## 1 Introduction  
Economic development necessitates and promotes the growth of cargo transportation. In practice, there are many large-sized products and packages, such as steel coils, prefabricated building components, and electromechanical devices. Due to constraints related to loading and unloading time, transportation space, and transportation costs, these packages often cannot be secured using fastener measures and are referred to as freestanding packages. During transportation, freestanding packages may exhibit various dynamic responses, including rocking, sliding, and jumping, in reaction to vehicle acceleration. Among these responses, rocking motion is particularly critical for transportation safety. If the rocking angle of a package exceeds a critical threshold, it may overturn, potentially leading to severe traffic accidents. Therefore, it is essential to determine the dynamic response states, analyze the rocking behavior, and assess the overturning risks of freestanding packages. The package is simplified and modeled as a rigid block, and the safety requirements for package transportation have prompted extensive research into determining motion states and analyzing the rocking motion of rigid blocks on moving bases. The possible motion states of a rigid block on a moving base have been analytically determined in [1]. Among these states, rocking and overturning are particularly detrimental to transportation safety. Considering the resurrection phenomenon and the weaving behavior of the rocking Incremental Dynamic Analysis (IDA) curve, the IDA method [2; 3] was refined and applied to evaluate the rocking behavior and overturning probability of freestanding rigid blocks [4]. In [5], numerical analysis was conducted to assess the statistical distribution and overturning probability of freestanding rigid blocks, revealing that the peak ground velocity is the most effective ground motion intensity measure for predicting rocking behavior. Similarly, in [6], various ground motion intensity measures were tested by fitting power-law models to evaluate their effectiveness in predicting rocking response characteristics and overturning risks. [PERSON] et al. [7] proposed a set of _I-\(\theta\)-\(p\)_ equations derived through nonlinear regression to predict rocking response characteristics, which can be used to assess the overturning risks of freestanding blocks under seismic excitation. Although the rocking block structure is simple, its response is piecewise smooth and involves impacts, leading to complex dynamic phenomena. Studies [8; 9] have revealed phenomena such as symmetric and asymmetric periodic responses, branching points, and chaotic motions in rigid block responses under harmonic excitation. To enhance the stability of rocking rigid blocks, several devices have been designed and implemented, including tuned mass dampers [10], pendulum absorbers [11], and moving compliant bases [12]. In [13], machine learning is utilized to predict the maximum seismic response of free-standing rigid blocks subjected to ground motion excitations. Large-amplitude shock waves during earthquakes or vehicle transportation pose significant risks to the rocking safety of freestanding rigid blocks. The rocking response and overturning fragility of such blocks have been extensively analyzed in [14; 15]. However, the analytical methods for predicting rocking responses are only valid for slender blocks (\(\alpha\leq 15^{\circ}\)). For non-slender blocks (\(\alpha>15^{\circ}\)), extensive numerical analysis is required to accurately determine rocking responses and overturning risks, making the process inefficient. Moreover, numerical analysis often fails to reveal the intrinsic relationships between block properties, excitation parameters, and rocking response characteristics. The computational demands of numerical analysis for freestanding package rocking responses can be mitigated by adopting soft computing techniques.