## 4 Discussion
Nevertheless, when calculating the maximum obtained frequency\(f_{\rm r1}\approx 85.6\) Hz in the magnetic field \(H\approx 240\) kA/m, it was necessary to assume in (1) that the magnetically induced MAE's Young's modulus of the order of 2.2 \(\times\) 10\({}^{9}\) Pa was an unreasonably high value for these composite materials. Thus, an increase in the resonance frequency of PEP-MAE composite structures occurred and was not only due to the magnetorheological effect. The reason why there was the giant increase in the bending resonance frequencies in an applied magneticfield remained unclear to us. The largest magnetic field-induced increase in the resonance frequency of more than 200% was observed for the fundamental bending resonance in sample S4. The fundamental resonance frequency of the bending oscillations \(f_{r1}\approx 23\) Hz in the absence of a magnetic field was obtained in two samples with the thickest MAE layers. It was somewhat lower than the value reported in [4]. Such a low value could be attributed to the larger mass added to the PEP cantilever. Substituting (2) and (3) into (1), one obtains that  
\[f_{ri}=\frac{k_{i}^{2}}{2\pi}\frac{t_{\mathrm{PEP}}}{L^{2}}\sqrt{\frac{Y_{ \mathrm{m}}\cdot V_{\mathrm{m}}+Y_{\mathrm{p}}\cdot V_{\mathrm{p}}}{12(\rho_{ \mathrm{m}}\cdot V_{\mathrm{m}}+\rho_{\mathrm{p}}\cdot V_{\mathrm{p}})}}. \tag{4}\]  
Because \(\left(Y_{\mathrm{m}}/Y_{\mathrm{p}}\right)\ll\left(\rho_{\mathrm{m}}/\rho_{ \mathrm{p}}\right)\), an increase in the MAE volume affected the denominator in (4) more than the nominator, leading to the decrease in a resonance frequency. Formula (4) makes it possible to estimate the resonance frequency for bending vibrations with good accuracy. A similar formula was used, for example, in [4]. Improvements could be achieved by using more advanced models and numerical calculations. A large number of works were devoted to the theoretical description of the ME effect by various methods. The calculation of ME stresses and frequencies of bending resonances and ME coefficients of composite structures depend on their mechanical and dielectric parameters that they were given, for example, in [28,29]. The calculation of mechanical stresses arising in a composite structure, as well as the modeling of the ME effect in it using the finite element method, is considered, for example, in [30,31]. However, previous theoretical approaches considered only solids and did not take into account the viscoelastic properties of constitutive materials, which should be important, e.g., for MAEs. The existence of the optimum bias field, where the ME coupling is at its maximum, is well known for conventional layered heterostructures. This optimum magnetic field should correspond to the particular magnetic field where the piezomagnetic coefficient \(q=\partial\lambda/\partial H\) has its maximum. For conventional ferromagnetic materials, such as metals and alloys, measurement of the piezomagnetic coefficient can be easily made, e.g., with the help of strain gauges. However, the measurement of magnetostriction \(\lambda\) of soft MAE layers is somewhat a challenge because stiff strain gauges would distort the measurements. To the best of our knowledge, the magnetostriction of MAE slabs had not been measured before. There is on-going research in this area in order to develop the measurement  
Figure 9: Comparison of theoretical (continuous lines) and experimental (dots) dependences of the resonance frequencies \(f_{r1}\) and \(f_{r2}\) on the thickness of the MAE layer. The inset schematically shows the bending modes. setup for accomplishing this problem. According to [32; 33; 34] flexible, environmentally friendly (lead-free) polymer-based ME materials with high ME coefficients, low remanent magnetization/coercive fields, and low leakage currents are one of the most promising research directions for creating magnetic field sensors, actuators, flexible displays, energy harvesting, and haptic devices.