# Dynamic Analytical Solution of a Piezoelectric Stack Utilized in an Actuator and a Generator
## 4 Comparison and Discussion  
In the previous sections, the exact solutions of the mechanical components and electric components of piezoelectric stacks were obtained. In order to gain a comprehensive understanding, the present theoretical results are compared with experimental results from other investigations. First, comparisons between the present theoretical results and experimental findings for a piezoelectric stack subjected to a voltage load were conducted. [PERSON] et al. [40] conducted an experimental study on the electric-mechanical behavior of piezoelectric stack actuators. In their investigation, the Physik Instrumente P-245.70 type stack actuator was adopted, whose geometry information and material parameters can be found in Table 1. The number of total layers and the thickness of each layer in the stack are 182 and 0.5 mm, respectively. In the present analysis, the analytical model and the parameters of the stack used by [PERSON] et al. [40] were adopted. Figures 2 and 3 give the magnitude and phase of the coupled impedance changing with the loading frequency, respectively. In the calculation, the elastic constant \(C_{33P}\) and the dielectric constant \(\kappa_{33}^{\sigma}\) are replaced with their complex forms, i.e., \(C_{33P}{}^{*}=C_{33P}(1+\eta_{m}j)\) and \(\kappa_{33}^{\sigma}{}^{*}=\kappa_{33}^{\sigma}(1+\eta_{e}j)\), to take into account the mechanical and dielectric losses. Here, \(\eta_{m}\) and \(\eta_{e}\) are the mechanical and dielectric loss factors, as listed in Table 1. From Figures 2 and 3, it can be found that the present analytical results match well with the experimental results for both the magnitude and phase of the coupled impedance. The difference is that the predicted resonance and anti-resonance frequencies are slightly lower than in the experimental data. Figure 4 gives the relationship between the output displacement amplitude and the external voltage frequency for different voltage amplitudes. It is obvious that the resonance frequency corresponds well with the above findings. In addition, the output displacement amplitude increases linearly with the increase of the external voltage amplitude. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Symbol** & **Actuator [40]** & **Generator [32]** \\ \hline Elastic constant (elastic modulus) & \(C_{33P}\left(\times 10^{9}\mathrm{N/m^{2}}\right)\) & 32.7 & 44 \\ Piezoelectric constant & \(d_{33}\left(\times 10^{-12}\mathrm{C/N}\right)\) & 427 & 650 \\ Dielectric constant & \(\kappa_{33}^{\prime}/\varepsilon_{0}\) (\(\mathrm{nF/m}\)) & 2203 & 6802 \\ Density & \(\rho_{P}\) (\(\mathrm{kg/m^{2}}\)) & 7800 & 5750 \\ Number of layers & \(N\) & 182 & 130 \\ Layer thickness & \(h_{P}\) (mm) & 0.5 & 0.123 \\ Stack area & \(S\) (mm\({}^{2}\)) & 78.5 & 25 \\ Stack length & \(L\) (mm) & 91 & 16 \\ Spring stiffness & \(k_{s}\left(\times 10^{6}\mathrm{N/m^{2}}\right)\) & 9.66 & \(\sim\) \\ Spring damping & \(c_{s}\) & 0.01 & \(\sim\) \\ Spring mass & \(m_{s}\) (g) & 7.3 & \(\sim\) \\ Output rod mass & \(m_{rad}\) (g) & 0.4 & \(\sim\) \\ Mechanical loss factor & \(\eta_{m}\) & \(8.33\times 10^{-3}\) & \(\sim\) \\ Dielectric loss factor & \(\eta_{e}\) & \(15\times 10^{-3}\) & \(\sim\) \\ \hline \hline \end{tabular} \(\varepsilon_{0}=8.85\times 10^{-12}\mathrm{F/m}\), permittivity of free space.  
\end{table} 
Table 1: Geometric and material parameters of the piezoelectric stack. Figure 3: Phase of coupled impedance versus loading frequency. Figure 2: Magnitude of coupled impedance versus loading frequency. When an external voltage with amplitude \(V_{0}=2\) V is applied, Figures 5 and 6 give the influence of the external voltage frequency on the distribution of stress \(\sigma_{zP}(z)\) and displacement \(u_{P}(z)\) along the axial direction, respectively. Here, it should be noted that when the voltage frequency is far from the resonance frequency, the mechanical loss, dielectric loss, as well as the spring damping can be neglected because their effects are very limited.