# Parametric and Experimental Modeling of Axial-Type Piezoelectric Energy Generator with Active Base
## 2 Description of Model Parameters and Electric Scheme
### Working Principle of the Model  
The proposed model works as follows: When the rigid base (8-10) is exposed to external mechanical forces such as shocks and vibrations, the vibrations are transmitted to the base plate (7), affecting the piezoelectric elements, in which alternating deformations of compression (in cylinders) and tension-compression (in plates) due to the reaction of the supports occur. Due to the direct piezoelectric effect on the electrodes of additional piezoelectric elements, AC voltage is generated and, therefore, additional electrical energy. The combined use of such elements allows you to increase the output power and conversion efficiency of the converter efficiency. This AC voltage and additional electrical energy can be converted using bridge rectifiers to DC voltage, which is stored in batteries using harvesting energy systems. Piezoelectric elements can be connected in parallel, in series or separately. The choice between the types of connections of elements depends on the device that needs to be powered: if a higher output voltage is required, then a serial connection should be selected, and if a higher output current is required, then a parallel connection. A schematic electrical diagram of a PEG connection with a resistive load is shown in Figure 2. The resistive load is supplied to each PE individually. Voltage is found at the contact points of the resistor. In numerical simulation, voltage is calculated as the difference of its amplitudes at the nodes of the FE element presented, with an option indicating the type of resistor. The added weight can vary from 3 to 25 g. In the experiment, we used the masses M = 3.71; 6.13 g. The proof mass was located at a distance of Lm = 150 to 230 mm to the left edge of the generator (Figure 1). Theoretical Description of the Model of Composite Elastic, Electroelastic and Acoustic Media by FE Simulation  
The energy storage PEG is a composite elastic and electroelastic body. It is assumed that the device performs elastic small oscillations in a moving coordinate system. The rectilinear vertical motion of this system in the area of fixation is given by the law for steady oscillations:  
\[u(t)=\overline{u}\ e^{i\omega t} \tag{1}\]  
Under these conditions, a sufficiently adequate mathematical model of the operation of the device is the initial-boundary value problem of the linear theory of electroelasticity [40,41]. In the general formulation, the equations for a piezoelectric medium are written as:  
\[ \begin{array}{c}\rho\ddot{u}_{i}+\alpha\rho\dot{u}_{i}-\sigma_{ijj}=f_{i}; \\ D_{i,i}=0;\\ \sigma_{ij}=\epsilon_{ijkl}\big{(}\epsilon_{kl}+\beta\dot{\epsilon}_{kl}\big{)} -\epsilon_{ijk}E_{k};\\ D_{i}+\zeta_{d}\dot{D}_{i}=\epsilon_{kl}\big{(}\epsilon_{kl}+\zeta_{d}\dot{ \epsilon}_{kl}\big{)}-\epsilon_{ik}E_{k};\\ \epsilon_{kl}=(u_{kl}+u_{lk})/2;\\ E_{k}=-\sigma_{jk};\end{array}  \tag{2}\]  
where \(\rho\) is the material density;  
\(u_{i}\) are the components of the displacement vector;  
\(\sigma_{ij}\) are the components of the stress tensor;  
\(f\) are the components of the vector of the density of mass forces;  
\(D_{i}\) are the components of the electric induction vector;\(c_{ijkl}\) are the components of the fourth rank tensor of the elastic moduli;  
\(e_{ikl}\) are the components of the third rank tensor of piezoelectric coefficients;  
\(\varepsilon_{kl}\) are the components of strain tensor;  
\(E_{k}\) are the components of the electric field vector;  
\(\varphi_{k}\) is the electric potential;  
\(\epsilon_{ik}\) are the components of the dielectric constants tensor;  
\(\alpha,\ \beta,\ \zeta\) are non-negative damping coefficients (the value of \(\zeta_{\text{d}}\) is used in ANSYS software). For elastic medium, we have:  
\[ \begin{array}{l}\rho\ddot{u_{i}}+\alpha\rho\dot{u_{i}}-\sigma_{ij}=f_{i};\\ \sigma_{ij}=c_{ijkl}(\varepsilon_{kl}+\beta\dot{\varepsilon}_{kl});\\ \varepsilon_{kl}=(u_{kl}+u_{lk})/2.\end{array}