Movement Equations 2

Michel BOREL, Retired since 2010, mechanics lecturer at CNAM Center in Versailles (1971-1999); associated professor at Petroleum and Motors National high School - ENSPM (1978-2008); engineer at Bertin Society (1970-1980); engineer at General Armement Direction - DGA (1980-2009). Georges VÉNIZÉLOS, Professor, Chair of Mechanical Systems Designing, Conservatoire National des Arts et Métiers (CNAM).

• Introduction xiTable of Notations  xiiiChapter 1. Vector Calculus  11.1. Vector space 11.1.1. Definition  11.1.2. Vector space – dimension – basis  21.1.3. Affine space 31.2. Affine space of dimension 3 – free vector 41.3. Scalar product a⋅b 51.3.1. Properties of the scalar product 61.3.2. Scalar square – unit vector  61.3.3. Geometric interpretation of the scalar product  71.3.4. Solving the equation a�� ⋅ x�� = 0  91.4. Vector product a ∧ b  91.4.1. Definition  91.4.2. Geometric interpretation of the vector product  101.4.3. Properties of vector product 111.4.4. Solving the equation a ∧ x = b  111.5. Mixed product (a ,b, c )  121.5.1. Definition  121.5.2. Geometric interpretation of the mixed product  121.5.3. Properties of the mixed product 131.6. Vector calculus in the affine space of dimension 3  151.6.1. Orthonormal basis 151.6.2. Analytical expression of the scalar product 161.6.3. Analytical expression of the vector product  161.6.4. Analytical expression of the mixed product  171.7. Applications of vector calculus  181.7.1. Double vector product 181.7.2. Resolving the equation a�� ⋅ x�� = b 221.7.3. Resolving the equation a ∧ x = b  231.7.4. Equality of Lagrange  251.7.5. Equations of planes 251.7.6. Relations within the triangle 271.8. Vectors and basis changes 281.8.1. Einstein’s convention  281.8.2. Transition table from basis (e) to basis (E) 301.8.3. Characterization of the transition table 32Chatper 2. Torsors and Torsor Calculus 352.1. Vector sets  352.1.1. Discrete set of vectors 352.1.2. Set of vectors defined on a continuum  362.2. Introduction to torsors  372.2.1. Definition 372.2.2. Equivalence of vector families  382.3. Algebra torsors  382.3.1. Equality of two torsors 382.3.2. Linear combination of torsors 392.3.3. Null torsors  392.3.4. Opposing torsor 402.3.5. Product of two torsors 402.3.6. Scalar moment of a torsor – equiprojectivity  412.3.7. Invariant scalar of a torsor 432.4. Characterization and classification of torsors  432.4.1. Torsors with a null resultant  432.4.2. Torsors with a no-null resultant  452.5. Derivation torsors  482.5.1. Torsor dependent on a single parameter q 492.5.2. Torsor dependent of n parameters qi functions of p  512.5.3. Explicitly dependent torsor of n + 1 parameters 52Chapter 3. Derivation of Vector Functions 553.1. Derivative vector: definition and properties  553.2. Derivative of a function in a basis  563.3. Deriving a vector function of a variable 573.3.1. Relations between derivatives of a function in different bases 573.3.2. Differential form associated with two bases  633.4. Deriving a vector function of two variables  653.5. Deriving a vector function of n variables 683.6. Explicit intervention of the variable p  703.7. Relative rotation rate of a basis relative to another  71Chapter 4. Vector Functions of One Variable Skew Curves  734.1. Vector function of one variable  734.2. Tangent at a point M  744.3. Unit tangent vector τ ( q)  764.4. Main normal vector ( ) q ν 774.5. Unit binormal vector ( ) q β 794.6. Frenet’s basis  804.7. Curvilinear abscissa  814.8. Curvature, curvature center and curvature radius 834.9. Torsion and torsion radius 844.10. Orientation in (λ) of the Frenet basis  87Chapter 5. Vector Functions of Two Variables Surfaces  915.1. Representation of a vector function of two variables 915.1.1. Coordinate curves 915.1.2. Regular or singular point – tangent plane – unit normal vector  935.1.3. Distinctive surfaces  955.1.4. Ruled surfaces 1015.1.5. Area element  1105.2. General properties of surfaces  1115.2.1. First quadratic form  1115.2.2. Darboux–Ribaucour’s trihedral 1145.2.3. Second quadratic form  1195.2.4. Meusnier’s theorems 1215.2.5. Geodesic torsion  1235.2.6. Prominent curves traced on a surface  1255.2.7. Directions and principal curvatures of a surface  127Chapter 6. Vector Function of Three Variables: Volumes 1356.1. Vector functions of three variables  1356.1.1. Coordinate surfaces 1356.1.2. Coordinate curves  1366.1.3. Orthogonal curvilinear coordinates 1366.2. Volume element 1376.2.1. Definition 1376.2.2. Applications to traditional coordinate systems 1386.3. Rotation rate of the local basis 1396.3.1. Calculation of the partial rotation rate 1δ (λ ,e)  1406.3.2. Calculation of the rotation rate  143Chapter 7. Linear Operators  1457.1. Definition 1457.2. Intrinsic properties  1457.3. Algebra of linear operators 1477.3.1. Unit operator 1477.3.2. Equality of two linear operators 1477.3.3. Product of a linear operator by a scalar 1477.3.4. Sum of two linear operators  1487.3.5. Multiplying two linear operators 1487.4. Bilinear form 1497.5. Quadratic form  1507.6. Linear operator and basis change 1507.7. Examples of linear operators  1527.7.1. Operation f = a ^ F 1527.7.2. Operation f = a ^ (a ^ F) 1527.7.3. Operation f = a(b ⋅ F)  1537.7.4. Operation f = a ^ (F ^ a) 1557.8. Vector rotation Ru��,a  1567.8.1. Expression of the vector rotation 1567.8.2. Quaternion associated with the vector rotation Ru��,a  1597.8.3. Matrix representation of the vector rotation  1607.8.4. Basis change and rotation vector 162Chapter 8. Homogeneity and Dimension 1658.1. Notion of homogeneity  1658.2. Dimension  1658.3. Standard mechanical dimensions 1668.4. Using dimensional equations 168Bibliography  171Index  173