## 5 Example  
To validate the rocking analysis method proposed in this study, we conducted an experiment on a rocking package excited by sinusoidal base acceleration. Additionally,  
Figure 3: Algorithm flowchart. numerical analysis of the package rocking response under random load excitation was performed. The experimental and numerical results were then compared with the response predicted by the algorithm proposed in this paper. An experimental system was designed to record the response of the rocking package, as illustrated in Figure 4. The system consists of a vibration exciter, a test table, a vibration controller, a rocking block, an acceleration data acquisition system, a high-speed camera, and other components. The vibration exciter is rotated and connected to the test table, while control parameters are set on the vibration controller to ensure that the block on the table is subjected to horizontal vibration with the desired characteristics. The block simulates the freestanding package, and the excitation acceleration on the test table is measured with an accelerometer and recorded by a computer through the acceleration data acquisition system. The high-speed camera is adjusted so that the axis of the camera lens is perpendicular to the plane of the block and passes through its centroid. The rocking motion of the package is recorded with the camera, and the rocking angle at each time instant is determined. The vibration controller is configured to apply sinusoidal excitation to the test table. Considering the transition of the excitation from the initial state to the sinusoidal state, the acceleration of the test table is expressed as:  
\[\ddot{u}_{g}(t)=A_{0}\Big{(}1-\mathrm{e}^{-\beta t}\Big{)}\sin\omega t \tag{44}\]  
The parameters of the excitation acceleration and the block are provided in Table 1. The acceleration of the test table is shown in Figure 5a. The algorithm proposed in this study, based on time transformation, is applied to analyze the rocking response of the block. The rocking response predicted by the algorithm is shown in Figure 5b, alongside the recorded rocking response obtained from the high-speed camera. From the results, it can be concluded that the proposed algorithm accurately analyzes the rocking response of the freestanding package. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Excitation amplitude \(A/\mathrm{g}\) & 2.0 \\ Angular frequency of excitation \(\omega/\mathrm{rad}/\mathrm{s}\) & 4\(\pi\) \\ Sinusoidal correction parameter \(\beta\) & 1.76 \\ Slenderness angle \(\alpha/^{\circ}\) & 20 \\ Block half-diagonal length \(R/\mathrm{m}\) & 0.6 \\ Block rocking frequency \(p/\mathrm{Hz}\) & 3.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of excitation acceleration and rocking block. Figure 4: Experimental system for freestanding package rocking under sinusoidal excitation. In addition to sinusoidal excitation, random vibration is applied to the rocking block to further validate the proposed algorithm. The recorded random acceleration of the vehicle during transportation is shown in Figure 6a. The algorithm proposed in this study is employed to predict the rocking response, with the time step \(\sigma_{1}\) set to 0.01 s. The resulting rocking response of the block is presented in Figure 6b. From the response, it is observed that the block begins to rock at \(t\) = 5.023 s. After 15 impacts between the rocking block and the vehicle, the rocking response becomes negligible. For comparison, the second-, third-, and fourth-order Runge-Kutta methods (with a time step \(\sigma\) = 0.001 s) are also used to analyze the rocking response under random vibration. The rocking response simulated by the Runge-Kutta method is convergent, the simulation error is \(O(\sigma^{p+1})\), and \(p\) is the order of the Runge-Kutta method. The response obtained from the Runge-Kutta method is included in Figure 6b.