## 3 Theory and Modeling
### Principal and Modeling of Piezoelectric Transducer Component  
The output signal of piezoelectric transducers is often difficult to apply in signal inversion monitoring due to uncertainties such as excitation frequency and damping. However, it could compensate for the limitations of electrostatic transducers, which have lower power density. The excitation frequency does not necessarily equal the vibration response frequency, and the magnetically actuated piezoelectric transducer utilizes this characteristic to achieve frequency up-conversion, which is commonly used in low-frequency situations where the excitation frequency does not easily reach the resonance frequency. As researched in [22; 23], the magnetically actuated structure can be simplified to state that when the permanent magnet approaches the cantilever beam, it provides a brief magnetic excitation based on magnetic force \(F\), causing the cantilever beam to undergo initial deformation and store maximum elastic potential energy. Once the magnet moves away, based on the principle of damped natural vibration, the excitation can be regarded as ceasing, and the cantilever beam will vibrate from the maximum amplitude, decaying exponentially at its natural frequency to stop, in the form of underdamped response, until the next magnetic excitation occurs. The motion process and output voltage have been confirmed to exhibit periodicity, while the natural frequency, which is also referred to as the first-order characteristic frequency, and the excitation frequency are the primary frequency components. Compared to resonant systems that operate at their characteristic frequency, magnetically actuated systems operate over a much wider frequency range but with lower output. To increase the output power of the magnetically actuated piezoelectric component, in addition to the most basic way that increase the number of permanent magnets to trigger the oscillation decay process based on natural frequency more frequently, changing the structure and dimensions of the piezoelectric cantilever beam is also a method. Studies [24; 25] show that replacing the rectangular cantilever beam with a variable cross-section cantilever beam structure will cause changes in the natural frequency of the piezoelectric cantilever beam, the surface area of the piezoelectric layer, and the stress or charge distribution on surface, thereby affecting the output of the piezoelectric cantilever beam. Thus, increasing the output of the magnetically actuated piezoelectric component based on the optimization method of non-uniform cross-section cantilever beams is chosen. The coupled structure uses four sets of piezoelectric double-layer cantilever beams in parallel, with a single component structure used for calculation and analysis shown in Figure 9.  
When the polarization directions of the upper and lower piezoelectric layers are the same, the polarities are opposite during the bending deformation of the cantilever beam, and they are connected in parallel. Assuming that the piezoelectric cantilever beam model is composed of linearly elastic materials with uniform distribution, small deformations occur within the linear elastic range to avoid damaging the fragile piezoelectric layer. Under the linear elastic deformation model, stress and strain are proportional and exhibit a uniform distribution. When the rotation positions the permanent magnet directly below the active end of the piezoelectric cantilever at a certain moment, the magnetic force \(F\) between the magnets is maximized at this point, which will cause the piezoelectric cantilever to undergo the greatest linear elastic deformation. Based on the above conditions, combining with the research in [24], the strain distribution \(S_{F}(x,\,z)\) at the maximum linear elastic deformation of the length direction \(x\) can be considered as Equation (8):  
\[ \begin{split}& S_{F}=\frac{M(x)z}{Y(x)}\\ &=\frac{F(l_{p}+0.5l_{m}-x)z}{b(x)\gamma}\end{split}