BURMESTER THEORY FOR FOUR PRECISION POSITIONS: AN EXTENDED DISCOURSE WITH APPLICATION TO THE DIMENSIONAL SYNTHESIS OF ARBITRARY PLANAR LINKAGES

Dimensional synthesis of a planar linkage consists of determining the link lengths of a pre-defined linkage topology to accomplish some pre-defined task. Burmester theory provides a well-known solution to the problem of determining vector pairs, called dyads, which will assemble at four arbitrarily prescribed orientations. Mechanisms are dimensionally synthesized using Burmester theory by prescribing the orientations for a combination of these vector pairs such that they can be assembled into a solution linkage modelled with these vectors.;Modelling mechanisms with dyads alone severely limits the linkage topologies which can be synthesized using the four precision position approach. Fortunately, classic Burmester theory can be expanded to synthesize strings of three vectors, called triads. The addition of this facility enables the synthesis of nearly any common planar linkage topology using the four precision position approach. A complete enumeration of all triad chains which can be reduced to synthesis using standard Burmester theory techniques is developed. The significance and practical value of the special points to triads are discussed.;Efficiently generating, representing, and interacting with a set of Burmester curves requires a knowlege of the curve properties which will result from an arbitrary set of precision positions. Methods for determining these properties using the complex number formulation are developed. These methods are particularly well suited for implementation using computer-assisted techniques.;The purpose of all the developments included here is to enable the synthesis of arbitrary planar mechanisms using the tools provided by Burmester theory. Therefore, a series of examples of how these tools are put to practical use is included. The applications range from all variations of the simple four-bar mechanism to complex multiloop mechanisms synthesized to meet demanding constraints.;A method for generating the Burmester curves using a complex number formulation of the four precision position synthesis problem is reviewed. A full discussion of special points which can occur on these curves is included. The discussion includes the effect of these points on the shape of the Brumester curves and useful mechanisms which can be constructed using these points.