## 6 Conclusions  
Under horizontal acceleration excitation in vehicles, we have analyzed the potential motion states of freestanding packages and established the conditions governing each state. Among these, particular attention has been given to the rocking response due to its critical importance for transportation safety. The key findings of this study are:  
1. Motion state analysis: Following the initiation of rocking motion and subsequent package-vehicle impact, four distinct motion states are possible: static equilibrium, bouncing, rocking, and detachment. The governing factors for these states are the ratio \(h/b\) and the restitution coefficient \(\epsilon_{N}\) at the impact point. A tangential force \(\Lambda_{T}\) develops at the pivot point, necessitating that the interfacial friction force exceeds \(\Lambda_{T}\) to prevent package sliding. 2. Numerical algorithm development: We have developed an efficient solution for the piecewise-smooth differential Equation (26) by modeling the boundary as a hyperplane in state space \(x\) and implementing a time transformation method (\(s=\psi(x(t))\), \(t=\alpha(s)\)). The algorithm precisely determines impact time instants \(t_{f}\) when \(s(t_{f})=0\), corresponding to boundary crossings in Equation (26). This approach partitions the solution into smooth rocking intervals between impacts, significantly improving computational efficiency while maintaining accuracy. 3. Error analysis: The accuracy of the proposed algorithm is primarily governed by the time step \(\sigma_{1}\) and order \(p\) of the Runge-Kutta method in [0, \(t_{k}\)], as well as time step \(\sigma_{2}\) and order \(q\) of the single-step Runge-Kutta method in [\(t_{k}\), \(t_{f}\)]. The dominant factor \(\sigma_{1}\) enables effective error control through appropriate step size selection. The proposed method offers substantial improvements over conventional Runge-Kutta and equivalent model approaches by precise determination of impact time instants, the identification of smooth response intervals, and enhanced computational efficiency while maintaining solution accuracy. This work provides a robust framework for analyzing freestanding package dynamics under some transportation conditions, with particular relevance for safety assessment and package design optimization. Future studies will utilize the proposed computational framework to systematically investigate the rocking dynamics of freestanding packages across a comprehensive parameter space, including varying geometric configurations and transportation-induced excitations. Through this parametric analysis, we will employ machine learning techniques to establish robust quantitative relationships between (1) package dimension parameters, (2) excitation spectra and intensity profiles, and (3) resultant rocking amplitude probability distributions. These data-driven correlations will facilitate the development of predictive safety metrics for \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Runtimes** & **MSE** \\ \hline Proposed method & 161 min & 0.0066 \\ Second-order Runge–Kutta & 317 min & 0.0268 \\ Third-order Runge–Kutta & 563 min & 0.0135 \\ Fourth-order Runge–Kutta & 837 min & 0.0054 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Methods comparison. evaluating package stability during transport, ultimately enabling optimized packaging design and risk mitigation strategies. Conceptualization, Z.Q. and D.Z. ; methodology, Z.Q. ; software, Z.Q. and D.Z. ; validation, Z.Q. and D.Z. ; formal analysis, D.Z. ; investigation, Z.Q. ; resources, D.Z. ; data curation, D.Z. ; writing--original draft preparation, Z.Q. ; writing--review and editing, Z.Q. ; visualization, Z.Q. ; supervision, D.Z. ; project administration, D.Z. ; funding acquisition, D.Z. All authors have read and agreed to the published version of the manuscript. This research was supported by the Gansu provincial Key Research and Development Project, under grant number 25 YFGA048, and the National Natural Science Foundation of China, grant number 51765028.