On the Snappability and Singularity-Distance of Frameworks with Bars and Triangular Plates

Abstract

In a recent article the author presented a method to measure the snapping capability -shortly called snappability -of bar-joint frameworks based on the total elastic strain energy by computing the deformation of all bars using Hooke's law and the definition of Cauchy/Engineering strain. Within the paper at hand, we extend this approach to isostatic frameworks composed of bars and triangular plates by using the physical concept of Green-Lagrange strain. An intrinsic pseudometric based on the resulting total elastic strain energy density cannot only be used for evaluating the snappability but also for measuring the distance to the closest singular configuration. The presented methods are demonstrated on the basis of the 3-legged planar parallel manipulator.

Multi-stable design of triangulated origami structures on cones of revolution

“…This is in accordance with known results on snapping frameworks (cf. [29,30]). The shaky pose R s can be used to compute the snappability of these structures (see Section 5.2.1).…”

“…Within the paper we use the analytic approach of the so-called snappability [29,30], which measures the snapping capability of the structure and is based on the following property: The snap between two realizations R + and R − has to pass a shaky configuration R s at the maximum state of deformation with respect to the total elastic strain energy (using the concept of Green-Lagrange stain), which is used to evaluate the snappability. For the triangulated cone the mentioned energy can be computed for two different interpretations; namely as bar-joint structure or as panel-hinge structure.…”

“…Based on Eq. (62) we can determine U total as the limit of a geometric series where r is the common 9 The interested reader is referred to [29,30] for the formulation of the panel-hinge model. ration between adjacent terms; i.e.…”

“…Finally the snappability index ς of the structure can be computed as U total /(EVol total ) according to [29,30].…”

On Homotopy Continuation Based Singularity Distance Computations for 3-RPR Manipulators

“…Terminology and notations used for 3-RPR manipulator as frameworks. According to [13], a configuration of a 3-RPR robot can be considered as a planar isostatic framework in the Euclidean plane 𝔼 2 . In this representation, the legs are modeled as bars (|) and the platform/base either as a triangular plate (▴) or a pin-jointed triangular bar structure (▵).…”