## 3 Results  
As an example, Figure 3 presents the frequency dependence of the ME voltage coefficient \(\alpha=u/\left(t_{\rm PE}h\right)\) for the sample S1 with a thickness of 0.85 mm at different values of the bias magnetic field \(H\). The results for other samples look similar. Two electromechanical resonances are clearly observed at frequencies \(f_{\rm r1},f_{\rm r2}\)(\(f_{\rm r1}<f_{\rm r2}\)). It was seen that these resonance frequencies depended on the applied bias magnetic field and that there seemed to be an optimum magnetic field at which the magnetoelectric voltage coefficient had the maximum value at a particular resonance frequency. In fact, a third (less pronounced) resonance could be speculated at resonance frequency \(f_{\rm r3}\) (\(f_{\rm r1}<f_{\rm r3}<f_{\rm r2}\)); however, it was only well pronounced for sample S2 (\(t_{\rm MAE}=1.05\) mm). Our theoretical estimates, presented below, showed that this resonance corresponded to another type of mechanical oscillation (bending oscillations along the structure's width) other than resonances 1 and 2 (bending oscillations along the structure's length). The quality factor \(Q\) was determined from the frequency dependence of the generated voltage, and it was defined as the ratio of the resonance frequency to the resonance width on the \(-3\) dB level. It should be noted that for all samples the \(Q\)-factor of the resonance at the frequency \(f_{\rm r1}\) increased with an increasing magnetic field, while the magnetic field did not significantly affect the \(Q\)-factor of the peak at the frequency \(f_{\rm r2}\) in comparison with the fundamental frequency \(f_{\rm r1}\) of bending resonance (Figure 4). It was also verified that the amplitude \(h\) was sufficiently small so that the linear ME effect could be investigated. An example can be seen in Figure 5. The sensitivity to the magnetic field \(u/h\) at frequencies \(f_{\rm r1}\) and \(f_{\rm r2}\) was 0.16 and 0.1 mV\(\cdot\)m/A, respectively. Figures 6 and 7 depict the field dependences of the resonance frequencies \(f_{\rm r1},f_{\rm r2}\) and the corresponding ME voltage coefficient \(\alpha\) at resonance. MAE/PEP heterostructures demonstrated pronounced dependence on the resonance eigenfrequency on the bias magnetic field. The resonance frequency increased with the increasing bias magnetic field. This effect was observed for all samples; it seemed to be in concordance with the field-induced stiffening (or, alternatively, the magnetorheological effect) of MAEs [24]. However, as will be discussed below, the variation range of the resonance frequencies was too high to be assigned solely to the field-induced stiffening. Another specific feature of MAE/PEP heterostructures is their hysteresis behavior in the second bending resonance frequency \(f_{\rm r2}\) and the corresponding ME voltage coefficient \(\alpha\) at resonance in their dependence on the bias magnetic field (Figure 7). The slightly different values of \(f_{\rm r1}\) and \(\alpha\) at this excitation frequency for ascending and descending magnetic field \(H\) were within the uncertainty of measurement (Figure 6). The hysteresis behavior of MAEs in response to an external magnetic field is a well-known phenomenon, see, e.g., [25; 26]. The hysteresis originated in different microstructures (relative arrangements) of ferromagnetic particles in the MAEs, depending on the magnetization history. This was also true for soft magnetic particles, as was shown in this work. The hysteresis of magnetic properties for the investigated MAE  
Figure 2: Schematic diagram of the experimental setup: 1—electromagnet’s poles; 2—excitation coil; 3—magnetic field sensor. Directions of the magnetization \(M\) in the MAE layer and the polarization \(P\) in the PEP are shown. material was demonstrated [20]. Therefore, it could be expected that the hysteresis could be observed in the resonance frequency and ME voltage coefficient dependences.