Precision-point synthesis problems for design of four-bar linkages have typically been formulated using two approaches. The exclusive use of path-points is known as “path synthesis,” whereas the use of poses, i.e., path-points with orientation, is called “rigid-body guidance” or the “Burmester problem.” We consider the family of “Alt–Burmester” synthesis problems, in which some combination of path-points and poses is specified, with the extreme cases corresponding to the classical problems. The Alt–Burmester problems that have, in general, a finite number of solutions include Burmester's original five-pose problem and also Alt's problem for nine path-points. The elimination of one path-point increases the dimension of the solution set by one, while the elimination of a pose increases it by two. Using techniques from numerical algebraic geometry, we tabulate the dimension and degree of all problems in this Alt–Burmester family, and provide more details concerning all the zero- and one-dimensional cases.

Several established methods determine if an RR dyad will pass through a set of finitely separated positions in order. The new method presented herein utilizes only the displacement poles in the fixed frame to assess whether a selected fixed pivot location will yield an ordered dyad solution. A line passing through the selected fixed pivot is rotated one-half revolution about the fixed pivot, in a manner similar to a propeller with infinitely long blades, to sweep the entire plane. Order is established by tracking the sequence of displacement poles intersected. With four or five positions, fixed pivot locations corresponding to dyads having any specified order are readily found. Five-position problems can be directly evaluated to determine if any ordered solutions exist. Additionally, degenerate four-position cases for which the set of fixed pivots corresponding to ordered dyads that collapse to a single point on the center point curve can be identified.

This paper considers single degree-of-freedom (DOF), closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has input singularities, that is, turning points with respect to the input angle, which break the motion curve into branches. Motion of the linkage along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. Allowing the design parameter to vary, the singularities form a curve called the critical curve, whose projection is the singularity trace. Many critical points are the singularities of the critical curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. This paper presents a general method to compute the singularity trace and its critical points. As an example, the method is used on a Stephenson III linkage, and a range of the design parameter is found where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with critical points that appear as cusps on the singularity trace.

This paper presents the development of variable geometry dies that enable the extrusion of plastic parts with a varying cross section. Extrusion accounts for 40% of all manufactured plastic parts because it is a relatively low-cost and high-production-rate process. Conventional polymer extrusion technology, however, is limited to fixed dies that produce continuous plastic products of constant cross section defined by the die exit profile. A shape-changing die allows the cross section of the extruded part to change over its length, thereby introducing the capacity to manufacture plastic faster and with lower tooling costs than injection molding. This paper discusses design guidelines that were developed for movable die features including revolute and prismatic joint details, land length, and the management of die leakage. To assess these guidelines, multiple dies have been designed and constructed to include an arbitrary four-sided exit profile where changes were made to the internal angles and length of sides as the extruder was operating. Experimental studies were conducted by using different extruder line settings and time between die movements. Test results are presented that include shape repeatability and the relationship between extrudate profile and die exit geometry.

This paper considers single-degree-of-freedom, closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has turning points (dead-input singularities), which break the motion curve into branches such that the motion along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. As the design parameter changes, the turning points sweep out a curve we call the “turning curve,” and the critical points are the singularities in this curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. We present a general method to compute the turning curve and its critical points. As an example, the method is used on a Stephenson II linkage. Additionally, the Stephenson III linkage is revisited where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with cusps on the turning point curve.