3,934 research outputs found
Generic Rigidity Matroids with Dilworth Truncations
We prove that the linear matroid that defines generic rigidity of
-dimensional body-rod-bar frameworks (i.e., structures consisting of
disjoint bodies and rods mutually linked by bars) can be obtained from the
union of graphic matroids by applying variants of Dilworth
truncation times, where denotes the number of rods. This leads to
an alternative proof of Tay's combinatorial characterizations of generic
rigidity of rod-bar frameworks and that of identified body-hinge frameworks
Iterative Universal Rigidity
A bar framework determined by a finite graph and configuration in
space is universally rigid if it is rigid in any . We provide a characterization of universally rigidity for any
graph and any configuration in terms of a sequence of affine
subsets of the space of configurations. This corresponds to a facial reduction
process for closed finite dimensional convex cones.Comment: 41 pages, 12 figure
From graphs to tensegrity structures: Geometric and symbolic approaches
A form-finding problem for tensegrity structures is studied; given an
abstract graph, we show an algorithm to provide a necessary condition for it to
be the underlying graph of a tensegrity in (typically )
with vertices in general position. Furthermore, for a certain class of graphs
our algorithm allows to obtain necessary and sufficient conditions on the
relative position of the vertices in order to underlie a tensegrity, for what
we propose both a geometric and a symbolic approach.Comment: 17 pages, 8 figures; final versio
Finite motions from periodic frameworks with added symmetry
Recent work from authors across disciplines has made substantial
contributions to counting rules (Maxwell type theorems) which predict when an
infinite periodic structure would be rigid or flexible while preserving the
periodic pattern, as an engineering type framework, or equivalently, as an
idealized molecular framework. Other work has shown that for finite frameworks,
introducing symmetry modifies the previous general counts, and under some
circumstances this symmetrized Maxwell type count can predict added finite
flexibility in the structure.
In this paper we combine these approaches to present new Maxwell type counts
for the columns and rows of a modified orbit matrix for structures that have
both a periodic structure and additional symmetry within the periodic cells. In
a number of cases, this count for the combined group of symmetry operations
demonstrates there is added finite flexibility in what would have been rigid
when realized without the symmetry. Given that many crystal structures have
these added symmetries, and that their flexibility may be key to their physical
and chemical properties, we present a summary of the results as a way to
generate further developments of both a practical and theoretic interest.Comment: 45 pages, 13 figure
- …