## 3 Theory and Modeling
### Principal and Modeling of Piezoelectric Transducer Component
When the rotation moves the permanent magnet away from the piezoelectric cantilever beam, it can be considered that the applied force disappears, and the cantilever beam vibrates with exponentially decaying amplitude at its natural frequency \(\omega_{n}\) to stop. Studies [22; 23] have shown that the output voltage of the magnetically actuated structure vary with the vibrational deformation of the cantilever beam, exhibiting the same exponential oscillatory decay based on the natural frequency \(\omega_{n}\), in the form of underdamped response. Therefore, the output voltage can be defined as \(U_{p}X(\omega_{n}t)\), where \(U_{p}\) represents amplitude, and \(X(\omega_{n}t)\) is the oscillatory function that characterizes the exponential oscillatory decay of the voltage based on the natural frequency as in Equation (11) under ideal conditions:  
\[X(\omega_{n}t)=e^{-\zeta\omega_{n}t}\cos(\omega_{n}t\sqrt{1-\zeta^{2}}+\phi) \tag{11}\]  
where \(\phi\) represents the initial phase, and \(\zeta\) is the damping ratio. For the simplified model of linear elastic small deformation, the piezoelectric cantilever beam was expected to respond well under magnetically actuated excitation without excessive damping; this model is designed and analyzed with light damping ratio \(\zeta\) of 0.045. Since it has been mentioned earlier that the internal equivalent capacitance \(C_{\rm p}\) is regarded constant during the vibrational deformation process, it can be inferred that the total surface charge \(Q_{p}\) will also follow the same pattern as the output voltage during vibration, which can be defined as \(Q_{p}X(\omega_{n}t)\). And the output current of the piezoelectric component is derived from the change in surface charge that flows through the circuit, with no current output in the static state. During the vibration process, the output current could be represented by \(dQ_{p}X(\omega_{n}t)/dt\) or \(C_{p}dU_{p}X(\omega_{n}t)/dt\) ideally in this simplified piezoelectric model, which is as a capacitance-like structure. Therefore, increasing the natural frequency \(\omega_{n}\) could increase the charge's rate of change and then enhance the output current amplitude of the piezoelectric component within one cycle. Based on research in [24; 25], a cantilever beam with a non-uniform cross-section has been approved, and it was found that increasing the width of the fixed end will enhance its bending stiffness and elevating the characteristic frequency. Conversely, reducing the width of the movable end decreases the stiffness and characteristic frequency. Thus, the method of achieving non-uniform cross-section piezoelectric cantilever beams by varying \(b_{1}\) or \(b_{2}\) can change the output voltage, and the output current also could be enhanced by the increase in natural frequency through this method. Based on this, the design of the piezoelectric cantilever beam's size and structure can be optimized to improve the output voltage or output current, thereby increasing the output power of the magnetically actuated piezoelectric component. A finite element simulation of an electromechanical coupled multi-physical distributed parameter model is established to calculate the open-circuit voltage and characteristic frequency of the non-uniform cross-section model resulting from varying the widths at both ends of the cantilever beam. Using the finite element simulation software COMSOL v.6.2, set the boundary conditions with the fixed-support auxiliary base at the left end of the cantilever beam and the voltage on the inner surface of the piezoelectric sheet in parallel to be zero.