As soft elements for force transmission, braided fiber ropes play important roles in many fields where the fiber ropes are used bent over sheaves, while the relevant experiments are time-consuming and expensive. Computational simulation is a promising choice for evaluating the performance of fiber ropes when bent over a sheave. This article presents two methods that could be employed to build a model of braided rope bent over a sheave. One is the mathematical method which deduces the exact mathematical equations of braiding curves based on the Frenet–Serret frame. The spatial equations, considering the phase difference of strands in the same direction and the difference of strands’ projection in different directions, are discussed carefully. The final equation of braided strands is confirmed by modeling the braided rope in Maple® 17. The other method, which is inspired by the analysis of braiding movements, is based on the intersection of surfaces of braiding surface and helical surface which are introduced and defined based on the motion analysis of bobbins and take-up roller. The SolidWorks® 2018 is successfully employed to realize the modeling process.

中文翻译：

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编织绳索在滑轮上弯曲的数学和几何建模

编织纤维绳作为传力的软元件，在纤维绳弯曲在滑轮上使用的许多领域中发挥着重要作用，而相关实验既费时又费钱。计算模拟是评估纤维绳在滑轮上弯曲时的性能的一种很有前景的选择。本文介绍了两种可用于构建弯曲在滑轮上的编织绳模型的方法。一种是基于Frenet-Serret框架推导编织曲线精确数学方程的数学方法。仔细讨论了考虑同一方向绞线相位差和不同方向绞线投影差异的空间方程。通过对 Maple® 17 中的编织绳进行建模，可以确认编织股线的最终方程。另一种方法受编织运动分析的启发，基于编织面和螺旋面的交集，这是基于线轴和卷取辊的运动分析引入和定义的。SolidWorks® 2018 成功地实现了建模过程。