This paper deals with the relation of the dynamic elastic Cosserat rod model and the Kirchhoff beam equations. We show that the Kirchhoff beam without angular inertia is the asymptotic limit of the Cosserat rod, as the slenderness parameter (ratio between rod diameter and length) and the Mach number (ratio between rod velocity and typical speed of sound) approach zero, i.e., low-Mach-number–slenderness limit. The asymptotic framework is exact up to fourth order in the small parameter and reveals a mathematical structure that allows a uniform handling of the transition regime between the models. To investigate this regime numerically, we apply a scheme that is based on a Gauss–Legendre collocation in space and an α-method in time.

中文翻译：

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弹性Cosserat杆的低马赫数和细长极限及其数值研究

本文讨论了动态弹性Cosserat杆模型与Kirchhoff梁方程的关系。我们表明，不带角惯性的基尔霍夫束是Cosserat杆的渐近极限，因为细长参数（杆直径和长度之间的比率）和马赫数（杆速度与典型声速之间的比率）接近零，即，低马赫数-细长度限制。渐近框架的小参数精确到四阶，并且揭示了一种数学结构，该结构允许对模型之间的过渡状态进行统一处理。为了对这种状态进行数值研究，我们应用了一种基于空间中高斯-勒格朗德搭配和时间上的α方法的方案。