## 2 Background and Driving Principles
### Structure of Typical PAMTs
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Technical Indicators** & **Numeric Value** & **Unit** \\ \hline Stiffness & 24.5 & \(\upmu\)m/N \\ Maximum displacement & 45 \(\pm\) 1 & \(\upmu\)m \\ Displacement loss ratio & 18.3\% \(\pm\) 0.6 & \\ Hinge minimum width d & 1 & mm \\ Hinge cutout radius R & 2 & mm \\ Span length L & 22 & mm \\ Spring recovery force & 1344 & N \\ Piezo minimum preload & 968 & N \\ Natural frequencies & 858.8 & Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Technical parameters of PAMT structure. Figure 1: Typical structure of PAMT. Analysis of Nonlinear Characteristics of Piezoelectric Actuators Under S-Shaped Trajectory Driving Voltage  
The dynamic behavior of piezoelectric actuators is jointly influenced by their electrical properties, mechanical properties, and the coupling effects between them. The electrical behavior of piezoelectric actuators can be modeled as an equivalent parallel circuit, where R represents the leakage resistance of the material, reflecting the dissipation of internal charge, and C represents the equivalent capacitance of the material, reflecting its ability to store charge. In this equivalent circuit, the time constant \(\tau=RC\) is referred to as the first-order relaxation time. The dielectric response of piezoelectric actuators is determined by the internal electric dipole moments. When an external electric field is applied, the field drives the rearrangement of the internal electric dipole moments, resulting in polarization. Additionally, the rearrangement of electric dipole moments requires overcoming lattice barriers or internal molecular resistance, leading to a time delay in the polarization process. This time delay is determined by the material's internal atomic structure and thermodynamic behavior, manifesting as a typical relaxation process. The dielectric relaxation behavior of piezoelectric ceramics can be described via the Debye model [29]. Its fundamental assumption is that the response of electric dipole moments follows a first-order relaxation process, satisfying the following dynamic equation:  
\[\frac{dP}{dt}+\frac{P}{\tau}=\epsilon_{0}\frac{dE}{dt} \tag{1}\]  
where \(P\) is the polarization intensity; \(E\) is the applied electric field intensity; \(\tau\) is the relaxation time; and \(\epsilon_{0}\) is the vacuum permittivity. By solving this equation, the complex dielectric constant of the material in the frequency domain is obtained:  
\[\varepsilon(\omega)=\varepsilon_{\infty}+\frac{\varepsilon_{s}-\varepsilon_{ \infty}}{1+(\omega\tau)^{2}}\quad\text{(Real part)} \tag{2}\]  
\[\varepsilon^{*}(\omega)=\varepsilon_{\infty}+\frac{(\varepsilon_{s}- \varepsilon_{\infty})\omega\tau}{1+(\omega\tau)^{2}}\quad\text{(Imaginary part)} \tag{3}\]  
where \(\varepsilon_{\infty}\) is the static dielectric constant at a high frequency; \(\varepsilon_{s}\) is the static dielectric constant at a low frequency; \(\omega\) is the angular frequency; and \(\tau\) is the relaxation time. When \(\omega\tau\ll 1\) (low frequency), the material has sufficient time to respond to the electric field, and the complex dielectric constant \(\varepsilon\approx\varepsilon_{s}\). When \(\omega\tau\gg 1\) (high frequency), the electric dipole moments cannot follow the changes in the electric field, and the complex dielectric constant \(\varepsilon\approx\varepsilon_{\infty}\). This phenomenon can be explained by the electromechanical dynamic coupling theory.