139,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
This volume is the focal point of the work undertaken in the previous volumes of this set of books: the statement of the fundamental principle of the dynamics whose implementation, according to two paths whose choice depends on the problem to be treated, leads to equations of motion.
In order to achieve this, it is treated first of all in the context of solids in their environment, as a prerequisite for the formulation of the fundamental principle. Then, in addition to its use in some exercises, the approach is illustrated by three particular cases.
The first is an example where it is developed end-to-end and addresses the two approaches that lead to the equations of motion. The two other examples deal with two classical but important subjects, the movement of the Earth according to the hypotheses that can be stated about it, and Foucault’s pendulum.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 121
Veröffentlichungsjahr: 2017
Ähnliche
Table of Contents
Cover
Title
Copyright
Introduction
Table of Notations
1 Fundamental Principle of Dynamics
1.1. The fundamental principle of dynamics and its scalar consequences
1.2. Secondary principles
1.3. Motion of a set in a given frame
λ
1.4. Motion of a non-deformable solid in a given frame
2 Solid in Space. Efforts and Links: Power
2.1. Degrees of freedom of a solid
2.2. Free solid
2.3. Linked solids and links
2.4. Virtual power developed on a material set
2.5. Power of the efforts exerted on a solid
2.6. Properties of power
3 Scalar Consequences and Movement Equations
3.1. Establishment principle of the movement equations
3.2. Movement equations of a solid
3.3. Movement equations of the free solid
3.4. Movement equations of the linked solid with configurable links
3.5. Energetic expression of the equations of analytical mechanics
3.6. Summary example
4 Particular Applications
4.1. Simulation of the motion of Earth
4.2. Foucault’s pendulum
5 Methodological Formulary
5.1. Reference outline on the motion of a solid
5.2. Kinematics of the solid
5.3. Principle of motion with fixed plane
5.4. Combination of motions
5.5. Kinetics of non-deformable solids
Bibliography
Index
End User License Agreement
List of Illustrations
1 Fundamental Principle of Dynamics
Figure 1.1.
Set in the universe
Figure 1.2.
The solar Galilean frame
Figure 1.3.
Relative position of the two frames
λ
and
μ
2 Solid in Space. Efforts and Links: Power
Figure 2.1.
Material set
Figure 2.2.
System of two opposing forces
3 Scalar Consequences and Movement Equations
Figure 3.1.
Exercise 1 – Situation of the system
Figure 3.2.
Exercise 1 – Situation of the system
Figure 3.3.
Exercise 1 – Representing points alignment
Figure 3.4.
Exercise 2 – Configuration of the solid
Figure 3.5.
Exercise 2 – Position of the center of rotation
Figure 3.6.
Exercise 3 – Orthogonal section of cylinder S
1
Figure 3.7.
Exercise 3 – Frame-solid set 1
Figure 3.8.
Exercise 3 – Angular parameters
Figure 3.9.
Configuration of the device
4 Particular Applications
Figure 4.1.
Spherical model of Earth
Figure 4.2.
Configuration of the rotation rate
Figure 4.3.
Diagrams of Euler’s representation
Figure 4.4.
Configurations of the rotation rate
Figure 4.5.
Model of Earth
Figure 4.6.
Trajectory of G
T
in the ecliptic plane
Figure 4.7.
Principle of Foucault’s pendulum
Figure 4.8.
Relative situation of the different frames
Figure 4.9.
Situation of the body in the new reference frame
Figure 4.10.
Identification of the point I
Figure 4.11.
Situation of Foucault’s pendulum
Figure 4.12.
Hypocycloidal trajectory of the point P
5 Methodological Formulary
Figure 5.1.
Principle of the Maxwell corkscrew
Figure 5.2.
Locating principle of a solid
Figure 5.3.
Cartesian coordinates
Figure 5.4.
Cylindro-polar coordinates
Figure 5.5.
Polar parameter
Figure 5.6.
Spherical coordinates
Figure 5.7.
Euler angles
Figure 5.8.
Situation of a solid onto the plane
Figure 5.9.
Trajectory of a material point
Figure 5.10.
Principle of plane motion on a plane
Figure 5.11.
Relative situation of two frames
Guide
Cover
Table of Contents
Begin Reading
Pages
C1
iii
iv
v
ix
x
xi
xiii
xiv
xv
xvi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
G1
G2
G3
G4
G5
G6
e1
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.
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 27-37 St George’s Road London SW19 4EU UKwww.iste.co.uk
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USAwww.wiley.com
© 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
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.
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.
Table of Notations
M
material point
t
time
m
H
mass of the sun – 2.10
30
kg
m
T
mass of the Earth – 6.10
24
kg
G
H
center of inertia of the Sun
G
T
center of inertia of the Earth
G
T
G
H
distance between the Sun and the Earth ~ 150.10
9
m
Universal gravitational constant 6,67.10
−11
m
3
kg
−1
S
−2
m
(
S
)
mass of a solid (
S
)
δ
ij
Kronecker symbol
ε
ijk
three-index permutation symbol
vector
basis
frame
ψ
,
θ
,
φ
Euler angles, specifically the precession, nutation and spin angles in order
plane of the two vectors and
plane of the two vectors and passing through point O
bipoint vector
situation bipoint or situation vector of point O
S
in relation to the point O
λ
of selected frame of reference
λ
angle of two vectors oriented from towards
norm of vector
scalar product of vectors and
vector product of vectors and
polar unit vector in cylindrical-polar coordinates
polar unit vector in spherical coordinates
vector rotation of angle
α
around the axis defined by vector
trajectory, in the frame
λ
, of material point M, during the time interval [
t
i
, t
f
]
velocity at time
t
of the material point M during its motion in the frame
λ
acceleration at time
t
of the material point M throughout its motion in the frame
λ
rotation vector or rotation rate of the solid (
S
) in its motion in relation to frame
λ
drive velocity of the material point M in the relative motion of the frame
μ
in relation to the frame
λ
drive acceleration of the material point M in the relative motion of the frame
μ
in relation to the frame
λ
Coriolis acceleration applied to the material point M during its relative motion of the frame
μ
in relation to the frame
λ
derivative in relation to time of the vector in the frame
λ
torsor characterized by its two reduction elements at point P
resultant of the torsor {} : 1
st
reduction element
moment at P of the torsor {} : 2
nd
reduction element
scalar invariant of the torsor {}, independent of point P
product of two torsors
velocity distributing torsor or kinematic torsor associated with the motion of the material point P
s
of the solid (
S
)
kinetic torsor associated with the motion of the solid (
S
) in the frame
λ
dynamic torsor associated with the motion of solid (
S
) in the frame
λ
I
O
s
(
S
|
m
)
inertia operator of the solid (
S
) provided the measure of mass
m
inertia drive torsor of the solid (
S
) in the relative motion of
λ
in relation to
g
inertia Coriolis torsor of solid (
S
) in the motion relative of
λ
in relation to
g
{Δ}
torsor of known effortsacceleration 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
torsor of unknown efforts
link acting upon a solid
torsor of link efforts applied to the solid (
S
)
power developed by the set of forces
F
acting upon the solid (
S
) throughout its motion
partial power relative to the variable
Q
α
, 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
(
L
α
)
Lagrange equation relative to the variable
Q
α
When the situation of the solid (S) in the frame λ is represented by the parameters Qα, we write : where
partial distributing torsor relative to the variable
Q
α
partial rotation rate relative to the variable
Q
α
, component of the variable of the rotation rate, such that
component of the variable of the velocity vector of the point O
S
, such that
