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Non-deformable Solid Mechanics Set

coordinated by
Abdelkhalak El Hami

Volume 2

Movement Equations 2

Mathematical and Methodological Supplements

Michel Borel

Georges Vénizélos

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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.