# Parametric and Experimental Modeling of Axial-Type Piezoelectric Energy Generator with Active Base
## 4 Modeling  
Within the framework of the presented studies, modal and harmonic analyses of PEG oscillations were carried out during modeling. When modeling, the value of the attached mass M= [0.05; 1; 3; 5; 7; 9] gr, as well as its position Lm = [150; 170; 190; 210; 230] mm on a flexible base plate is used. The value of the active load was taken as R = 1000 \(\Omega\). At the stage of harmonic analysis, the vibration amplitudes were recorded at a displacement of the PEG rigid fastening zone by [0.001...0.01] mm. The next task was to consider the harmonic modeling of PEG oscillations with a constant displacement of the generator housing attachment zone by 0.01 mm and variations in material properties (duralumin: E\({}_{1}\) = 0.7 10\({}^{11}\), Pa, \(\rho_{1}\) = 2600 kg/m\({}^{3}\), \(\
u^{1}\) = 0.33; and fiberglass: E\({}_{2}\) = 0.06 10\({}^{11}\), Pa, \(\rho_{2}\) = 1600 kg/m\({}^{3}\), \(\
u_{2}\) = 0.33) and corresponding thickness options h = [1; 2] mm base plate. In this case, the attached mass was fixed in the position L\({}_{\rm m}\) = 150 mm and its value was M = 0.5 gr, 30 gr. As an active load in the form of resistance, R for each piezoelectric element was taken, equal to 1000 \(\Omega\). The natural frequencies, the oscillation amplitude of the center of the flexible base plate of the generator and the output voltage taken from the piezoelectric elements were considered as output parameters. The simulation was carried out in FE software ANSYS. Figure 3 shows the FE model of the PEG. PEG consists of a base in the form of a strip, clamped on one of the sides. During the modeling, elements of the SOLID92 type with a tetrahedral structure were used, which facilitated the partition of the model. When modeling the thin walls of the planks, it was assumed not to consider deformations over the thickness of the structure. As a result, the value of the smallest edge of the FE in the form of a tetrahedron was taken, equal to the wall thickness of the structural element. Modeling of piezoelectric elements was carried out using FE type SOLID5. Modeling of elements in the form of active resistance was carried out using the FE CIRCU94 with the resistor option. The PE of a three-dimensional structure in the form of piezocylinders (PE PC) was divided by a FE mesh with an edge size multiple of 0.15 of the piezocylinder height. In a PE with a planar structure (PE PS), the size of the partitioning of the FE mesh by thickness was a multiple of the value equal to the thickness of the PE. The division was in the form of triangular prisms. The direction of polarity in a cylindrical PE was taken along the main axis of the PE, coaxially with the PEG, in a plate-shaped PE along the thickness. The polarization vector in the bimorph for the upper and lower PEs was directed along the normal to the surface. The number of FE elements when splitting the model was more than 33,000 nodes greater than 62,500. Figure 3: FE-model of PEG of axial type and boundary conditions. At the first stage, modal PEG analysis was carried out. Figure 4 shows the four vibration modes of the model. With this configuration, the model had a first design frequency of 259 Hz. The vibration mode for the first mode had a prevailing bending character for the generator warp bar. Vibration modes 1, 4 and 8 correspond to (1,2,3) flexural vibration modes of the PEG base plank in the vertical direction. The second mode shape was obtained at 525 Hz. It was assumed that bending deformations of the bar in the smallest plane of rigidity (in the vertical direction) would excite the highest output stress in all PEs. Investigations of oscillations of axial-type PEG under harmonic action were carried out. The harmonic action was calculated under the action of a uniformly applied acceleration of 10 m/s\({}^{2}\) for all structural units at the corresponding harmonics. For the first and second vibration modes, the dependences of their value on the mass and location of the load were calculated.