Movement Equations 2 Mathematical and Methodological Supplements

The formalism processing of unbuckled solids mechanics involves several mathematical tools which are to be mastered at the same time. This volume collects the main points which take place in the course of the formalism, so that the user immediately finds what he needs without looking for it. Furthermore, the book contains a methodological formulary to guide the user in his approach.

Georges VÉNIZÉLOS, Professor, Chair of Mechanical Systems Designing, Conservatoire National des Arts et Métiers (CNAM).

Introduction xi

Table of Notations  xiii

Chapter 1. Vector Calculus  1

1.1. Vector space 1

1.1.1. Definition  1

1.1.2. Vector space – dimension – basis  2

1.1.3. Affine space 3

1.2. Affine space of dimension 3 – free vector 4

1.3. Scalar product a⋅b 5

1.3.1. Properties of the scalar product 6

1.3.2. Scalar square – unit vector  6

1.3.3. Geometric interpretation of the scalar product  7

1.3.4. Solving the equation a;; ⋅ x;; = 0  9

1.4. Vector product a ∧ b  9

1.4.1. Definition  9

1.4.2. Geometric interpretation of the vector product  10

1.4.3. Properties of vector product 11

1.4.4. Solving the equation a ∧ x = b  11

1.5. Mixed product (a ,b, c )  12

1.5.1. Definition  12

1.5.2. Geometric interpretation of the mixed product  12

1.5.3. Properties of the mixed product 13

1.6. Vector calculus in the affine space of dimension 3  15

1.6.1. Orthonormal basis 15

1.6.2. Analytical expression of the scalar product 16

1.6.3. Analytical expression of the vector product  16

1.6.4. Analytical expression of the mixed product  17

1.7. Applications of vector calculus  18

1.7.1. Double vector product 18

1.7.2. Resolving the equation a;; ⋅ x;; = b 22

1.7.3. Resolving the equation a ∧ x = b  23

1.7.4. Equality of Lagrange  25

1.7.5. Equations of planes 25

1.7.6. Relations within the triangle 27

1.8. Vectors and basis changes 28

1.8.1. Einstein’s convention  28

1.8.2. Transition table from basis (e) to basis (E) 30

1.8.3. Characterization of the transition table 32

Chatper 2. Torsors and Torsor Calculus 35

2.1. Vector sets  35

2.1.1. Discrete set of vectors 35

2.1.2. Set of vectors defined on a continuum  36

2.2. Introduction to torsors  37

2.2.1. Definition 37

2.2.2. Equivalence of vector families  38

2.3. Algebra torsors  38

2.3.1. Equality of two torsors 38

2.3.2. Linear combination of torsors 39

2.3.3. Null torsors  39

2.3.4. Opposing torsor 40

2.3.5. Product of two torsors 40

2.3.6. Scalar moment of a torsor – equiprojectivity  41

2.3.7. Invariant scalar of a torsor 43

2.4. Characterization and classification of torsors  43

2.4.1. Torsors with a null resultant  43

2.4.2. Torsors with a no-null resultant  45

2.5. Derivation torsors  48

2.5.1. Torsor dependent on a single parameter q 49

2.5.2. Torsor dependent of n parameters qi functions of p  51

2.5.3. Explicitly dependent torsor of n + 1 parameters 52

Chapter 3. Derivation of Vector Functions 55

3.1. Derivative vector: definition and properties  55

3.2. Derivative of a function in a basis  56

3.3. Deriving a vector function of a variable 57

3.3.1. Relations between derivatives of a function in different bases 57

3.3.2. Differential form associated with two bases  63

3.4. Deriving a vector function of two variables  65

3.5. Deriving a vector function of n variables 68

3.6. Explicit intervention of the variable p  70

3.7. Relative rotation rate of a basis relative to another  71

Chapter 4. Vector Functions of One Variable Skew Curves  73

4.1. Vector function of one variable  73

4.2. Tangent at a point M  74

4.3. Unit tangent vector τ ( q)  76

4.4. Main normal vector ( ) q ν 77

4.5. Unit binormal vector ( ) q β 79

4.6. Frenet’s basis  80

4.7. Curvilinear abscissa  81

4.8. Curvature, curvature center and curvature radius 83

4.9. Torsion and torsion radius 84

4.10. Orientation in (λ) of the Frenet basis  87

Chapter 5. Vector Functions of Two Variables Surfaces  91

5.1. Representation of a vector function of two variables 91

5.1.1. Coordinate curves 91

5.1.2. Regular or singular point – tangent plane – unit normal vector  93

5.1.3. Distinctive surfaces  95

5.1.4. Ruled surfaces 101

5.1.5. Area element  110

5.2. General properties of surfaces  111

5.2.1. First quadratic form  111

5.2.2. Darboux–Ribaucour’s trihedral 114

5.2.3. Second quadratic form  119

5.2.4. Meusnier’s theorems 121

5.2.5. Geodesic torsion  123

5.2.6. Prominent curves traced on a surface  125

5.2.7. Directions and principal curvatures of a surface  127

Chapter 6. Vector Function of Three Variables: Volumes 135

6.1. Vector functions of three variables  135

6.1.1. Coordinate surfaces 135

6.1.2. Coordinate curves  136

6.1.3. Orthogonal curvilinear coordinates 136

6.2. Volume element 137

6.2.1. Definition 137

6.2.2. Applications to traditional coordinate systems 138

6.3. Rotation rate of the local basis 139

6.3.1. Calculation of the partial rotation rate 1δ (λ ,e)  140

6.3.2. Calculation of the rotation rate  143

Chapter 7. Linear Operators  145

7.1. Definition 145

7.2. Intrinsic properties  145

7.3. Algebra of linear operators 147

7.3.1. Unit operator 147

7.3.2. Equality of two linear operators 147

7.3.3. Product of a linear operator by a scalar 147

7.3.4. Sum of two linear operators  148

7.3.5. Multiplying two linear operators 148

7.4. Bilinear form 149

7.5. Quadratic form  150

7.6. Linear operator and basis change 150

7.7. Examples of linear operators  152

7.7.1. Operation f = a ^ F 152

7.7.2. Operation f = a ^ (a ^ F) 152

7.7.3. Operation f = a(b ⋅ F)  153

7.7.4. Operation f = a ^ (F ^ a) 155

7.8. Vector rotation Ru;;,a  156

7.8.1. Expression of the vector rotation 156

7.8.2. Quaternion associated with the vector rotation Ru;;,a  159

7.8.3. Matrix representation of the vector rotation  160

7.8.4. Basis change and rotation vector 162

Chapter 8. Homogeneity and Dimension 165

8.1. Notion of homogeneity  165

8.2. Dimension  165

8.3. Standard mechanical dimensions 166

8.4. Using dimensional equations 168

Bibliography  171

Index  173

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