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Structural Rigidity
In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges. Definitions Rigidity is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility. In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected by flexible hinges. A structure is rigid if it cannot flex; that is, if there is no continuous motion of the structure that preserves the shape of its rigid components and the pattern of their connections at the hinges. There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount. Infinitesimal rigidity means that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Rigidity Basic Examples
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as organism, biological organisms, minerals and chemical substance, chemicals. Abstract structures include data structures in computer science and musical form. Types of structure include a hierarchy (a cascade of one-to-many relationships), a Complex network, network featuring many-to-many Link (geometry), links, or a lattice (order), lattice featuring connections between components that are neighbors in space. Load-bearing Buildings, aircraft, skeletons, Ant colony, anthills, beaver dams, bridges and salt domes are all examples of Structural load, load-bearing structures. The results of construction are divided into buildings and nonbuilding structure, non-building structures, and make up the infrastructure of a human society. Built str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Theorem (geometry)
Cauchy's theorem is a theorem in geometry, named after Augustin-Louis Cauchy, Augustin Cauchy. It states that convex polytopes in three dimensions with congruence (geometry), congruent corresponding faces must be congruent to each other. That is, any Net (polyhedron), polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron. For instance, if six squares are connected in the pattern of a cube, then they must form a cube: there is no convex polyhedron with six square faces connected in the same way that does not have the same shape. This is a fundamental result in rigidity theory (structural), rigidity theory: one consequence of the theorem is that, if one makes a physical model of a convex polyhedron by connecting together rigid plates for each of the polyhedron faces with flexible hinges along the polyhedron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting On Frameworks
''Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures'' is an undergraduate-level book on the mathematics of structural rigidity. It was written by Jack E. Graver and published in 2001 by the Mathematical Association of America as volume 25 of the Dolciani Mathematical Expositions book series. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion by undergraduate mathematics libraries. Topics The problems considered by ''Counting on Frameworks'' primarily concern systems of rigid rods, connected to each other by flexible joints at their ends; the question is whether these connections fix such a framework into a single position, or whether it can flex continuously through multiple positions. Variations of this problem include the simplest way to add rods to a framework to make it rigid, or the resilience of a framework against the failure of one of its rods. To study this question, Graver has organized ''Cou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebychev–Grübler–Kutzbach Criterion
The Chebychev–Grübler–Kutzbach criterion determines the number of degrees of freedom of a kinematic chain, that is, a coupling of rigid bodies by means of mechanical constraints. These devices are also called linkages. The Kutzbach criterion is also called the ''mobility formula'', because it computes the number of parameters that define the configuration of a linkage from the number of links and joints and the degree of freedom at each joint. Interesting and useful linkages have been designed that violate the mobility formula by using special geometric features and dimensions to provide more mobility than predicted by this formula. These devices are called overconstrained mechanisms. Mobility formula The mobility formula counts the number of parameters that define the positions of a set of rigid bodies and then reduces this number by the constraints that are imposed by joints connecting these bodies.J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish physicist and mathematician who was responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and light as different manifestations of the same phenomenon. Maxwell's equations for electromagnetism achieved the Unification (physics)#Unification of magnetism, electricity, light and related radiation, second great unification in physics, where Unification (physics)#Unification of gravity and astronomy, the first one had been realised by Isaac Newton. Maxwell was also key in the creation of statistical mechanics. With the publication of "A Dynamical Theory of the Electromagnetic Field" in 1865, Maxwell demonstrated that electric force, electric and magnetic fields travel through space as waves moving at the speed of light. He proposed that light is an undulation in the same medium that is the cause of electric and magnetic phenomena. (Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Bracing
In construction, cross bracing is a system utilized to reinforce building structures in which diagonal supports intersect. Cross bracing is usually seen with two diagonal supports placed in an X-shaped manner. Under lateral force (such as wind or seismic activity) one brace will be under tension while the other is being compressed. In steel construction, steel cables may be used due to their great resistance to tension (although they cannot take any load in compression). The common uses for cross bracing include bridge (side) supports, along with structural foundations. This method of construction maximizes the weight of the load a structure is able to support. It is a usual application when constructing earthquake-safe buildings. Cross bracing can be applied to any rectangular frame structure, such as chairs and bookshelves. Its rigidity for two-dimensional grid structures can be analyzed mathematically as an instance of the grid bracing problem. Cross bracing may employ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Grid
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane consisting of four squares around every vertex. John Horton Conway called it a quadrille. Structure and properties The square tiling has a structure consisting of one type of congruent prototile, the square, sharing two vertices with other identical ones. This is an example of monohedral tiling. Each vertex at the tiling is surrounded by four squares, which denotes in a vertex configuration as 4.4.4.4 or 4^4 . The vertices of a square can be considered as the lattice, so the square tiling can be formed through the square lattice. This tiling is commonly familiar with the flooring and game boards. It is self-dual, meaning the center of each square connects to another of the adjacent tile, forming square tiling itself. The square tiling acts transitively on the ''flags'' of the tiling. In this case, the flag consists of a mutually incident vertex, edge, and tile of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grid Bracing
In the mathematics of structural rigidity, grid bracing is a problem of adding cross bracing to a rectangular grid to make it into a rigid structure. If a two-dimensional grid structure is made with rigid rods, connected at their ends by flexible hinges, then it will be free to flex into positions in which the rods are no longer at right angles. Cross-bracing the structure by adding more rods across the diagonals of its rectangular or square cells can make it rigid. The problem can be translated into graph theory by constructing a graph in which the vertex (graph theory), graph vertices represent rows and columns of the grid, and each edge (graph theory), edge represents a cross-braced cell in a given row and column. The grid is rigid if and only if the resulting graph is a connected graph. Every minimal system of cross-braces that makes the grid rigid corresponds to a spanning tree of a complete bipartite graph. The graph-theoretic solution to the grid bracing problem has been g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Volume
Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The definition of length and height (cubed) is interrelated with volume. The volume of a container is generally understood to be the capacity of the container; i.e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. By metonymy, the term "volume" sometimes is used to refer to the corresponding region (e.g., bounding volume). In ancient times, volume was measured using similar-shaped natural containers. Later on, standardized containers were used. Some simple three-dimensional shapes can have their volume easily calculated using arithmetic formulas. Volumes of more complicated shapes can be calculated with integral calculus if a formula exists for the shape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bellows Conjecture
In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex (this is also true in higher dimensions). Examples The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by . They are self-intersecting surfaces isometric to an octahedron. The first example of a flexible non-self-intersecting surface in \mathbb^3, the Connelly sphere, was discovered by . Steffen's polyhedron is another non-self-intersecting flexible polyhedron derived from Bricard's octahedra. Bellows conjecture In the late 1970s Connelly and D. Sullivan formulated the bellows conjecture stating that the volume of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere by using elimination theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Connelly
Robert Connelly (born July 15, 1942) is a mathematician specializing in discrete geometry and rigidity theory. Connelly received his Ph.D. from University of Michigan in 1969. He is currently a professor at Cornell University. Connelly is best known for discovering embedded flexible polyhedra. One such polyhedron is in the National Museum of American History. His recent interests include tensegrities and the carpenter's rule problem. In 2012 he became a fellow of the American Mathematical Society. Asteroid 4816 Connelly, discovered by Edward Bowell at Lowell Observatory 1981, was named after Robert Connelly. The official was published by the Minor Planet Center The Minor Planet Center (MPC) is the official body for observing and reporting on minor planets under the auspices of the International Astronomical Union (IAU). Founded in 1947, it operates at the Smithsonian Astrophysical Observatory. Funct ... on 18 February 1992 (). Author Connelly has authored or co-a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
