Dynamic Analysis and Efficient Numerical Algorithm for Rocking Response of Freestanding Packages Under Transportation Excitations

The transportation of freestanding packages, such as steel coils and prefabricated building components, poses significant safety challenges due to their susceptibility to dynamic responses like rocking, sliding, and overturning under vehicle acceleration. This study focuses on the critical rocking motion, which can lead to catastrophic accidents if the rocking angle exceeds a critical threshold. The package is modeled as a rigid block on a moving base, and its motion states, including static equilibrium, bouncing, rocking, and detachment, are analytically determined, with specific conditions identified for each state. A novel numerical algorithm, combining a time transformation method with the Runge–Kutta method, is proposed to efficiently determine collision time instants and analyze the piecewise-smooth rocking response. This approach partitions the solution into smooth intervals between impacts, significantly improving computational efficiency while maintaining accuracy. Experimental validation under sinusoidal and numerical analysis with random excitations confirms the algorithm’s effectiveness in predicting rocking behavior. The results highlight the importance of the slenderness ratio and restitution coefficient in governing motion states and demonstrate the algorithm’s superiority over traditional methods in terms of precision and efficiency. This work provides a robust framework for assessing transportation safety and optimizing package design, with implications for reducing overturning risks in practical scenarios.