## 5 Case Study
### Specimens and Frequency Response Test of the CLD Plate  
Figure 7 shows the aluminum plate specimens before and after damping treatment. The geometry and material parameters of the substrate, constraining layer, and viscoelastic core layer are listed in Table 1. It should be noted that the lengths listed in Table 1 are the effective length used in dynamic analysis and the total length of this aluminum plate is 331.2 mm, so the clamping length of the cantilever constraint is 31 mm. All the geometry parameters are measured by a vernier caliper. The density, storage modulus, and Poisson's ratio of the substrate, and constraining layer and the Poisson's ratio of the viscoelastic core layer, are known, and the density of the ZN_1 viscoelastic material can be obtained according to the definition of density. However, the storage modulus and loss factor should be identified by the method developed in this study. In the actual test, the aluminum plate was fixed on the vibration shaker in cantilever status and the tightening torque of bolts was 39 N.m. The testing instruments used in this test are all shown in \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Type of Materials** & \begin{tabular}{c} **Length** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Width** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Thickness** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Storage** \\ **Modulus (GPa)** \\ \end{tabular} & \begin{tabular}{c} **Density** \\ **(kg/m3)** \\ \end{tabular} & \
\begin{tabular}{c} **Poisson's** \\ **Ratio** \\ \end{tabular} &
\begin{tabular}{c} **Loss** \\ **Factor** \\ \end{tabular} \\ \hline Aluminum plate substrate & 300.2 & 199.9 & 1.9 & 6.89 & 2700 & 0.3 & 0 \\ Steel constraining layer & 300.2 & 199.9 & 0.3 & 209 & 7890 & 0.3 & 0 \\ Viscoelastic core layer ZN\_1 & 300.2 & 199.9 & 0.5 & - & 789.5 & 0.49 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometry and material parameters of each layer of the CLD plate. Figure 7: Aluminum plate specimens. (**a**) Before damping treatment; (**b**) After damping treatment. Table 2, and the corresponding photograph is shown in Figure 8. The laser vibrometer was used to pick up the vibration signal in the process of the experiment and the coordinate of the pick vibration point was (60 mm, 60 mm), which is in accordance with the theoretical analysis in Section 3. According to the requirement of the parameter identification of viscoelastic material, a hammer test was carried out for the aluminum plate without damping treatment. The half-power bandwidth method was adopted to obtain the modal damping ratio of the plate without damping treatment, and the relevant results are listed in Table 3. Subsequently, using a vibration shaker, a segmented frequency sweeping test was carried out for the aluminum plate with damping treatment, according to the determined frequency sweeping interval which contains the resonance frequency. The frequency responses in the first, third, and fifth order resonance region were chosen to identify the parameters of the viscoelastic core layer. The corresponding level of excitation was 1 g and the relevant results are shown in Figure 9.  
Figure 8: Actual photograph of the test. Figure 9: Vibration response obtained by the experiment and theoretical calculations. (**a**) The first order resonance region; (**b**) The third order resonance region; (**c**) The fifth order resonance region. In order to demonstrate the rationality of the identification results, the fractional derivative model of viscoelastic materials was written in the form of the complex modulus and substituted into the analysis model. The frequency response of the CLD plate needs to be calculated at each frequency point (the calculation step is 1 Hz). Firstly, the storage modulus and loss factor of the viscoelastic material were determined according to each frequency value.