Singular Conditions and Solutions of the Standard Triad Displacement Equation

Abstract

In the dimensional synthesis of a standard planar triad, if the prescribed angular displacements of two of the three links are the same at all precision positions or the prescribed angular displacements of one link are zero at all precision positions, the standard triad displacement equation will not yield a practical solution: a triad having all links with finite lengths. The displacement equation may also fail to generate a practical solution triad in case that the freely chosen angular displacement of one link equals to zero, or equals to the prescribed angular displacement of another link at the same precision position. In this paper, the special cases in the dimensional synthesis of a standard triad are discussed in detail, the types of triads which can generate the motions corresponding to the special cases are listed, and the solution methods to solve the displacement equations of the special types of triads are developed. Finally a numerical example is given to show the application of one of the special types of triad in the dimensional synthesis of an eight-bar linkage.