发表机构
Indian Institute of Technology Kanpur; Indian Institute of Technology Palakkad（坎普尔印度理工学院; 帕拉克德印度理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究推导了无质量弦公式有效的渐近条件，发现轨迹悖论实为边界层现象，通过奇异摄动分析揭示了内解与外解的本质联系。
AI 中文摘要
本研究推导了无质量拉伸弦公式在惯性弦携带移动点质量的等效动力学中有效的渐近条件。因此，在包含弦惯性的情况下，这些条件下的控制方程可以写成一个奇异摄动偏微分方程。当忽略弦的惯性时，控制方程简化为一个非齐次超几何常微分方程，控制移动质量的横向运动。该超几何方程在终止边界处具有不连续解，即所谓的轨迹悖论。然而，数值求解实际方程得到的是在边界附近快速变化的连续解。完整解类似于内解，而超几何常微分方程导致外解，揭示了轨迹悖论是一个边界层。
英文摘要
This study derives the asymptotic conditions under which the massless stretched string formulation is valid for the equivalent dynamics of the inertial string carrying a moving point mass. Consequently, the governing equation under such conditions can be written as a singularly perturbed partial differential equation when the inertia of the string is included. When the inertia of the string is ignored, the governing equation reduces to a non-homogeneous hypergeometric ordinary differential equation governing the transverse motion of the moving mass. This hypergeometric equation has a discontinuous solution at the terminating boundary known as the trajectory paradox. However, numerically solving the actual equation yields a continuous solution with a rapid variation near the boundary. The complete solution resembles the inner solution as well, whereas the hypergeometric ordinary differential equation leads to an outer solution, revealing that the trajectory paradox is a boundary layer.
