where F mi is the position-dependent modal magnetic coupling force from the magnetic charge model. Because this truncation yields the cubic nonlinearities, the oscillator model is the 2-DOF Duffing-type oscillator. The cross-sectional dimensions of the metal substrate and the piezoelectric layers are 10 mm × 0.3 mm and 10 mm × 0.052 mm, respectively. For the metal substrate, density and Young’s modulus are set to be 7850 kg/m 3 and 200 GPa. For the piezoelectric layer, 1780 kg/m 3 of density and 3 GPa of Young’s modulus are used. Piezoelec- tric constant and permittivity are − 23 pm/V and 110 pF/m, respectively. The magnets are made of neodymium with 900 kA/m of magnetization. The dimensions of the magnets are 10 mm × 6 mm × 2 mm (width × thick- ness × length). The damping ratios are chosen as 0.015. The length of Beam 1 is 70 mm, and the length of Beam 2 is varied as summarized in Table 1. Note that the parameter values for the dimensions and properties of the system are chosen in physically realistic ranges, referring to the previous report where our oscillator model was validated experimentally 17 .