悬臂梁弹射系统的动态响应分析与多目标优化研究

Dynamic Response Analysis and Multi-Objective Optimization Study of Cantilever Beam Catapult Systems

摘要: 基于2025年青年物理学家锦标课题的尺子加农炮问题，本文构建了一个融合动态能量传递与稳定性约束的悬臂梁理论模型，通过理论建模、参数化仿真与多目标优化分析，系统探究了影响投射物初速度的关键参数。研究建立了考虑摩擦损耗与梁自身动能的动态能量平衡方程，揭示了施力点位置通过模态激发机制影响能量转化效率的规律。进一步，结合梁–柱理论分析了压弯组合变形下的屈曲稳定性，提出了基于精确挠度解的势能修正方法。最终，通过构建综合优化系数曲面，实现了稳定性与动态效率的协同优化，量化了结构稳定性与发射效率之间的权衡关系，确定了最优设计参数域为该类弹性动力系统的参数设计提供了理论依据。

Abstract: Based on the ruler cannon problem from the 2025 Young Physicists’ Tournament, this paper constructs a cantilever beam theoretical model integrating dynamic energy transfer and stability constraints. Through theoretical modeling, parametric simulation, and multi-objective optimization analysis, the key parameters affecting the initial velocity of the projectile are systematically investigated. A dynamic energy balance equation considering friction loss and the beam’s own kinetic energy is established, revealing the law that the position of the force application point influences energy conversion efficiency through the modal excitation mechanism. Furthermore, combined with beam-column theory, the buckling stability under combined compression-bending deformation is analyzed, and a potential energy correction method based on the exact deflection solution is proposed. Finally, by constructing a comprehensive optimization coefficient surface, the coordinated optimization of stability and dynamic efficiency is achieved, the trade-off relationship between structural stability and launch efficiency is quantified, and the optimal design parameter domain is determined, providing a theoretical basis for the parameter design of such elastic dynamic systems.