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E-raamat: Frameworks, Tensegrities, and Symmetry

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(University of Cambridge), (Cornell University, New York)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 27-Jan-2022
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781009234221
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Frameworks, Tensegrities, and SymmetrySuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 27-Jan-2022
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781009234221
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This introduction to the theory of rigid structures explains how to analyze the performance of built and natural structures under loads, paying special attention to the role of geometry. The book unifies the engineering and mathematical literatures by exploring different notions of rigidity - local, global, and universal - and how they are interrelated. Important results are stated formally, but also clarified with a wide range of revealing examples. An important generalization is to tensegrities, where fixed distances are replaced with 'cables' not allowed to increase in length and 'struts' not allowed to decrease in length. A special feature is the analysis of symmetric tensegrities, where the symmetry of the structure is used to simplify matters and allows the theory of group representations to be applied. Written for researchers and graduate students in structural engineering and mathematics, this work is also of interest to computer scientists and physicists.

This introduction to the theory of rigid structures explains how to analyze the performance of built and natural structures under loads, paying special attention to the simplifying role of symmetry. Written for researchers and graduate students in structural engineering and mathematics, and of interest to computer scientists and physicists.

Arvustused

'Rigidity theory mathematicians and structural engineers are like two branches of a tribe that separated long ago. In the intervening time, the language and knowledge of each group has evolved to where concepts no longer align and common terms no longer have common meanings. As a result, when they interact today, confusion reigns. Frameworks, Tensegrities and Symmetry is a guide that both groups can use to understand the other.' William F. Baker, Skidmore, Owings & Merrill 'The authors promise 'an attempt to build a bridge between two cultures' and they have done a remarkable job of this unenviable task. Requiring only a minimum of mathematical and engineering prerequisites the book develops intuitively, and rigorously, the rigidity theory of both bar frameworks and tensegrity frameworks and applies this theory to analyse built structures. Two masters of the field have carefully designed the book to move seamlessly between the analysis and synthesis of specific structures and providing the general, generic and symmetric theories.' Anthony Nixon, Lancaster University 'This excellent book unifies the engineering and mathematical literature by exploring different notions of rigidity - local, global and universal - and how they are interrelated. A lot of revealing examples accompany the mathematically precise description, hence the book can be recommended to researchers and students of mathematics, structural engineering, physics and computer science alike.' Andras Recski, Mathematical Reviews

Muu info

Why don't things fall down? Engineering meets mathematics in this introduction to the geometry of rigid and flexible structures.
Preface xi
1 Introduction
1(6)
1.1 Prerequisites
6(1)
1.2 Notation
6(1)
Part I The General Case 7(178)
2 Frameworks and Rigidity
9(8)
2.1 Introduction
9(1)
2.2 Definition of a Framework
9(1)
2.3 Definition of a Flex
10(1)
2.4 Definitions of Rigidity
11(5)
2.5 Exercises
16(1)
3 First-Order Analysis of Frameworks
17(45)
3.1 Introduction
17(1)
3.2 Kinematics
17(10)
3.3 Statics
27(8)
3.4 Static/Kinematic Duality
35(2)
3.5 Graphical Statics
37(4)
3.6 First-Order Stiffness
41(1)
3.7 Example: Structural Analysis of a Pin-Jointed Cantilever
42(5)
3.8 The Basic Rigidity Theorem
47(2)
3.9 Another Example of Infinitesimal Rigidity
49(6)
3.10 Projective Transformations
55(4)
3.11 Exercises
59(3)
4 Tensegrities
62(16)
4.1 Introduction
62(2)
4.2 Rigidity Questions
64(1)
4.3 Infinitesimal Rigidity
64(2)
4.4 Static Rigidity
66(1)
4.5 Elementary Forces
67(1)
4.6 Farkas Alternative
68(1)
4.7 Equivalence of Static and Infinitesimal Rigidity
69(2)
4.8 Roth-Whiteley Criterion for Infinitesimal Rigidity
71(1)
4.9 First-Order Stiffness
72(1)
4.10 Application to Circle Packings
73(3)
4.11 Exercises
76(2)
5 Energy Functions and the Stress Matrix
78(32)
5.1 Introduction
78(1)
5.2 Energy Functions and Rigidity
78(3)
5.3 Quadratic Energy Function
81(1)
5.4 Equilibrium
81(3)
5.5 The Principle of Least Energy
84(2)
5.6 The Stress Matrix
86(3)
5.7 The Configuration Matrix
89(1)
5.8 Universal Configurations Exist
90(2)
5.9 Projective Invariance
92(2)
5.10 Unyielding and Globally Rigid Examples
94(1)
5.11 Universal Tensegrities
95(1)
5.12 Small Unyielding Tensegrities
95(3)
5.13 Affine Motions Revisited
98(3)
5.14 The Fundamental Theorem of Tensegrity Structures
101(6)
5.15 Exercises
107(3)
6 Prestress Stability
110(25)
6.1 Introduction
110(1)
6.2 A General Energy Function
110(9)
6.3 Quadratic Forms
119(2)
6.4 Reducing the Calculation
121(1)
6.5 Second-Order Rigidity
121(4)
6.6 Calculating Prestressability and Second-Order Rigidity
125(2)
6.7 Second-Order Duality
127(3)
6.8 Triangulated Spheres
130(2)
6.9 Exercises
132(3)
7 Generic Frameworks
135(20)
7.1 Introduction
135(1)
7.2 Definition of Generic
135(1)
7.3 Infinitesimal Rigidity is a Generic Property
136(3)
7.4 Necessary Conditions for Being Generically Rigid
139(1)
7.5 Generic Rigidity in the Plane
140(3)
7.6 Pebble Game
143(3)
7.7 Vertex Splitting
146(1)
7.8 Generic Global Rigidity
147(6)
7.9 Applications
153(1)
7.10 Exercises
153(2)
8 Finite Mechanisms
155(30)
8.1 Introduction
155(1)
8.2 Finite Mechanisms Using the Rigidity Map
156(1)
8.3 Finite Mechanisms Using Symmetry
157(4)
8.4 Algebraic Methods for Creating Finite Mechanisms
161(1)
8.5 Crinkles
162(3)
8.6 A Triangulated Surface that is a Finite Mechanism
165(5)
8.7 Carpenter's Rule Problem
170(6)
8.8 Algebraic Sets and Semi-Algebraic Sets
176(5)
8.9 Exercises
181(4)
Part II Symmetric Structures 185(84)
9 Groups and Representation Theory
187(18)
9.1 Introduction
187(1)
9.2 What is Symmetry?
187(1)
9.3 What is a Group?
188(5)
9.4 Homomorphisms and Isomorphisms of Groups
193(2)
9.5 Representations
195(10)
10 First-Order Symmetry Analysis
205(29)
10.1 Internal and External Vector Spaces
205(1)
10.2 Decomposition of Internal and External Vector Spaces
206(3)
10.3 Internal and External Vector Spaces as RG-Modules
209(9)
10.4 Symmetry Operations, Equilibrium and Compatibility - RG-Homomorphisms
218(4)
10.5 Decomposition of Internal and External RG-Modules
222(7)
10.6 Irreducible Submodules
229(5)
11 Generating Stable Symmetric Tensegrities
234(35)
11.1 Symmetric Tensegrities
234(1)
11.2 Some Group Definitions
234(3)
11.3 Irreducible Components
237(3)
11.4 Groups for 3-Dimensional Examples
240(1)
11.5 Presentation of Groups
241(1)
11.6 Representations for Groups of Interest
241(22)
11.7 Non-Transitive Examples
263(2)
11.8 Comments
265(4)
Appendix A Useful Theorems and Proofs 269(2)
A.1 Basic Rigidity
269(1)
A.2 Proof for the Cusp Mechanism
270(1)
References 271(9)
Index 280
Robert Connelly is professor of mathematics at Cornell University and a pioneer in the study of tensegrities. His research focuses on discrete geometry, computational geometry, and the rigidity of discrete structures and its relations to flexible surfaces, asteroid shapes, opening rulers, granular materials, and tensegrities. In 2012 he was elected a fellow of the American Mathematical Society. Simon D. Guest is Professor of Structural Mechanics in the Structures Group of the Department of Engineering at the University of Cambridge, and Head of the Civil Engineering Division. His research straddles the border between traditional structural mechanics and the study of mechanisms, and includes work on 'morphing' and 'deployable' structures.
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