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


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.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

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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