Mechanics, tensors & virtual works:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Internat. Science Publ.
2003
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 460 S. |
| ISBN: | 1898326118 |
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| adam_text | Titel: Mechanics, tensors virtual works
Autor: Talpaert, Yves R
Jahr: 2003
CONTENTS
PREFACE .
CONTENTS
Chapter 0. REQUIREMENTS 1
1. POINT SPACE AND VECTOR SPACE.......................... 1
1.1 Point space (or affine space).................................... i
1.2 Frame of reference and basis....................................2
2. DYNAMS....................................................3
2.1 Dynam definition and reduction elements.........................4
Dynam definition.............................................. 4
Representation of dynams ....................................... 6
2.2 Properties and operations on dynams............................ 6
Equality of dynams.............................................6
Operations on dynams.......................................... 7
Equiprojective fields of moments.................................10
Invariants....................................................U
Reduction of a vector system and dynam...........................12
2.3 Dynam of velocities...........................................17
Velocity field................................................ 17
Dynam of velocities........................................... 20
2.4 Acceleration vectors.......................................... 21
2.5 Sliding velocity.............................................. 26
3. EXERCISES.................................................27
Chapter 1. STATICS 33
1. CLASSIC METHOD......................................34
1.1 Mechanical actions.......................................34
Definitions...................................................34
Forces......................................................34
Moment of a force.............................................34
Dynam of a mechanical action...................................35
1.2 Classification of forces........................................36
External forces............................................... 36
Internal forces................................................37
13 Equilibrium conditions....................................... 38
Definitions and conditions...................................... 38
Particular collections of forces applied to a rigid body.................43
vii
Contents
1.4 Types of equilibrium of rigid bodies and structures................45
1.5 Stress and contact dynam .....................................47
.................................47
Stress...................................................
Contact dynam............................................... ^8
Dry friction and Coulomb laws.................................. 50
1.6 Types of constraints.......................................... 54
Punctual constraint...........................................55
Rectilinear constraint.......................................... 56
Annular-linear constraint....................................... 58
Ball-and-socket joint.......................................... 59
Plane support................................................ 60
Sliding pivot.................................................61
Sliding guide................................................ 63
Screw joint.................................................. 64
Pivot........................................................65
Embedding or welded joint..................................... 67
1.7 Free-body diagram...........................................68
2. METHOD OF VIRTUAL WORK...............................70
2.1 Number of degrees of freedom and generalized coordinates.........71
Number of degrees of freedom...................................71
Generalized coordinates........................................72
Types of constraints........................................... 75
2.2 Virtual displacements and virtual velocities.......................78
Generalized coordinates........................................78
Definition and expression of virtual displacements ...................79
Virtual velocity and examples................................... 81
Virtual fields and dynams....................................... 91
2.3 Virtual work................................................ 94
Definitions, rigid body and ideal constraint.........................94
Principle of virtual work (First expression)......................... 98
Principle of virtual work (Second expression)......................106
3. EXERCISES
114
Chapter 2. TENSORS
1.1
135
FIRST STEPS WITH TENSORS............................ 135
Multilinear forms........................................... 135
Linear mapping..............................................135
Multilinear form............................................. 136
1.2 Dual space, vectors and covectors..............................137
Dual space..................................................137
Expression of a covector.......................................137
Einstein summation convention..................................138
Change of basis and cobasis.................................... 140
1.3 Tensors and tensor product...................................143
Contents ix
Tensor product of multilinear forms..............................143
Tensor of type (°) ............................................144
Tensor of type (J,) ............................................145
Tensor of type (£) ............................................146
Tensor of type (J) ............................................149
Tensoroftype (}) ............................................150
Tensor of type ($) ........................................... 151
Symmetric and antisymmetric tensors............................ 153
2. OPERATIONS ON TENSORS................................156
2.1 Tensor algebra..............................................156
Addition of tensors...........................................156
Multiplication by a scalar......................................157
Tensor multiplication......................................... 157
2.2 Contraction and tensor criteria................................158
Contraction..................................................158
Tensor criteria...............................................162
3. EUCLIDEAN VECTOR SPACE.............................. 164
3.1 Pre-Euclidean vector space................................... 164
Scalar multiplication and pre-Euclidean space......................164
Fundamental tensor...........................................164
3.2 Canonical isomorphism and conjugate tensor....................166
Canonical isomorphism........................................166
Conjugate tensor and reciprocal basis.............................167
Covariant and contravariant representations of vectors...............170
Representation of tensors of order 2 and contracted products.......... 172
3.3 Euclidean vector spaces......................................174
4. EXTERIOR ALGEBRA......................................178
4.1 p-forms....................................................178
Definition of a p-form.........................................178
Exterior product of 1-forms.................................... 179
Expression of a p-form........................................ 181
Exterior product of p-forms.................................... 184
Exterior algebra..............................................185
4.2 vectors...................................................188
5. POINT SPACES............................................192
5.1 Point space and natural frame.................................192
Point space..................................................192
Coordinate system and frame of reference.........................192
Natural frame................................................194
5.2 Tensor fields and metric element.............................. 197
Transformations of curvilinear coordinates........................ 197
X
Contents
Tensor fields..................................
Metric element .................................
5.3 Christoffel symbols............................
Definition of Christoffel symbols..................
Ricci identities and Christoffel formulae............
5.4 Absolute differential, Covariant derivative, Geodesic
Absolute differential of a vector and covariant derivatives............207
Absolute differential of a tensor and covariant derivatives........................209
Geodesic and Euler s equations..................................................................211
Parallel transport........................................................................................213
Absolute derivative of a vector..................................................................214
5.5 Volume form and adjoint........................................................................216
Volume form..............................................................................................216
Adjoint................................................218
5.6 Differential operators..............................................................................220
Gradient......................................................................................................220
Divergence.................................................225
Curl............................................................................................................228
Laplacian...............................................230
6. EXERCISES............................................................................................232
Chapter 3 MASS GEOMETRY AND INERTIA TENSOR 263
1. MASS DISTRIBUTION AND INTEGRALS.................... 263
1.1 Density................................................... 263
1.2 Integrals of real-valued functions and vector functions........... 265
2. CENTER OF MASS........................................ 267
2.1 Definitions................................................ 267
2.2 Subdivision................................................269
23 Theorems of Guldin (and Pappus)...................,......... 271
3. INERTIA TENSOR.........................................272
3.1 Moments and products of inertia .............................. 272
3.2 Inertia tensor.............................................. 274
4. INERTIA ELLIPSOID...................................... 276
4.1 Moment of inertia about an axis.............................. 276
4.2 Equation of the quadric......................................278
4.3 Nature of the quadric................................................................279
4.4 Radius of gyration..................................................................^ 280
5.
PRINCIPAL AXES
281
Contents xi
5.1 Fundamental theorem about a symmetric tensor.................281
5.2 Equal eigenvalues...........................................284
5.3 Inertia ellipsoid and principal axes.............................286
5.4 Material symmetries........................................ 287
6. THEOREM OF STEINER (and HUYGENS)....................288
7. EXERCISES...............................................291
Chapter 4 KINETICS AND DYNAMICS OF SYSTEMS 299
1. NEWTON S POSTULATES..................................299
1.1 Experimental laws.......................................... 300
1.2 Postulates..................................................301
1.3 Galilean relativity and inertial frames..........................303
2. KINETICS.................................................307
2.1 Kinetic dynam..............................................307
2.2 Kinetic energy..............................................308
3. THEOREMS OF MECHANICS OF SYSTEMS..................309
3.1 First integrals of a system of particles.......................... 309
3.2 Linear momentum theorems..................................310
Linear momentum theorem.....................................310
Theorem of conservation of mass................................311
Theorem of motion of mass center...............................311
Special case of rigid bodies.....................................312
3.3 Angular momentum theorems.................................316
Angular momentum theorem....................................316
Relation between kinetic dynam and dynam of forces................317
Conservation of angular momentum..............................318
Special case of rigid bodies.....................................319
3.4 Kinetic energy theorems......................................323
Kinetic energy theorem........................................323
Special case of rigid bodies.....................................326
4. EXERCISES...............................................331
Chapter 5 LAGRANGIAN DYNAMICS AND VARIATIONAL PRINCIPLES 339
1. LAGRANGIAN DYNAMICS.................................339
1.1 Holonomic and scleronomic systems............................340
1.2 D Alembert-Lagrange principle............................... 342
^ Contents
344
13 Lagrange s equations.......................................
Lagrange s equations in the general case.........................
Lagrange s equations for conservative forces ......................
Lagrange s equations with undetermined multipliers................ 35U
1.4 Configuration space and Lagrange s equations.................. 354
1.5 Adjoint Lagrangian and first integrals......................... 358
2. VARIATIONAL CALCULUS AND PRINCIPLES...............360
2.1 Euler s equations............................................361
A variational problem and variations.............................361
Euler s equations............................................ 364
2.2 Hamilton s variational principle...............................368
Hamilton s postulate..........................................368
Hamilton s principle and motion equations........................369
2.3 Jacobi s form of the principle of least action of Maupertuis........ 371
3. EULER-NOETHER THEOREM..............................374
3.1 One-parameter group of diffeomorphisms...................... 374
3.2 Euler-Noether theorem......................................376
3. EXERCISES............................................... 379
Chapter 6 HAMILTONIAN MECHANICS 393
1. N-BODY PROBLEM AND CANONICAL EQUATIONS......... 393
2. CANONICAL EQUATIONS AND HAMILTONIAN............. 397
2.1 Legendre transformation and Hamiltonian..................... 397
23 Canonical equations.........................................401
23 First integrals and cyclic coordinates...........................404
2.4 Liouville s theorem in statistical mechanics......................406
3. CANONICAL TRANSFORMATIONS.........................409
3.1 Lagrange and Poisson brackets................................409
Preservation of canonical form and Poisson bracket..................409
Poisson bracket and symplectic matrix............................411
Lagrange and Poisson brackets..................................415
3.2 Canonical transformations....................................416
Canonical transformations and brackets...........................417
Canonical transformations and generating functions.................419
4. HAMILTON-JACOBI EQUATION........................... 425
4.1 Hamilton-Jacobi equation and Jacobi theorem.................. 425
4.2 Separability..........................................................................43Q
5. EXERCISES.....................
BIBLIOGRAPHY
Contents
INDEX
|
| any_adam_object | 1 |
| author | Talpaert, Yves |
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| discipline | Physik |
| edition | 1. publ. |
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| spelling | Talpaert, Yves Verfasser aut Mechanics, tensors & virtual works Yves R. Talpaert 1. publ. Cambridge Cambridge Internat. Science Publ. 2003 XIII, 460 S. txt rdacontent n rdamedia nc rdacarrier Calculus of tensors Mechanics, Analytic Theoretische Mechanik (DE-588)4185100-6 gnd rswk-swf Theoretische Mechanik (DE-588)4185100-6 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012858381&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Talpaert, Yves Mechanics, tensors & virtual works Calculus of tensors Mechanics, Analytic Theoretische Mechanik (DE-588)4185100-6 gnd |
| subject_GND | (DE-588)4185100-6 |
| title | Mechanics, tensors & virtual works |
| title_auth | Mechanics, tensors & virtual works |
| title_exact_search | Mechanics, tensors & virtual works |
| title_full | Mechanics, tensors & virtual works Yves R. Talpaert |
| title_fullStr | Mechanics, tensors & virtual works Yves R. Talpaert |
| title_full_unstemmed | Mechanics, tensors & virtual works Yves R. Talpaert |
| title_short | Mechanics, tensors & virtual works |
| title_sort | mechanics tensors virtual works |
| topic | Calculus of tensors Mechanics, Analytic Theoretische Mechanik (DE-588)4185100-6 gnd |
| topic_facet | Calculus of tensors Mechanics, Analytic Theoretische Mechanik |
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