Non-deformable Solid Mechanics Set
coordinated by
Abdelkhalak El Hami
Volume 3
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:
ISTE Ltd
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© ISTE Ltd 2017
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: 2017947280
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-034-8
With this third volume, the series on non-deformable solids reaches its acme; this is where we introduce and enlarge on the movement equations of non-deformable solids, which was always the initial goal.
The first volume of the series served to prepare the material necessary for writing these equations, that is how best to situate a solid in space to study its motion, how to describe its kinematics, the velocity and acceleration fields that drive it, how to characterize a solid through its inertial and kinetic configurations, and determine the energy statement of its motion.
But the development of this material, to arrive at the movement equations, requires various mathematical tools which the authors thought useful to remind rather than letting research them individually. This is the point of Volume 2.
With this third volume, readers are ready to touch on the core of the matter, the fundamental principle of dynamics and its application to cases where solids are free, or considered to be linked when there are bonds restricting their motion.
Chapter 1 of the book proposes a global vision of the fundamental principle and the conditions for its use, in particular the case where the observation frame of the motion of a solid is non-Galilean. The frame from which the motion of a solid is observed is crucial as it is this environment which exerts efforts upon it, affecting its progression.
The efforts, whether they are known or unknown (the links), have on the motion energetic consequences which we will evaluate by applying the fundamental principle. Chapter 2 places the solid in its environment, identifies the efforts and characterizes the power and energetic aspects they put into play throughout the motion.
The data for the problem are therefore acquired through the two first chapters, that means the following one, Chapter 3, is then in a position to begin applying the fundamental principle by presenting and enlarging on the scalar consequences that result from it and which produce the movement equations. Chapter 3 then ends with an example which serves to look through the different forms of these scalar consequences, knowing that the one which eventually is chosen depends essentially on the problem at hand.
Chapter 4 proposes two interesting cases for the application of the fundamental principle and shows how movement equations are used in various complex problems the solutions to which can only be obtained from hypotheses and simplifications without which the problem would not be treatable. These two cases are the motion of the Earth using inertial assumptions, and Foucault’s pendulum according to the study by Michel Cazin in Sciences magazine in July 2000 where he bases himself on simplifying hypotheses to propose a credible explanation to the observed motion.
Chapter 5, which is the final chapter, plays a completely different role. Developing applications of the fundamental principle and establishing its scalar consequences require being familiar with the elements which contribute to its formulation, as they are presented in the first entry in the series. To grant readers with autonomy when using this book, a methodological formulary has been included, which recaps all essential points from Volume 1. This is the purpose of Chapter 5.
Arriving at this point, it is interesting to continue exploring certain individual cases through the ways they are used. This will be the subject of the fourth and fifth books in this series; the first among them will focus on the study of equilibrium situations for non-deformable solids and on small motions (or oscillations) that they experience around them; the final entry in the series will look at the motions of solid systems including cases of equilibrium and oscillations, with an introduction to robotics.
With this present volume and with the ones that preceded it and will follow it, the authors wished to explore the motion of non-deformable solids, and provide professional or student users with a structured mathematical approach. The lessons they have been giving at the CNAM since the 1970s has convinced them of the benefits of using such an approach and encouraged them to create this series.
| M | material point |
| t | time |
| mH | mass of the sun – 2.1030 kg |
| mT | mass of the Earth – 6.1024 kg |
| GH | center of inertia of the Sun |
| GT | center of inertia of the Earth |
| GT GH | distance between the Sun and the Earth ~ 150.109 m |
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Universal gravitational constant 6,67.10−11 m3kg−1S−2 |
| m(S) | mass of a solid (S) |
| δij | Kronecker symbol |
| εijk | three-index permutation symbol |
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vector |
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basis |
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frame |
| ψ, θ, φ | Euler angles, specifically the precession, nutation and spin angles in order |
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plane of the two vectors  and  |
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plane of the two vectors  and  passing through point O |
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bipoint vector |
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situation bipoint or situation vector of point OS in relation to the point Oλ of selected frame of reference  λ |
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angle of two vectors oriented from  towards  |
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norm of vector  |
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scalar product of vectors  and  |
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vector product of vectors  and  |
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polar unit vector in cylindrical-polar coordinates |
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polar unit vector in spherical coordinates |
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vector rotation of angle α around the axis defined by vector  |
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trajectory, in the frame λ, of material point M, during the time interval [ti, tf] |
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velocity at time t of the material point M during its motion in the frame λ |
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acceleration at time t of the material point M throughout its motion in the frame λ |
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rotation vector or rotation rate of the solid (S) in its motion in relation to frame λ |
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drive velocity of the material point M in the relative motion of the frame μ in relation to the frame λ |
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drive acceleration of the material point M in the relative motion of the frame μ in relation to the frame λ |
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Coriolis acceleration applied to the material point M during its relative motion of the frame μ in relation to the frame λ |
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derivative in relation to time of the vector  in the frame λ |
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torsor characterized by its two reduction elements at point P |
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resultant of the torsor { } : 1st reduction element |
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moment at P of the torsor { } : 2nd reduction element  |
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scalar invariant of the torsor { }, independent of point P |
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product of two torsors |
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velocity distributing torsor or kinematic torsor associated with the motion of the material point Ps of the solid (S) |
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kinetic torsor associated with the motion of the solid (S) in the frame λ |
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dynamic torsor associated with the motion of solid (S) in the frame λ |
| IOs (S|m) | inertia operator of the solid (S) provided the measure of mass m |
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inertia drive torsor of the solid (S) in the relative motion of λ in relation to g |
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inertia Coriolis torsor of solid (S) in the motion relative of λ in relation to g |
| {Δ} | torsor of known efforts acceleration of Earth’s gravity ~ 9.80665 ms−2 (9.81 on average) |
| g | depending on the location and latitude of the body which is subject to |
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torsor of unknown efforts |
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link acting upon a solid |
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torsor of link efforts applied to the solid (S) |
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power developed by the set of forces F acting upon the solid (S) throughout its motion |
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partial power relative to the variable Qα, developed by the set of forces F acting upon the solid (S) throughout its motion |
| T(λ)(S) | kinetic energy of the solid (S) throughout its motion in relation to the frame λ |
| (Lα) | Lagrange equation relative to the variable Qα |
When the situation of the solid (S) in the frame λ is represented by the parameters Qα, we write : 
where
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partial distributing torsor relative to the variable Qα |
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partial rotation rate relative to the variable Qα, component of the variable  of the rotation rate, such that  |
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component of the variable  of the velocity vector of the point OS, such that  |