One brick at a time: a survey of inductive constructions in rigidity theory

We present a survey of results concerning the use of inductive constructions to study the rigidity of frameworks. By inductive constructions we mean simple graph moves which can be shown to preserve the rigidity of the corresponding framework. We describe a number of cases in which characterisations of rigidity were proved by inductive constructions. That is, by identifying recursive operations that preserved rigidity and proving that these operations were sufficient to generate all such frameworks. We also outline the use of inductive constructions in some recent areas of particularly active interest, namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar frameworks. We summarize the key outstanding open problems related to inductions.Comment: 24 pages, 12 figures, final versio

A characterisation of generically rigid frameworks on surfaces of revolution

A foundational theorem of Laman provides a counting characterisation of the finite simple graphs whose generic bar-joint frameworks in two dimensions are infinitesimally rigid. Recently a Laman-type characterisation was obtained for frameworks in three dimensions whose vertices are constrained to concentric spheres or to concentric cylinders. Noting that the plane and the sphere have 3 independent locally tangential infinitesimal motions while the cylinder has 2, we obtain here a Laman-Henneberg theorem for frameworks on algebraic surfaces with a 1-dimensional space of tangential motions. Such surfaces include the torus, helicoids and surfaces of revolution. The relevant class of graphs are the (2,1)-tight graphs, in contrast to (2,3)-tightness for the plane/sphere and (2,2)-tightness for the cylinder. The proof uses a new characterisation of simple (2,1)-tight graphs and an inductive construction requiring generic rigidity preservation for 5 graph moves, including the two Henneberg moves, an edge joining move and various vertex surgery moves.Comment: 23 pages, 5 figures. Minor revisions - most importantly, the new version has a different titl

Generic rigidity with forced symmetry and sparse colored graphs

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of minimally rigid periodic frameworks with fixed-area fundamental domain and fixed-angle fundamental domain.Comment: 21 pages, 2 figure

The rigidity of infinite graphs

A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in R^d with respect to the classical l^p norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and Henneberg combinatorial characterisations of generic infinitesimal rigidity for finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation of the rigidity of generic finite body-bar frameworks in d-dimensional Euclidean space is generalised to the non-Euclidean l^p norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit of an inclusion tower of finite graphs for which the inclusions satisfy a relative rigidity property. For d>2 a countable graph which is rigid for generic placements in R^d may fail the stronger property of sequential rigidity, while for d=2 the equivalence with sequential rigidity is obtained from the generalised Laman characterisations. Applications are given to the flexibility of non-Euclidean convex polyhedra and to the infinitesimal and continuous rigidity of compact infinitely-faceted simplicial polytopes.Comment: 51 page

The orbit rigidity matrix of a symmetric framework

A number of recent papers have studied when symmetry causes frameworks on a graph to become infinitesimally flexible, or stressed, and when it has no impact. A number of other recent papers have studied special classes of frameworks on generically rigid graphs which are finite mechanisms. Here we introduce a new tool, the orbit matrix, which connects these two areas and provides a matrix representation for fully symmetric infinitesimal flexes, and fully symmetric stresses of symmetric frameworks. The orbit matrix is a true analog of the standard rigidity matrix for general frameworks, and its analysis gives important insights into questions about the flexibility and rigidity of classes of symmetric frameworks, in all dimensions. With this narrower focus on fully symmetric infinitesimal motions, comes the power to predict symmetry-preserving finite mechanisms - giving a simplified analysis which covers a wide range of the known mechanisms, and generalizes the classes of known mechanisms. This initial exploration of the properties of the orbit matrix also opens up a number of new questions and possible extensions of the previous results, including transfer of symmetry based results from Euclidean space to spherical, hyperbolic, and some other metrics with shared symmetry groups and underlying projective geometry.Comment: 41 pages, 12 figure

Rigidity of frameworks on expanding spheres

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterisations are established for the rigidity of generic frameworks for $d=1$ with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterisation of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in 1D and 2D and to some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.Comment: 22 pages, 2 figures, updated reference

Biporous Metal-Organic Framework with Tunable CO2/CH4 Separation Performance Facilitated by Intrinsic Flexibility.

In this work, we report the synthesis of SION-8, a novel metal-organic framework (MOF) based on Ca(II) and a tetracarboxylate ligand TBAPy4- endowed with two chemically distinct types of pores characterized by their hydrophobic and hydrophilic properties. By altering the activation conditions, we gained access to two bulk materials: the fully activated SION-8F and the partially activated SION-8P with exclusively the hydrophobic pores activated. SION-8P shows high affinity for both CO2 ( Qst = 28.4 kJ/mol) and CH4 ( Qst = 21.4 kJ/mol), while upon full activation, the difference in affinity for CO2 ( Qst = 23.4 kJ/mol) and CH4 ( Qst = 16.0 kJ/mol) is more pronounced. The intrinsic flexibility of both materials results in complex adsorption behavior and greater adsorption of gas molecules than if the materials were rigid. Their CO2/CH4 separation performance was tested in fixed-bed breakthrough experiments using binary gas mixtures of different compositions and rationalized in terms of molecular interactions. SION-8F showed a 40-160% increase (depending on the temperature and the gas mixture composition probed) of the CO2/CH4 dynamic breakthrough selectivity compared to SION-8P, demonstrating the possibility to rationally tune the separation performance of a single MOF by manipulating the stepwise activation made possible by the MOF's biporous nature

The effects of localized damping on structural response

The effect of localized structural damping on the excitability of higher order normal modes of the large space telescope was investigated. A preprocessor computer program was developed to incorporate Voigt structural joint damping models in a NASTRAN finite-element dynamic model. A postprocessor computer program was developed to select critical modes for low-frequency attitude control problems and for higher frequency fine-stabilization problems. The mode selection is accomplished by ranking the flexible modes based on coefficients for rate gyro, position gyro, and optical sensors, and on image-plane motions due to sinusoidal or random power spectral density force and torque inputs

Exploring the flexibility of MIL-47(V)-type materials using force field molecular dynamics simulations

The flexibility of three MIL-47(V)-type materials (MIL-47, COMOC-2, and COMOC-3) has been explored by constructing the pressure versus volume and free energy versus volume profiles at various temperatures ranging from 100 to 400 K This is done with first-principles-based force fields using the recently proposed QuickFF parametrization protocol. Specific terms were added for the materials at hand to describe the asymmetry of the one-dimensional vanadium oxide chain and to account for the flexibility of the organic linkers. The force fields are used in a series of molecular dynamics simulations at fixed volumes but varying unit cell shapes. The three materials show a distinct pressure-volume behavior, which underlines the ability to tune the mechanical properties by varying the linkers toward different applications such as nanosprings, dampers, and shock absorbers