# Research on the Design Method of Cushioning Packaging for Products with Unbalanced Mass
## 3 Analysis of Response for Damping Material to Center-of-Mass Offset System  
Since damping materials have good corrosion-resistance, vibration-damping, and anti-buffering properties, damping vibration-isolation materials are used to correct the vibration response of a mass-unbalanced product in this paper. When four counterweight blocks are adopted and the center of the left sides or right sides are symmetrically installed, then the center of mass can be derived from its geometrical center, \(\lambda_{x}\) = 19.79 mm. Similarly, when eight counterweight blocks are asymmetrically installed, the center of mass is offset by \(\lambda_{x}\) = 35.88 mm. When 16 blocks are installed, the eccentric distance is \(\lambda_{x}\) = 83.69 mm. First, the different theoretical models corresponding to different offset distances of the center of mass are simulated and analyzed by applying workbench [29; 30]. Following this, a harmonic response analysis is carried out [31; 32]. In addition, the peak acceleration response is discussed according to the frequency acceleration curves. In this paper, the frequency is set to be from 3 to 21 Hz and the acceleration is set to 1 m/s\({}^{2}\) in \(y\) direction. The theoretical model is shown in Figure 4, and the material parameters are listed in Table 1.  
Figure 3: Comparison of vibration response of product with balanced and unbalanced mass. (**a**) Mass-balanced. (**b**) Mass-unbalanced. A harmonic response is added to simulate the effect of a cushioning material on the cushioning characteristics of a mass-eccentric system. The effect of changing relevant parameters of the same cushioning packaging material, i.e., the area, thickness and density of the damping material, on the cushioning characteristics of the mass-eccentric system is analyzed when the quality of product packaging is maintained at a certain level. The results of the dynamic response of the mass-eccentric system, with different parameters that have the same cushioning-damping packaging material, are shown in Table 2.  
Here, the ratio in Table 2 is the ratio between the acceleration amplitude of the mass-unbalanced product with different damping-installation methods to the maximum acceleration with the same balanced mass without damping-optimized installation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Materials** & \begin{tabular}{c} **Density** \\ **(kg/m3)** \\ \end{tabular} & \
\begin{tabular}{c} **Young’s Modulus** \\ **(MPa)** \\ \end{tabular} &
\begin{tabular}{c} **Poisson’s Ratio** \\ **Poisson’s Ratio** \\ \end{tabular} \\ \hline Object & Aluminum alloy & 2770 & \(7.1\times 10^{4}\) & 0.33 \\ Counterweight block & Structural steel & 7850 & \(2.0\times 10^{5}\) & 0.30 \\ Vibration Isolators & ASI304 & 8000 & \(1.9\times 10^{11}\) & 0.29 \\ Damping material & EPE & 16 & \(5.5\times 10^{5}\) & 0.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of material. Figure 4: Theoretical model. 1—object block. 2—spring vibration isolator. 3—shaker tabletop. 4—clamp plate. 5—angular aluminum fixings. 6—damping blocks. The parameters in Table 2 are based on commonly used buffered-packaging design parameters for experimental design. In this study, we studied the vibration response of products with unbalanced mass by changing the thickness, width, height and density of damping materials. The data in Table 2 show the vibration response of the mass unbalanced product with the same damping-material characteristics. It can be observed from the data in Table 2 that there are differences in the maximum acceleration under different thickness conditions. Comparing the data at larger thicknesses (50 mm) and smaller thicknesses (40 mm and 30 mm), it can be find that the maximum acceleration at larger thicknesses is relatively small. This may be due to the fact that the larger thickness provides greater structural stability and reduces the energy loss from vibration conduction.