Space-curve Cartan matrix and exact differentiability of the curvature and torsion
Space-curve Cartan matrix and exact differentiability of the curvature and torsion
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DOI：
10.1080/15397734.2022.2038198
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发表时间：
2022-03
影响因子：
3.9
通讯作者：
A. Shabana
中科院分区：
文献类型：
--
作者：
A. Shabana
在梁振动方程的建立中，空间曲线的曲率被用来定义应变能和弹性力。只有在特殊情况下，曲率和挠率才能与角的导数联系起来。此外，弯曲扭曲是耦合的面内和面外弯曲模式的结果。虽然这种模式耦合可以由两个旋转来表示，但是曲线曲率或扭转通常不能与单个旋转相关联。曲率和挠率，在其最一般的形式，定义使用反对称Cartan矩阵，这导致Serret-Frenet方程。本文利用两种不同的旋转序列讨论了曲率和挠率的精确可微性，并证明了曲线挠率一般不能定义为关于曲线切向量的精确定义的角的导数。本文利用Frenet角导出了曲线Cartan矩阵元素的简单通用表达式。分析和结果表明，Bishop剪切角，这是不唯一的，不进入曲线几何的定义之间的根本区别;和Frenet银行的角度，这是唯一的，进入曲线几何的定义。
Abstract In formulation of beam-vibration equations, curvature of space curves is used to define the strain energy and elastic forces. Only in special cases, the curvature and torsion can be associated with derivatives of angles. Furthermore, curve twist is result of coupled in-plane and out-of-plane bending modes. While this mode coupling can be represented by two rotations, curve curvature or torsion cannot, in general, be associated with single rotation. Curvature and torsion, in their most general forms, are defined using skew-symmetric Cartan matrix, which leads to the Serret-Frenet equations. This paper uses two different sequences of rotation to discuss exact differentiability of curvature and torsion and demonstrate that curve torsion cannot, in general, be defined as derivative of uniquely-defined angle performed about curve tangent vector. Frenet angles are used to develop simple and general expressions for elements of curve Cartan matrix. The analysis and results presented show the fundamental difference between Bishop shear angle, which is not unique and does not enter into definition of curve geometry; and Frenet bank angle, which is unique and enters into definition of curve geometry.