## 6 Numerical Simulation of Harvesters with ITDs
### Open Circuit Voltage  
First, the FRFs of the harvesters equipped with ITDs were calculated to assess the trimming capabilities of the ITDs. The open circuit condition was considered, since analytical results and experimental tests on prototypes showed that the resistive load causes only minor frequency shifts of the peaks. Figure 13a shows that ITD1 trims the main resonance to 89.7 Hz with a peak value (1.6 V/ms\({}^{-2}\)), a bit smaller than the PPA 1001 alone. The second resonance peak, which appears at 154.0 Hz, is much lower than the main peak and is caused by the excitation of the second mode of vibration generated by ITD1. For comparison, Figure 13a shows the effect of a large tip mass (0.59 g). In this case the trimming frequency is 88.1 Hz and the peak value is 2.7 V/ms\({}^{-2}\). Figure 12: Numerical models of the harvester with (**a**) ITD1, (**b**) ITD2. Figure 13: Numerical results, FRFs of the harvester with ITDs (integrated trimming devices), with tip mass and alone; open circuit condition. (**a**) ITD1, (**b**) ITD2. Figure 13b shows that ITD2 is able to lower the frequency of the main resonance peak to 89.7 Hz and to create a second resonance peak at double frequency (179.5 Hz). The heights of the two peaks are 1.5 V/ms\({}^{-2}\) and 0.57 V/ms\({}^{-2}\) respectively. When the frequency of base excitation coincides with one of the resonance frequencies and the deformed shape is dominated by the mode of vibration excited in resonance, the strain distribution inside the PZT layer gives useful information about the effect of the ITD on the harvester. Figure 14 shows the strain in the upper plane of the PZT layer of the harvester with ITD1. Edges of structural material surround this active layer. The strain component in the longitudinal direction is represented, since it is the most important strain component caused by harvester bending. The color maps show both the amplitude and phase of strain. Strain node lines [37], which are the lines on the surface of the PZT layer where the strain changes sign, are represented by light lines in the phase plots. At the first resonance the ITD moves in phase with the harvester. The amplitude of strain decreases rather regularly from the clamped-end to the free-end. There are changes in the sign of strain near the corners of the free-end of the layer. This is a 3D effect, since the ITD is attached at the middle of the free-end of the harvester, see Figure 12a. With this strain distribution cancellations in generated voltage [37] are very small and the first peak is high. At the second resonance the ITD moves in phase opposition with respect to the harvester tip (Figure 14d). The strain distribution in this condition is very different with a strain node line at about 6/10 of the harvester's length. There is a partial cancellation in the generated voltage, which leads to a small resonance peak. The strain distributions in the PZT layer of the harvester equipped with ITD2 are represented in Figure 15. At the first resonance there is only a strain node line near the central part of the free-end of the layer, because the ITD is attached to the harvester in correspondence of the side edges of structural material that surround the active layer (Figure 15a). Owing to the presence of this strain node line, a cancellation in the generated voltage takes place, but this effect is small, because the negative strain area is small and negative strains have small amplitudes. When the harvester with ITD2 is excited at the frequency of the second resonance peak, the side beams of ITD2 alter the strain distribution inside the PZT layer and there is a partial cancellation in generated voltage. Figure 14: Numerical results, strains in the piezo layer of the harvester with ITD1. (**a**) Modal shape at the first resonance (89.8 Hz). (**b**) Strain modulus at the first resonance (89.9 Hz). (**c**) Strain phase at the first resonance (89.8 Hz). (**d**) Modal shape at the second resonance (154.0 Hz). (**e**) Strain modulus at the second resonance (154.0 Hz).