## 2 Theoretical Background of Piezolectric Actuators
This curve is crucial for understanding the charging of the piezoelectric stack actuator with a typical electrical capacitance \(C\). The variables in Equation (1) are as follows: \(x_{0}\) is the stress-free displacement [m/V], \(U_{\text{pzt}}\) is the supply voltage [V], \(k_{\text{pzt}}\) is stiffness [N/m], and \(F_{\text{pzt}}\) is the generated force [N]. It is important to note that the performance of piezoelectric actuators depends on the piezoelectric material used and the effective stiffness, as highlighted in [17]. Additionally, the characteristics of piezoelectric stacks can vary under different working conditions, such as temperature and the type of AC or DC signal used for testing, as explained in [18]. \[x_{pzt}=x_{0}\cdot U_{pzt}-\frac{1}{k_{pzt}}\cdot F_{pzt} \tag{1}\]  
Under no-load conditions, the piezoelectric stack actuator does not exert any force, and its displacement \(x_{\text{pzt}}\) corresponds to the free displacement \(x_{\text{free}}\). The free displacement varies linearly with the applied electrical voltage \(U_{\text{pzt}}\) (illustrated in Figure 2 as examples U1, U2, and Umax) and is defined by Equation (2). When the piezoelectric stack actuator operates under a very large load (exceeding the generated force), its output displacement \(x_{\text{pzt}}\) becomes zero, and the actuator generates a blocking force \(F_{\text{block}}\), as described by Equation (3). Additionally, when the piezoelectric stack actuator operates at the maximum voltage \(U_{\text{max}}\), Equation (1) can be rearranged into the form presented in Equation (4), where the generated force is reduced by a portion of the actuator's displacement [15,16]. \[x_{free}=x_{0}\cdot U_{pzt} \tag{2}\]  
\[F_{block}=k_{pzt}\cdot x_{0}\cdot U_{pzt} \tag{3}\]  
\[F_{pzt}=-k_{pzt}\cdot x_{pzt}+k_{pzt}\cdot x_{0}\cdot U_{\text{max}} \tag{4}\]  
Figure 1: Real picture of linear multilayer piezoelectric stack actuator (**a**) and operational principles (**b**). A key consideration when using a piezoelectric stack actuator in practical applications is the type of preload applied to the actuator. As detailed in [15; 16], a piezoelectric actuator behaves as an elastic body with a specific stiffness, denoted as \(k_{\text{pzt}}\). Mechanically, it can be subjected to two distinct types of loading: the first is when the load remains constant during the extension (displacement) process and the second is when the load on the piezo stack varies during the extension process. In our investigation, we focus on the second scenario. Figure 3 illustrates the piezoelectric stack actuator loaded with a spring of stiffness \(k_{\text{s}}\). The free displacement of the unloaded piezoelectric stack actuator is given by the simplified Equation (5) (case A), where \(F_{\text{pzt}}\) represents the generated force of the piezoelectric stack actuator and \(k_{\text{pzt}}\) denotes its stiffness. The free displacement of the spring-loaded piezoelectric stack actuator (case B) is described by Equation (6). A spring load affects the free displacement capability of the piezo stack, reducing it by \(\Delta x\), as shown in Equation (7). Furthermore, the free displacement of the loaded piezoelectric stack actuator is expressed as a function of the original free displacement, as described in Equation (8) [11; 15]. \[x_{freeA}=\frac{F_{pzt}}{k_{pzt}} \tag{5}\]  
\[x_{freeB}=\frac{F_{pzt}}{k_{pzt}+k_{s}} \tag{6}\]  
\[\Delta x=x_{freeA}\cdot\left(1-\frac{k_{pzt}}{k_{pzt}+k_{s}}\right) \tag{7}\]  
Figure 3: Effect of a spring preload on a piezoelectric stack actuator. Figure 2: Force–displacement curve for a piezoelectric stack actuator. \[x_{free}=x_{0}\cdot U_{pzt} \tag{2}\]0