A novel phenomenological model for duffing nonlinearity in magnetoelectric resonators

Mechanical magnetoelectric (ME) resonators, which integrate piezomagnetic and piezoelectric materials, exhibit interesting magnetic-field-dependent nonlinear dynamics due to the dominant nonlinearity of the piezomagnetic component. Despite experimental observation of phenomena like spring-hardening and -softening in ME resonators, a corresponding analytical model has been lacking. Here, we present a novel phenomenological model for magnetic-field-dependent Duffing nonlinearity in voltage-driven ME resonators, where different orders of nonlinearities can be flexibly accessed. First, the length-extensional ME resonator is equivalent to a mass-spring-damper system. In this simplified model, the nonlinear spring constants are introduced by a Taylor series expansion of Young's modulus of piezomagnetic phase. Based on the Zheng-Liu magneto-mechanical constitutive relation, the equivalent spring constants are determined and substituted into the governing equation of the resonator. The numerical solutions reveal transitions among hardening, softening, and mixed nonlinear regimes, depending on the bias-magnetic-field and excitation amplitude. Furthermore, the analytical amplitude-frequency relation at H ≈ 18 Oe is obtained by approximating the governing equation as a cubic-quintic Duffing equation. Subsequently, the proposed model is validated by reproducing the reported experimental results. It is confirmed that the pronounced hardening behavior experimentally observed in the ME resonator under the optimal magnetic bias originates from the elastic nonlinearity of the piezomagnetic phase, and as the bias-magnetic-field varying, the associated jump and hysteresis response in the mechanically-mediated ME coupling can be predicted. This work is anticipated to advance the understanding of the nonlinear effects in ME resonator-based devices, particularly acoustically actuated antennas, providing guidance for engineering nonlinearities to enhance frequency stability.