrease occurs in 𝜙_max2 between 𝐿=100 and 620 mm, and an abrupt increase occurs near 𝐿=970 mm. Both 𝜙_max1 and 𝜙_max2 shift between 0 and 180 deg with changing 𝐿 and intersect at approximately 90 deg and near the length at which 𝐿_𝑝𝑣 peaked in Figure 6. 𝜙_min1 and 𝜙_min2 behave exactly alike but opposite to 𝜙_max1 and 𝜙_max2.

Figure 8 

Comparison between theoretical and experimental results for phase difference.

In Figure 8, the theoretical results for 𝜙_max where 𝐿_𝑝𝑣 peaks and the experimental results for 𝜙_exp as 𝐿_𝑝1 and 𝐿_𝑝2 are maximized at each 𝐿 are also plotted. 𝜙_exp ranges greatly between in-phase and out-of-phase and 𝜙_max exists in the process where 𝜙_exp changes abruptly. Then, 𝜙_exp lies in the light yellow areas surrounded by 𝜙_max1 and 𝜙_max2 in the 𝐿 ranges longer than the lengths when the sound pressure level peaks and occurs in the yellowish green areas surrounded by 𝜙_min1 and 𝜙_min2 on the other side. In other words, since vibroacoustic coupling is gradually weakened with increasing 𝐿 after the peaks of 𝐿_𝑝1 and 𝐿_𝑝2, the acoustic mode involved in coupling shifts to that having the next order 𝑞.

Figure 9 shows the vibration levels 𝐿_𝑣1 and 𝐿_𝑣2 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. 𝐿_𝑣1 is smaller than 𝐿_𝑣2 in the ranges of 100 to 610 mm, 800 to 1200 mm, and 1300 to 1820 mm, so that 𝐿_𝑣1 and 𝐿_𝑣2 intersect at a number of 𝐿 and the intersections take place around the lengths where 𝐿_𝑝𝑣 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 𝐿_𝑝1 and 𝐿_𝑝2. As a result, the relative relationship of 𝛼₁ and 𝛼₂ becomes opposite to that of 𝐿_𝑣1 and 𝐿_𝑣2 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 𝐿_𝑝1 and 𝐿_𝑝2 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 Figure 10, the experimental phase difference 𝜙_exp, at which 𝐿_𝑝1 and 𝐿_𝑝2 are maximized, is compared with the theoretical phase differences 𝜙_max1, 𝜙_max2, 𝜙_min1, and 𝜙_min2 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. 𝜙_exp shifts between 0 and 180 deg with changing 𝐿 and corresponds approximately with the 𝜙_max1 and 𝜙_max2 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. Figure 11 shows the relationship between the electric power 𝑃_𝑒 and mechanical power 𝑃_𝑚 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 𝑃_𝑒 is considerably smaller than 𝑃_𝑚, their relationship is directly proportional. Here, the relationship between 𝑃_𝑒 and 𝑃_𝑚 is defined as 𝑃_𝑒𝑚=𝑃_𝑒𝑃_𝑚. (28)

Figure 11 

Relationship of electric power and mechanical power.

Figure 12 shows variations in 𝑃_𝑒𝑚 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 𝑃_𝑒𝑚 of plate 1 remains almost constant over the entire range of 𝐿, 𝑃_𝑒𝑚 of plate 2 increases greatly at 𝐿=615, 1275, and 1900 mm. It is natural that the relationship between 𝑃_𝑒𝑚 of plate 1 and 2 is derived from the behavior of 𝛼₁ and 𝛼₂ in Figure 9. 𝑃_𝑒𝑚 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 𝑃_𝑒𝑚 for plates 1 and 2 correspond to those of 𝛼₁ and 𝛼₂ and the values of 𝑃_𝑒𝑚 are substantially small, that is, the value of 𝑃_𝑒 is extremely small in comparison with that of 𝑃_𝑚, 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, although 𝑃_𝑅 is taken as the ratio of 𝑃_𝑒𝑚 measured with and without the cylinder, 𝑃_𝑒𝑚 measured with the cylinder is obtained from the total value 𝑃_𝑒 with respect to plates 1 and 2. Figure 13 shows variations in 𝑃_𝑅 with 𝐿; the results of the excitation of both ends are also indicated to study the effect of the excitation method. 𝑃_𝑅 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 𝑃_𝑅 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, 𝑃_𝑅 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 𝑃_𝑒 and mechanical power 𝑃_𝑚. As a result, 𝑃_𝑅 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 efficiency 𝑃_𝑒𝑚, 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.