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# Preface of "Mechanics of Rigid Bodies"

The objective of this book is to develop the fundamental statements of the Mechanics of Rigid Bodies. The text is designed for undergraduate courses of Mechanical Engineering. The basic mathematical concepts are covered in the first part, thereby making the book self-contained.
The different parts of the book are carefully developed to provide continuity of the concepts and theories. Finally the text has been established so as to construct chapter after chapter a unified procedure for analyzing any mechanical system constituted of rigid bodies.

The first part, Mathematical Basics, introduces the usual concepts needed in the study of mechanical systems: vector space R3, geometric space, vector derivatives, curves. A chapter is devoted to torsors whose concept is the key of the book. The general notion of "measure center" is introduced in this chapter.

The second part, Kinematics, begins with the analysis of the motion of a point (kinematics of point). Particular motions are next considered, with a chapter related to motions with central acceleration. Next, is studied the kinematics of a rigid body:
parameter of situation, kinematic torsor, analysis of particular motions. The change of reference system, which introduces the notion of "training" has been excluded deliberately from this part. The notion of "training" is not really assimilated by the students at this level of the text. In fact this notion is implicitly introducing by using the concept of kinematic torsor. The change of reference system will be considered as a whole within the frame of Kinetics (Part 4). The last chapter analyses the kinematics
of rigid bodies in contact.

The third part, Mechanical Actions, introduces first the general concepts on the mechanical actions exerted on a rigid body or on a system of rigid bodies. Represented by torsors, the mechanical actions have general properties which are derived from the concepts considered previously for torsors. Thus, mechanical actions are classified as forces, couples and arbitrary actions. Gravitation and gravity are analyzed. A chapter is devoted to the mechanical actions involved by the connections between rigid bodies, whose concept is the basis of the technological design of mechanical systems. The introduction of the power developed by a mechanical action simplifies greatly the restrictions imposed in the case of perfect connections (connections without friction). In the last chapter, the investigation of some problem of Statics will grow the reader familiar with the analysis of mechanical actions exerted on a body or a system of bodies.

The fourth part, Kinetics of Rigid Bodies, introduces the tools needed to analyze the problems of Dynamics: operator of inertia, kinetic torsor, dynamic torsor and kinetic energy. Next, the problem of the change of reference system is considered.

At this step, the reader has acquired the whole elements needed to analyze the problems of Dynamics of a rigid body or a system of rigid bodies. This analysis is developed in the fifth part Dynamics of Rigid Bodies. First, the general process for analyzing a problem of Dynamics is established. Next, particular problems are considered. The process of analysis is always the same: kinematic analysis, kinetic analysis, investigation of the mechanical actions, deriving the equations of Dynamics as a consequence of the fundamental principle of dynamics, assumptions on the physical nature of connections between bodies, solving the equations of motion and the equations of connections. The designer will have to take an interest in the parameters of the motion as well as in the mechanical actions exerted at the level of connections to design the mechanical systems. The application of the of the fundamental principle of dynamics allow to derive the whole equations of dynamics (equations of motion and equations of mechanical actions at the level of connections). However, designer which takes an interest only in the equations of motion needs a systematic tool for deriving these equations: the Lagrange’s equations which are considered in the last chapter of part V.

In general, the equations of motions of a body or of a system of rigid bodies are complex, and most of these equations cannot be solved using an analytical process. Now, mechanical engineers dispose of numerical tools (numerical processes and microcomputers) needed to solve the motion equations, whatever the complexity of these equations may be. The sixth part, Numerical procedures for the Resolution of Motion Equations, is an introduction to the numerical processes used to solve equations of motion. Examples are considered.

The correction of the exercises is reported at the end of the textbook. The writing has been developed extensively and structured in such a way to improve the capacity of the comprehension of the reader. At the end of the textbook, the designer will have all the elements which allow him to implement a complete and structured analysis of mechanical
systems.

Vallouise, France                                            Jean-Marie BERTHELOT

September 2015