Teodorescu P.P. Mechanical Systems, Classical Models Volume II: Mechanics of Discrete and Continuous Systems
Подождите немного. Документ загружается.
Mechanical Systems, Classical Models
MATHEMATICAL AND ANALYTICAL
TECHNIQUES WITH APPLICATIONS
TO ENGINEERING
Series Editor
Alan Jeffrey
The importance of mathematics in the study of problems arising from the real
world, and the increasing success with which it has been used to model situations
ranging from the purely deterministic to the stochastic, in all areas of today's
Physical Sciences and Engineering, is well established. The progress in applicable
mathematics has been brought about by the extension and development of many
important analytical approaches and techniques, in areas both old and new,
frequently aided by the use of computers without which the solution of realistic
problems in modern Physical Sciences and Engineering would otherwise have been
impossible. The purpose of the series is to make available authoritative, up to date,
and self-contained accounts of some of the most important and useful of these
analytical approaches and techniques. Each volume in the series will provide a
detailed introduction to a specific subject area of current importance, and then will
go beyond this by reviewing recent contributions, thereby serving as a valuable
reference source.
For other titles published in this series, go to
www.springer.com/series/7311
Mechanical Systems,
Classical Models
Volume II:
123
Mechanics of Discrete
by
Petre P. Teodorescu
Faculty of Mathematics,
University of Bucharest,
Romania
and Continuous Systems
© 2002 EDITURA TEHNICÃ
This translation of “Mechanical Systems, Classical Models” (original title: Sisteme mecanice. Modele
clasice, Published by: EDITURA TEHNIC , Bucuresti, Romania, 1984 2002), First Edition, is
published by arrangement with EDITURA TEHNICÃ, Bucharest, RO MANIA.
-
Translated into E nglish, revised and extended by Petre P. Teodorescu
Ã
Romania
Printed on acid-free paper
987654321
springer.com
Prof. Dr. Petre P. Teodorescu
Str. Popa Soare 38
023984 Bucuresti 20
ISBN: 978-1-4020-8987-9 e-ISBN: 978-1-4020-8988-6
Library of Congress Control Number: 2007943834
c
2009 Springer Science+Business Media B.V.
No part of this work may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming, recording
or otherwise, without written permission from the Publisher, with the exception
of any material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work.
Cover Design: eStudio Calamar S.L.
Contents
Preface vii
11. Dynamics of Discrete Mechanical Systems 1
11.1 Dynamics of Discrete Mechanical Systems with Respect to an
Inertial Frame of Reference 1
11.1.1 Introductory Notions 1
11.1.2 General Theorems. Conservation Theorems 10
11.2 Dynamics of Discrete Mechanical Systems with Respect to a
Non-inertial Frame of Reference 50
11.2.1 Motion of a Discrete Mechanical System with Respect
to a Koenig Frame of Reference 50
11.2.2 Motion of a Discrete Mechanical System with Respect
to an Arbitrary Non-inertial Frame of Reference 58
12.
Dynamics of Continuous Mechanical Systems 79
12.1 General Considerations 79
12.1.1 Introductory Notions. General Principles 79
12.1.2 General Theorems. Conservation Theorems 94
12.2 One-dimensional Continuous Mechanical Systems 101
12.2.1 Motion of Threads 101
12.2.2 Motion of a Straight Bar 119
13.
Other Considerations on Dynamics of Mechanical Systems 129
13.1 Motions with Discontinuity 129
13.2 Dynamics of Mechanical Systems of Variable Mass 169
13.2.1 Discrete Mechanical Systems 169
13.2.2 Applications 179
13.2.3 Continuous Mechanical Systems 189
14.
Dynamics of the Rigid Solid 193
14.1 Motion of a Free or Constrained Rigid Solid 193
14.1.1 Motion of the Free Rigid Solid 193
14.1.2 Motion of the Rigid Solid Subjected to Constraints 229
v
13.1.1 Phenomenon of Collision in Case of a Discrete
Mechanical System 129
13.1.2 Elastic and Plastic Collisions of Discrete Mechanical
Systems 155
MECHANICAL SYSTEMS, CLASSICAL MODELS
vi
14.2 Motion of a Rigid Solid about a Fixed Axis. Plane-parallel
Motion of the Rigid Solid 239
14.2.1 Motion of the Rigid Solid about a Fixed Axis 239
14.2.2 Plane-parallel Motion of the Rigid Solid 253
15.
Dynamics of the Rigid Solid with a Fixed Point
15.1.1 General Results
15.2 The Case in which the Ellipsoid of Inertia is of Rotation. Other
Cases of Integrability
15.2.2 The Sonya Kovalevsky Case
15.2.3 Other Cases of Integrability
16.
Other Considerations on the Dynamics of the Rigid Solid
16.1 Motions of the Earth
16.1.1 Euler’s Cycle. The Regular Precession
16.1.2 Free Nutation. Pseudoregular Precession
16.2 Theory of the Gyroscope
16.2.1 General Results
16.2.2 Applications
16.3 Dynamics of the Rigid Solid of Variable Mass
16.3.1 Variational Methods of Calculation
Variable Mass
16.3.3 The Motion of the Aircraft Fitted Out with Jet
Propulsion Motors
17. Dynamics of Systems of Rigid Solids
17.1 Motion of Systems of Rigid Solids
17.1.1 General Results
17.1.2 Contact of Two Rigid Solids
17.2 Motion with Discontinuities of the Rigid Solids. Collisions
17.2.1 Percussion of Two Rigid Solids
17.2.2 Motion of a Rigid Solid Subjected to the Action of a
Percussive Force
17.3 Applications in Dynamics of Engines
17.3.1 General Results
17.3.2 Applications
Bibliography
Subject Index
Name Index
15.1 General Results. Euler-Poinsot case
15.1.2 The Euler-Poinsot Case
15.2.1 The Lagrange-Poisson Case
16.3.2 Applications to Dynamics of the Rigid Solid of
279
279
279
300
342
342
355
364
381
381
381
390
396
396
420
436
436
441
444
459
459
459
471
498
498
512
519
520
526
533
553
557
As it was already seen in the first volume of the present book, its guideline is precisely
the mathematical model of mechanics. The classical models which we refer to are in
fact models based on the Newtonian model of mechanics, on its five principles, i.e.: the
inertia, the forces action, the action and reaction, the parallelogram and the initial
conditions principle, respectively. Other models, e.g., the model of attraction forces
between the particles of a discrete mechanical system, are part of the considered
Newtonian model. Kepler’s laws brilliantly verify this model in case of velocities much
smaller than the light velocity in vacuum. The non-classical models are relativistic and
quantic.
Mechanics has as object of study mechanical systems. The first volume of this book
dealt with particle dynamics. The present one deals with discrete mechanical systems
for particles in a number greater than the unity, as well as with continuous mechanical
systems. We put in evidence the difference between these models, as well as the
specificity of the corresponding studies; the generality of the proofs and of the
corresponding computations yields a common form of the obtained mechanical results
for both discrete and continuous systems. We mention the thoroughness by which the
dynamics of the rigid solid with a fixed point has been presented. The discrete or
continuous mechanical systems can be non-deformable (e.g., rigid solids) or deformable
(deformable particle systems or deformable continuous media); for instance, the
condition of equilibrium and motion, expressed by means of the “torsor”, are necessary
and sufficient in case of non-deformable systems and only necessary in case of
deformable ones.
Passing by non-significant details, one presents some applications connected to
important phenomena of the nature and one gives also the possibility to solve problems
presenting interest from technical, engineering point of view. In this form, the book
becomes – we dare say – a unique outline of the literature in the field; the author wishes
to present the most important aspects connected with the study of mechanical systems,
mechanics being regarded as a science of nature, as well as its links to other sciences of
nature. Implications in technical sciences are not neglected.
Concerning the mathematical tool, the five appendices contained in the first volume
give the book an autonomy with respect to other works, special previous mathematical
knowledge being not necessary. The numeration of the chapters follows that of the first
volume, to which one makes reference for various results (theorems, formulae etc.).
The book covers a wide number of problems (classical or new ones), as one can see
from its contents. It uses the known literature, as well as the original results of the
author and his more than fifty years experience as a Professor of Mechanics at the
Preface
v i i
MECHANICAL SYSTEMS, CLASSICAL MODELS
viii
University of Bucharest. It is devoted to a large circle of readers: mathematicians
(especially those involved in applied mathematics), physicists (particularly, those
interested in mechanics and its connections), chemists, biologists, astronomers,
engineers of various specialities (civil, mechanical engineers etc., who are scientific
researchers or designers), students in various domains etc.
7 January 2008
P.P. Teodorescu
Chapter 11
Dynamics of Discrete Mechanical Systems
In dynamics one studies the motion of open mechanical systems, which are subjected to
the action of given external forces (input) and which exert certain actions upon other
systems (output); closed mechanical systems (without input and output, which are
moving only due to the interaction of the component particles) will be considered too.
We begin the study by the motion with respect to an inertial (Galilean) frame of
reference, passing then to the case of a non-inertial (non-Galilean) frame; we consider
thus discrete mechanical systems, free or subjected to constraints. We present also the
differential principles of mechanics.
First of all, some introductory notions are given: general theorems and conservation
theorems of discrete mechanical systems, free or subjected to constraints, with respect
to an inertial frame of reference, which lead to first integrals; the motion with respect to
a non-inertial frame is taken into consideration too. We mention also applications to
various important problems, e.g. the problem of n particles.
11.1.1 Introductory Notions
In what follows we introduce the notions of momentum, moment of momentum, kinetic
and potential energy, work and power, with respect to an inertial frame, in case of a
discrete mechanical system, extending the notions corresponding to a particle (see
Chap. 6, Sect. 1.1). We mention also the formulation of the problem of mechanical
systems of free particles.
11.1.1.1 Moment. Moment of Momentum. Torsor of Momentum
Let be a mechanical system
S of geometric support Ω. The momentum (linear
momentum) of the system with respect to a given fixed frame of reference (considered
to be inertial) is given by
ddmm
ΩΩ
==
∫∫
Hv r
,
(11.1.1)
the integral being a Stieltjes one, while the mass
()mm= r is a distribution.
Introducing the unit mass (1.1.71–1.1.71''), we may write
11.1 Dynamics of Discrete Mechanical Systems with Respect
to an Inertial Frame of Reference
P.P. Teodorescu, Mechanical Systems, Classical Models,
© Springer Science+Business Media B.V. 2009
1