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Файл:Integrational mechanics. Lecture and exercises
.pdf
D. F. Polishchuk, E. G. Krylov
Integrational mechanics.
Lectures and exercises
МоскваИжевск
2019
УДК 531.1
ББК 22.21
П50
Polishchuk D. F., Krylov E. G.
Integrational mechanics. Lecture and exercises. — Москва–Ижевск: Институт
компьютерных исследований; НИЦ «Регулярная и хаотическая динамика»,
2019. — 148 с.
In the book the basic ideas of integrational mechanics with reference to a brief course
of classical mechanics are considered. The unity of mathematics, physics and applied
philosophy allows to study compactly fundamentals of classical mechanics including
vibration, stability and impact. Ten problems on dynamics with the analysis of typical
receptions of creativity are solved in detail. The book is intended for students and the
engineers interested in studying classical mechanics in English.
ISBN 978-5-4344-0730-4
c
Polishchuk D. F., Krylov E. G., 2019
ББК 22.21
Contents
Introduction ............................... 5
Chapter 1. CLASSICAL MECHANICS AS A SYSTEM THEORY .7
1.1. Structureofthecourseofclassicalmechanics........... 7
1.2. Unity of mathematics, physics, and philosophy in Newton’s me-
chanics ............................... 8
1.3. Methods of creation in integrational mechanics . . . . . ..... 8
1.4. Classification of mechanics problem according to the type of
nonlinearity............................. 19
Chapter 2. SYSTEM APPROACH IN STATICS AND KINEMATICS 22
2.1. Informationoperatorofnullactionandaxiomsofstatics..... 22
2.2. Systemofconcurrentforces .................... 23
2.3. Moment of a force about a point and an axis . . . . . . ..... 24
2.4. Reduction of two parallel forces. Couple . ............ 26
2.5. Thebasictheoremofstatics .................... 27
2.6. Coplanar force system. Varygnon’s theorem ............ 28
2.7. Statically determinate and statically indeterminate problems . . . 29
2.8. Center of gravity of bodies . .................... 29
2.9. Invariantsofforcesystem ..................... 29
2.10. Statics paradoxes . . . . . . .................... 30
2.11. Peculiarities of kinematics as an ideal theory . . . . . . ..... 30
2.12. Specification of a particle motion and the information compres-
sionprinciple ............................ 31
2.13. Differentiation of a vector of unit length and the analogy principle 31
2.14. The information operator and velocity and acceleration diagrams
for a body moving with a general plane motion . . . . . ..... 32
2.15. Graphical method of successive analysis of velocity and acceler-
ation in a rigid body plane motion . ................ 34
2.16. A system way to derive Coriolis acceleration . . . . . . ..... 38
2.17. The analogy principle and compound rotational motions of a
rigid body . . . ........................... 42
4 Contents
Chapter 3. DYNAMICS ........................ 44
3.1. Newton’slawssystematization................... 44
3.2. Informational compact of Newton’s vector dynamics . . ..... 46
3.3. The basic information compact of dynamics problems . ..... 49
3.4. Compact of dynamics problems (resonance) ............ 58
3.5. Energymechanics.......................... 72
3.6. ElementsofLagrange’sanalyticalmechanics ........... 79
3.7. Compact“Impactphenomenainmechanics” ........... 82
3.8. Compact “Linear and nonlinear problems in dynamics” ..... 89
3.8.1. Classificationsofvibrationproblems............ 89
3.9. Compact “Stability” . . . . . ....................100
3.10.Analyticalmechanicsasan“ideal”theory.............107
3.11.Systemclassificationofforces...................112
3.12. Classification of “physical” bodies . ................115
Chapter 4. EXAMPLES OF PROBLEM ANALYSIS ......... 119
4.1. Kinematicsofmass–pointparticle.Analogies..........119
4.2. Dynamicsofmass–pointparticle ................122
4.3. Dynamics of translation motion of a system of rigid bodies . . . 124
4.4. Dynamics of rotation of a system of rigid bodies . . . . .....130
4.5. Motion of a body in potential field ................133
4.6. Distribution of inertia forces of rigid body being in general plane
motion ...............................136
4.7. Bearingreactions ..........................139
4.8. Differentialequationofmotionofamechanism..........143
Bibliography .............................. 146
Introduction
Teaching the fundamentals of classical mechanics using methods of creativity allows to apply reception if creativity for various educational and scientific
engineering disciplines.
In the first chapter the basic typical receptions (procedures) of creativity
and system operators are in detail considered, the description of analytical and
design algorithm is presented.
In the second chapter statics and kinematics are considered. All axioms
and theorems of statics are essentially compressed by means of the application
of informational operator of null action. In kinematics updating of a graphical
method of academician Kotelnikov A. P. is given. The method allows to find
accelerations in two — dimensional problems with evident check of obtained results. the convenient method of derivation of a formula for Coriolis acceleration
ia given.
In the third chapter it is considered the system aspect on Newton’s laws,
compact of Newton’s dynamics, and compact of Lagrange’s dynamics. The basic compact of dynamics problems includes the check of conformity of laws of
classical mechanics to experimental data, setting up the equations of motion, the
analysis of the initial equations, solving the basic equations. This compact incorporates also special compacts on vibration and damping,stability and impact.
It is paid special attention to skill “to solve a problem, not solving it” by the example of a classical problem of vibration of the simple pendulum with different
equilibrium positions in a vertical plane.
Special classification of “physical bodies” is offered on the base of the
general operator of infirmation. It distinguishes three groups of bodies. the first
group illustrates distinctions between classical mechanics,continuum mechanics,
Hertz contact theory, stereo-mechanical theory of impact,strength of materials.
The second group is devoted to the methods of setting up and solving of the
equations of motion. The third group represents system of bodies (impact of two
bodies in the stereo-mechanical theory, impact in the Hertz’ contact theory, etc.),
and also the methods of creation of linearized theories with essential change of
physics.
In compact of vibration and damping the concert of resonance, and typical
receptions of damping of vibration are given. This compact concerns also the
6 Introduction
distinctions between linear and nonlinear vibration. The concept of the united
theory of spatial vibration is given by the example of the helical thin bar. This
concept is based on know idea of classical mechanics about two simple motions
of a body (translation and rotation) and its analogue in the classical theory of
wave in continuum mechanics (a wave process can be reduced to the combination of the wave of expansion and the wave of shear, as it had been stated by
Poisson).
In the fourth chapter solutions of ten typical problems from the textbook
after Meshersky I. V. with the analysis of typical receptions of creativity are
given.
First three chapters represent lectures which were delivered in English by
Polishchuk D. F. in Izhevsk State Technical University (IzSTU). Final proofreading of the English text of first three chapters is executed by senior lecturer
Krylov E. G. who is author of chapter four.
In the list of literature manuals in English written by Krylov E. G., Zhichkina E. S., Pirozhkova L. N. are resulted. Besides the of literature includes books
by Polishchuk D. F. devoted to integrational mechanics.
Authors are grateful to publishing house “Institute of coputer researches”
and personally to editor-in-chief of edition “Regular and chaotic dynamics”
Pashkina S. S. who was among the first students of a course of the classical
mechanics in English delivered by authors of book in IzSTU. Authors are grateful to technical editor Shirobokov A. V.
We hope that foreign students in Russia will ideas of integrational mechanics. These ideas were reported at the University of Frendship between
Nations (Moscow) and the name of integrational mechanics was offered by Galiullin A. S., professor of this university.
Chapter 1
CLASSICAL MECHANICS
AS A SYSTEM THEORY
1.1. Structure of the course of classical mechanics
The classical course of mechanics involves three parts: statics, kinematics,
dynamics.
Statics studies a rest and equilibrium. It deals with forces and moments of
forces but not with the results of their action — translation and rotation of bodies.
In kinematics motion appears but force parameters disappear.
Dynamics is the unity of statics and kinematics but the latter two are essentially changed since statics is combined with motion, and motion in kinematics
is combined with forces. Mechanics like mathematics has its own definitions,
notions and axioms of the rigid-body statics and dynamics.
The structure of the course is from simple to complex. Statics studies:
a system of concurrent forces, moment of force about a point, moment about an
axis, reduction of a given force system to the simplest one, condition of force
system equilibrium, invariants and particular cases of force system.
Kinematics deals: with basic notions of kinematics of mass-point particle,
simple motions of a rigid body, compound motions of a particle, general-plane
motion of a rigid body, rotation of a rigid body about a fixed point, a general
case of a body motion, kinematics of compound motions of a rigid body.
The same approach is also in dynamics: dynamics axioms, dynamics of
a mass-point particle, dynamics of a relative motion of a particle, geometry of
masses, common theorems of dynamics of a particle and of a system of particles,
fundamentals of analytical mechanics, dynamics of a rigid body having one fixed
point, notions of theory of vibration, impact, motion of a particle with variable
mass.
The structure of the course of mechanics is taken from the textbook [1], it is
traditional and in accordance with a well-known “consistent” teaching. A system
teaching, which the author supports, supposes the study of the course in integrity,
from the common to particular, and in integrity again etc.
8 Chapter 1
1.2. Unity of mathematics, physics, and philosophy in
Newton’s mechanics
It is known that Archimed derived the mathematical formulas for calculations of volumes and areas of geometrical bodies, using mechanics. He used
mechanics as a “secret method”. Understanding that mechanics is a part of the
natural philosophy, Isaak Newton (with the participation of Leibnitz) worked
out the differential calculus and formalized the notions of Galileo – Descart’s
mechanics.
The mathematical theory of vector analysis and also the theory of stability
arose from statics. Moreover, the mechanics laws became the first general laws.
Descart, Leibnitz, Newton were the philosophers in proper sense. One of the
first Descart published a treatise concerning the methods of cognition theory, he
also stated the priority of a system momentum conservation principle. Cartesian
coordinates allowed to simplify greatly the determining of a body compound
motion in space. Lagrange’s desire was to intensify the mathematical basis
in mechanics and formalize the deriving of initial equations which resulted in
analytical mechanics. As the analytical mechanics has a virtual displacement
principle, which was transformed by N. E. Zhukovsky into a virtual velocity
principle.
D’Alembert’s principle allows to solve dynamics problems by the use of
statics methods. Introduction of the notion of ideal constraints enables us to
see each reaction of constraint in reality but as a single reaction of one constraint. The sum of works of all ideal constraint reactions done on all virtual
displacements results in zero.
It is obvious from these brief remarks that classical mechanics is a system
one. But system operators were not formulated distinctly in classical mechanics,
that is why statics axioms, equilibrium conditions, virtual displacement principle
appeared, which describe system equilibrium condition in different ways. The
system interpretation of classical mechanics was not completed. So this chapter
is devoted to extending classical mechanics on the base of further generalization,
finding the support information, compressing the information using the operators of system mechanics. But the consideration of mechanics as the unity of
mathematics, physics, philosophy is to some extent a backward movement.
1.3. Methods of creation in integrational mechanics
Methods of creation in integrational mechanics are presented by the typical
procedures, analytical and design algorithm, and by system operators [2].
1.3. Methods of creation in integrational mechanics 9
Typical procedures of integrational mechanics are:
1. Consideration of each problem peculiarities.
2. Usage of a predetermined information.
3. Supervision of problem solving process using an additional information.
4. Usage of analogies.
5. Multiple usage of the same procedure.
6. Usage of some familiar procedures (superposition principle, dividing a complex problem into a set of simple ones, a “fork” method etc.).
7. Inversion (to do inversely, to turn harm into benefit).
8. Solution of various problems using different degrees of mathematical rigorous and different physical models.
9. Development of optimal qualitatively different methods at each stage.
10. Development of the advanced solutions in mechanics and the approximated
solutions in mathematics.
11. Generation of a module for control of a complex problem solving process;
development of the model of “ideal theory”.
12. Complex procedure of system analysis.
Many algorithms and procedures had been worked out at the field of techni-
cal creation. But in integrational mechanics the analytical and design algorithm
is used.
This algorithm consists of six stages [2]:
• Problem statement.
• Organization of an analytical solving of the problem.
• Generation of the experimental results “field”.
• Search for the analytical solution.
• Developing the design solutions.
• Working out the practical recommendations.
10 Chapter 1
The analytical and design algorithm is adapted for the problems of dynam-
ics and strength of machines.
At the stage of the problem statement a particular technical problem should
be formulated in terms of “ideal engineering solution”. For example, it may be
formulated as decreasing overall dimensions, increasing dynamical performance
of the process, and increasing the durability of an elastic element in a spring
mechanism. In this case we have an apparent engineering contradiction because
increasing overall dimensions of an elastic element does not contribute to a
durability increase.
The next stage is realizing the necessity of an analytical solution. The
necessity of the analytical solution of a given problem deals with the search
for an analytical overcoming of engineering contradiction. It is justified by
a complex study of vibration and stability, vibration and stress, statical and
dynamical loading of various spring mechanisms.
Stages of analytical solution organization:
1. Search for support analytical solutions.
2. Analysis of existing support analytical solutions using system analysis procedures.
3. Formulation of new analytical approaches after the analysis of the support
analytical solutions.
4. Justification of choice of a new model for the problem solution.
5. Distinguishing of basic module problems from others among expected synthesized module problem.
6. Analysis of characteristic property of module problem solutions on the basis
of a new model.
7. Choice of an approximate scheme of a module problem interconnection.
8. development of a system inspection of the expected analytical solution.
Stages of creating experimental results “field”:
1. Development of an experimental results “field” on the base of support analytical problem solution.
2. Experimental validation of hypotheses used in analytical solutions.
3. Analysis of interconnection of experimental results.
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