# Optimizing Energy Conversion in a Piezo Disk Using a Controlled Supply of Electrical Load
## 2 Non-Axisymmetric Planar Oscillations of a Piezoceramic Thin Disk
Non-zero elements of the displacement vector are  
\[ \begin{cases}u_{r}=-\frac{1}{k_{1}^{2}}\bigg{(}\frac{d\Phi_{0}}{dr}+\sum_{k=1} ^{\infty}(\frac{d\Phi_{k}}{dr}-\frac{k(1-\
u)}{2r}\psi_{k})\cos k\varphi\bigg{)} \exp^{i\omega t},\\ \\ u_{\varphi}=-\frac{1}{k_{1}^{2}}\bigg{(}\sum_{k=1}^{\infty}(\frac{k}{r}\Phi_ {k}+\frac{1-\
u}{2}\frac{d\Psi_{k}}{dr}+\frac{k}{r}d_{31}(1+\
u)a_{k})\sin k \varphi\bigg{)}\exp^{i\omega t}.\end{cases}  \tag{15}\]  
Components of the stress tensor are  
\[ \begin{cases}\sigma_{r}=\frac{1}{s_{11}^{E}}\bigg{(}(\frac{\
u}{1-\
u}\Phi_{0} +\frac{du_{0}}{dr}-\frac{d_{31}(1+\
u)a_{0}}{1-\
u})+\sum_{k=1}^{\infty}(\frac {\
u}{1-\
u}\Phi_{k}+\frac{du_{k}}{dr}-\frac{d_{31}(1+\
u)a_{k}}{1-\
u})\cos k \varphi\bigg{)}\exp^{i\omega t},\\ \\ \sigma_{\varphi}=\frac{1}{s_{11}^{E}}\bigg{(}(\frac{\
u}{1-\
u}\Phi_{0}+\frac {u_{0}}{r}-\frac{d_{31}(1+\
u)a_{0}}{1-\
u})+\sum_{k=1}^{\infty}(\frac{\
u}{ 1-\
u}\Phi_{k}+\frac{u_{k}}{r}-\frac{d_{31}(1+\
u)a_{k}}{1-\
u})\cos k\varphi \bigg{)}\exp^{i\omega t},\\ \\ \tau_{r\varphi}=\frac{1}{s_{11}^{E}}\bigg{(}\sum_{k=1}^{\infty}(2\frac{dv_{k}} {dr}-\psi_{k})\sin k\varphi\bigg{)}\exp^{i\omega t}.\end{cases}  \tag{16}\]  
Components of the strain tensor are  
\[ \begin{cases}\varepsilon_{r}=\frac{\partial u_{r}}{\partial r}=\bigg{(} \varepsilon_{0}^{1}+\sum_{k=1}^{\infty}\varepsilon_{k}^{1}\cos k\varphi \bigg{)}\exp^{i\omega t},\\ \\ \varepsilon_{\varphi}=\frac{1}{r}\frac{\partial u_{\varphi}}{ \partial\varphi}+\frac{u_{r}}{r}=\bigg{(}\varepsilon_{0}^{2}+\sum_{k=1}^{ \infty}\varepsilon_{k}^{2}\cos k\varphi\bigg{)}\exp^{i\omega t},\\ \\ \varepsilon_{r\varphi}=\frac{\partial u_{\varphi}}{\partial r}-\frac{u_{ \varphi}}{r}+\frac{1}{r}\frac{\partial u_{r}}{\partial\varphi}=\bigg{(}\sum_{ k=1}^{\infty}\gamma_{k}\sin k\varphi\bigg{)}\exp^{i\omega t}.\end{cases}  \tag{17}\]  
Expressions of quantities \(\varepsilon_{k}^{1}\), \(\varepsilon_{k}^{2}\) 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  
\[D_{z}=\bigg{(}\varepsilon_{33}^{T}(1-k_{p}^{2})a_{0}+\frac{d_{31}}{s_{11}^{E} (1-\
u)}\Phi_{0}+\sum_{k=0}^{n}(\varepsilon_{33}^{T}(1-k_{p}^{2})a_{k}+\frac{ d_{31}}{s_{11}^{E}(1-\
u)}\Phi_{k})\cos k\varphi\bigg{)}\exp^{i\omega t}. \tag{18}\]Here, \(k_{p}^{2}=\dfrac{d_{31}^{2}}{s_{11}^{E}\varepsilon_{33}^{T}}\dfrac{2}{1-\
u}\) is a static planar coefficient of electromechanical coupling, \(\varepsilon_{33}^{T}\) 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 \((k_{1}a)_{(k,l)}\)[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 \(\tau_{p\varphi}\) tend to the center of the disk.