# Vibration Control of an Aero Pipeline System with Active Constraint Layer Damping Treatment
## 2 Materials and Methods
### FEM Model of ACLD Patches  
A pipeline with ACLD patch is modeled by the modeling sub-module in the preprocessing of ANSYS, and the three-dimensional model of each layer structure is obtained by the CYLIND command. The geometric model is shown in Figure 4. The uppermost layer is a piezoelectric constraint layer; the middle layer is a viscoelastic layer; and the third one is the upper surface of a pipeline. The properties of each layer are shown in Table 2.  
The adjacent surfaces of each layer structure are bonded to make it a whole using the bonding command in ANSYS while giving different material properties to different bodies. The force and deformation transfer between the layers can correspond with the actual situation. The meshing method is carried out by sweeping command: the unit size is 0.001 mm. The finite element model shown in Figure 5 consists of 60,034 nodes and 18,013 units. The viscoelastic layer and the pipeline are modeled using a Solid186 unit, which is a three-dimensional solid structure with 20 nodes (each node have three translational degrees of freedom UX, UY, UZ). Since the piezoelectric constraint layer is an electromechanical device, the unit type must support the piezoelectric coupling analysis. The unit types that support coupled field  
Figure 4: Pipeline with ACLD patch. Figure 5: Finite element model of active constrained layer damping pipeline. analysis in ANSYS are Plane13, Solid5, Solid98 and so on. The Solid5 unit can be used to analyze electric fields, magnetic fields, piezoelectric fields, structural fields, thermal fields and to form limited coupling between fields. The Solid5 unit has 8 nodes while each node has 6 degrees of freedom. In this research, 4 degrees of freedom are used, which are the translational degrees of freedom in the X, Y, and Z axis (UX, UY, UZ) and VOLT degree of freedom. Fixed constraints are applied at both ends of the pipeline. The upper and lower surfaces of the piezoelectric constraint layer need to be applied with a voltage. Through the node coupling command, the minimum node number is obtained. Then a voltage of 50 V is applied on the node. The boundary conditions are shown in Figures 6 and 7. When the base of the installed pipeline vibrates, the vibration energy is transmitted to the pipeline through the support. When the vibration frequency of the base is close to the modal frequency of the pipeline, a large vibration of the tubular body is caused. The finite element analysis software is used to analyze the transient response of the pipeline with the ACLD patches at the root, and the response of the intermediate node of the pipeline is considered when different voltages are applied. The damping performance of the ACLD patches is investigated. By using a sinusoidal excitation of the piping system and extracting its frequency domain response, the loss factor \(\eta\) or damping ratio \(\zeta\) of the piping system can be identified using the half power bandwidth method. The structural damping of the pipeline can be regarded as Rayleigh damping,  
\[C=\alpha M+\beta K, \tag{20}\]  
where, \(\alpha\) and \(\beta\) are the Rayleigh damping coefficients and meet the equation below with the damping ratio \(\zeta\) of the pipeline. \[\frac{\alpha}{2\times\omega_{0}}+\frac{\beta\omega_{0}}{2}=\zeta, \tag{21}\]  
The Rayleigh damping coefficient calculated by Equation (21) is: \(\alpha=14.903\), \(\beta=2.4\times 10^{-6}\). Figure 6: Fixed boundary. Figure 7: Voltage boundary. The selected pipeline structural parameters and material parameters are as Tables 1 and 3. The basic excitation is applied to the entire pipeline structure, using a constant acceleration amplitude excitation, \(a=0.2\times g\times\sin(2\times\pi\times 375\times t)\), the voltage on piezoelectric constraint layer is U = 0 V, and the pipeline is fixed at both ends to constrain all node degrees of the end face of the pipeline. The integral time step is \(\Delta t=0.000125\)s. The first load is a step load, and the subsequent loads are gradient loads.