关于刚接三维框架的完整图形静力学。第三部分：循环的运动学
A Complete Graphic Statics for Rigid-Jointed 3D Frames. Part 3: Loops for Kinematics
AI总结：
本文基于循环形式理论，探讨三维框架结构的运动学行为，通过四维空间中的循环投影面积表示位移和旋转，结合虚拟功作为四维体积积分，实现运动学与平衡的统一。
摘要：
在本文中，三维框架结构的运动学行为通过四维空间中的循环形式理论来描述。在第二部分中，用于描述平衡的几何工具（如多边形、多面体和多胞形）被更一般的CW复形概念所取代。任何杆件截面的应力结果的六个分量通过四维空间中循环的六个投影的定向双矢量面积表示，其中三个分量代表力，三个分量代表力矩。在本文中，四维空间中的循环投影面积将代表运动学变量，其中三个投影面积表示框架上一点的位移，另外三个投影面积表示该点结构的旋转。该理论的四维设定包括通常的三个物理空间维度以及一个用于应力函数的第四维度。虚拟功在此四维设定中表现为一个四维体积形式（一个定向的四维体积），即结构上局部平衡和运动学变量的楔积积分。
英文摘要：
In Part 3 of this sequence of papers, the kinematic behaviour of 3D frame structures is described using the loop formalism that was developed in Part 2 to describe equilibrium. There, the notions of polygons, polyhedra and polytopes that form the geometric toolbox underlying graphic statics were replaced by the more general concept of CW-complexes from algebraic homology. The six components of the stress resultant acting on any cut face of a bar in a rigidly-jointed framework were represented by the oriented bivector areas of the six projections of a loop in a 4D-space, with three components representing the force and three components representing the moment. In this paper, projected areas of loops in 4D will represent kinematic variables, with three projected areas representing the displacement of a point on the frame, and three other projected areas representing the rotation of the structure at that point. The 4D setting for the theory consists of the usual three dimensions of physical space together with a fourth dimension for the stress function. Virtual Work then manifests as a top form (an oriented 4-volume) in this 4D setting, being the integral over the structure of the wedge product of bivectors representing the local equilibrium and kinematic variables.
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