Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userParallelogram frameworks and flexible quasicrystals
68
0
0.0
(
0
)
Authors
S. C. Power
Ask ChatGPT about the research
No Arabic abstract
The first-order flex space of the bar-joint framework $G_P$ of a parallelogram tiling $P$ is determined in terms of an explicit free basis. Applications are given to braced parallelogram frameworks and to quasicrystal frameworks associated with multigrids in the sense of de Bruijn and Beenker. In particular we characterise rigid bracing patterns, identify quasicrystal frameworks with finite dimensional flex spaces, and define a zero mode spectrum.
rate research
Read More
Symmetry equations are obtained for the rigidity matrix of a bar-joint framework in R^d. These form the basis for a short proof of the Fowler-Guest symmetry group generalisation of the Calladine-Maxwell counting rules. Similar symmetry equations are obtained for the Jacobian of diverse framework systems, including constrained point-line systems that appear in CAD, body-pin frameworks, hybrid systems of distance constrained objects and infinite bar-joint frameworks. This leads to generalised forms of the Fowler-Guest character formula together with counting rules in terms of counts of symmetry-fixed elements. Necessary conditions for isostaticity are obtained for asymmetric frameworks, both when symmetries are present in subframeworks and when symmetries occur in partition-derived frameworks.
We formulate and prove a periodic analog of Maxwells theorem relating stressed planar frameworks and their liftings to polyhedral surfaces with spherical topology. We use our lifting theorem to prove deformation and rigidity-theoretic properties for planar periodic pseudo-triangulations, generalizing features known for their finite counterparts. These properties are then applied to questions originating in mathematical crystallography and materials science, concerning planar periodic auxetic structures and ultrarigid periodic frameworks.
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint frameworks in a variety of such contexts. The main results are combinatorial characterisations for (i) frameworks restricted to surfaces with both Euclidean and geodesic distance constraints, (ii) frameworks in the plane with Euclidean and non-Euclidean distance constraints, and (iii) direction-length frameworks in the non-Euclidean plane.
We extend the mathematical theory of rigidity of frameworks (graphs embedded in $d$-dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call almost-rigidity; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost-rigidity. We briefly discuss how the theory may be applied to tensegrities.
In mathematical crystallography and computational materials science, it is important to infer flexibility properties of framework materials from their geometric representation. We study combinatorial, geometric and kinematic properties for frameworks modeled on sodalite.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
