# Optimizing Energy Conversion in a Piezo Disk Using a Controlled Supply of Electrical Load
## 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:  
\[E_{x}=E(\varphi)\exp^{i\omega t}. \tag{1}\]  
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:  
\[\Psi\mid_{z\pm h}=\pm V_{0}(\varphi)\exp^{i\omega t}. \tag{2}\]Figure 1 provides visual representation of the problem. Here, \(V_{k}\exp^{i\omega t},k=1,\ldots,n\), are harmonic time difference of the potentials applied to the upper and lower parts of the electrode coating of the \(k\)th 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  
\[\frac{2}{1-\
u}\text{grad}(\text{div}\vec{u})-2d_{31}\frac{1+\
u}{1-\
u}\text{ grad}E_{z}-\text{rot}(\text{rot}\vec{u})=2\rho s_{11}^{E}(1+\
u)\frac{\partial^{2} \vec{u}}{\partial t^{2}}, \tag{3}\]  
where \(\
u\)--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:  
\[\sigma_{r}\mid_{r=a}=0;\qquad\tau_{\varphi}\mid_{r=a}=0. \tag{4}\]  
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:  
\[\vec{u}=\vec{e}_{r}u_{r}+\vec{e}_{\varphi}u_{\varphi},\quad\vec{u}=(\vec{e}_{ r}\vec{\tau}_{r}(r,\varphi)+\vec{e}_{\varphi}\vec{\tau}_{\varphi}(r, \varphi))\exp^{i\omega t}. \tag{5}\]  
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  
\[E_{z}=\left(a_{0}+\sum_{k=1}^{\infty}a_{k}\cos k\varphi+b_{k}\sin k\varphi \right)\exp^{i\omega t}. \tag{6}\]  
Here, \(a_{0}\), \(a_{k}\), \(b_{k}\)--Fourier coefficients for function \(E(\varphi)\). Then, according to the choice of a coordinate system, we find the solution of Problems (3)-(4) in the form  
\[ \begin{array}{l}u_{r}=\big{(}u_{0}(r)+\sum_{k=1}^{\infty}(u_{k}\cos k\varphi +u_{k}^{1}\sin k\varphi)\big{)}e^{i\omega t},\\ u_{\varphi}=\big{(}v_{0}(r)+\sum_{k=1}^{\infty}(v_{k}\sin k\varphi+v_{k}^{1} \cos k\varphi)\big{)}e^{i\omega t}.\end{array}  \tag{7}\]  
Figure 1: Scheme of electrical load supply. 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  
\[u_{r}=\left(u_{0}(r)+\sum_{k=1}^{\infty}u_{k}(r)\cos k\varphi\right),\quad u_{ \varphi}=\sum_{k=1}^{\infty}\
u_{k}(r)\sin k\varphi. \tag{8}\]  
Let us proceed to the solution of Problems (3)-(4). We consider the form of differential operators of a vector field in cylindrical coordinates  
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