Table of Contents

Cover

Title

Introduction

Table of Notations

1 Vector Calculus

1.1. Vector space
1.2. Affine space of dimension 3 – free vector
1.3. Scalar product
1.4. Vector product
1.5. Mixed product
1.6. Vector calculus in the affine space of dimension 3
1.7. Applications of vector calculus
1.8. Vectors and basis changes

2 Torsors and Torsor Calculus

2.1. Vector sets
2.2. Introduction to torsors
2.3. Algebra torsors
2.4. Characterization and classification of torsors
2.5. Derivation torsors

3 Derivation of Vector Functions

3.1. Derivative vector: definition and properties
3.2. Derivative of a function in a basis
3.3. Deriving a vector function of a variable
3.4. Deriving a vector function of two variables
3.5. Deriving a vector function of n variables
3.6. Explicit intervention of the variable p
3.7. Relative rotation rate of a basis relative to another

4 Vector Functions of One Variable Skew Curves

4.1. Vector function of one variable
4.2. Tangent at a point M
4.3. Unit tangent vector
4.4. Main normal vector
4.5. Unit binormal vector
4.6. Frenet’s basis
4.7. Curvilinear abscissa
4.8. Curvature, curvature center and curvature radius
4.9. Torsion and torsion radius
4.10. Orientation in (λ) of the Frenet basis

5 Vector Functions of Two Variables Surfaces

5.1. Representation of a vector function of two variables
5.2. General properties of surfaces

6 Vector Function of Three Variables: Volumes

6.1. Vector functions of three variables
6.2. Volume element
6.3. Rotation rate of the local basis

7 Linear Operators

7.1. Definition
7.2. Intrinsic properties
7.3. Algebra of linear operators
7.4. Bilinear form
7.5. Quadratic form
7.6. Linear operator and basis change
7.7. Examples of linear operators
7.8. Vector rotation

8 Homogeneity and Dimension

8.1. Notion of homogeneity
8.2. Dimension
8.3. Standard mechanical dimensions
8.4. Using dimensional equations

Bibliography

Index

End User License Agreement

List of Illustrations

1 Vector Calculus

Figure 1.1. Representation of the concept of vector in the affine space
Figure 1.2. Angle of vectors in the scalar product
Figure 1.3. Orthogonal projection of a vector on an axis
Figure 1.4. Orthogonal projection of a vector on an axis
Figure 1.5. Maxwell’s corkscrew rule
Figure 1.6. Geometric interpretation of the vector product
Figure 1.7. Geometric interpretation of the mixed product
Figure 1.8. Illustration of the double vector product
Figure 1.9. Projection of a vector on a plane
Figure 1.10. Resolving the equation
Figure 1.11. Resolving the equation
Figure 1.12. Plane normal to a vector and passing through a point
Figure 1.13. Plane defined by two vectors and through a point
Figure 1.14. Configuration of a triangle

2 Torsors and Torsor Calculus

Figure 2.1. Equiprojectivity of a torsor
Figure 2.2. Synthesized representation of null torsor
Figure 2.3. Synthetic representation of a couple torsor
Figure 2.4. Centerline of a torsor

4 Vector Functions of One Variable Skew Curves

Figure 4.1. Trajectory of a point in the affine space
Figure 4.2. Vector tangent to a skew curve
Figure 4.3. Normal plane to a skew curve
Figure 4.4. Normal to a skew curve, osculatory circle
Figure 4.5. Frenet’s basis (also called Frenet’s trihedral)
Figure 4.6. Frenet’s basis planes
Figure 4.7. Curvature center and radius of a skew curve

5 Vector Functions of Two Variables Surfaces

Figure 5.1. Coordinate curve clusters
Figure 5.2. Local frame associated with coordinate curves
Figure 5.3. Skew curve Γ
Figure 5.4. Cylindrical surface structure
Figure 5.5. Special case of cylindrical surface
Figure 5.6. Conical surface structure
Figure 5.7. Structure of revolving surface
Figure 5.8. Property of a revolving surface
Figure 5.9. Prominent planes associated with a ruled surface
Figure 5.10. Tangent plane to an undevelopable ruled surface
Figure 5.11. Singular point on a non-developable surface
Figure 5.12. Turning edge of a developable ruled surface
Figure 5.13. Skew curve and developable ruled surface
Figure 5.14. Area element
Figure 5.15. Darboux–Ribaucour trihedral
Figure 5.16. Rotation between the Frenet and Darboux–Ribaucour bases
Figure 5.17. Application setting for Meusnier’s theorems

6 Vector Function of Three Variables: Volumes

Figure 6.1. Volume element in the vicinity of the point M

7 Linear Operators

Figure 7.1. Vector rotation
Figure 7.2. Frame associated with the vector rotation
Figure 7.3. Diagram showing the projection of vector rotation

First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Michel Borel and Georges Vénizélos to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016953254

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ISBN 978-1-78630-033-1

Introduction

This second volume of the Non-deformable Solid Mechanics Set, which we have undertaken, takes a break in the progression toward the equations of motion that are our ultimate focal point.

Indeed, the development of these equations utilizes many mathematical tools that are not always easy to master when the need arises. With their many years of teaching experience, the authors of this Set intended to compile in this second volume the mathematical statements used to support the development of mechanical formalism.

Chapter 1 goes back over vectors, since they are the basic language in this formalism. Remember the rules of employment and the operations that they operate must facilitate their quasi-permanent usage.

Then, in Chapter 2, the torsors that are predominant in the process of developing the equations of motion come into question. As they can be used to synthesize a set of vectors, to simplify the writing of vector expressions along the course, to condense into a symbol the complementary aspects of a concept of mechanics, for example the equations of motion, their use is described in detail.

The movement of a mechanical system is by its very nature evolving over time; it is the same for all vector quantities involved in its description. Expressing their variations based on the parameters that describe a problem, notably time, brings us to consider the derivation of vectors and vector functions. This question is the subject of Chapter 3.

Now, if we consider the definition of the machining of a workpiece with a numerical control, the need of having to proceed along an appropriate curve imposes to prescribe the position of the tool in relation to this curve, with its own localized spotting. The study of vector functions of one, two or three variables that represent the skew curves, surfaces and volumes, and the local frames relating to it, is the subject of Chapters 4–6.

Many vector operations are performed in the formalism of rigid solid mechanics, apply to vectors, and have also vector results. It is therefore vector operators, often linear, which are expressed in matrix form in the formalism. The properties of these operators and their use in matrices are described in Chapter 7.

The formulas and equations in accordance with the development of the mechanical formalism must imperatively be homogeneous, that is to say all their additive terms shall have the same dimension, as well as both members of an equality. In addition, some of these terms are sometimes quite complex and ensuring their dimension is a precaution that must be instinctive to the engineer. Chapter 8 strives to instill a few straightforward rules to minimize the risk of errors throughout the process of developing equations of motion.

Finally, as a result of their countless lessons, how many times did the authors of this Set have to remind mathematics to their audience? They realized that the mathematical concepts not practiced regularly are very volatile and refreshing them on occasion was a necessity. This is what they wanted to do in this second volume of the Non-deformable Solid Mechanics Set, before going with the course of their presentation in Volume 3.

Movement Equations 2

Mathematical and Methodological Supplements

Introduction