Movement Equations 3: Dynamics and Fundamental Principle

185 Pages·2017·2.64 MB·English

by  Michel Borel, Georges Vénizélos

Movement Equations 3
Non-deformable Solid Mechanics Set
coordinated by
Abdelkhalak El Hami
Volume 3
Movement Equations 3

Dynamics and Fundamental Principle

Michel Borel
Georges Vénizélos
First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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Library of Congress Control Number: 2017947280

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A CIP record for this book is available from the British Library
ISBN 978-1-78630-034-8
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Table of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Chapter 1. Fundamental Principle of Dynamics . . . . . . . . . . 1
1.1. The fundamental principle of dynamics and
its scalar consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1. Fundamental principle of dynamics . . . . . . . . . . . . . . . 1
1.1.2. Choosing a frame . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3. Preferred time scale . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2. Secondary principles . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1. First secondary principle of the separation of effects . . . . 7
1.2.2. Second secondary principle of effort generators . . . . . . . 8
1.2.3. Third secondary principle of effort receivers . . . . . . . . . 8
1.3. Motion of a set (D) in a given frame 〈λ〉 . . . . . . . . . . . . . . 9
1.3.1. Presentation of the context . . . . . . . . . . . . . . . . . . . . 9
1.3.2. Combination of accelerations . . . . . . . . . . . . . . . . . . 10
1.3.3. Coriolis inertial torsor . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4. Drive inertial torsor . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.5. Relation between the dynamic torsors
in the two frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.6. Applying the fundamental principle . . . . . . . . . . . . . . 13
1.4. Motion of a non-deformable solid in a given frame . . . . . . . . 14
1.4.1. Coriolis inertial torsor . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.2. Drive inertial torsor . . . . . . . . . . . . . . . . . . . . . . . . 16
vi Movement Equations 3
Chapter 2. Solid in Space. Efforts and Links: Power . . . . . . . 19
2.1. Degrees of freedom of a solid . . . . . . . . . . . . . . . . . . . . . 19
2.2. Free solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1. Velocity distributing torsor . . . . . . . . . . . . . . . . . . . . 20
2.2.2. Kinetic torsor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.3. Dynamic torsor . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.4. Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.5. Applying the fundamental principle of dynamics . . . . . . . 22
2.3. Linked solids and links . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1. Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2. Configurable links . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3. Linked solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4. Virtual power developed on a material set (D) . . . . . . . . . . . 28
2.5. Power of the efforts exerted on a solid . . . . . . . . . . . . . . . . 30
2.5.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.2. Discrete force field . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.3. Non-deformable mechanical set . . . . . . . . . . . . . . . . . 31
2.5.4. Continuous mechanical set . . . . . . . . . . . . . . . . . . . . 31
2.6. Properties of power . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6.1. Powers developed in two distinct frames . . . . . . . . . . . . 32
2.6.2. Case of a system of forces equivalent to zero acting
on a solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6.3. Case of a system of forces equivalent to zero acting on a
deformable mechanical set . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6.4. Partial powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Chapter 3. Scalar Consequences
and Movement Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1. Establishment principle of the movement equations . . . . . . . . 35
3.1.1. Vector projection . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.2. Torsor products . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.3. Choice of representative scalar consequences . . . . . . . . . 38
3.2. Movement equations of a solid . . . . . . . . . . . . . . . . . . . . 39
3.2.1. Scalar consequences via vector projection . . . . . . . . . . . 39
3.2.2. Scalar consequences of the analytic mechanics
of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2.3. Linear independence of torsors . . . . . . . . . . . . . . . . . . 58
3.2.4. Exercise 3 – Scalar consequences using
analytical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3. Movement equations of the free solid . . . . . . . . . . . . . . . . 72
Contents vii
3.4. Movement equations of the linked solid with
configurable links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.1. Velocity distributing torsor and partial
distributing torsors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.2. Case of configurable links independent of time . . . . . . . . 76
3.4.3. Case of configurable links dependent on time. . . . . . . . . 77
3.4.4. Perfect configurable links . . . . . . . . . . . . . . . . . . . . 78
3.5. Energetic expression of the equations of analytical
mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.1. Case of configurable links explicitly
independent of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5.2. Case of configurable links explicitly
dependent on time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6. Summary example . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6.1. Locating the solid . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.6.2. Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.6.3. Solid kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.6.4. Kinetics of the solid . . . . . . . . . . . . . . . . . . . . . . . . 85
Chapter 4. Particular Applications . . . . . . . . . . . . . . . . . . . 97
4.1. Simulation of the motion of Earth . . . . . . . . . . . . . . . . . . 97
4.1.1. Application of the fundamental principle . . . . . . . . . . . 97
4.1.2. Theorem of dynamic moment at G . . . . . . . . . . . . . . . 99
4.1.3. Theorem of dynamic resultant . . . . . . . . . . . . . . . . . . 108
4.2. Foucault’s pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.2.1. Observation of the phenomenon . . . . . . . . . . . . . . . . . 114
4.2.2. Analyzing the phenomenon . . . . . . . . . . . . . . . . . . . 116
Chapter 5. Methodological Formulary . . . . . . . . . . . . . . . . . 127
5.1. Reference outline on the motion of a solid . . . . . . . . . . . . . 127
5.1.1. Representation of a frame . . . . . . . . . . . . . . . . . . . . 127
5.1.2. Reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.1.3. Situation of the solid . . . . . . . . . . . . . . . . . . . . . . . 130
5.1.4. Notion of basis in a frame . . . . . . . . . . . . . . . . . . . . 133
5.2. Kinematics of the solid . . . . . . . . . . . . . . . . . . . . . . . . 133
5.2.1. Kinematics of a material point M . . . . . . . . . . . . . . . . 134
5.2.2. Kinematics of non-deformable solids . . . . . . . . . . . . . . 139
5.3. Principle of motion with fixed plane. . . . . . . . . . . . . . . . . 142
5.3.1. Kinematics of a solid . . . . . . . . . . . . . . . . . . . . . . . 142
viii Movement Equations 3
5.3.2. Fixed and mobile centroids in a motion
with fixed plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.4. Combination of motions . . . . . . . . . . . . . . . . . . . . . . . . 145
5.4.1. Combination of velocities . . . . . . . . . . . . . . . . . . . . . 145
5.4.2. Combination of accelerations . . . . . . . . . . . . . . . . . . . 147
5.5. Kinetics of non-deformable solids . . . . . . . . . . . . . . . . . . 148
5.5.1. Center of measure . . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.5.2. Linear momentum and kinetic torsor . . . . . . . . . . . . . . 149
5.5.3. Dynamic torsor . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.5.4. Kinetic energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Introduction
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.
x Movement Equations 3
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