The Modelling and Analysis of the Mechanics of Ropes

128 Pages·2014·4.076 MB·English

by  C.M. Leech (auth.)

Solid Mechanics and Its Applications
C. M. Leech
The Modelling
and Analysis of
the Mechanics
of Ropes
Solid Mechanics and Its Applications
Volume 209
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The median level of presentation is the first year graduate student. Some texts
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C. M. Leech
The Modelling and Analysis
of the Mechanics of Ropes
123
C. M.Leech
Windsor House
Congleton, Cheshire
UK
ISSN 0925-0042 ISSN 2214-7764 (electronic)
ISBN 978-94-007-7840-5 ISBN 978-94-007-7841-2 (eBook)
DOI 10.1007/978-94-007-7841-2
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To my wife, Brenda and to my daughters,
Andrea, Helen, Nicola and Suzanne
Preface
Linear fibre structures, typified by the many configurations of ropes, cables, stays
and the smaller components, yarns and strands, have been the subject of analysis
over many years.This isanattempt toformalise the subjectandtoemphasisethe
exploitation of these developments in the various engineering industries.
Aswellasconsideringtheconventionaltheoriesoftwistedassemblyextension,
thetheoriesofloadingofbraidedandplaitedstructures,bending,inter-component
friction and component dilation and distortion are introduced. Also introduced is
the modelling of wear or abrasion and of component heating.
Various modelling techniques have also evolved, typically the finite element
theories and their associated finite element codes. These have been developed so
that they form the regular tool in engineering offices, through their application to
design and engineering assessment. However, they do use some element of con-
tinua even though they may account for many different components. Fibre struc-
tures on the other hand contain typically many million components, and it is not
really appropriate to use such packages. This text considers these assemblies, the
various sub-assemblies and their interaction behaviour. A notable inclusion is the
frictionbetweenthecomponents,andthevariousmodesthroughwhichfrictionis
applicable.
This text is solely focused on linear structures where the effect and consider-
ation of the third dimension, length, is somewhat secondary to the primary
dimensions, diameter and angular disposition. It is essentially a quasi-static
investigation, although there is some small reference to general dynamics.
Finally there are other types of fibre assemblies, areas which include textiles
and cloth, and volumetric assemblies which typically include bulk assemblies;
these are not considered here.
Acknowledgments
The author would like to thank the various colleagues at UMIST and TTI for
various suggestions and inspiration that have surfaced over the years of contact.
The author is also indebted to Mr. Gerry Needham for the preparation of various
illustrations that are used in this text.
vii
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline of Following Chapters . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Closure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Fibre Geometry and Fibre Mechanics. . . . . . . . . . . . . . . . . . . . . . 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Fibre Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Fibre Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Fibre Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Fibre Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.3 Fibre Material Stress–Strain Behaviour . . . . . . . . . . . . . 10
2.4 Primary Deformation, Extension . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Polynomial Approximations. . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Strain Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Anelasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.1 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.2 Linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.3 Nonlinear Viscoelasticity. . . . . . . . . . . . . . . . . . . . . . . 20
2.5.4 Large Strain Rate Modelling for Polymer Fibres. . . . . . . 23
2.5.5 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.6 Creep and Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Secondary Deformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.6.1 Fibre Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6.2 Fibre Flexure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.7 Strain Energy or the Energy of Deformation. . . . . . . . . . . . . . . 35
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 Component Path Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1 Curves in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 General Path Geometry . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.2 Geometry Preserving Structures . . . . . . . . . . . . . . . . . . 39
ix
x Contents
3.2 The Helical Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 The Helix Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Extensional and Twist Deformations. . . . . . . . . . . . . . . 41
3.2.3 Flexure in Helical Components. . . . . . . . . . . . . . . . . . . 42
3.2.4 The Partitioning of Strain Energy. . . . . . . . . . . . . . . . . 43
Reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Transversely Continuous Structures. . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Geometrically Preserving Structures. . . . . . . . . . . . . . . . . . . . . 48
4.1.1 The Assembly of Transversely Continuous Structures. . . 48
4.1.2 The Straining of Transversely Continuous Structures. . . . 50
4.1.3 The Loading of Transversely Continuous Structures . . . . 53
4.1.4 An Analytic Result for Strain Energy . . . . . . . . . . . . . . 53
4.1.5 The Principle of Virtual Work for Transversely
Continuous Structures . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.6 The Progressive Failure of Transversely
Continuous Structures . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2 Parallel Structures, Variability. . . . . . . . . . . . . . . . . . . . . . . . . 57
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Hierarchical Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.1 Component Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Normalisation of Hierarchical Structures . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6 Transversely Discrete Structures . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.1 Numerous Small Component Structures. . . . . . . . . . . . . . . . . . 67
6.1.1 The Hierarchical Tree . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.1.2 The Assembly of Transversely Discrete Structures . . . . . 69
6.1.3 The Straining of Transversely Discrete Structures. . . . . . 70
6.1.4 The Principle of Virtual Work for Transversely
Discrete Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Component Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2.1 The Assembly of Large Component Structures. . . . . . . . 73
6.2.2 The Loading of Large Component Structures. . . . . . . . . 77
6.3 Tubular Woven Geometries: Braiding and Plaits. . . . . . . . . . . . 78
6.3.1 The Geometry of Braids and Plaits. . . . . . . . . . . . . . . . 80
6.4 Bending of Helical Structures. . . . . . . . . . . . . . . . . . . . . . . . . 83
6.4.1 General Kinematics Associated with Bent Structures. . . . 83
6.4.2 No Slip (Friction) Bending. . . . . . . . . . . . . . . . . . . . . . 84
6.4.3 Geodesic (No Friction) Bending. . . . . . . . . . . . . . . . . . 85
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Contents xi
7 Contact Force and Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.1 Contact Forces and Pressures . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.2 Friction and Relative Slip: Inter Modes . . . . . . . . . . . . . . . . . . 94
7.2.1 Mode 1: Axial Slip. . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.2.2 Mode 2: Component Twist. . . . . . . . . . . . . . . . . . . . . . 98
7.2.3 Mode 3: Scissoring. . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.2.4 Mode 4: Sawing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.2.5 Transition Between the Sliding Modes . . . . . . . . . . . . . 105
7.3 Component Transverse Deformation and Set: Intra Modes. . . . . 106
7.3.1 Dilation Measure, N and Packing Factor P . . . . . . . . . . 107
f
7.3.2 Distortion Measure, X. . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3.3 The Set in Distortion and Dilation . . . . . . . . . . . . . . . . 108
7.3.4 The Energy of Dilation and Distortion. . . . . . . . . . . . . . 111
7.3.5 The Continuum Model. . . . . . . . . . . . . . . . . . . . . . . . . 112
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8 Component Wear, Life and Heating. . . . . . . . . . . . . . . . . . . . . . . 115
8.1 The Work Done by Relative Motion Between Components . . . . 115
8.2 The Friction Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
8.3 Wear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
8.4 Steady State or Equilibrium Temperature, Surface
Heat Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123