## 2 Motion States of Freestanding Package  
During the transportation of freestanding packages, vehicle horizontal acceleration \(\ddot{u}_{g}\) is potentially induced by vehicle braking, speed variations, and train marshalling. This excitation may cause the package to rock or slide on the base. Additionally, impacts between the bottom of the package and the vehicle during rocking can lead to the package jumping. The motion states and parameter definitions of a freestanding package under horizontal acceleration are illustrated in Figure 1. Here, \(2h\) and \(2b\) represent the height and width of the package, respectively, \(m\) denotes the package mass, and \(\alpha\) is the slenderness angle of the package, where \(\tan\alpha=b/h\). The state of the package is described by the vector \(\mathbf{q}=[y_{h}y_{v}.\theta]^{\mathrm{T}}\), where \(y_{h}\) and \(y_{v}\) are the horizontal and vertical displacements of the package, and \(\theta\) is its rotation angle. The corresponding normal displacements of points 1 and 2 are denoted by \(g_{N1}\) and \(g_{N2}\). The interaction between the package and the vehicle can be modeled as a multibody motion problem with unilateral contact. The motion equations of rocking package can be derived using the Newton-Euler equations:  
\[M\ddot{\mathbf{q}}-\mathbf{f}(\mathbf{C},\mathbf{K},\ddot{u}_{g}(t))-\mathbf{W}_{N}\mathbf{\lambda}_{ N}-\mathbf{W}_{T}\mathbf{\lambda}_{T}=\mathbf{0} \tag{1}\]  
Here, \(\ddot{u}_{g}\) represents the vehicle acceleration, while \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) denote the mass, damping, and stiffness matrices of the package-vehicle impact system, respectively. The vector \(\mathbf{f}\) includes external forces, damping forces, and elastic forces. \(\mathbf{W}_{N}\) and \(\mathbf{W}_{T}\) are direction matrices of contact force and constraint in the normal and tangential directions, respectively. \(\mathbf{\lambda}_{N}\) and \(\mathbf{\lambda}_{T}\) represent the normal and tangential contact force vectors of the impact system. For the package-vehicle impact system illustrated in Figure 1, the matrices and vectors in Equation (1) are defined as follows:  
\[\mathbf{M}= \begin{bmatrix}m&0&0\\ 0&m&0\\ 0&0&I_{G}\end{bmatrix} ,\mathbf{f}= \begin{bmatrix}-m\ddot{u}_{g}(t)\\ -m\mathfrak{g}\\ 0\end{bmatrix} ,\ \mathbf{W}_{N}= \begin{bmatrix}0&0\\ 1&1\\ -b&b\end{bmatrix} ,\ \mathbf{W}_{T}= \begin{bmatrix}1\\ 0\\ h\end{bmatrix}  \tag{2}\]  
where \(I_{G}\) is moment of inertia of the package about the center of gravity, \(I_{G}=(1/3)mR^{2}\), and \(R^{2}=h^{2}+b^{2}\). Figure 1: Schematic of freestanding package motion under horizontal base excitation. Denoting the time instants immediately before and after the impact as \(t^{-}\) and \(t^{+}\), respectively, Equation (1) is integrated over the time interval [\(t^{-}\),\(t^{+}\)]. During the integration of Equation (1) over the infinitesimal time interval [\(t^{-}\), \(t^{+}\)], we consider only the impulsive forces as contributing to the impact dynamics, in accordance with standard impact theory Vector \(f\) only contains the non-impulsive forces; then, the impulses are simply the time integrals of contact forces \(\lambda_{N}\) and \(\lambda_{T}\).