# Vibration Control of an Aero Pipeline System with Active Constraint Layer Damping Treatment
## 2 Materials and Methods
### Transform of Piezoelectric Parameters Matrix  
The structural and performance parameters of each layer are shown in the Tables 1 and 2 respectively. As shown in the Figure 3 arc piezoelectric ceramics are adopted to fit the pipeline well and polarized along the thickness. The parameters of piezoelectric ceramics adopted in this paper are shown in Table 3. \begin{table}
\begin{tabular}{c c c c c} \hline  & **External Diameter (mm)** & **Thickness (mm)** & **Length (mm)** & **Radian (rad)** \\ \hline Pipeline & 18 & 1.5 & 500 & \(2\pi\) \\ Viscoelastic layer & 18.52 & 0.26 & 100 & \(\pi/2\) \\ Active constraint layer & 20.52 & 1 & 100 & \(\pi/2\) \\ \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of the pipeline with ACLD patches. \begin{table}
\begin{tabular}{c c c c} \hline
**Material** & **Elasticity Modulus \(E\) (Pa)** & **Poisson’s Ratio \(\
u\)** & **Density (kg/m3)** \\ \hline Pipeline & \(2.01\times 10^{11}\) & 0.3 & \(8.03\times 10^{3}\) \\ Viscoelastic layer & \(4.5\times 10^{5}\) & 0.499 & 980 \\ Active constraint layer & \(\underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{ \underline{ \underline{ }}}}}}}}}}}}}}\) & 0.3 & 7400 \\ \hline \end{tabular}
\end{table} 
Table 2: Performance parameter of the pipeline with ACLD patches. Figure 3: Piezoelectric coefficient coordinate system. The anisotropic stiffness matrix of the piezoelectric ceramics (polarized in \(Z\) direction) is as follows:  
\[C=\left[ \begin{array}{cccccc}c_{11}&c_{12}&c_{13}&0&0&0\\ c_{21}&c_{11}&c_{23}&0&0&0\\ c_{31}&c_{32}&c_{33}&0&0&0\\ 0&0&0&c_{44}&0&0\\ 0&0&0&0&c_{55}&0\\ 0&0&0&0&0&c_{66}\end{array} \right]=\left[ \begin{array}{cccccc}13.9&7.78&7.43 &\\ 7.78&13.9&7.43&\\ 7.43&7.43&11.5&\\ 2.56&\\ &&&&2.56\\ &&&&3.06\end{array} \right]\times 10^{10} \tag{11}\]  
The electrical parameters of piezoelectric materials include: resistivity, dielectric constant and piezoelectric constants. The dielectric constants of piezoelectric materials used in this paper are as follows:  
\[\varepsilon_{11}^{s}=\varepsilon_{11}^{s},\,\varepsilon_{0}=370\times 8.84\times 1 0^{-12}=3.27\times 10^{-9}, \tag{12}\]  
\[\varepsilon_{22}^{s}=\varepsilon_{22}^{s},\,\varepsilon_{0}=370\times 8.84 \times 10^{-12}=3.27\times 10^{-9}, \tag{13}\]  
\[\varepsilon_{33}^{s}=\varepsilon_{33r}^{s},\,\varepsilon_{0}=635\times 8.84 \times 10^{-12}=5.613\times 10^{-9}. \tag{14}\]  
E-Type piezoelectric equation is adopted in the piezoelectric coupling analysis in this paper:  
\[\left\{ \begin{array}{c}T=c^{E}S-eE\\ D=eS+\varepsilon^{S}E\end{array} \right., \tag{15}\]  
where \(T\) is the mechanical stress, \(S\) is the mechanical strain, \(D\) is the electrical displacement, and \(E\) is the electric field intensity. Piezoelectric parameters \(e_{ij}\) is used to calculate the \(e\) matrix, the parameters are entered as a datasheet, and the parameters in the matrix will change according to the different order of the polarization direction. The piezoelectric stress of piezoelectric ceramics adopted in this paper is as follows:  
\[e=\left[ \begin{array}{cccc}&e_{31}\\ &e_{31}\\ &e_{33}\\ &e_{15}\\ e_{15}\end{array} \right]=\left[ \begin{array}{cccc}&-5.2\\ &-5.2\\ &15.1\\ &12.7&\\ 12.7&\\ \end{array} \right]. \tag{16}\]  
The arc piezoelectric material is polarized in radial direction, while the section of pipeline and piezoelectric materials are defined as X-Y plane when modeling. Therefore, the matrix needs to be transformed to the X-Direction polarized state. The final piezoelectric parameter matrix of the piezoelectric material is thus shown below. Elastic constant matrix (X-Direction polarized):  
\[C=\left[ \begin{array}{cccccc}c_{33}&c_{13}&c_{13}&0&0&0\\ c_{13}&c_{11}&c_{12}&0&0&0\\ c_{13}&c_{12}&c_{11}&0&0&0\\ 0&0&0&c_{44}&0&0\\ 0&0&0&0&c_{66}&0\\ 0&0&0&0&0&c_{44}\end{array} \right]=\left[ \begin{array}{cccccc}11.5&7.43&7.43 &\\ 7.43&13.9&7.78&\\ 7.43&7.78&13.9&\\ &2.56&\\ &&2.56\end{array} \right]\times 10^{10}.