# Actuating Performance Analysis of a New Smart Aggregate Using Piezoceramic Stack
## 2 Basic Theory of SAPS
### Basic Equations  
The constitutive equations of the \(i\)th PZT disc and elastic materials including the epoxy and protective shell are given by [53]  
\[\left\{ \begin{array}{ll}\sigma_{pzti}=C_{33\varepsilon t}\varepsilon_{pzti}- e_{33}E_{i}\\ D_{i}=e_{33}\varepsilon_{pzti}+\kappa_{33}^{\varepsilon}E_{i}\\ E_{i}=-\frac{\partial\phi_{i}}{\partial z}\\ \frac{\partial D_{i}}{\partial z}=0\end{array} \right.\qquad i=1,\;2,\cdots,\; n\;(\text{Piezoelectric materials}), \tag{1}\]  
\[\left\{ \begin{array}{ll}\sigma_{si}=C_{33\varepsilon}\varepsilon_{si}\\ \sigma_{ei}=C_{33\varepsilon}\varepsilon_{ei}\end{array} \right.\qquad i=1,\; 2,\;\cdots,\;n\;(\text{Elastic materials}), \tag{2}\]  
Figure 1: The Structure of SAPS. where \(\sigma_{pzti}\) and \(\varepsilon_{pzti}\) are the stress and strain of the \(i\)th PZT disc in the \(z\)-axis direction, respectively; \(\sigma_{ei}\) and \(\varepsilon_{ei}\) are the stress and strain of the epoxy in the clearance between the protective shell and the PZT disc in the \(z\)-axis direction, respectively; \(\sigma_{si}\) and \(\varepsilon_{si}\) are the stress and strain of the protective shell in the \(z\)-axis direction, respectively; \(\phi_{i}\), \(E_{i}\) and \(D_{i}\) are the electric potential, electric field, and electric displacement of the PZT disc, respectively; \(C_{33e}\), \(C_{33s}\), and \(C_{33 pzt}\) are the elastic moduli of the epoxy, the protective shell, and the PZT disc, respectively; \(e_{33}=C_{33 pzt}d_{33}\), \(\kappa_{33}^{e}=\kappa_{33}^{e}-C_{33 pzt}d_{33}^{2}\), where \(d_{33}\) and \(\kappa_{33}^{e}\) are the piezoelectric coefficient and permittivity coefficient, respectively. Moreover, the kinematic equation at the \(i\)th composite layer of the protective shell, the \(i\)th PZT disc, and the epoxy, as shown in Figure 3, can be expressed as  
\[\frac{\partial F_{ci}}{\partial z}=\big{(}\rho_{pzt}S_{pzt}+\rho_{e}S_{e}+ \rho_{s}S_{s}\big{)}\frac{\partial^{2}u_{ci}}{\partial t^{2}}\ \ i=1,\ 2,\ \cdots,\ n, \tag{3}\]  
where \(S_{pzt}=\pi d_{pzt}^{2}/4\), \(S_{s}=\pi\big{(}d_{o}^{2}-d_{in}^{2}\big{)}/4\), \(S_{e}=\pi\Big{(}d_{pzt}^{2}-d_{in}^{2}\Big{)}/4\), \(F_{ci}\) and \(u_{ci}\) is, respectively, the normal force and the displacement at the \(i\)th composite layer in the \(z\)-axis direction; \(\rho_{s}\), \(\rho_{pzt}\), and \(\rho_{e}\) are the densities of the protective shell, PZT disc, and epoxy, respectively; \(d_{pzt}\), \(d_{o}\), and \(d_{in}\) are the diameter of the PZT disc and the outer and inner diameters of the protective shell, respectively; \(i\) is the time. Additionally, we have the following relationships:  
\[\left\{ \begin{array}{l}F_{ci}=\sigma_{pzti}S_{pzt}+\sigma_{ei}S_{e}+\sigma_{ si}S_{s}\\ \varepsilon_{pzt}=\varepsilon_{si}=\varepsilon_{ei}=\frac{\partial u_{ci}}{ \partial z}\end{array} \right.\ \ i=1,\ 2,\ \cdots,\ n, \tag{4}\]  
Figure 3: The composite layer of the protective shell, the \(i\)th PZT disc, and epoxy. Figure 2: The simplified one-dimensional axial model of SAPS. Combining Equations (1)-(4) gives the following expressions:  
\[\left\{ \begin{array}{l}\chi_{0}\frac{\partial^{2}u_{ci}}{\partial z^{2}}=\chi_{1 }\frac{\partial^{2}u_{ci}}{\partial t^{2}}\\ \frac{\partial E_{i}}{\partial z}=-\chi_{2}\frac{\partial^{2}u_{ci}}{\partial z ^{2}}\end{array} \right.\quad i=1,\;2,\;\cdots,\;n, \tag{5}\]  
where the constants are \(\chi_{0}=S_{pzt}(C_{33 pzt}+\chi_{3})+S_{s}C_{33s}+S_{e}C_{33e}\), \(\chi_{1}=S_{pzt}\theta_{pzt}+S_{s}\theta_{s}+S_{e}\theta_{e}\), \(\chi_{2}=e_{33}/\kappa_{33}^{e}\) and \(\chi_{3}=e_{33}^{2}/\kappa_{33}^{e}\). Suppose that a harmonic voltage is loaded to the SAPS actuator and is expressed by  
\[U(t)=U_{0}e^{j2\pi ft} \tag{6}\]  
where \(U_{0}\) and \(f\) are the amplitude and frequency of the inputted harmonic voltage, respectively; and \(j\) is the unit imaginary number. Therefore, the displacement \(u_{ci}\), electric potential \(\phi_{i}\), and normal force \(F_{ci}\) for the harmonic and steady vibration can be obtained as  
\[\left\{