Displacement Analysis of the General N-Bar, Single-Loop, Spatial Linkage. Part 1: Underlying Mathematics and Useful Tables

In order to carry out the displacement analysis of the general n-bar, single-loop, spatial linkage, it was imperative to introduce a rationalization in the mathematical procedure. This rationalization includes the establishment of special symbology and conventions, as well as the setting up of some novel mathematical tools. Among the latter, we find the definitions and properties of cyclic sums and cyclic products, the presentation of an indicial way of handling trigonometric expressions, called Indicial Trigonometry, the definitions and properties of multiple displacement and linkage functions, and the operations of index underlining, index overlining, and index dotting. Tensor algebra is also used to a great extent. Moreover, a method of separating any selected rotation angle from a displacement function leads—in most cases—to linear expressions in the sine and cosine of the selected angle or to second-degree polynomials in the tangent of half the angle. A set of tables is included setting forth the aforesaid expressions for an important number of displacement functions. These tables are a very valuable aid when performing a kinematic analysis.