Figure 1 shows the analytical model that consists of a cavity with plates 1 and 2.

It is assumed that both plates are supported by translational and rotational springs. These springs are distributed at constant intervals and the support conditions are determined by the translational spring stiffness T and the rotational spring stiffness R. The plates whose radius and thickness are denoted by r_c and h_c, respectively, have a Young’s modulus E_c and a Poisson’s ratio ν_c. Piezoelectric elements are installed at the centers of the plates and have radius r_p, thickness h_p, Young’s modulus E_p, and Poisson’s ratio ν_p. Then an electrode plate that is made of brass is sandwiched between the above plate and piezoelectric element and has radius r_b, thickness h_b, Young’s modulus E_b and Poisson’s ratio ν_b. The suffixes c, p, and b denote the circular plate, piezoelectric element, and electrode plate. On the other hand, the cylinder has the same radius as that of the plates and the sound field is formed inside the cavity via coupling with the plate vibrations. The length L is varied widely because the resonance frequency depends on the length. The boundary conditions are regarded as structurally and acoustically rigid at the lateral wall except the plates. The representative coordinates are radius r, angle θ between the planes of the plates and the cross-sectional plane of the cavity, and distance z along the cylinder axis. Plate 1 is excited by the periodic point force F at distance r₁ and angle θ₁ and its natural frequency is employed as the excitation frequency.

Because the suffixes 1 and 2 indicate Plates 1 and 2, the flexural displacements of the Plates 1 and 2 are denoted by w_c1 and w_c2 and those of the piezoelectric elements installed on the plates are denoted by w_p1 and w_p2, respectively. Their displacements are expressed by Equation (1) as suitable trial functions and have the plate mode X n m s of Equation (2). The flexural displacements of the piezoelectric elements are identical to those of the plates, respectively, because it is assumed that the piezoelectric elements adhere completely to each circular plate through the electrode plate.

w c 1 = w p 1 = ∑ s = 0 1 ∑ n = 0 ∞ ∑ m = 0 ∞ X n m s A 1 n m s e j ( ω t + α 1 ) , w c 2 = w p 2 = ∑ s = 0 1 ∑ n = 0 ∞ ∑ m = 0 ∞ X n m s A 2 n m s e j ( ω t + α 2 ) , (1)

X n m s = sin ( n θ + s π / 2 ) ( r / r c ) m , (2)

where n and m are the circumferential and radial orders and s is symmetry index with respect to the vibration mode. A 1 n m s and A 2 n m s are coefficients to be determined and correspond to the amplitude of the flexural displacement. ω is the angular frequency of the harmonic point force acting on the plate and t is the elapsed time. α₁ and α₂ are the phases of the respective plate vibrations, here, α₁ is set to 0˚, and α₂ ranges from 0˚ to 180˚.

To explain modelling of only the piezoelectric part of Plate 1, this section takes note of its stress σ p 1 , strain ε_p1, electric displacement D₁, and electric field E₁ and their relationships are as follows:

{ σ p 1 D 1 } = [ E p E − e T e γ ε ] { ε p 1 E 1 } . (3)

E p E signifies Young’s modulus when a electric field is constant and γ ε indicates the dielectric constant when a strain is constant. Because electrical and mechanical characteristics of a piezoelectric element are related by the piezoelectric coupling coefficient e, these relationships are represented by Equation (3). The piezoelectric coupling coefficient becomes as follows:

e = d 31 E P E , (4)

where d₃₁ is the piezoelectric strain constant, and here, electric field occurs in the normal direction of the in-plane strain.

Then the electric field E₁, which occurs in the above direction of the piezoelectric element, is expressed as follows:

E 1 = Y n m v 1 = − R c q ˙ 1 = j ω B 1 n m s e j ( ω t + α 1 ) . (5)

v₁ is the voltage that occurs in the electric field. The electric potential across the piezoelectric element is constant since it is assumed that it does not reach the plate, so that Y_nm is defined as Equation (6). R_c is the overall resistance value in an electricity generation circuit. The magnitude ofthe electric charge q₁ depends on the coefficient B 1 n m s that is determined in this analysis as with A 1 n m s .

Y n m = { − 1 / h p         h c / 2 < z < h c / 2 + h b + h p , 0                         h c / 2 < z < h c / 2 + h b + h p , (6)

q 1 = B 1 n m s e j ( ω t + α 1 ) . (7)

To make the electro-mechanical equation concise, the elements M 1 n m m ′ s and K 1 n m m ′ s of the mass and stiffness matrices can be defined as

M p 1 n m m ′ s = ∫ V p 1 ρ p X n m s X n m ′ s d V p 1 , (8)

K p 1 n m m ′ s = ∫ V p 1 z 2 X n m s " E p E X n m ′ s " d V p 1 . (9)

The index m' has a transposed relation to a radial order m. The respective elements of electro-mechanical coupling and capacitance matrices are denoted by β₁ and C_p1 and they are defined as follows:

β 1 = − ∫ V p 1 z ρ p X n m s e Y n m ′ d V p 1 , (10)

C p 1 = ∫ V p 1 Y n m γ ε Y n m ′ s d V p 1 . (11)

Here, electro-mechanical coupling is based on the above several relationships, and then mechanical-acoustic coupling is also induced from the relationships between the vibrations of both plates and the sound field into the cylindrical cavity. Resultantly, this electricity generation phenomenon takes place under the situation of electro-mechanical-acoustic coupling, so that the plate vibrations, the sound field, and the electricity generation are governed by the following Equations (12) and (13), respectively [9] :

∑ m ′ = 0 ∞ [ K c 1 n m m ′ s ( 1 + j η c ) + K p 1 n m m ′ s ( 1 + j η p ) + K b 1 n m m ′ s ( 1 + j η b )   − ω 2 ( M c 1 n m m ′ s + M p 1 n m m ′ s + M b 1 n m m ′ s )   + r c F s n { T + ( m r c ) ( m ′ r c ) R } ] A 1 n m ′ s e j α 1 − ∑ m ′ = 0 ∞ β 1 v 1 − B 1 n m ′ s e j α 1 = F n m s e j α 1 − S 1 n m s , (12)

∑ m ′ = 0 ∞ [ K c 2 n m m ′ s ( 1 + j η c ) + K p 2 n m m ′ s ( 1 + j η p ) + K b 2 n m m ′ s ( 1 + j η b )   − ω 2 ( M c 2 n m m ′ s + M p 2 n m m ′ s + M b 2 n m m ′ s )   + r c F s n { T + ( m r c ) ( m ′ r c ) R } ] A 2 n m ′ s e j α 2 − ∑ m ′ = 0 ∞ β 2 v 2 − B 2 n m ′ s e j α 2 = S 2 n m s . (13)

K c 1 n m m ′ s , K b 1 n m m ′ s and, K b 2 n m m ′ s are stiffness matrix elements in the end plates and M c 1 n m m ′ s , M b 1 n m m ′ s and M c 2 n m m ′ s , M b 2 n m m ′ s are mass matrix elements in the electrode plates, respectively. These are elements of the symmetrical matrices, since the index m' has a transposed relation to m, as with M p 1 n m m ′ s and K p 1 n m m ′ s . η_c, η_p, and η_b signify the structural damping factors of the end plate, piezoelectric element, and electrode plate, respectively. Then F_sn is a coefficient that is determined by the indices n and s and F n m s is a load vector that is applied to Plate 1 as the point force. On the other hand, S 1 n m ′ s and S 2 n m ′ s matrices are the acoustic excitation terms, which signify coupling between each plate vibration and the internal sound field, and both matrices are including A 1 n m s and A 2 n m s . The details of F_sn and the respective elements F n m s , S 1 n m s , and S 2 n m s of the vector and matrices are as follows:

F s n = { π ,           at   n ≠ 0 , 0 ,             at   n = 0   and   s = 0 , 2 π ,     at   n = 0   and   s = 1 , (14)

F n m s = ∫ A 1 F δ ( r − r 1 ) δ ( θ − θ 1 ) X n m s d A 1 , (15)

S 1 n m s = ∫ A 1 P s X n m s d A 1 ,     S 2 n m s = ∫ A 2 P s X n m s d A 2 . (16)

Here, the point force is expressed by performing Equation (12) including the delta function δ on Plate 1, whose area is denoted by A₁. P_s is the sound pressure at an arbitrary point on both plates, and A₂ signifies the area of Plate 2.

On the other hand, the electricity generation behaviors of the piezoelectric elements are also governed by the following Equations (17) and (18), respectively:

∑ m ′ = 0 ∞ C p 1 − 1 β 1 A 1 n m ′ s = ∑ m ′ = 0 ∞ ( j ω R p + C p 1 − 1 ) B 1 n m ′ s , (17)

∑ m ′ = 0 ∞ C p 2 − 1 β 2 A 2 n m ′ s = ∑ m ′ = 0 ∞ ( j ω R p + C p 2 − 1 ) B 2 n m ′ s . (18)

The relationships between A 1 n m s and B 1 n m s and between A 2 n m s and B 2 n m s are applied to Equations (12) and (13), respectively, and these coefficients can be derived by solving the above simultaneous equations. However, Plate 1 is excited by the point force of its natural frequency and the behavior is assumed to be harmonic, so that e j ω t could be eliminated in Equations (12) and (13). Moreover, e j ( ω t + α 1 ) and e j ( ω t + α 2 ) could be also eliminated in Equations (17) and (18), respectively.

In the case of the electricity generation by means of only the vibration of Plate 1, i.e., in the condition of removing the cylinder and Plate 2 in Figure 1, A 1 n m s and B 1 n m s are obtained from solving simultaneously Equations (12) and (17). However, the acoustic excitation term S 1 n m ′ s is deleted in Equation (12) because of not taking into consideration mechanical-acoustic coupling. Then the flexural displacement w_c1 is calculated by substituting A 1 n m s for Equation (1) and the mechanical power P_m to be supplied to Plate 1 is obtained from the relationships between its flexural displacement and the force at the excited point. The voltage v₁ is calculated by substituting B 1 n m s for Equations (5) and (7), when the piezoelectric element generates electricity by the vibration of Plate 1. Although the electric power P_e is obtained from the product of its voltage v₁ and the electric current corresponding to v₁, the derivation of its electric current is explained in Section 4.2.

Plate 1 must receive the reaction force from the circumambient medium, being excited by the point force. In other words, this means that the radiation impedance of Plate 1 increases by means of the amount of the reaction. Such an increase is taken up in this section. To simplify the problem, it is assumed that the acoustic radiation is caused by a circular piston source, which is derived from Plate 1 excited by the velocity amplitude U₀. First, velocity potential df at Point P on the x-axis of Plate 1 is considered, and it is derived based on assuming that it is caused by plate vibration on an arbitrary minute area dA₁, as shown in Figure 1. Then, the velocity potential f is obtained from the surface integral in the entire area of Plate 1 as follows:

ϕ = U 0 e j ω t 2 π ∫ 0 2 π d ψ ∫ 0 H e − j k h d h , (19)

H = x cos ψ + r c 2 − x 2 sin 2 ψ . (20)

Here, k is the wavelength constant, and the segment H can be expressed as Equation (20), so that Equation (19) becomes the following equation.

ϕ = j U 0 e j ω t k π ∫ 0 π { e − j k ( x cos ψ + r c 2 − x 2 sin 2 ψ ) − 1 } d ψ , (21)

x cos ψ = μ ,     r c 2 − x 2 sin 2 ψ = χ . (22)

In addition, according to variables transformations of Equation (22), the integrand of Equation (21) is expanded in the power series of (μ + χ), and f becomes the following equation.

ϕ = U 0 e j ω t π ∫ 0 π [ { ( μ + χ ) 1 ! − k 2 ( μ + χ ) 3 3 ! + k 4 ( μ + χ ) 5 3 ! − ⋯ }     − j { k ( μ + χ ) 2 2 ! − k 3 ( μ + χ ) 4 4 ! + k 5 ( μ + χ ) 6 6 ! − ⋯ } ] d ψ , (23)

P s = j ω ρ a ϕ , (24)

where P s is the sound pressure, and ρ_a is the medium density.

Because f is calculated from the integral of each term in Equation (23), P_s at Point P is obtained by substituting it into Equation (24). Because this sound pressure behaves as a reaction force to Point P, the total reaction forces on the respective surfaces are induced from the surface integral of P_s in the entire area of Plate 1, as shown in Figure 1. For instance, F_ra is expressed by the following equation.

F r a e j ( ω t + α a ) = ρ s c U 0 e j ( ω t + α a ) [ π r c 2 { ( 2 k r c ) 2 2 × 4 − ( 2 k r c ) 4 2 × 4 2 × 6 + ( 2 k r c ) 6 2 × 4 2 × 6 2 × 8 − ⋯ }     + j k 2 { ( 2 k r c ) 3 1 × 3 − ( 2 k r c ) 5 1 × 3 2 × 5 + ( 2 k r c ) 7 1 × 3 2 × 5 2 × 7 − ⋯ } ] = π r c 2 ρ s c U 0 e j ( ω t + α a ) [ { 1 − J 1 ( 2 k r c ) k r c } + j S 1 ( 2 k r c ) k r c ] , (25)

where α_a is the phase and J₁ and S₁ are the Bessel function of the first kind and the Struve function, respectively. Because the sound fields are considered symmetric on both sides of Plate 1, F_rb can be expressed by Equation (25) where the phase α_b is adopted instead of α_a.

By taking up the analytical model in which the cylinder and Plate 2 are removed (i.e., the analytical model of only Plate 1), and reflecting the above reaction forces to the motion equation of Plate 1, Equation (12) can be rewritten as follows:

∑ m ′ = 0 ∞ [ K c 1 n m m ′ s ( 1 + j η c ) + K p 1 n m m ′ s ( 1 + j η p ) + K b 1 n m m ′ s ( 1 + j η b )   − ω 2 ( M c 1 n m m ′ s + M p 1 n m m ′ s + M b 1 n m m ′ s )   + r c F s n { T + ( m r c ) ( m ′ r c ) R } ] A 1 n m ′ s e j α 1 − ∑ m ′ = 0 ∞ β 1 v 1 − B 1 n m ′ s e j α 1 = F n m s e j α 1 − F r a e j α a + F r b e j α b . (26)

To estimate the influence of these reaction forces on electricity generation characteristics, the above equation is solved simultaneously with Equation (17).

The plate and piezoelectric element were the same as the experimental apparatus in the respective dimensions, and had the mechanical and electrical properties shown in Table 1. The plate radii r_c were constant at 153 mm, and the plate thickness h_c ranged from 1.5 to 4.5 mm. The support conditions of the plates, which had flexural rigidity D [ = E h c 3 / { 12 ( 1 − ν 2 ) } ] , are expressed by the nondimensional stiffness parameters T n ( = T r c 3 / D ) and R n ( = R r c / D ) . These values were identical for both plates. If R_n ranged from 0 to 10⁸ when T_n was 10⁸, the support condition could be assumed from a simple support to a clamped support. Although almost all conditions adopted T_n = 10⁸ and R_n = 10¹ to get closer to the experimental support condition, R_n changed in a part of the analysis. Plate 1 was subjected to the point force F that was set to 1 N, as with the excitation experiment. Then, the position was almost located at r₁/r_c = 0.4, whereas, r₁/r_c changed in a part of the analysis.

With respect to the electricity generation system of only Plate 1, which does not have the cylinder and Plate 2 in Figure 1, the theoretical voltage v₁ is indicated with changing the excitation frequency f in Figure 4, and it peaked at f = 279 Hz, which corresponds to the natural frequency of Plate 1. The experimental results are also exhibited in this figure, and they were measured by changing the length L_S of a stick exciting Plate 1. Both voltages had each peak taking place at f = 245 and 231 Hz when L_S was 13.6 and 75.0 mm, respectively. Because the exciting stick existed between the load cell and Plate 1, the influence of the additional mass on the natural frequency could not be avoided. Therefore, the peak frequency shifted to the lower-frequency region if the stick became long. To convert vibration energy into usable electric energy via the piezoelectric element, a suitable resistor that can efficiently consume the electric energy should be chosen. Figure 5 shows the experimental v₁ as a function of the resistance R_c. Because it can be ensured that v₁ is saturated with increasing R_c from approximately 100 kΩ, varying rapidly in the initial range of R_c, the resistance of 97.5 kΩ is valid.

Figure 6 shows the theoretical v₁ as a function of nondimensional excitation position r₁/r_c. Here, v₁ changes with r₁/r_c and it reaches the maximum value at approximately r₁/r_c = 0.45. Because it is natural that v₁ depends strongly on the vibration characteristics of Plate 1, the acceleration a₁ at the excitation position is also indicated in the same figure. Further, a₁ is maximized around r₁/r_c = 0.35, changing with r₁/r_c as with changes in v₁. Although it is remarkable that v₁ and a₁ are not identical in the excitation position, it is more significant that r₁/r_c = 0.4 is valid.

Because it was difficult to apply the additional mass to this theoretical procedure, a theoretical analysis was attempted by shifting the natural frequency of Plate 1, which was caused by changing R_n. Figure 7 shows the theoretical v₁ and a₁ as functions of R_n. R_n ranged from 1.222 to 137.4, and its variation corresponded to f = 200 − 320 Hz. Further, v₁ was maximized at approximately R_n = 1.7 and decreased with increasing R_n beyond the maximum value. Moreover, a₁ behaved with a tendency similar to v₁, whereas the maximum value appeared in the larger range of R_n in comparison with v₁. The theoretical and experimental situations in this study can be understood from these results.

Here, the power P_m supplied from the vibrator was calculated from the product between F and w_c1. Figure 8 shows the theoretical P_m as a function of the plate thickness h_c, which ranged from 1.5 to 4.5 mm. P_m decreased gradually with increasing h_c and was proportional to the square of w_c1, because w_c1 is proportional to F. However, the electric power P_e at the electricity generation is also exhibited in the same figure. Further, P_e had behavior similar to P_m, because v₁ had strong correlation to w_c1. Because the piezoelectric element was assumed to be incorporated in the electrical circuit of the energy-harvesting device and to be connected in series with the resistance for power consumption, it is natural that an electrical current was in-phase to the voltage. Therefore, P_e was also proportional to the square of v₁ and could be obtained from only the voltage because of Ohm’s law.

A close relationship between P_e and P_m was confirmed from these results. In addition, to consider this relationship specifically, the electricity generation efficiency P_em was defined as follows:

P e m = P e P m × 100 [ % ] . (27)

Figure 9 shows P_em derived from the above P_e and P_m as a function of h_c, so not only the theoretical but also the experimental P_em is indicated. In the case of only Plate 1, the theoretical P_em increased gradually with increasing h_c, whereas the experimental P_em presented a totally reverse tendency, where it decreased gradually with increasing h_c. In this manner, it is recognized that the discrepancy between the theoretical and experimental data expanded with increasing h_c.

In the analytical model of only Plate 1, the medium contacted on Plate 1 was not assumed (i.e., there is Plate 1 in vacuo). Therefore, the radiated sound field was not formed by the vibration of Plate 1, and the acoustic radiation power was almost spent in the electricity generation, so it is supposed that the above discrepancy took place, because the vibration of Plate 1 formed the sound field. As mentioned in Section 2.4, the radiation impedance Z_r consisted of the radiation resistance R_r of its real part and the radiation reactance X_r of its imaginary part. With respect to R_r, because the sound pressure P_s and the vibration velocity U₀e^jωt were in phase, R_r contributed the acoustic radiation of Plate 1. However, because P_s and U₀e^jωt that relate to X_r had a phase shift of 90˚, X_r did not take part in the acoustic radiation, whereas it functioned in Plate 1 as additional mass. Figure 10 shows variations in the radiation resistance and reactance with the plate thickness h_c; however, the normalized R_rn and X_rn were adopted as a substitute for R_r and X_r. They increased with h_c, with the result that the rate of vibration energy that was directly spent on the acoustic radiation increased with h_c. Then, because the additional mass meant that the surrounding medium was involved in the plate vibration, the vibration energy was consumed further with increasing h_c. Therefore, the discrepancy between the theoretical and experimental P_em in Figure 9 was caused by the acoustic radiation and additional mass, and expanded with increasing h_c because of the behaviors of the radiation resistance and reactance.

To estimate the effect of Z_r on the electricity generation characteristics, an analysis that involved the reactions F_ra and F_rb derived from Z_r was carried out using Equation (26). Here, F_ra and F_rb were in an out-of-phase state, not having a phase shift, so that they were mutually canceled and did not contribute to the electricity generation characteristics. Hence, the phase shift from being out of phase is denoted by Δα, and Figure 11 shows theoretical P_em as functions of Δα with changing h_c. Although P_em suddenly varied in the initial range of Δα, it converged on a constant value beyond the initial range. Then, the tendency shifted to the lager range of Δα with increasing h_c. Because the appropriate Δα could not be decided in this analysis, the theoretical P_em was estimated from Δα, which was measured by the acoustic experiment based on Figure 3, and it is also plotted by each circle in this figure. Further, Δα increased with h_c, with the result that P_em decreased rapidly with increasing h_c from 2.0 mm and converged in a greater range than 2.5 mm. These data are also reflected in Figure 9, whereas they are significantly less than the experimental data. This is because the circular piston vibration is assumed as the acoustic radiation source, and the effect of the reactions is overestimated. However, the theoretical and experimental results are similar and verified that the radiation impedance affects the electricity generation characteristics considerably.

According to the results in the previous section, the application of the radiation power is a useful way to improve electricity generation characteristics. The improvement was attempted by using mechanical-acoustic coupling, which was brought to this electricity generation system by adding the cylinder and Plate 2, as shown in Figure 1 and Figure 2. The thicknesses of Plates 1 and 2 were the same, and the point force was set to 1 N, as with that of only Plate 1. Figure 12 shows the sound pressure levels L_pv, L_p1, and L_p2 as functions of the cylinder length L. The theoretical level L_pv was averaged over the entire volume of the cavity and was maximized at a specific L. This was when the plate thickness h_c and the excitation frequency f were set to 3 mm and 280 Hz, respectively, and the phase α₂ of the vibration of Plate 2 ranged from 0˚ to 180˚. The theoretical level L_pv peaked at 610 mm. The peaks were caused by the promotion of mechanical-acoustic coupling between the plate vibration and acoustic mode. Then, the acoustic modes were the (0, 0, 1) mode, whose plane modal shape is similar to that of the plate vibration mode (0, 0). The experimental levels L_p1 and L_p2, measured near Plates 1 and 2, corresponded to L_pv in the occurrence of those peaks, so that it obviously promoted coupling at approximately the above length.

The electric power P_e and the supplied power P_m with coupling were the results under such a situation involving the (0, 0, 1) mode. Although the electricity generation efficiency P_em was obtained from these powers, P_e was the total electric power generated by both plate vibrations. Figure 13 shows the theoretical and experimental P_em as functions of h_c, and P_em without coupling (i.e., by means of only the plate vibration), and is also indicated to estimate the effect of coupling. The theoretical P_em increased gradually with increasing h_c, and was very close to that of only the plate vibration without coupling. The experimental P_em was somewhat less than the theoretical result, whereas it tended to vary similarly

to theoretical result and the difference from the theoretical result was reduced in comparison with that of only the plate vibration without coupling. This is because the vibration of Plate 1 contributed to the formation of the internal sound field and the vibration of Plate 2, via mechanical-acoustic coupling. Moreover, the expansion between the experimental values with increasing h_c demonstrates the validity of the application of mechanical-acoustic coupling to an electricity generation system with piezoelectric elements.