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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
The set of books on Mechanical Engineering and Solid Mechanics, of which this book is the first volume, is an essential tool for those looking to develop a rigorous knowledge of the discipline, whether students, professionals (in search of an approach to a problem they are dealing with), or anyone else interested.
This volume deals with the elements required for establishing the equations of motion when dealing with solid bodies. Chapter 1 focuses on the systems of reference used to locate solid bodies relative to the observer, and demonstrates how to describe their position, orientation, and evolution during their motion. Chapter 2 introduces descriptors of motion such as velocity and acceleration, and develops the concept of torsor notation in relation to these descriptors. Finally, Chapter 3 concerns the notions of mass and inertia, as well as the kinetic torsor and dynamic torsor which consolidate the kinematic and kinetic aspects in a single concept.
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Table of Contents
Cover
Title
Copyright
Introduction
Table of Notations
1 Location of Solid Bodies
1.1. The notion of system of reference
1.2. Frame of reference
1.3. Location of a solid body
1.4. Positioning of a system of reference connected to a solid
1.5. Vector rotation
1.6. Other exercises
2 Solid Kinematics
2.1. Generalities on moving solids
2.2. Kinematics of a material point
2.3. Velocity field associated with the motion of the rigid solid
2.4. Acceleration field of the rigid solid
2.5. Motion with fixed plane
2.6. Combining motions within a mobile frame of reference
2.7. Relative motion of two rigid solids in contact
2.8. Other exercises
3 Kinetics of Solid Bodies
3.1. The mass of a continuous mechanical set
3.2. Center of the measure of
μ
on
3.3. Interpretation of the notion of center of measure
3.4. Kinetic torsor of a mechanical set
3.5. Dynamic torsor of a mechanical set (D)
3.6. Kinetic energy of a mechanical set
3.7. Partition of a continuous mechanical set
Bibliography
Index
End User License Agreement
List of Illustrations
1 Location of Solid Bodies
Figure 1.1.
Reference trihedron
Figure 1.2.
Projection parallel to an axis
Figure 1.3.
Illustration of how Maxwell’s corkscrew rule is applied
Figure 1.4.
The principle of Foucault’s pendulum
Figure 1.5.
The principle of locating a solid body
Figure 1.6.
The Cartesian coordinates
Figure 1.7.
Cylindrical-polar coordinates
Figure 1.8.
Principle of the notation
Figure 1.9.
Spherical coordinates
Figure 1.10.
Example of general coordinates
Figure 1.11.
The Cartesian system of reference
Figure 1.12.
Cylindrical-polar system of reference
Figure 1.13.
Spherical system of reference
Figure 1.14.
Diagram of the Frenet trihedron
Figure 1.15.
Diagram of the Darboux–Ribaucour trihedron
Figure 1.16.
Orientation of one basis relative to the other
Figure 1.17.
General diagram of the passage from one basis to another
Figure 1.18.
Euler angles diagram
Figure 1.19.
Euler angles, location of the systems of reference
Figure 1.20.
Precession
Figure 1.21.
Nutation
Figure 1.22.
Spin
Figure 1.23.
Vector rotation
Figure 1.24.
System of reference associated with the vector rotation
Figure 1.25.
Diagram of projection in the vector rotation
2 Solid Kinematics
Figure 2.1.
Rigid mechanical system
Figure 2.2.
Identification of a solid
Figure 2.3.
Trajectory of a material point
Figure 2.4.
Relative observations of a trajectory
Figure 2.5.
The spherical coordinates
Figure 2.6.
Motion with fixed plane
Figure 2.7.
Relative situation of the two systems
Figure 2.8.
Illustration of the effect of Coriolis acceleration
Guide
Cover
Table of Contents
Begin Reading
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G1
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Non-deformable Solid Mechanics Set
coordinated by
Abdelkhalak El Hami
Volume 1
Movement Equations 1
Location, Kinematics and Kinetics
Michel Borel
Georges Vénizélos
First published 2016 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
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2016
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: 2016946047
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-032-4
Introduction
The idea of mechanics is probably one of the first scientific thoughts that captivated the human mind when, for example, man wanted to use a lever to lift the stone that he needed for building a shelter. Then, with the passing of time, as the mechanics developed and enabled the description of the motion of bodies, man used it to conceive the functioning of machines, which at first were an aid in fulfilling his tasks, and then even replaced him.
Moreover, man entertained this idea, though unconsciously, when he took an interest in the cycle of the seasons, as it had occurred to him that he could provide for his subsistence by cultivating the land. Subsequently, astral phenomena started occupying his thoughts more and more, since he had always sensed they were influencing his life; and little by little, throughout the centuries he learnt how to analyze, comprehend and formalize these phenomena.
Then, Newton came along and formulated his laws, which led to the equations of motion! Their unquestionable power − though the resulting path equations are far more complex − was once more illustrated recently by the Rosetta spacecraft, which traveled for several years covering phenomenal distances through space before entering orbit around the Chury comet, on which it delivered the Philae lander, whose mission was to observe and analyze the comet. And, the challenge was met! A further recent proof of this power of the equations is the “theoretical discovery” of a planet on the frontier of the solar system, a result of the observation of the paths of neighboring astral bodies that were obviously influenced by its presence. It was in this way that Neptune was discovered in 1845.
There is something fascinating, even mysterious, about the equations of motion of a solid body, though they have nothing to do with magic, but simply with man-made science.
At the French National Conservatory of Arts and Crafts (Conservatoire National des Arts et Métiers – CNAM), this science is taught to very diverse audiences. Professor Michel Cazin, who led the chair of mechanics for over two decades, was very keen to provide the audience with clear and mathematically well-structured teaching, which was always well received, whether in Paris or in the centers throughout the country. Those of us who have worked by his side and have drawn inspiration from his ideas and approach, which he continuously sought to improve, have wished to preserve this manner of conceiving the teaching of mechanics, to the benefit of generations to come. This is the ambition that gives birth to this set of five volumes on Non-deform able Solid Mechanics.
In order to clearly distinguish the essential stages in the development of the formalism leading to the establishment of the equations of motion, which is in a way the core of the discipline, with its major types of application, Volumes 1 to 3 out of the Set will focus specifically on movement equations, the first of which is the present book.
This Volume 1 relates to the elements required for the establishment of the equations of motion, namely how to position in space the solid body whose motion is to be described, how to formulate its kinematics, which expresses this motion in terms of velocity and acceleration fields, and how to characterize its mass, inertia and energy properties.
Volume 2 is special. Throughout the development of the formalism, the mechanics scientist uses various mathematical tools − vectors, torsors, for example − that he is required to handle proficiently. In many real-life instances, having to a certain extent forgotten these tools, the practitioner needs information on how to properly handle some of these tools and achieve the desired result; it may happen that the information he finds is not always in a form that is readily usable, requiring time to adapt and properly use it. Volume 2 gathers a certain number of useful mathematical tools in a form that makes them ready to use in the application of the present work.
Volume 3 is dedicated first to the positioning of the solid body in its environment, while taking into account the forces acting on it, then to the introduction of the fundamental law of dynamics, and to the equations of motion that result from it either as a direct expression or as a consequence of the energy dimension of motion. This presentation is preceded by a methodological form that resumes the main formulae developed in Volume 1, aiming to guide the mechanics scientist in preparing the material required for setting up the equations of motion.
Throughout this third volume and the next ones, these equations will be used in the study of the small motions and vibrations of a solid body or of the conditions of stationary motion, as well as to express the motion of systems of solid bodies and introduce robotics.
Now, regarding the present volume, it consists of three main chapters, illustrated by a certain number of exercises that are presented either within each chapter, allowing for the content developed to be readily applied, or at the end of the chapter, if going further into the use of its content may be useful.
With a focus on locating solid bodies, the first chapter identifies the main types of systems of reference that can be used and shows how to identify the solid body in relation to them, how to describe its position, orientation and also evolution during the motion. The method for determining local systems of reference connected to curves or surfaces will be developed in the corresponding chapters of Volume 2. This chapter also presents the case where a rotation about an axis should be taken into account in order to define the location of a solid body; however, vector rotation will be more extensively developed in Volume 2.
The second chapter introduces motion descriptors such as velocity and acceleration, and then provides an extensive description of how the fields of velocity and acceleration of a solid body in motion can be expressed in torsor form, given that the torsor notation proves to be a particularly beneficial tool in the mechanics of the solid body. It examines the formulation of these fields in the specific cases of motion with fixed plane, with the very physical notions of fixed centroid and mobile centroid, the combination of motions when the systems of reference are moving relative to each other, and when the solid bodies are in contact with one another during their motion, by characterizing the nature of this contact in terms of displacements and velocities.
The third chapter introduces the kinetic properties of a solid that play an essential role in how its motion unfolds, namely mass and inertia. It introduces the two essential notions of kinetic torsor and dynamic torsor that amalgamate the two kinematic and kinetic aspects in the same concept; the expression of these two torsors is one of the last stages leading to the fundamental law of dynamics, the last being the energetic dimension of motion. This chapter approaches in its last part the energetic aspect by defining and developing the concept of kinetic energy of a moving solid body, by insisting on the torsor form of its expression, which will facilitate its subsequent use.
To conclude the introduction to this volume, the Non-deformable Solid Mechanics set such as described above is intended to be an essential tool for readers who want to gain quite rigorous knowledge of the discipline, such as students or those who, in the exercise of their profession, feel the need to develop their own approach to the problem they are dealing with. Digital technologies can certainly help engineers to solve the problems they are faced with, but among a practitioner’s preoccupations there may also be a need to go beyond the framework defined by software, hence the importance of having a tool to guide his efforts.
This is the spirit in which we have conceived this set and the feedback given by the CNAM audience throughout the years of teaching has been a great encouragement for us.
Table of Notations
M
material point
t
time
[
t
i
,
t
f
]
time interval
mass of a continuous mechanical set
ρ
(M)
density of the material point M
measure on a continuous mechanical set
δ
ij
Kronecker symbol
ε
ijk
three-index permutation symbol
average of the scalar or vector function
ϕ
on the continuous mechanical set
vector
basis
system of reference
table of passage from basis to basis
matrix of passage from basis to basis
x
,
y
,
z
Cartesian coordinates of a point
r
,
α
,
z
cylindrical-polar coordinates of a point
R
,
α
,
β
spherical coordinates of a point
ψ
,
θ
,
φ
Euler angles: precession, nutation and spin, respectively
plane of the two vectors plane of the two vectors passing through the point O
vector from point O (tail) to point M (head)
position vector of the point O
S
relative to the origin O
λ
of the chosen system of reference ‹
λ
›
angle between two vectors, oriented from
norm of vector
scalar product of two vectors
vector product of two vectors
polar vector in cylindrical-polar coordinates
polar vector in spherical coordinates
vector rotation by angle
α
about the axis defined by vector
trajectory, in the system of reference ‹
λ
›, of the material point M, during the time interval[
t
i
,
t
f
]
displacement, in the system of reference ‹
λ
›, of the material point M during the time interval[
t
i
,
t
f
]
velocity at time
t
of physical point M as it moves through reference system ‹
λ
›
acceleration at time
t
of physical point M as it moves through reference system ‹
λ
›
rotation vector of solid (
S
) as it moves in relation to reference system ‹
λ›
drive velocity of physical point M in relation to the movement of reference system ‹
μ
› relative to reference system
‹λ›
drive acceleration of physical point M in relation to the movement of reference system ‹
μ
› relative to reference system
‹λ›
Coriolis acceleration as applied to physical point M in relation to the movement of reference system ‹
μ
› relative to reference system
‹λ›
derivative with respect to time of vector in the system of reference
‹λ›
torsor characterized by its two reduction elements in point P
sum of torsor {}: 1st reduction element of the torsor
moment in P of the torsor {}: 2nd element of reduction of the torsor for which the relation is verified ∀Q
scalar invariant of torsor {}, independent of point P
product of two torsors
torsor distributor of velocities or kinematic torsor associated to the motion of the material point P
s
of the solid (
S
)
kinetic torsor associated with the motion of the solid (
S
) in the system of reference
‹λ›
dynamic torsor associated with the motion of solid (
S
) in the system of reference
‹λ›
I
O
S
(
S|m
)
inertia operator of the solid (
S
) having the measure of mass
m
matrix representation in the basis (
S
) of the inertia operator of the solid (
S
) in one point of the solid, taken as reference point, here O
S
A
,
B
,
C
moments of inertia of the solid (
S
) relative to the three axes of the system of reference (
S
) passing through the reference point of the operator, products of inertia of the solid (
S
)
D
,
E
,
F
relative to the three planes formed by the axes of the system of reference (
S
) and passing through the reference point of the operator
T
(
λ
)
(
S
)
kinetic energy of the solid (
S
) in its motion relative to the system of reference
‹λ›
2
T
(
λ
)
(
S
)
vis viva
of the solid (
S
) in its motion relative to the system of reference
‹λ›
If the location of the solid (S) relative to the system of reference ‹λ› is represented by the parameters qi with i ≤ 6, we write: where:
partial torsor distributor relative to the variable
q
i
partial rotation quantity relative to variable
q
i
, component of the variable of rotation quantity such as
component of the variable of the velocity of the point O
S
, such that
