Using the singularity trace to understand linkage motion characteristics


Lin Li

Date of Award


Degree Name

M.S. in Mechanical Engineering


Department of Mechanical and Aerospace Engineering


Advisor: Andrew P. Murray


This thesis provides examples of a new method used to analyze the motion characteristics of single-degree-of-freedom, closed-loop linkages with a designated input angle and one or two design parameters. The method involves the construction of a singularity trace, which is a plot that reveals changes in the number of geometric inversions, singularities, and changes in the number of branches as a design parameter is varied. This thesis applies the method to planar linkages such as the Watt II, Stephenson III and double butterfly, and spatial linkages such as spherical four-bar and Revolute-Cylindrical-Cylindrical-Cylindrical (RCCC) linkages. Results from this investigation include the following. Special instances of the singularity trace for the Watt II linkage include multiple coincident projections of the singularity curve and symmetric characteristics of the singularity trace for special design parameters. For the Stephenson III and double butterfly linkages, instances where the input angle is able to rotate more than one revolution between singularities have been identified. This characteristic demonstrates a net-zero, singularity free activation sequence that places the mechanism into a different geometric inversion. Additional observations from the examples show that net-zero, singularity free activation sequences can occur in the motion curves with U " or " S " shapes. Cases are shown where subtle changes to two design parameters of a Stephenson III linkage drastically alter the motion. Additionally, isolated critical points are found to exist for the double butterfly, where the linkage becomes a structure and looses the freedom to move. This thesis also applies this method to spatial linkages to analyze a spherical mechanism. The structure of the loop closure equations was modified to accommodate a third, spatial coordinate. An example is provided that uses a spherical four-bar linkage. The singularity map has been found to be similar to a comparable planar four-bar case. Lastly, loop closure and singularity equations have been formulated to address a truly spatial linkage, the RCCC. A solution method has been identified. The analysis is listed in future work."


Links and link-motion Mathematical models, Machinery, Kinematics of Mathematical models, Singularities (Mathematics), Mathematics; mechanical engineering; Watt II; Stephenson III; double butterfly; singularity; mathematical modeling; motion curve; singularity trace

Rights Statement

Copyright 2013, author