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Graver J.E. — Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures

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Название: Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures

Автор: Graver J.E.

Аннотация:

Rigidity theory is a body of mathematics developed to aid in designing structures. Consider a scaffolding that is constructed by bolting together rods and beams. The ultimate question is: "Is the scaffolding sturdy enough to hold the workers and their equipment?" There are several features of the structure that have to be considered in answering this question. The rods and beams have to be strong enough to hold the weight of the workers and their equipment, and to withstand a variety of stresses. Also important in determining the strength of the scaffolding is the way the rods and beams are put together. And, although the bolts at the joints must hold the scaffolding together, they are not expected to prevent the rods and beams from rotating about the joints. So, ultimately, the sturdiness of the scaffolding depends on the way it is braced. Just how to design a properly braced scaffolding (or the structural skeleton of any structure) is the problem that motivates rigidity theory.

The purpose of this book is to develop a mathematical model for rigidity.
Three distinct models are developed. The structures studied in these models, are represented by a framework: a configuration of straight line segments joined together at their end points. It is assumed that the joints are flexible and a question is posed that asks if the configuration can be deformed without changing the lengths of the segments. A triangle is a simple rigid framework in the plane while the square is not rigid - it deforms into a rhombus The first model presented here is for rigidity, the degrees of freedom model, is very intuitive and very easy to use with small frameworks. But, at the outset, this model lacks a rigorous foundation and actually fails to give a correct prediction in some cases. The next model is based on systems of quadratic equations: each segment yielding the quadratic equation which states that the distance between the joints at the end of the segment must equal the length of the segment. It is easy to see that this model will give the correct answer to our question in all cases. Hence, this is the standard model of rigidity. Unfortunately, this model is not so easy to use even with small frameworks. The third model constructed is equally accurate but is based on a slightly different definition of rigidity, called infinitesimal rigidity. In this model, the quadratic equations are replaced with linear equations and it is, therefore, much easier to use. Ultimately we show that all three models agree except for very few very special frameworks. The final chapter of the book is devoted to using these models to understand the structure of linkages, geodesic domes and tense grity structures.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2001

Количество страниц: 192

Добавлена в каталог: 11.04.2008

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Предметный указатель
 1-skeleton      131 3-double-edge-split      125 Adjacent      28 Bipartite graph      39 Birigid      49 Bracing, tension      158 Circuit      42 Circuit, m-circuit      161 Combinatorial properties      31 Complete graph      39 components      42 Connected      38 Connected components      42 Connected strongly      159 Constricted frameworks      72 Convex      129 Convex strictly      129 Deformation      12 35 Deformation infinitesimal      73 Directed graph      159 Directed path      159 Disconnected      38 Dome      140 Dome geodesic      142 Double-edge-split      125 Dual map      61 e(u)      46 EDGE      27 Edge-split: 3-double-      125 Edge-split: 3-double-m-      110 Embedding      31 Embedding generic      104 Endpoints      28 Extension: 1-, 49; m-      109 Forest      43 Framework: constricted      72 Framework: constricted linear      12 26 Framework: constricted m-dimensional      30 Framework: constricted normal      72 Framework: constricted planar      12 26 Framework: constricted plate and hinge      131 Framework: constricted spatial      12 26 Framework: constricted tensegrity      161 Fullerene      153 General position      88 Generic: embedding      104 Generic: embedding framework      104 Generic: embedding intuitive definition      99 Geodesic dome      142 Geometric properties      31 Graph      27 Graph directed      159 Graph plane      59 Implied edge      48 Infinitesimal: rigidity      72 Infinitesimal: rigidity rotation      86 Infinitesimal: rigidity translation      85 Isolated vertex      38 Isomorphic graphs      39 Isostatic framework      48 Joint, removable      9 Length: of a circuit      42 Length: of a circuit of a path      40 Linkages      134 M-circuit      161 M-edge-split      110 M-extension      109 M-rigid      100 M-tree      107 MAP      59 Map dual      61 Map structure      140 Motion      35 Motion infinitesimal      73 Motion infinitesimal rigid      73 Motion rigid      35 Nanotubes      153 Normal frameworks      72 Path      40 Path directed      159 Peaucellier's cell      137 Pendant vertex      38 Planar graph      55 Plane graph      59 Plate and hinge framework      131 Platonic map      62 Platonic solids      62 Removable joint      9 Resolvable stresses, set of      165 Rigid      35 Rigid framework      13 Rigid infinitesimally      73 Rigid motion      12 Rigid, m-rigid      100 Rigidity matrix      101 Rigidity, infinitesimal      72 Rotation, infinitesimal      86 Skeleton, 1-skeleton      131 Space of infinitesimal motions      80 Space of infinitesimal rigid motions      80 Spanning subgraph      39 Stresses      165 Strictly convex      129 Strongly connected      159 Structure graph      31 Structure map      140 Subgraph      39 Tensegrity framework      161 Tension bracing      158 Translation, infinitesimal      85 TREE      42 Tree, m-tree      107 Valence of a face      58 Valence of a vertex      38 Vertex      27
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