Optimizing Energy Conversion in a Piezo Disk Using a Controlled Supply of Electrical Load

Abstract

Piezoceramic products are actively used in modern technical devices and appliances. Disk piezoelectric devices are widely used in elements of information systems: in wireless communication, elements of satellite communication, global positioning systems. Among such devices, microwave piezo motors on traveling waves of various types are distinguished, including a motor built on an electromechanical disk converter, which performs non-axisymmetric oscillations under the action of a harmonic electric load. In this study, we perform separation of the electrode coating of the plate with thin diametrical dielectric sections and introduce a time-varying electric potential difference of different amplitude or phase to individual parts of the plate. We study the problem of non-axisymmetric planar oscillations of a piezoceramic disk with thickness polarization and the problem of optimization of the electromechanical coupling coefficient (EMCO) value as a function of an infinite number of parameters in space $l_2$ of square summable sequences.

1. Introduction

Different methods and approaches are often used when setting up and solving applied natural or socio-economic problems. Many of these methods are based on a pre-defined set of axioms and laws. Based on this set, a mathematical model of the applied problem and methods of its solution are formulated. On the other hand, an extreme approach can be used to formulate the problem model. For example, in mechanics, Lagrange equations of the second kind can be obtained both by classical methods (using Newton’s three laws and axioms) and using variational principles of mechanics. Applying the methods of variational calculus, we obtain Euler’s differential equation as in the classical problem of variational calculus, which is characterized by the replacement of the Lagrange function with the integral function of the functional, which has certain restrictions formulated in Euler’s theorem.

The maximum principle is the main method of solving the problem of optimal control. Its formulation includes construction of a system of differential equations for phase variables and an additional system of functions and search for the maximum of the Hamiltonian at fixed values of its arguments in addition to the control functions. Canonical equations of material system’s motion contain the same system of differential equations as in the maximum principle. We replace the auxiliary vector function with a system of generalized impulses leading to transformation of the Hamiltonian (in case of holonomic stationary constraints) into the full mechanical energy of the material system.

From a physical standpoint, we can say that all physical processes have both physical and extreme nature. This is confirmed by the presence of a large number of variational principles that describe phenomena of a different nature closely related to each other.

For example, Maupertuy–Euler’s variational principle, which describes the movement of a material point in potential force field, or Fermat’s variational principle, which describes the movement of a light beam, state that trajectories are the same and velocities are reversed in magnitude.

In most applied problems, the task of selecting all or part of the input parameters to optimize the required output is not set. On the other hand, there is a number of classes of problems in various fields of science, in which the achievement of optimal results is required to improve the quality of the processes.

The class of such problems includes problems of the mechanics of bound fields: magnetoelasticity, thermoelasticity (thermodynamics), and electroelasticity, in which external actions are most often not external mechanical forces. External actions in these problems use influences of a different physical nature. Also, these problems consider structural elements made of materials in which magnetoelastic or electroelastic phenomena are observed. It is necessary to obtain a given result that is optimal.

For the problems of electroelasticity, the main issue is to determine the coupled electromechanical field and to determine the ratio of the received mechanical energy capable of inversion to the supplied electrical energy. Meson’s empirical formula [1] is used for simple one-dimensional problems. This formula gives approximation of the value of the measure of efficiency of energy conversion at resonant frequencies of oscillations of the piezo element—the coefficient of electromechanical coupling (EMCO). Still, it does not provide an answer at other frequencies.

There is another approach to determine EMCO—the so-called energy criterion [2]. EMCO is determined by definition, directly calculating the ratio of energies. The question arises about the form of the function of the electric load, at which the value of EMCO reaches its maximum at a certain frequency of oscillations. The formulation and solution of the problem requires the use of the physical and extreme approaches.

An example of solving such a problem for a piezoceramic rod is the work in [3]. In this paper, the possibility of increasing the efficiency of energy conversion at certain resonance frequencies is studied along with the possibility of turning off some frequencies from the working spectrum of the element with the help of a controlled energy input. Mathematically, we have the problem of determining the largest value of a non-negative function and its zero on a segment.

A brief overview of model situations: a piezoceramic body with an internal elliptical crack, a piezoceramic body with an external elliptical crack, and a piezoceramic body with a spherical cavity are investigated in [4]. The dependencies of stresses and electrical displacements on the shape of the cavity are obtained for the cases of electrical and solely mechanical loads.

Paper [5] considered the simulation of a Rosen-type ring piezoelectric transformer. The developed model is based on the Hamiltonian approach, which allows obtention of the main parameters and estimation of the efficiency for the first radial vibrational modes. The methodology and presentation of the final results are described in detail. The input bandwidth and the distribution of the electric potential on the surface of the secondary part are compared with numerical and experimental data.

The electromechanical model of the disk piezoelectric transformer (PT) is studied in works [6,7]. To analyze models of this type, the vibration characteristics of the piezoelectric disc with free boundary conditions are preliminarily analyzed. According to the results of the vibration analysis of the piezoelectric disk, the operating frequency and vibration mode of the PT are selected. Then, the electromechanical equations of motion for the PT can be obtained based on Hamilton’s principle. Hamilton’s principle can be used to model the coupled electromechanical system for a transformer.

Papers [8,9] experimentally deal with the radial in-plane vibration characteristics of disk-shaped piezoelectric transducers. The radial in-plane motion, which is induced due to Poisson’s ratio in the piezoelectric disk polarized in the thickness direction, is measured using an in-plane laser vibrometer, and the natural frequencies are measured using an impedance analyzer. The experimental results are compared with theoretical predictions obtained by simplified theoretical and finite-element analyses. It appears that the fundamental mode of a piezoelectric disk transducer is a radial mode, and its radial displacement distribution from the center to the perimeter is not monotonic but shows a maximum slightly apart from the perimeter.

Actuators based on piezoelectric materials have gained high popularity in the development of microrobotic systems due to their high throughput, high resolution, and high force density. However, one of the main disadvantages is the low relative stroke. One of the methods of optimizing the topology, which is used to maximize the drive stroke, is studied in [10]. Along with the optimization of the material density, the optimization of the polarity of the electrodes is considered. This approach allows combination of both material expansion and compression to increase the output volume of the actuator.

Optimization of the topology of a node consisting of a piezoelectric layer attached to a plate with a support is considered in [11]. The object of optimization is the piezoelectric layer. The SIMP (Solid Isotropic Material with Penalization) method with forced oscillations by harmonic electrical excitation is considered and applied. The maximum dynamic displacement is reached. It is proven that the considered objective function can be used under certain boundary conditions to optimize sound radiation. The vibration models obtained as a result of the optimization are analyzed in comparison with modes from the analysis of eigenvalues. Multi-frequency optimization is achieved using adaptive weighted sums. As a second optimization criterion, a flat frequency response is included in the optimization process.

The application of model situations for the optimization of technological processes related to energy management and conversion is considered in works [12,13,14,15]. In works [16,17], methods and solutions of mathematical models described by the corresponding systems of equations are investigated.

Disk piezoelectric devices are widely used in elements of information systems: in wireless communication, elements of satellite communication, global positioning systems. Among such devices, microwave piezo motors on traveling waves of various types are distinguished, including a motor built on an electromechanical disk converter, which performs non-axisymmetric oscillations under the action of a harmonic electric load. In this study, we perform separation of the electrode coating of the plate with thin diametrical dielectric sections and introduce a time-varying electric potential difference of different amplitude or phase to individual parts of the plate. We study the problem of non-axisymmetric planar oscillations of a piezoceramic disk with thickness polarization and the problem of optimization of the EMCO value as a function of an infinite number of parameters in space $l_2$ of square summable sequences.

2. Non-Axisymmetric Planar Oscillations of a Piezoceramic Thin Disk

The problem of non-axisymmetric planar oscillations of a piezoceramic thin disk with thickness polarization under the action of a time-harmonic electric load is considered. Time-harmonic electric load given by the function of the electric field strength is constant in thickness [2] and in the direction of the radial coordinate:

$$E_x=E\left(\phi \right){exp}^{i\omega t}.$$

(1)

Plate dimensions: radius $r=a$, thickness—$2h$.

The front surfaces of the disc have an electrode coating, which ensures the formal fulfillment of the boundary conditions for the electric potential function:

$$\Psi {\mid }_{z\pm h}=\pm V_0\left(\phi \right){exp}^{i\omega t}.$$

(2)

Figure 1 provides visual representation of the problem. Here, $V_k{exp}^{i\omega t},k=1,\cdots ,n,$ are harmonic time difference of the potentials applied to the upper and lower parts of the electrode coating of the kth segment of the piezoceramic disk, $V_k$—its amplitude.

The vector differential equation of motion, which describes the planar oscillations of thin piezoceramic plates with thickness polarization, has the form

$$\frac{2}{1-\nu }\mathrm{grad}\left(\mathrm{div}\overrightarrow{u}\right)-2d_{31}\frac{1+\nu }{1-\nu }\mathrm{grad}{E}_z-\mathrm{rot}\left(\mathrm{rot}\overrightarrow{u}\right)=2\rho s_{11}^E(1+\nu )\frac{{\partial }^2\overrightarrow{u}}{\partial t^2},$$

(3)

where $\nu$—Poisson’s ratio; $s_{11}^E$–-elastic compliance; $\rho$–-density of piezoceramics; $d_{31}$—piezoelectric constant of connection between electric and mechanical fields.

The mechanical boundary conditions of the free edge recorded for the normal and shear components of the stress tensor must be fulfilled:

$${\sigma }_r{\mid }_{r=a}=0;{\tau }_{r\phi }{\mid }_{r=a}=0.$$

(4)

We suppose the load changes over time in the form of amplitude and time multipliers. Then, the displacement vector can be written in the following form:

$$\overrightarrow{u}={\overrightarrow{e}}_r{u}_r+{\overrightarrow{e}}_{\phi }{u}_{\phi },\overrightarrow{u}=({\overrightarrow{e}}_r{\overrightarrow{\tau }}_r(r,\phi )+{\overrightarrow{e}}_{\phi }{\overrightarrow{\tau }}_{\phi }(r,\phi )){exp}^{i\omega t}.$$

(5)

Further considerations should be continued by choosing the form of the electric load amplitude function with respect to the coordinate grid lines. It means the parity or oddity of this function on the circular coordinate. This function can have a general form and can have a finite number of discontinuities of the first kind. The last fact is connected from a technical point of view with the need to divide the electrode coating into parts with thin dielectric gaps to obtain the required form of the electric load function. From a technical point of view, we actually have a piecewise constant function in the circumferential coordinate.

In a general case, the existence of the Fourier distribution in the circular coordinate is required for function

$$E_z=\left(a_0+\sum _{k=1}^{\infty }{a}_{k}cosk\phi +b_{k}sink\phi \right){exp}^{i\omega t}.$$

(6)

Here, $a_0$, $a_k$, $b_k$—Fourier coefficients for function $E\left(\phi \right)$.

Then, according to the choice of a coordinate system, we find the solution of Problems (3)–(4) in the form

$$\begin{array}{c}{u}_r=\left(u_0\left(r\right)+\sum _{k=1}^{\infty }(u_{k}cosk\phi +u_k^{1}sink\phi )\right)e^{i\omega t},\hfill \\ u_{\phi }=\left({\nu }_0\left(r\right)+\sum _{k=1}^{\infty }({\nu }_{k}sink\phi +{\nu }_k^{1}cosk\phi )\right)e^{i\omega t}.\hfill \end{array}$$

(7)

In what follows, we limit ourselves to the case of an even function. The calculations are similar for an odd function. For a function of the general form, it is possible to write a superposition of solutions.

Then, we obtain

$$u_r=\left(u_0\left(r\right)+\sum _{k=1}^{\infty }{u}_k\left(r\right)cosk\phi \right),u_{\phi }=\sum _{k=1}^{\infty }{\nu }_k\left(r\right)sink\phi .$$

(8)

Let us proceed to the solution of Problems (3)–(4). We consider the form of differential operators of a vector field in cylindrical coordinates

$$\begin{array}{c}\mathrm{grad}\psi ={\overrightarrow{e}}_r\frac{\partial \psi }{\partial r}+\frac{{\overline{e}}_{\phi }}{r}\frac{\partial \psi }{\partial \phi }+\overrightarrow{k}\frac{\partial \psi }{\partial z},\hfill \\ \mathrm{div}\overrightarrow{u}=\frac{1}{r}\frac{\partial u_{\phi }}{\partial r}+\frac{1}{r}\frac{\partial r{u}_r}{\partial r}+\frac{\partial u_z}{\partial z},\hfill \\ \mathrm{rot}\overrightarrow{u}={\overrightarrow{e}}_r\left(\frac{1}{r}\frac{\partial u_z}{\partial \phi }-\frac{\partial u_{\phi }}{\partial z}\right)+{\overrightarrow{e}}_{\phi }\left(\frac{\partial u_r}{\partial z}-\frac{\partial u_z}{\partial r}\right)+\overrightarrow{k}\left(\frac{1}{r}\frac{\partial \left(r{u}_{\phi }\right)}{\partial r}-\frac{1}{r}\frac{\partial u_k}{\partial \phi }\right).\hfill \end{array}$$

(9)

Let us write vector Equation (3) in the form of a system of two scalar differential equations with respect to the divergence $\mathrm{div}\overrightarrow{u}$ of displacements and nonzero component ${\omega }_z$ of the vortex field of displacements:

$$\frac{2}{1-\nu }\frac{\partial }{\partial r}\left(\mathrm{div}\overrightarrow{u}\right)-\frac{1}{r}\frac{\partial }{\partial \phi }{\omega }_z-\frac{\partial }{\partial r}\left(2d_{31}\frac{1+\nu }{1-\nu }{E}_z\right)=2\rho s_{11}^E(1+\nu )\frac{{\partial }^2{u}_r}{\partial t^2},$$

(10)

$$\frac{2}{(1-\nu )r}\frac{\partial \left(\mathrm{div}\overrightarrow{u}\right)}{\partial \phi }-\frac{\partial {\omega }_z}{\partial r}-\frac{1}{r}\frac{\partial \left(2d_{31}\frac{1+\nu }{1-\nu }{E}_z\right)}{\partial \phi }=2\rho s_{11}^E(1+\nu )\frac{{\partial }^2{u}_{\phi }}{\partial t^2}.$$

(11)

Then, from (7) and (9), we obtain

$$\begin{array}{c}\mathrm{div}\overrightarrow{u}=\left[\left(\frac{d{u}_0}{dr}+\frac{u_0}{r}\right)+\sum _{k=1}^{\infty }\left(\frac{d{u}_k}{dr}+\frac{u_k}{r}+k\frac{{\nu }_k}{r}\right)cosk\phi \right]{exp}^{i\omega t}\hfill \\ =\left[{\displaystyle {\Phi }_0\left(r\right)+\sum _{k=1}^{\infty }{\Phi }_k\left(r\right)cosk\phi }\right]{exp}^{i\omega t},\hfill \\ {\omega }_z=\left[\sum _{k=1}^{\infty }\left(\frac{d{u}_k}{dr}+\frac{u_k}{r}-k\frac{{\nu }_k}{r}\right)sink\phi \right]{exp}^{i\omega t}=\left[\sum _{k=1}^{\infty }{\Psi }_k\left(r\right)\right]{exp}^{i\omega t}.\hfill \end{array}$$

We perform time differentiation in the right-hand side of (10)–(11), and after reducing the non-zero factors ${exp}^{i\omega t}$, $sin\left(k\phi \right)$, $cosk\phi$, we obtain the following system of ordinary differential equations:

$$\left\{\begin{array}{c}{\displaystyle \frac{2}{1-\nu }\frac{d{\Phi }_k}{dr}-\frac{k}{r}{\psi }_k=-2\rho s_{11}^E(1+\nu ){\omega }^2{u}_k,}\hfill \\ \\ {\displaystyle -\frac{2}{1-\nu }\frac{k}{r}{\Phi }_k-\frac{d{\Psi }_k}{dr}+d_{31}\frac{1+\nu }{1-\nu }{a}_k=-2\rho s_{11}^E(1+\nu ){\omega }^2{\nu }_k.}\hfill \end{array}\right.$$

(12)

To determine the unknown potentials ${\Phi }_k$ and ${\Psi }_k$, we obtain a system of two Bessel ordinary differential equations

$$\left\{\begin{array}{c}{\displaystyle r^2\frac{d^2{\Phi }_k}{d{r}^2}+r\frac{d{\Phi }_k}{dr}+\left({\left(k_{1}r\right)}^2-k^2\right){\Phi }_k=-k^2{d}_{31}(1+\nu )a_k=A_k,}\hfill \\ \\ {\displaystyle r^2\frac{d^2{\Psi }_k}{d{r}^2}+r\frac{d{\Psi }_k}{dr}+\left({\left(k_{2}r\right)}^2-k^2\right){\Psi }_k=0.}\hfill \end{array}\right.$$

(13)

In the case of odd function $E\left(\phi \right)$, the first equation is homogeneous, and the second—heterogeneous. In the case of the general form of the electric load function, one can consider the superposition of solutions, or the solution of two inhomogeneous Bessel differential equations. In these equations, respectively, radial and circular wave numbers are denoted with

$$k_1=\frac{\omega }{c},c=\frac{1}{\sqrt{\rho s_{11}^E(1-{\nu }^2)}},k_2=\sqrt{\frac{2}{1-\nu }}{k}_1.$$

The general integrals of Equation (13) have the following form:

$$\left\{\begin{array}{c}{\displaystyle {\Phi }_k=C_k{J}_k(k,r)+A_k\frac{\pi }{2}\left(Y_k(k,r){\int }_0^r{J}_k\left(k_1\rho \right)\frac{d\rho }{\rho }+J_k(k,r){\int }_0^r{Y}_k\left(k_1\rho \right)\frac{d\rho }{\rho }\right)}\hfill \\ \\ {\displaystyle {\Psi }_k=D_k{J}_k\left(k_{2}r\right)}\hfill \end{array}\right..$$

(14)

The absence of a Bessel function of the second kind in the solutions of (12) is due to the presence of a feature of this function at $r=0$.

It is necessary to determine the components of the coupled electromechanical field through the obtained potentials. Non-zero elements of the displacement vector are

$$\left\{\begin{array}{c}{\displaystyle u_r=-\frac{1}{k_1^2}\left({\displaystyle \frac{d{\Phi }_0}{dr}+\sum _{k=1}^{\infty }(\frac{d{\Phi }_k}{dr}-\frac{k(1-\nu )}{2r}{\psi }_k)cosk\phi }\right){exp}^{i\omega t},}\hfill \\ \\ {\displaystyle u_{\phi }=-\frac{1}{k_1^2}\left({\displaystyle \sum _{k=1}^{\infty }(\frac{k}{r}{\Phi }_k+\frac{1-\nu }{2}\frac{d{\Psi }_k}{dr}+\frac{k}{r}{d}_{31}(1+\nu )a_k)sink\phi }\right){exp}^{i\omega t}.}\hfill \end{array}\right.$$

(15)

Components of the stress tensor are

$$\left\{\begin{array}{c}{\displaystyle {\sigma }_r=\frac{1}{s_{11}^E}\left({\displaystyle (\frac{\nu }{1-\nu }{\Phi }_0+\frac{d{u}_0}{dr}-\frac{d_{31}(1+\nu )a_0}{1-\nu })+\sum _{k=1}^{\infty }(\frac{\nu }{1-\nu }{\Phi }_k+\frac{d{u}_k}{dr}-\frac{d_{31}(1+\nu )a_k}{1-\nu })cosk\phi }\right){exp}^{i\omega t},}\hfill \\ \\ {\displaystyle {\sigma }_{\phi }=\frac{1}{s_{11}^E}\left({\displaystyle (\frac{\nu }{1-\nu }{\Phi }_0+\frac{u_0}{r}-\frac{d_{31}(1+\nu )a_0}{1-\nu })+\sum _{k=1}^{\infty }(\frac{\nu }{1-\nu }{\Phi }_k+\frac{u_k}{r}-\frac{d_{31}(1+\nu )a_k}{1-\nu })cosk\phi }\right){exp}^{i\omega t},}\hfill \\ \\ {\tau }_{r\phi }=\frac{1}{s_{11}^E}\left({\displaystyle \sum _{k=1}^{\infty }(2\frac{d{\nu }_k}{dr}-{\psi }_k)sink\phi }\right){exp}^{i\omega t}.\hfill \end{array}\right.$$

(16)

Components of the strain tensor are

$$\left\{\begin{array}{c}{\displaystyle {\epsilon }_r=\frac{\partial u_r}{\partial r}=\left({\epsilon }_0^1+\sum _{k=1}^{\infty }{\epsilon }_k^{1}cosk\phi \right){exp}^{i\omega t},}\hfill \\ \\ {\displaystyle {\epsilon }_{\phi }=\frac{1}{r}\frac{\partial u_{\phi }}{\partial \phi }+\frac{u_r}{r}=\left({\epsilon }_0^2+\sum _{k=1}^{\infty }{\epsilon }_k^{2}cosk\phi \right){exp}^{i\omega t},}\hfill \\ \\ {\displaystyle {\epsilon }_{r\phi }=\frac{\partial u_{\phi }}{\partial r}-\frac{u_{\phi }}{r}+\frac{1}{r}\frac{\partial u_r}{\partial \phi }=\left(\sum _{k=1}^{\infty }{\gamma }_{k}sink\phi \right){exp}^{i\omega t}.}\hfill \end{array}\right.$$

(17)

Expressions of quantities ${\epsilon }_k^1$, ${\epsilon }_k^2$ and ${\gamma }_k$ through potentials are from (14).

The thickness component of the electric field intensity vector was previously determined by Formula (6). For the thickness component of the electric induction vector, we have

$$D_z=\left({\epsilon }_{33}^T(1-k_p^2)a_0+\frac{d_{31}}{s_{11}^E(1-\nu )}{\Phi }_0+\sum _{k=0}^n({\epsilon }_{33}^T(1-k_p^2)a_k+\frac{d_{31}}{s_{11}^E(1-\nu )}{\Phi }_k)cosk\phi \right){exp}^{i\omega t}.$$

(18)

Here, ${\displaystyle k_p^2=\frac{d_{31}^2}{s_{11}^E{\epsilon }_{33}^T}\frac{2}{1-\nu }}$ is a static planar coefficient of electromechanical coupling, ${\epsilon }_{33}^T$ is a dielectric constant.

The terms of the zero index in Formulas (15)–(18) correspond to the case of axisymmetric radial oscillations of a piezoceramic disc [2].

To determine the constants of integration, we use mechanical boundary conditions (4) and stress expressions (16). For each value k, we have a system of two linear algebraic equations with respect to the unknown constants of integrations $C_k$ and $D_k$. Equating the main determinants of these systems to zero, we obtain the so-called resonance equations, each of which have an infinite number of solutions in the form of dimensionless frequencies ${\left(k_{1}a\right)}_{(k,l)}$ [2].

Figure 2 shows the results of numerical calculations for normal stresses and tangential stresses in the plate plane. The case of sector angles with the correspondingly selected direction of the polar axis, which coincides with the bisector of one of the sectors, is shown. The maximum values of the normal stresses are concentrated on the lines of symmetry of each of the four segments of the disk. In places where the electrode coating is cut, the stresses are zero. For tangential stresses, as well as for normal stresses, zero values are obtained in places of the cut of the electrode coating. The difference between the obtained results is that the maximum stresses ${\tau }_{\rho \phi }$ tend to the center of the disk.

3. Investigation of the Problem of Optimizing the EMCO Value

To determine the measure of the efficiency of energy conversion—the dynamic coefficient of electromechanical coupling (EMCO)—there is an energy criterion that allows determination of this coefficient at the resonant frequency. According to this criterion, the EMCO is defined as the ratio of the electrical (mechanical) energy accumulated in the volume of the body capable of rotation to the total mechanical (electrical) energy supplied to the body [2]:

$$k_d^2=\frac{U_{\epsilon }^{rot}}{U^n}=\frac{U_p-U_k}{(U_p-U_k)+U_k},$$

(19)

where $U_k$ is energy in the volume of the body with short-circuited electrodes under the condition $E_z=0$, $U_p$ is full energy in the body with open electrodes provided

$${\int \int }_{S^{+}}{D}_{z}d{S}^{+}=0.$$

(20)

In both cases, the total energy in the body is calculated on the same constant deformation field.

The general view of the expression of the total electromechanical energy in the volume of the piezoceramic body is

$$U=\frac{1}{2}{\int \int \int }_V\left({\sigma }_x{\epsilon }_x+{\sigma }_y{\epsilon }_y{\sigma }_z{\epsilon }_z+{\sigma }_{xy}{\epsilon }_{xy}+{\sigma }_{xz}{\epsilon }_{zx}+{\sigma }_{yz}{\epsilon }_{yz}+D_x{E}_x+D_y{E}_y+D_z{E}_z\right)dV.$$

(21)

Having applied the transformations corresponding to the applied hypotheses of electroelasticity, analogous to the Kirchhoff–Leav hypotheses of the theory of plates and shells in Expression (20), we exclude zero factors, including non-zero deformation ${\epsilon }_{33}$.

We have the following estimate:

$${\epsilon }_z=-\frac{s_{13}^E}{s_{11}^E}\frac{1}{1-\nu }div\overrightarrow{u}+\left(d_{33}+2d_{31}\frac{1}{1+\nu }\frac{s_{13}^E}{s_{11}^E}\right)E_z.$$

(22)

The obtained estimate explains the effect of the electric load during the thickness polarization of the plate and the implicit consideration of this non-zero deformation in this problem. We write down the total electromechanical energy in the plate in a simplified form for the selected type of body deformation:

$$U=\frac{1}{2}{\int \int \int }_V\left({\sigma }_r{\epsilon }_r+{\sigma }_{\phi }{\epsilon }_{\phi }+{\tau }_{r\phi }{\epsilon }_{r\phi }+D_z{E}_z\right)dV.$$

(23)

Then, for the amplitude of the energy difference in Numerator (19) after passing to the repeated integral, taking into account the orthogonality of the system of functions, we obtain

$$\begin{array}{c}{U}_p-U_k=\frac{\pi h}{2}{\int }_0^a\left(2({\sigma }_0^{1\left(p\right)}-{\sigma }_0^{1\left(k\right)}){\epsilon }_0^1+2({\sigma }_0^{2\left(p\right)}-{\sigma }_0^{2\left(k\right)}){\epsilon }_0^2\right)\rho d\rho \hfill \\ +\frac{\pi h}{2}{\int }_0^a\sum _{k=1}^a\left(({\sigma }_k^{1\left(p\right)}-{\sigma }_k^{1\left(k\right)}){\epsilon }_k^1+({\sigma }_k^{2\left(p\right)}-{\sigma }_k^{2\left(k\right)}){\epsilon }_k^2+(t_k^p-t_k^k)t_k\right)\rho d\rho .\hfill \end{array}$$

The amplitude of the total mechanical energy accumulated in a plate with short-circuited electrodes is determined by the following integral:

$$U_k=\frac{\pi h}{2}{\int }_0^a\left(2({\sigma }_0^{1\left(k\right)}{\epsilon }_0^1+{\sigma }_0^{2\left(k\right)}{\epsilon }_0^2)+\sum _{k=1}^a({\sigma }_k^{1\left(k\right)}{\epsilon }_k^1+{\sigma }_k^{2\left(k\right)}{\epsilon }_k^2)\right)\rho d\rho .$$

To analyze the integral, we consider the expressions of radial, circumferential, and shear stresses due to deformations and the nonzero component of the electric field intensity. Obviously, it follows from (15) that the tangential stresses ${\tau }_{r\phi }$ do not depend on the intensity of the electric field; therefore, $t_k^p-t_k^k=0$, $k=1,2,3,\dots$ Thus, the corresponding term of the integrand is equal to zero. Expressions for the corresponding normal breakdown voltages in the case of short-circuited electrodes do not contain electrical quantities. For open electrodes, they contain an addition with a thickness component of electrical induction $E_z^{\left(p\right)}$, different from that caused by the initial load.

The following expressions for the amplitude differences hold:

$${\sigma }_k^{1\left(p\right)}-{\sigma }_k^{1\left(k\right)}={\sigma }_k^{2\left(p\right)}-{\sigma }_k^{2\left(k\right)}=-\frac{d_{31}}{s_{11}^E(1-\nu )}{a}_k^{\left(p\right)}cosk\phi .$$

To determine coefficients $a_k^{\left(p\right)}$, we use Condition (20), which from the form of Expansion (19) is fulfilled identically. This condition must be fulfilled for an arbitrary sector of the breakdown of the electrode coating (taking into account the assumption of parity of the electric load function) of the face surfaces of the disc. Condition (18) must be fulfilled separately for each sector.

Then, we have

$${\int \int }_{S_i^{+}}\left({\epsilon }_{33}^T(1-k_p^2)a_0^{\left(p\right)}+\frac{d_{31}}{s_{11}^E(1-\nu )}{\Phi }_0+\sum _{k=0}^n\left({\epsilon }_{33}^T(1-k_p^2)a_k^{\left(p\right)}+\frac{d_{31}}{s_{11}^E(1-\nu )}{\Phi }_k\right)cosk\phi \right){exp}^{i\omega t}d{S}_i^{+}=0,i=\overline{0,n-1},$$

and, as a consequence, for each $k\ge 0$,

$${\epsilon }_{33}^T(1-k_p^2)a_k^{\left(p\right)}\frac{a^2}{2}=-\frac{d_{31}}{s_{11}^E(1-\nu )}{\int }_0^a{\Phi }_k\rho d\rho ,$$

where

$$a_k^{\left(p\right)}=-\frac{1}{a^2}\frac{d_{31}}{s_{11}^E{\epsilon }_{33}^T}\frac{2}{1-\nu }\frac{1}{1-k_p^2}{\int }_0^a\Phi \rho d\rho =-\frac{1}{a^2}\frac{1}{d_{31}}\frac{k_p^2}{1-k_p^2}{\int }_0^a{\Phi }_k\rho d\rho ,k\ge 0.$$

To calculate energy difference $U_p-U_k$, we have

$$U_p-U_k=\frac{\pi h}{2a^2}\frac{1}{s_{11}^E(1-\nu )}\frac{k_p^2}{1-k_p^2}{\int }_0^a\left(2{\Phi }_0^2+\sum _{k=1}^{\infty }{\Phi }_k^2\right)\rho d\rho .$$

For the dynamic coefficient of electromechanical coupling, we have

$$k_d^2=\frac{K_1}{K_1+k_2},$$

(24)

where

$$\begin{array}{c}{K}_1=\frac{1}{2a^2}\frac{1}{s_{11}^E(1-\nu )}\frac{k_p^2}{1-k_p^2}{\int }_0^a\left(2{\Phi }_0^2+\sum _{k=1}^{\infty }{\Phi }_k^2\right)\rho d\rho ,\hfill \\ K_2={\int }_0^a\left(2({\sigma }_0^{1\left(k\right)}{\epsilon }_0^1+{\sigma }_0^{2\left(k\right)}{\epsilon }_0^2)+\sum _{k=1}^a({\sigma }_k^{1\left(k\right)}{\epsilon }_k^1+{\sigma }_k^{2\left(k\right)}{\epsilon }_k^2)\right)\rho d\rho .\hfill \end{array}$$

For each specific breakdown of the electrode coating of the plate, we can immediately write out the values of coefficients $a_k$ and calculate the value of the Dynamic EMCO. But the question of the rationality of the chosen electrical load connection scheme remains open.

Fixing the appropriate frequency of electrical load $\omega$, which is included as a factor in the dimensionless propagation velocities of radial and circumferential waves in the piezoceramic material, (23) can be considered functional. This functionality should be explored to the maximum (supremum), depending on the choice of parameters $a_k$, $k=0,1,2,\dots$ in space $l_2$ of square summable sequences, taking into account the additional restriction $\underset{k\to \infty }{lim}{a}_k=0$, which is imposed on unknown variables.

We finish the section with an example of numerical calculations of dynamic EMCO for the first ${\left(k_{1}a\right)}_{(2,1)}$ and second ${\left(k_{1}a\right)}_{(2,2)}$ resonant frequencies. The results are summarized in

Table 1. Comparison of the application of two approaches for the first and second resonant frequencies.

${\mathit{k}}_{\mathit{d}}^2$	(2,1)	(2,2)
Mason’s formula	0.0008	0.0298
Energy criterion	0.0020	0.0289

below.

The results obtained are in accordance with the calculations of the EMCO using Mason’s formula [1].

4. Conclusions and Future Research

For the considered electromechanical system, the boundary value problem of electroelasticity is formulated. General solutions are obtained in an explicit form. Analytical expressions of the components of the coupled electromechanical field are obtained. Expressions of total energy in the body with open- and short-circuited electrodes are recorded. The problem of optimization of the EMCO value as a function of an infinite number of parameters in space $l_2$ is investigated.

The obtained results make it possible to determine a rational scheme for supplying electrical energy to the electrode coating. We plan to continue simulation studies in the future research.