## 2 Background and Driving Principles
### Structure of Typical PAMTs
The mechanical dynamic response of piezoelectric actuators can be characterized by the electromechanical coupling constitutive equations--specifically, the following:  
\[S=sT+d^{T}E \tag{4}\]  
\[D=dT+\varepsilon E \tag{5}\]  
where \(S\) is the strain; \(T\) is the stress; \(E\) is the applied electric field intensity; \(D\) is the electric displacement; \(s\) is the compliance matrix (elastic compliance under a constant electric field); \(d\) is the piezoelectric constant matrix; \(d^{T}\) is the transpose of the piezoelectric constant matrix; and \(\varepsilon\) is the dielectric constant matrix (dielectric constant under constant stress). By introducing a frequency-dependent dielectric constant model, as shown in Equations (6) and (7), the suppression effect of dielectric relaxation on the charge quantity under variable frequency conditions can be revealed. Specifically, the charge quantity \(Q(\omega)\) decreases as the capacitance decreases with increasing frequency. \[C(\omega)=\frac{\epsilon}{1+(\omega\tau)^{2}} \tag{6}\]  
\[Q(\omega)=C(\omega)\cdot V=\frac{\epsilon V}{1+(\omega\tau)^{2}} \tag{7}\]  
where \(C(\omega)\) is the capacitance and \(V\) is the voltage. Furthermore, the impedance effect imposes additional limitations on the electrical and mechanical behavior under variable frequency conditions. Under such conditions, the equivalent impedance of piezoelectric ceramics is determined by the coupling of electrical and mechanical impedance, expressed as  
\[Z(\omega)=R+j\omega M-\frac{1}{j\omega C} \tag{8}\]  
At high frequencies, the capacitive term \(\frac{1}{j\omega C}\) becomes small, while the mechanical impedance term \(j\omega M\) increases significantly, leading to changes in the overall impedance and suppressing the rapid accumulation of charge. Simultaneously, the mechanical dynamic behavior is significantly influenced by inertia and damping effects, described by the dynamic equation  
Figure 1:0  
The frequency response of the displacement \(u(\omega)\) is given by  
Figure 1:1  
At high frequencies, the inertia term \(-\omega^{2}m\) and the damping term \(j\omega c\) dominate the system's behavior, leading to a significant reduction in mechanical deformation. Combining the above analysis, the charge quantity and deformation of the piezoelectric actuator under variable frequency conditions can be uniformly expressed as follows:  
Figure 1:2  
The impedance effect further suppresses the mechanical response by influencing the charge quantity and electric field. Overall, when the electric field intensity is constant, under high-frequency conditions, the combined effects of dielectric relaxation, impedance changes, and mechanical inertia and damping result in a significant decrease in both the charge quantity and deformation of piezoelectric actuators with increasing frequency. Due to the frequency dependence of the nonlinear characteristics of piezoelectric actuators in the high-frequency range, four voltage frequencies were selected in this paper to avoid the influence of frequency dependence. In addition, this is also in line with the general operating conditions of PAMTs. By applying the same maximum voltage but at different frequencies for the S-shaped curve trajectory driving voltage on the piezoelectric actuator, the nonlinear characteristics can be observed, as shown in Figure 1:3  
The specific derivation formula of the S-shaped trajectory driving voltage is shown in Appendix A. The red line represents the voltage boost phase, and the blue line represents the voltage back phase. As can be seen from Figure 2, the nonlinear hysteresis loops at different frequencies are all counterclockwise, the voltage boost phase presents a concave-then-convex curve shape, and the voltage back section presents a completely convex curve shape, which conforms to the general law of the deformation speed of the voltage boost phase, and presents the same trend.