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 |  موضوع: كتاب Computational Modeling of Tensegrity Structures السبت 14 نوفمبر 2020, 11:16 am | |
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أخوانى فى الله
أحضرت لكم كتاب
Computational Modeling of Tensegrity Structures
Art, Nature, Mechanical and Biological Systems
Buntara Sthenly Gan
و المحتوى كما يلي :
Contents
1 All About Tensegrity . 1
1.1 Definition 1
1.2 Computational Tensegrity . 2
1.3 Where Can We Find Tensegrity? 3
1.3.1 Tensegrity in Nature and Biology . 3
1.3.2 Tensegrity in Art and Architecture . 4
1.3.3 Tensegrity in Mechanical Engineering . 5
1.4 Mathematical Modeling of a Pin-Joint Structure 5
1.5 Classification of Tensegrity 8
1.6 Creating a Class I Tensegrity: A Workshop 9
1.6.1 Material . 10
1.6.2 Triangular Icosahedron Tensegrity . 10
1.6.3 Truncated Tetrahedron Tensegrity . 14
1.6.4 Rhombicuboctahedron Tensegrity . 14
1.6.5 Skew Hexagonal Cylindrical Tensegrity 16
References 20
2 Linear Algebra for Tensegrity . 23
2.1 First-Order Linear Equation in Matrix Form . 23
2.1.1 Homogeneous Matrix . 24
2.1.2 Zero Determinant of a Matrix 24
2.2 Rectangular Matrix 26
2.2.1 Vertical Matrix 27
2.2.2 Horizontal Matrix 28
2.3 A Solution of an Indefinite Rectangle Matrix 29
2.4 Rank of a Matrix 29
2.4.1 Full Rank Matrix 30
2.4.2 Deficient Rank Matrix 31xii
2.5 Eigenvalues Decomposition (EVD) 31
2.5.1 Various Transformation Matrix . 33
2.5.2 Various Transformation Matrices’ Program Listing . 35
2.6 Singular Value Decomposition (SVD) 37
2.6.1 An Inverse of a Matrix Using SVD 38
2.6.2 Inverse of Rectangular Matrices’ Program Listing 40
Further Reading . 41
3 Structural Computations by Using SVD . 43
3.1 Structural Analysis of a System of Structure . 43
3.2 Classification of a Structural System . 43
3.3 A Physical Interpretation of SVD of an Equilibrium Matrix . 44
3.4 Force Density Method (FDM) 46
3.4.1 Equilibriums at a Node . 46
3.4.2 Connectivity Matrix 47
3.4.3 Equilibrium Matrix of a Structural System 48
3.5 Equilibrium of a Structural System 49
3.6 Solving Structural System Examples . 51
3.6.1 Class I Structural System 52
3.6.2 Class I Structural System Program List . 53
3.6.3 Class II Structural System . 55
3.6.4 Class II Structural System Program List 57
3.6.5 Class III Structural System 59
3.6.6 Class III Structural System Program List . 61
3.6.7 Class IV Structural System 63
3.6.8 Class IV Structural System Program List . 65
References 67
4 Form-Finding of a Tensegrity . 69
4.1 Rank Condition and Nullity of a Tensegrity . 69
4.1.1 Flowchart for Form-Finding of a Tensegrity . 70
4.2 Initial Stiffness of a Tensegrity . 70
4.2.1 Triplex Tensegrity Example 72
4.2.2 Calculations of Tension Coefficient 76
4.2.3 Form-Finding Conditions 77
4.2.4 TriplexFormFindingExample Program List 79
4.3 Truncated Cone Tensegrity Example . 82
4.3.1 Calculations of Nodal Coordinates 82
4.3.2 Calculations of Tension Coefficient 85
4.3.3 TconeFormFindingExample Program List 90
4.4 Triangular Icosahedron Tensegrity Example . 92
4.4.1 Calculations of Tension Coefficient 95
4.4.2 Form-Finding Conditions 96
4.4.3 Icosahedron Form-Finding Example Program List 99
References 103
Contentsxiii
5 Designing a Tensegrity . 105
5.1 Introduction 105
5.2 Tensegrity Structures . 106
5.3 Numerical Form-Finding Method . 106
5.3.1 Form-Finding of Tensegrity Using Force
Density Prototype 106
5.3.2 Approximation of Coordinates from Tension
Coefficients 107
5.3.3 Approximation of Force Densities from Coordinates 108
5.4 Genetic Algorithm (GA) 108
5.4.1 A Practical Example of Using GA . 109
5.4.2 Example Using GA Program List . 114
5.5 GA for Form-Finding of a Tensegrity 116
5.5.1 Solution Procedure . 117
5.5.2 Encoding Scheme for Individual Population . 118
5.5.3 Fitness and Penalty Functions 118
5.6 Numerical Examples . 121
5.6.1 Six-Node Irregular Tensegrity Structure 121
5.6.2 Eight-Node Irregular Tensegrity Structure . 121
5.6.3 Ten-Node Irregular Tensegrity Structure 122
5.6.4 IrregularFormFindingUsingGA Program List 122
5.7 Design of a Real Eight-Node Irregular Tensegrity Structure . 135
5.7.1 Form-Finding . 135
5.7.2 Sizing of Members . 136
5.7.3 Structural Analysis . 138
5.7.4 Loadings and Stress/Stability Response Evaluation . 138
References 140
6 Tensegrity in Art and Architecture: Geometrical Works . 141
6.1 Tensegrities Inside a Sphere Geometry . 141
6.2 Cylindrical Tensegrity 142
6.2.1 N-Plex Cylindrical Tensegrity Generator Program List . 144
6.3 Truncated-Cone (T-Cone) Tensegrity 148
6.3.1 N-Plex T-Cone Tensegrity Generator Program List 153
6.4 Conic Tensegrity 155
6.4.1 N-Plex Conic Tensegrity Generator Program List . 157
6.5 Paraboloid Tensegrity . 158
6.6 Various Stacked Cylindrical Tensegrity . 158
6.6.1 Layered N-Plex Cylindrical Tensegrity Program List 160
6.7 3D Geometric Transformation 162
6.8 Arch Tensegrity . 163
6.8.1 Arch Tensegrity Program List 164
6.9 Spiral Tensegrity 166
6.9.1 Spiral Tensegrity Program List . 167
Contentsxiv
7 Tensegrity for Mechanical Application: Vibration . 171
7.1 Introduction 171
7.2 Dynamics of Axially Prestressed Cables 172
7.3 Equation of Motion 173
7.4 Spectral Element Modeling 174
7.4.1 Governing Equations in the Frequency Domain 174
7.4.2 Spectral Nodal DOFs, Forces, and Moments . 175
7.5 Weak Form of Governing Equation 177
7.5.1 Spectral Element Equation . 177
7.6 Local to Global Axis Transformation and Assembling . 179
7.7 Vibration of a Uniform Prestressed Cable Example . 182
7.8 Vibration of a Uniform Prestressed Cable Example
Program List . 183
7.8.1 Vibration of a Triplex Tensegrity Example 185
References 190
8 Tensegrity in Biological Application: Cellular Tensegrity . 193
8.1 Introduction to Cell Mechanics . 193
8.2 Models for Cytoskeletal Mechanics 193
8.3 Nucleated Cell of Tensegrity . 194
8.3.1 NucleatedCellTensegrity Program List . 195
8.4 Enumeration of Triangular Icosahedron Tensegrity 198
8.4.1 EnumerationTIcosahedronTensegrityGenerator
Program List 199
8.5 Models for Red Blood Cell 201
8.5.1 Enumeration Ten-Plex Cylindrical Tensegrity
Program List 202
8.6 Tensegrity Models for DNA 204
8.6.1 DNA Strands of Tensegrity Module Program List . 204
References 207
Index .
A
Arch tensegrity, 4, 5, 163
Art, 4, 5
Axially prestressed structures, 172
B
Biological application
cytoskeletal mechanics, 193, 194
nucleated cell, 194
red blood cell, 201
triangular icosahedron tensegrity, 198
Biology, 3, 4
C
Cell mechanic, 193
Class I tensegrity
material, 10
rhombicuboctahedron, 14, 16
skew hexagonal cylindrical, 16, 20
triangular icosahedron tensegrity, 10–14
truncated tetrahedron, 14
Computational cost and speed, 3
Computational tensegrity, 2, 3
Conic tensegrity, 155–158
Constant matrix, 24
Coordinate transformation and assembling
processes, tensegrity member,
179–182
Cylindrical tensegrity
enumeration of 3-9 plex, 145–146
MATLAB code, N-plexCylindricalTensegr
ityGenerator.m., 144–147
shifting/rotation angle, 142
struts and braces, 143
tension coefficients, 144
Cytoskeleton, 4, 193, 194
D
Deficient rank matrix, 31
Degree of freedom (DOF), 8
Deoxyribonucleic acid (DNA), 204
Design
real eight-nodes irregular tensegrity
structure
loadings and stress/stability response
evaluation, 138, 139
numerical form-finding process
algorithm, 135–136
sizing of members, 136
structural analysis, 138
Discrete Fourier transform
(DFT), 172
Dynamic shape functions, 171
Dynamic stiffness method (DSM), 171
E
Eigenvalues decomposition (EVD)
linear algebra, 31
program listing, various transformation
matrices, 35, 37
singular values, 32
singular vectors, 32
square matrix, 33
transformation matrix, 33, 34, 36
Eight-node irregular tensegrity structure, 121,
124, 125
Index210
Enumeration ten-plex cylindrical tensegrity,
202–204
Equilibrium matrix
decomposition, 59, 64
FDM, 55
FGM, 51
physical interpretation of SVD, 44, 45
SEM, 186
structural system, 43, 48–51
SVD, 70, 108
F
Finite element method (FEM)
assembling process, 8
discretization, 7
eight-node tensegrity object, 138
elements, assembled, 7
harp, 7
idealization of physical system, 6
First order linear equation
constant matrix, 24
zero determinant of matrix, 24–26
Floating point operations per second (FLOPS),
2
Force density method (FDM)
cables network problems, 46
connectivity matrix, 47
equilibrium matrix, structural system, 48,
49, 52, 59
equilibriums at a node, 46, 47
tensegrity structure, 46
tension coefficients, 85, 95
Form-finding
cable connections chart, 142
flowchart, 70, 71
GA (see Genetic algorithm (GA))
irregular tensegrity, MATLAB code,
122–134
iteration process, 106
stochastic procedure and numerical
optimization algorithm, 105
structural configurations, self-stresses, 105
T-cone tensegrity, 82–92
triangular icosahedron tensegrity, 96–102
triplex tensegrity, 72–79
Free vibration
axially prestressed triplex tensegrity,
185–186
equation of motion, 173, 174
uniform prestressed cable, 182–183
Frequency domain, 172, 175, 177
Full rank matrix, 30
G
Generalized stress, 49
Genetic algorithm (GA)
connectivity matrix, 117
crossover, 109, 113
eliteness, 109
fitness functions, 109, 114
form-finding
connectivity matrices and prototype
force density vector, 117
encoding scheme, 118, 119
fitness and penalty functions, 118, 120,
121
irregular tensegrity structure, 117
global minimum value, polynomial
function, 111
MATLAB code, maximum value of
polynomial function, 114–116
mutation, 109, 113
numerical optimization algorithm, 108
pairing of parents, 112
polynomial function, minimum values, 110
population, 112
H
Homogeneous matrix, 24
Horizontal matrix equations, 29
Hybrid tensegrity bridge, 5
I
Inconsistent matrix equations, 25, 27, 29, 31
Indefinite matrix equations, 26, 30
Indefinite rectangle matrix, 29
Initial tangent stiffness matrix, 72
Irregular tensegrity structure, 117
L
Layered N-plex cylindrical tensegrity, 160,
161
Least squares solution, 37
Local coordinate system, 179
M
Mathematical modeling
FEM (see Finite element method (FEM))
harp, FEM, 7
physical system, 6
Mechanical engineering, 5
Million instructions per second (MIPS), 2
Index211
N
N-polygon cylindrical tensegrity, 141
Nucleated cell, 194–198
Null-space matrix, 72, 78, 88, 99
Null-space vector, 30, 33
Numerical form-finding method
approximation of force densities, 108
approximation of nodal coordinates, 107
force density prototype, 106
Numerical technique
eight-node irregular tensegrity structure,
121, 124, 125
MATLAB code, irregular tensegrity,
122–134
six-node irregular tensegrity structure,
121–123
ten-node irregular tensegrity structure, 122,
126–128
P
Paraboloid tensegrity, 158, 159
Probability-based mutation, 121
Pseudoinverse, 37, 39
R
Rank
conditions, 69, 70, 108
deficiency, 78, 88, 98, 107, 120
deficient rank matrix, 31, 56, 59, 64
full rank matrix, 30
Rectangular matrix
horizontal, 28, 29
vertical, 27, 28
Red blood cell, 201
Rhombicuboctahedron tensegrity, 14, 16
Rule of connection, 7
S
Self-stressing, 1
Singular value decomposition (SVD)
characteristics of matrices, 37
inverse matrix, 38, 39
InverseOfRectangularMatrices.m, 40
physical interpretation, equilibrium matrix,
44, 45
pseudoinverse, 37
vertical matrix, 38
Singular values, 32, 37, 50
Singular vectors, 32, 37, 44
Six-node irregular tensegrity structure,
121–123
Skew hexagonal cylindrical tensegrity, 16, 20
Skylon, 4
Spectral analysis method (SAM), 171
Spectral element matrix, 179, 180
Spectral element method (SEM), 172
Spectral element modeling
dynamic shape function, 175, 176
governing equations, frequency domain, 174
spectral element equation, 177–179
spectral nodal DOFs, forces and moments,
175
weak form, governing equation, 177
Sphere geometry, 141
Spiral tensegrity, 166–169
Stacked cylindrical tensegrity, 158–162
Structural analysis, 43, 138
Structural system
Class I, 52–55
Class II, 55–59
Class III, 59–63
Class IV, 63–67
classification, 43, 44
configuration, 43
equilibrium concept, 49–51
equilibrium matrix, 48, 49
Struts and cables, 8
T
Ten-node irregular tensegrity structure, 122,
126–128
Tensegrity
art and architecture, 4
classification, 8, 9
definition, 1
initial stiffness, 70, 72
mechanical engineering, 5
nature and biology, 3, 4
rank condition and nullity, 69, 70
struts and cables, 9
3D geometric transformation, 162, 163
Triangular icosahedron tensegrity, 10–14,
198–199
Cartesian coordinate axes, 92
Cartesian coordinates of nodes, 96
coordinate system, 93
form finding conditions, 96, 98–100
Lagrange multiplier, 94
MATLAB code, form finding, 99–102
tension coefficient, calculations, 95
Index
Triplex
Cartesian coordinates of nodes, 75
creation, triangular prism, 72
form finding conditions, 77–79
MATLAB code, form finding, 79, 81
nodal coordinates, 72, 74, 75
polar coordinates of nodes, 73
symmetrical constraints, 74
tension coefficients, calculations, 76, 77
Truncated cone (T-cone) tensegrity
Cartesian coordinates of nodes, 85
connectivity configurations, 150, 151
constrained optimization problem, 83
creation, square cone, 82
enumeration of 3-9 plex, 151
form finding conditions, 87–89
MATLAB code, form finding, 90, 92
nodal coordinates, 148, 151
N-plexTconeTensegrityGenerator program
list, 153–154
polar coordinates of nodes, 83
shifting/rotation angle, 149
struts and braces, 148
tension coefficient, calculations, 85
tension coefficients, 85, 86, 150
Truncated tetrahedron tensegrity, 14
V
Vertical matrix equation, 27, 28
Vibration modes, 171
W
Workshop
material, 10
rhombicuboctahedron tensegrity, 14, 16
skew hexagonal cylindrical tensegrity, 16,
20
triangular icosahedron tensegrity,
10–14
truncated tetrahedron tensegrity, 14
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