## 4 Discussion  
The results above prove that significant resonance-enhanced ME coupling can be achieved in MAE/PEP multiferroic heterostructures in conventional geometry (Figure 2). Since the employed materials are much softer than metals or ceramics the resonance frequencies in the absence of a magnetic field are low (\(f_{\rm r1}<50\) Hz, \(f_{\rm r2}<300\) Hz) and, as should be expected, they increase in an increasing bias field due to the stiffening of the MAE material. Let us estimate the resonance frequencies of the heterostructures. The frequency of bending vibrations of the structure was determined by the formula [27]:  
\[f_{\rm ri}=\frac{k_{i}^{2}}{2\pi}\frac{t_{\rm PEP}}{L^{2}}\sqrt{\frac{Y}{12 \rho}}, \tag{1}\]  
where \(k_{i}\) is some constant, \(i\) is the number of the oscillation mode (\(i=1,\,2,\,3\)), \(t_{\rm PEP}\) is the thickness of the PEP layer, \(L\) is the length of the free part of the cantilever, \(Y\) is the effective Young's modulus of the composite structure, \(\rho\) is the effective density of the composite structure. In this case, since the MAE layer was soft and did not expand over the entire structure (the cantilever was clamped only over the PEP layer) so it was considered as a certain mass distributed on the cantilever surface. Its influence was taken into account in (1) through the values of the effective Young's modulus and effective density, while the correction for the frequency value created by the cantilever deflection as a result of the action of the distributed force from the MAE layer was small [27]. The values of \(Y\) and \(\rho\) were calculated by the following formulas:  
\[Y=\frac{\gamma_{\rm m}\cdot V_{\rm m}+\gamma_{\rm p}\cdot V_{\rm p}}{V_{\rm m} +V_{\rm p}}, \tag{2}\]  
\[\rho=\frac{\rho_{\rm m}\cdot V_{\rm m}+\rho_{\rm p}\cdot V_{\rm p}}{V_{\rm m} +V_{\rm p}} \tag{3}\]  
where the subscripts \(m\) and \(p\) denote the ferromagnetic (i.e., MAE) and PEP layers, respectively. \(V_{m}=t_{\rm MAE}\cdot L_{\rm MAE}\cdot w\) and \(V_{\rm p}=t_{\rm PEP}\cdot L\cdot w\) are the volumes of the magnetic and PEP layers, correspondingly. Since the PEP layer was a composite structure, the equivalent Young's modulus and density of the PEP were calculated according to formulas similar to (2) and (3). Substituting the values \(L=20\) mm, \(t_{\rm PEP}=220\) um, \(Y_{\rm m}\approx 210\) kPa, \(Y_{\rm PVDF}\approx 2.1\) GPa, \(Y_{\rm Polyester}\approx 3\) GPa, \(\rho_{\rm m}\approx 3250\) kg/m\({}^{3}\), \(P_{\rm PVDF}\approx 1800\) kg/m\({}^{3}\), \(\rho_{\rm Polyester}\approx 1400\) kg/m\({}^{3}\) and, taking that for a rod fixed at one end, \(k_{1}=1.875\), \(k_{2}=4.694\), we obtained the values for the resonance frequencies in the absence of a magnetic field. The theoretical results shown in Figure 9 were calculated for an MAE layer length \(L_{\rm MAE}\) of 16.2 mm and a width \(w\) of 14.2 mm. Reasonable agreement between theory and experiment was seen. Therefore, we believe that we observed resonance enhancement of ME coupling due to bending resonances. Remarkably, the predicted resonance eigenfrequency of the lowest bending mode for two thin MAE layers (\(f_{\rm r1}\approx 40\) Hz) agreed well with the frequency of voltage oscillations in [19], while for the thickest MAE layer it agreed well with oscillation frequencies observed in [20] in another (T-T) geometry. For example, substituting the above parameters for sample S1 in the field \(H=0\) kA/m, the calculated frequency value\(f_{\rm r1}\approx 47.2\) Hz and \(f_{\rm r2}\approx 296.1\) Hz was obtained. The data obtained were in good agreement with the measurement results (the discrepancy for the \(f_{\rm r1}\) and \(f_{\rm r2}\) frequencies was \(\approx\)9.6% and less than 1%, respectively).