# Research on the Design Method of Cushioning Packaging for Products with Unbalanced Mass
## 2 Comparative Analysis of Vibration Response of Products with Balanced and Unbalanced Masses  
The buffer material on the bottom of a product can provide support, fixation, and isolate vibrations. In practical engineering applications, the structure of the buffer material cannot be arbitrarily adjusted, and it is constrained by the shape and size of the bottom of the product. The same buffer material is often used to ensure the relative level of the bottom of the product. Therefore, the base of the buffered packaging system is often imbued with the same stiffness and the same vibration-damping materials to simplify the system. First, a type of dimensional model of the cushioned packaging part is established as illustrated in Figure 1. \(G^{\prime}\) represents the center of gravity of the product. The coordinate system is established with the center of gravity as the origin, and its direction is marked as the left direction. Symbols \(l_{1},l_{2},h_{1},h_{2},w_{1}\) and \(w_{2}\) represent the distance from center of mass \(G^{\prime}\) to each side of the part. If \(w_{1}=w_{2}\) and \(h_{1}=h_{2}\), the center of mass and the body center are the same. Otherwise, the center of mass deviates from the body center and the product is in mass non-equilibrium, which produces a center-of-mass offset. Since the analysis is based on total energy, the displacement of the four support points in the direction of each coordinate axis must be calculated to obtain the forces of the spring-damper model. Taking center of mass \(G^{\prime}\) as the origin to establish the coordinate axes, the length of each side is shown in Figure 1. The displacement of the center of mass along the coordinate axis directions are \(x\), \(y\), and \(z\), respectively, and the angle of rotation around the coordinate axes are \(\alpha\), \(\beta\), and \(\gamma\), respectively. After analysis, the coordinates of the four-support points \(A\), \(B\), \(C\), and \(D\) on the bottom surface are as follows. \[ \begin{array}{l}x_{a}=x+l_{2}\beta+h_{2}\gamma,\;y_{a}=y-l_{2}\alpha-w_{1} \gamma,z_{a}=z+w_{1}\beta-h_{2}\alpha\\ x_{b}=x+l_{2}\beta+h_{2}\gamma,\;y_{b}=y-l_{2}\alpha+w_{2}\gamma,z_{b}=z-w_{2 }\beta-h_{2}\alpha\\ x_{c}=x-l_{1}\beta+h_{2}\gamma,\;y_{c}=y+l_{1}\alpha+w_{2}\gamma,z_{c}=z-w_{2 }\beta-h_{2}\alpha\\ x_{d}=x-l_{1}\beta+h_{2}\gamma,\;y_{d}=y+l_{1}\alpha-w_{1}\gamma,z_{d}=z+w_{1 }\beta-h_{2}\alpha\end{array}  \tag{1}\]  
The modeling process is performed by attaching counterweight blocks on both sides of the product, which causes the center of mass to deviate from the geometrical center. The simplified model is shown in Figure 2. The parameters \(k_{i}\) (\(i\) = 1, 2, 3, 4) and \(c_{j}\) (\(j\) = 1, 2, 3, 4) indicate spring stiffness and damping coefficient, respectively. Figure 1: Size model. Simplified models of products with balanced mass and unbalanced mass are all shown in Figure 2. An energy analysis is performed using the same initial conditions. In the case of the mass-balanced product, only translational motion in the \(y\)-direction occurs in the cushioning packaging system. In the case of mass non-equilibrium product, translational motion occurs in the \(y\)-direction as well as rotation in the \(z\)-direction. The kinetic energy, elastic potential energy, dissipative energy and total energy of packing system with balanced mass are set as \(T_{1}\), \(V_{1}\), \(U_{1}\), and \(E_{1}\), respectively, and those of packing system with unbalanced mass are set as \(T_{2}\), \(V_{2}\), \(U_{2}\) and \(E_{2}\), respectively.